![1
Concentration and crowding effects on
protein stability from a coarse-grained
model
jason k. cheung, vincent k. shen, jeffrey
r. errington, and thomas m. truskett
Introduction
Most of what we know about protein folding comes from experiments
on polypeptides in dilute solutions [1–4] or from theoretical models of isolated
proteins in either explicit or implicit solvent [5–12]. However, neither biologi-
cal cells nor protein solutions encountered in biopharmaceutical development
generally classify as dilute. Instead, they are concentrated or “crowded” with
solutes such as proteins, sugars, salts, DNA, and fatty acids [13–15]. How does
this crowding affect native-state protein stability? Are all crowding agents cre-
ated equal? If not, can generic structural or chemical features forecast their
effects?
To investigate these and other related questions with computer simulations
requires models rich enough to capture three parts of the folding problem:
the intrinsic free energy of folding of a protein in solvent, the main structural
features of the native and denatured states, and the connection between protein
structure and effective protein–protein interactions. The model must also be
simple enough to allow for the efficient simulation of hundreds to thousands of
foldable protein molecules in solution, which precludes the use of atomistically
detailed descriptions of either the proteins or the solvent.
We recently developed a coarse-grained modeling strategy that satisfies these
criteria. It is not optimized to describe any specific protein solution. Rather, it
is a general tool for understanding experimental trends regarding how concen-
tration or crowding impact the thermodynamic stability of globular proteins.
Statistical Mechanics of Cellular Systems and Processes, ed. Muhammad H. Zaman. Published by Cambridge
University Press. © Cambridge University Press 2009.
1](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-19-320.jpg)
![2 J. K. Cheung et al.
To date, the approach has been used to study how protein concentration affects
the folding transition [16], how solution demixing phase transitions (e.g., liquid–
liquid phase separation) couple to protein denaturation [17], and how surface
anisotropy of the native proteins relates to their unfolding and self-assembly
behaviors in solution [18]. In this chapter, we review the modeling strategy and
some key insights it has produced.
Coarse-grained modeling strategy
Intrinsic protein stability
A two-state protein molecule in a pure solvent has a temperature- and
pressure-dependent thermodynamic preference for either its native (folded)
or denatured (unfolded) form [19–24]. The free energy difference between
these states G0
f quantifies the driving force for folding in the absence of
protein–protein or protein–solute interactions. It also determines the equilib-
rium probability (1+exp[G0
f /kBT])−1 associated with observing the native state
in an infinitely dilute solution, where kB is the Boltzmann constant and T is the
temperature.
Interactions that influence this “intrinsic” stability of the native state include,
but are not limited to, intra-protein hydrogen bonding, electrostatics, disulfide
bonds, and London–van der Waals interactions, as well as effective forces due
to excluded volume, chain conformational entropy, and hydrophobic hydra-
tion [25]. Here, we focus exclusively on the last three, since they are relevant
not only to protein folding [26–29] but also to other self-assembly processes
in aqueous solutions [30–32]. Chain conformational entropy and intra-protein
excluded volume interactions favor the more expanded denatured state of a
protein, while the ability to bury hydrophobic residues in a largely water-free
core favors the compact native fold. Intrinsic stability characterizes how these
factors for a protein in the infinitely dilute limit balance at a given temperature
and pressure.
The coarse-grained modeling strategy we review here [16] calculates G0
f
under the assumption that a foldable protein can be represented as a collapsible
heteropolymer. The effective inter-segment and segment–solvent interactions
of the heteropolymer are chosen to qualitatively reflect the aqueous-phase
solubilities of the amino acid residues in the protein sequence [33, 34].
One advantage of heteropolymer collapse (HPC) models is that they derive
from independently testable principles of polymer physics and hydration
thermodynamics. A second advantage is that their behaviors can often be
predicted by approximate analytical theories or elementary numerical tech-
niques, which allow them to be efficiently incorporated into multiscale](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-20-320.jpg)
![Protein stability in concentrated and crowded solutions 3
simulation strategies such as the one discussed here. Although HPC theo-
ries are descriptive rather than quantitative in nature, the combination of
their simplicity and their ability to reproduce experimental folding trends
of globular proteins [34] makes them particularly attractive for use in model
calculations.
HPC theories often reflect a balance of structural detail and mathematical
complexity [9, 35–37]. In our preliminary studies, we use a basic, physically
insightful approach introduced by Dill and co-workers [34, 38]. This theory mod-
els each protein of Nr amino acid residues as a heteropolymer of Ns = Nr/1.4
hydrophobic and polar segments. As is explained in Ref. [38], the factor of 1.4
enters due to a lattice treatment of the protein in which the chain is parti-
tioned into cubic polymer segments. The amino acids in a globular protein
can be represented as occupying cubic volumes with an average edge length
of 0.53 nm [38], whereas the separation of α-carbons in an actual protein is
about 0.38 nm (0.53/0.38 ≈ 1.4). The inputs to the theory include temperature
T (and, more generally, pH and ionic strength [35]), the number of residues in
the protein sequence Nr, the fraction of those residues that are hydrophobic
(e.g., based on their aqueous solubilities [33, 34]), and the free energy per
unit kBT associated with hydrating a hydrophobic polymer segment χ. A sim-
ple parameterization for χ is available that captures experimental trends for
the temperature-dependent partitioning of hydrophobic molecules between a
nonpolar condensed phase and liquid water at ambient pressure [16]. Although
in this chapter we focus exclusively on thermal effects, we have previously
introduced a statistical mechanical method for extending the parameterization
for χ to also account for hydrostatic pressure [39].
To compute G0
f using this HPC model, one first constructs an imaginary two-
step thermodynamic path that reversibly connects the denatured (D) and native
(N) states [34]. In step 1, the denatured state with radius of gyration RD collapses
into a randomly condensed configuration with radius of gyration, RN. The the-
ory assumes that the fraction of solvent-exposed residues that are hydrophobic
in both the denatured and the randomly condensed states is , the sequence
hydrophobicity of the protein. In step 2, the native state is formed from the ran-
domly condensed state via residue rearrangement at constant radius of gyration,
so that the fractional surface hydrophobicity of the protein changes from to
. By independently minimizing the free energies of the native and denatured
states in this analysis, HPC theory predicts the values of both RD/RN and . The
intrinsic free energy of folding is the sum of the contributions from the two
steps along the imaginary folding path, G0
f = G0
1 +G0
2. Approximate analyt-
ical solutions for this HPC theory describe cases where the hydrophobic residues
have uniform [34] or patchy [18] spatial distributions on the protein surface. We](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-21-320.jpg)
![4 J. K. Cheung et al.
discuss below how these solutions can, in turn, be used to infer approximate
nondirectional and directional protein–protein interactions, respectively.
Non-directional protein–protein interactions
While intrinsic thermodynamic stability governs whether an isolated
protein favors the native or denatured state, protein–protein interactions
play a role in stabilizing or destabilizing the native state at finite protein
concentrations.
Protein–protein interactions reflect protein structure. Since HPC theories
provide only coarse information about structure, the effects we discuss here
are the most basic, generally pertaining to how protein size and surface chem-
istry couple to their interactions. We first examine the case where proteins
display a virtually uniform spatial distribution of solvent-exposed hydropho-
bic residues, so that protein–protein interactions are, to first approximation,
isotropic. We also limit our discussion to systems where the driving force of
proteins to desolvate their hydrophobic surface residues by burying them into
hydrophobic patches on neighboring proteins dominates the attractive part
of the effective protein–protein interaction. The repulsive contribution to the
inter-protein potential accounts for the volume that each protein statistically
excludes from the centers of mass of other protein molecules in the solution.
As should be expected, the structural differences between folded and unfolded
protein states translate into distinct native–native NN, native–denatured ND,
and denatured–denatured DD protein–protein interactions.
HPC theory correctly predicts that denatured proteins generally exclude
more volume to other proteins (RD RN) as compared to their native-state
counterparts as shown in Fig. 1.1a [34]. Moreover, denatured proteins exhibit
a greater fractional surface hydrophobicity than folded molecules ( ).
Mean-field approximations [16, 17] predict that the magnitudes of the average
“contact” attractions between two isotropic proteins scale as
εND =
Nsχ(T)kBT
12
fe(ρ∗
s )
[1 + ρ∗
s
−1/3
]2
+
fe(1)
[1 + ρ∗
s
1/3
]2
(1.1)
εNN =
Nsχ(T)fe(1)2kBT
24
(1.2)
εDD =
Nsχ(T)fe(ρ∗
s )2kBT
24
(1.3)
where ρ∗
s is the effective polymer segment density, fe(ρ∗
s ) = 1 − fi(ρ∗
s ) is
the fraction of residues in the denatured state that are solvent exposed, and
fi(ρ∗
s ) = [1 − (4πρ∗
s /{3Ns})1/3]3 is the fraction of residues that are on the interior
of the protein. A detailed derivation of the above equations can be found in
Refs. [16, 18].](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-22-320.jpg)
![Protein stability in concentrated and crowded solutions 5
1
1.5
2
R
D
/R
N
ε
DD
/
ε
NN
330 340 350 360 370
Temperature [K]
3
4
5
Φ = 0.400
Φ = 0.500
Φ = 0.500
Φ = 0.400
Φ = 0.455
Φ = 0.455
(a)
(b)
Fig. 1.1 Comparison of the (a) radius of gyration of the denatured protein relative
to the native protein, RD/RN, and (b) the effective magnitude of the DD attraction
relative to the NN attraction, εDD/εNN, for proteins of Nr = 154 residues and
sequence hydrophobicity = 0.400 (solid), 0.455 (dash), and 0.500 (dot).
Given (1.1–1.3) and , it follows that contact attractions involving dena-
tured proteins will generally be stronger than those involving the native state
(Fig. 1.1b). This is why denaturation often leads to aggregation and precipitation
in protein solutions. Along these lines, attractions between proteins increase in
strength with the hydrophobic content of the underlying protein sequence .
In our coarse-grained strategy, the interprotein exclusion diameters,
σDD/σNN ≈ RD/RN and σND/σNN ≈ (RN + RD)/2RN, and the contact energies of
(1.1–1.3), all of which are derived from HPC theory [34], serve as inputs into
an effective protein–protein potential Vij(r) that qualitatively captures many](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-23-320.jpg)
![6 J. K. Cheung et al.
aspects of protein solution thermodynamics and phase behavior (see, e.g., Refs.
[40, 41]):
Vij(r) = ∞ r σij
Vij(r) =
ij
625
1
[( r
σij
)2 − 1]6
−
50
[( r
σij
)2 − 1]3
r ≥ σij (1.4)
where ij corresponds to the type of interaction NN, ND, or DD.
Directional protein–protein interactions
In Dill and co-workers’ original development of this HPC theory,
they assume that there are no spatial correlations between solvent-exposed
hydrophobic residues in either the denatured or the native state [34]. One way
to relax this assumption is to allow for segregation of hydrophobic residues on
the surface of the native state. For example, consider the hypothetical scenario
where two symmetric “patches” form on the surfaces of native proteins during
folding. The patches are distinguishable because they have a different hydropho-
bic residue composition than the rest of the solvent-exposed “body.” As shown
in Fig. 1.2, the size of each patch is defined by the polar angle α. The fractional
patch hydrophobicity p and body hydrophobicity b are expressed as
p =
fph
1 − cos α
b =
(1 − fph)
cos α
(1.5)
Θ
p
α
Θp
Θb
Fig. 1.2 Schematic of two anisotropic native-state proteins. The patch (shaded) and
body (white) regions have different hydrophobic residue compositions. The size of
the patch is defined by the angle α. The hydrophobicities of the patch p and body
b are determined by (1.5). Since the dashed line connecting the protein centers
passes through a patch region on each molecule, these two proteins are currently
in a patch–patch alignment. Adapted from Ref. [18].](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-24-320.jpg)
![Protein stability in concentrated and crowded solutions 7
where fph quantifies the fraction of the surface hydrophobic residues that
are sequestered into the patch regions on the native protein. Increasing fph
increases p, which results in higher surface anisotropy and, as we see below,
stronger patch–patch attractions. Knowledge of native-state structure would,
in principle, allow one to formulate an approximate patch model for a given
protein [42], but predicting this structure directly from sequence information
using HPC theory is still not generally possible. In other words, fph, α, and patch
location, along with protein sequence, are still knowledge-based inputs for the
coarse-grained strategy.
The directional dependencies of the contact attractions of patchy proteins
are approximated as follows [18]:
εND =
Nsχ(T)mkBT
12
fe(ρ∗
s )
[1 + ρ∗
s
−1/3
]2
+
fe(1)
[1 + ρ∗
s
1/3
]2
(1.6)
εNN =
Nsχ(T)fe(1)mnkBT
24
(1.7)
εDD =
Nsχ(T)fe(ρ∗
s )2kBT
24
. (1.8)
Here, m and n denote the apparent surface hydrophobicities associated with
different orientational states of interacting native molecules m and n, respec-
tively. For example, to compute the value of m for molecule m of a given pair
interaction, one only needs to know the orientation of molecule m relative to
that of the imaginary line connecting its center of mass to that of the other par-
ticipating protein. If this line passes through a patch on molecule m’s surface
(see Fig. 1.2), then m = p; otherwise m = b, and so on. Equations (1.6–1.8)
reduce to (1.1–1.3) for the isotropic (uniform surface hydrophobicity) case (i.e.,
p = b = ).
As an illustration, we examine below aqueous solutions of two model pro-
teins of molecular weight Ns = 110 (i.e., Nr = 154) and hydrophobic residue
composition = 0.4, parameters typical for medium-sized, single-domain
globular proteins [43]. The difference between the two models is that their
native states display distinct surface residue distributions, which in turn lead
to different protein–protein interactions: “nondirectional” (i.e., no patches) and
“strongly directional” ( fph = 0.75, α = π/6). For simplicity, we refer to these
models by their names shown above in quotes, rather than by the fph and α
parameters that define them. As we discuss below, the behaviors of these two
model systems provide insights into the mechanisms for stability in several
experimental protein solutions.
Figure 1.3 shows the effect of native protein surface anisotropy on the
strength of protein–protein attractions. The patch–patch attractions for the](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-25-320.jpg)
![8 J. K. Cheung et al.
330
Temperature [K]
3
6
9
12
ε
N
N
k
/
B
T
Patch–Patch
0.1
0.2
0.3
0.4
0.5
ε
N
N
k
/
B
T
Body–Body
(b)
(a)
Nondirectional
Strongly directional
340 350 360 370
Fig. 1.3 Comparison of the contact attraction εNN relative to kBT for (a) body–body
alignment and (b) patch–patch alignment of Nr = 154, = 0.4 proteins with
strongly directional interactions (dash). The contact attraction for Nr = 154, = 0.4
native proteins with nondirectional interactions (solid) is also shown.
strongly directional protein are more than an order of magnitude larger than the
other attractions. Pairs of directional proteins can desolvate a higher number of
hydrophobic residues by self-associating (as compared to the nondirectional pro-
teins), but only if they do so with their hydrophobic patches mutually aligned,
which in turn imposes an entropic penalty. This balance between favorable
hydrophobic interactions and unfavorable entropy yields the possibility of con-
tinuous equilibrium self-assembly transitions involving the native state [18].
Given the symmetric patch geometry of the native state model studied here, the
morphology of the self-assembled “clusters” would resemble linear polymeric
chains.
The interactions discussed above are similar in spirit to a “two-patch”
description that was recently introduced to model the native–native protein
interactions of the sickle cell variant of hemoglobin [44] and also to other
semi-empirical anisotropic potentials developed for native proteins [42, 45–50].](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-26-320.jpg)
![Protein stability in concentrated and crowded solutions 9
However, the coarse-grained strategy described here differs significantly from
these earlier models in two ways: it explicitly accounts for the possibility of
protein denaturation, and it estimates the intrinsic properties of the native
and denatured states using a statistical mechanical theory for heteropolymer
collapse. This link to the polymeric aspect of the protein is crucial because
it allows our model to be used as a tool to investigate how native-state pro-
tein anisotropy affects folding equilibria, self-assembly, and the global phase
behavior of protein solutions.
Reducing protein stability to a classic chemical engineering problem
It is worth emphasizing that the coarse-grained model described above
represents an effective binary mixture of folded and unfolded proteins (the
aqueous solvent only entering through χ) connected via the protein folding
“reaction.” Links between the intrinsic native-state stability of the proteins,
G0
f , the physical parameters defining the protein sequence (Nr, ), the native-
state surface morphology (, α, fph), the interactions of hydrophobic residues
with aqueous solvent χ, and the protein–protein interactions ( ij, σij) are estab-
lished through the HPC model [16, 17]. As in experimental protein solutions,
the fraction of proteins in the native state generally depends on both temper-
ature and protein concentration. This fact, often neglected in other modeling
approaches which ignore the polymeric nature of proteins or protein–protein
interactions, arises because temperature affects the intrinsic stability of the
native state G0
f , and both temperature and protein concentration influence the
local structural and energetic environments that native and denatured proteins
sample in solution.
In short, the coarse-grained approach frames the stability of concentrated
protein solutions in terms of a classic chemical engineering problem: mapping
out the equilibrium states of a reactive, phase-separating mixture [51]. In the
next section of this chapter, we discuss how advanced Monte Carlo methods
designed to efficiently solve the latter problem can also be used to address the
former.
Simulation methods
The properties of the coarse-grained protein model described above
can be readily studied using transition-matrix Monte Carlo (TMMC) simulation.
TMMC is a relatively new simulation technique that has emerged in recent years
as a highly efficient method for investigating the thermophysical properties of
fluids. It is useful for a variety of applications ranging from the precise calcula-
tion of thermodynamic properties of pure and multicomponent fluids in bulk](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-27-320.jpg)
![10 J. K. Cheung et al.
and confinement [52–62], surface tension of pure fluids and mixtures [52, 63–65]
and Henry’s constants [66] to the investigation of wetting transitions [67, 68] and
adsorption isotherms [69, 70]. The basic goal of TMMC is to calculate the free
energy of a system along some order parameter path by determining the order
parameter probability distribution. To do this in a conventional simulation, a
histogram is constructed by simply counting the number of times the system
takes on, or visits, a given order parameter value. In a transition-matrix-based
approach, the distribution is determined by accumulating the transition proba-
bilities of the system moving from one order parameter value to another during
the course of a Monte Carlo simulation, and subsequently applying a detailed
balance condition over the explored region of order parameter space. To facili-
tate sufficient and uniform sampling of order parameter space, a self-adaptive
biasing scheme is often employed. The reader is referred to Refs. [52, 56–58,
63, 71–75] for further details. For this coarse-grained protein model, we use a
particular TMMC implementation designed for multicomponent systems [57].
To obtain thermodynamic and structural quantities of interest, we perform
TMMC simulations within the grand-canonical ensemble. Under these condi-
tions (fixed chemical potentials, volume, and temperature), an appropriate
choice of order parameter is the total number of molecules in the system. While
the order-parameter probability distribution is unique to the chemical poten-
tials used in the simulation, histogram reweighting can be used to determine the
distribution at other chemical potentials [76]. Because the coarse-grained pro-
tein model can be regarded as a binary mixture of native and denatured proteins
where the components can undergo a unimolecular chemical reaction (folding),
chemical equilibrium requires that the chemical potentials of the native and
denatured proteins be identical. Thus, only a single chemical potential needs
to be specified in a TMMC simulation of the protein solution. The free-energy
change of the reaction, that is the intrinsic free energy of folding G0
f , enters
as an activity difference between the folded and unfolded proteins, a treatment
which implicitly assumes that the protein’s intermolecular and intramolecular
degrees of freedom are separable.
Although the main output of a TMMC simulation is the total protein num-
ber (concentration) distribution, other system properties can be also calculated.
This is done in a straightforward way by collecting isochoric averages during the
course of a simulation. Combining these statistics with the above-mentioned
histogram-reweighting technique, the fluid-phase properties of the coarse-
grained protein model can be determined over a wide range of concentrations
from a single TMMC simulation. In our studies, we calculate, along with other
properties, the following quantities: the overall fraction of folded proteins fN,
the fraction of clustered or aggregated proteins fclust, and the average fraction](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-28-320.jpg)
![Protein stability in concentrated and crowded solutions 11
of clustered proteins that are folded fNc [18]. Since this is a binary system, fN is
tantamount to the composition. The midpoint unfolding transition in a protein
solution occurs at fN = 0.5. The quantity fclust measures the fraction of proteins
in the system that are in clusters. This has also been referred to as the extent
of polymerization in other contexts [77]. Here, two proteins are considered to
be in the same cluster if the magnitude of their effective pairwise attraction is
greater than 80% of the interprotein potential minimum ij. Finally, the quantity
fNc characterizes the overall average protein cluster composition.
Besides the conventional assortment of Monte Carlo trial moves (e.g., par-
ticle displacements, insertions/deletions, and identity changes), we perform
so-called aggregation-volume bias trial moves [78–80]. These specialized trial
moves are typically required because strong (relative to kBT) inter-protein inter-
actions can otherwise result in persistent bonded configurations which prevent
adequate sampling of phase space. Aggregation-volume bias moves circum-
vent this difficulty by preferentially performing trial protein displacements and
protein insertions or deletions in the immediate vicinity, called the “bonding
region,” of a randomly chosen molecule, thereby promoting the formation and
destruction of bonded configurations. Phase space sampling is further enhanced
by combining this suite of moves with multiple first-bead trial insertions and
configurational-bias Monte Carlo moves [80–82]. Finally, since we are interested
in the liquid-state properties of the coarse-grained protein model, a constraint
is imposed to prevent the system from crystallizing. This is required because,
for the one-component (all native or all denatured) system, the liquid state
is metastable with respect to the solid at intermediate and high concentra-
tions [40]. Specifically, we apply a constraint based upon a bond-orientational
metric capable of distinguishing between amorphous (liquid) and solid con-
figurations [83–86]. The interested reader is referred to Ref. [17] for further
details.
Effects of protein concentration: uniform vs. patchy proteins
In this section, we examine the predictions of our grand-canonical
TMMC simulations and coarse-grained modeling strategy with a focus on eluci-
dating how the surface anisotropy of the native state affects denaturation and
equilibrium self-assembly behaviors of proteins in solution. In particular, we
compare simulated equilibrium unfolding curves ( fN) and self-association mea-
sures ( fclust and fNc) for the nondirectional and the strongly directional model
proteins described above.
We first explore the physics that govern the stability of the nondirectional
protein in solution as a function of its concentration (Fig. 1.4a). The effects can
be seen most clearly by simulating along the G0
f = 0 isotherm of the protein,](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-29-320.jpg)
![12 J. K. Cheung et al.
0
Protein concentration
0.2
0.4
0.6
0.8
1
Strongly directional
0
0
0.2
0.4
0.6
0.8
1
Fraction
Fraction
Nondirectional
fNc
fN
fclust
0.1 0.2
0.02 0.04 0.06
(a)
(b)
Fig. 1.4 Fraction of folded proteins fN (solid), “clustered” proteins fclust (dotted),
and “clustered” proteins in the native state fNc (dashed), as a function of protein
concentration for the (a) nondirectional and (b) strongly directional model proteins
(Nr = 154, = 0.4) at their respective infinite-dilution midpoint folding
temperatures. Protein concentration is measured as the dimensionless density
(i.e., Nσ3
NN/V) where N is the total number of proteins and V is the simulation box
volume. To provide a comparison to experimental concentrations, a 100 mg/ml
solution of ribonuclease A would have a dimensionless density Nσ3
NN/V of
approximately 0.28, assuming σNN = 4.0 nm and a molecular weight of 13.7 kDa.
Adapted from Ref. [18].
i.e., its infinite dilution unfolding temperature. As we show below, the existence
of patches on a protein serve to modify this baseline behavior.
The minimum in the folded fraction fN curve for the nondirectional protein
in Fig. 1.4a can be understood as a balance between two factors: destabilizing
protein–protein attractions involving denatured species and stabilizing protein
crowding effects. At finite protein concentrations, a marginally stable protein](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-30-320.jpg)
![Protein stability in concentrated and crowded solutions 13
can favorably denature in solution if it meets two criteria: (a) its current envi-
ronment affords enough local free volume to accommodate the transition to the
more expanded denatured state, and (b) it can form enough new inter-protein
hydrophobic contacts with neighboring proteins upon denaturing so that it
overcomes its intrinsic free energy penalty for unfolding. The latter destabi-
lizing effect is facilitated by the fact that attractions involving the denatured
state are generally much stronger than those involving native nondirectional
proteins. Concentrating the protein solution decreases the probability of (a) but
increases the likelihood of (b).
To our knowledge, the coarse-grained strategy outlined here is the first to predict
this minimum in folded fraction versus protein concentration, a nontrivial trend that is
observed experimentally in, e.g., monoclonal antibody solutions [87]. Proteins
with high hydrophobicity are predicted by the coarse-grained model to show
more pronounced concentration-induced destabilization at low and intermedi-
ate protein concentrations [16, 17], which also appears to be in agreement with
available experimental trends [88].
Does clustering or self-assembly occur in solutions of nondirectional
proteins? Figure 1.4a shows that the fraction of clustered nondirectional
proteins fclust increases with protein concentration. Clustering of this sort can
have two potential origins. On one hand, it can occur for trivial “packing” rea-
sons alone at high protein concentrations, conditions for which pairs of proteins
are literally forced to adopt near-contact configurations that satisfy the above
geometric criteria for fclust. On the other hand, the more interesting case is
when protein clustering arises due to strong inter-protein attractions, which
are relevant even at low protein concentrations. For the nondirectional protein
discussed above, it is known that interprotein attractions are relatively weak
(Fig. 1.3). Thus, clustering for the nondirectional protein is due to geometric
“packing” effects. This is also reflected in the fact that the native composition of
the clusters fNc essentially tracks the fraction folded fN at higher protein concen-
trations, where most of the proteins are clustered. If, instead, clustering were
the result of highly favorable, native-stabilizing inter-protein attractions, then
one would expect to find fNc fN, even at relatively low protein concentrations.
The behaviors exhibited by the nondirectional protein discussed above are in
stark contrast to that of the strongly directional protein presented in Fig. 1.4b.
Notice that the relevant protein concentration range for the strongly directional
protein is an order of magnitude less than that for the nondirectional proteins.
Under these dilute conditions, the packing effects which drive geometric clus-
tering of the proteins are absent. Increases in fclust are therefore a result of
different physics: the highly energetically favorable patch–patch association.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-31-320.jpg)
![14 J. K. Cheung et al.
The fact that the folded fraction within the clusters fNc rises above the average
folded fraction fN indicates that the self-assembly behavior or “clustering” stabi-
lizes the patchy native state of the directional protein relative to the denatured
state.
In Fig. 1.5, we plot what might be referred to as “stability diagrams” for
the (a) nondirectional and (b) strongly directional proteins discussed above. The
shaded regions of these diagrams indicate temperature and concentrations that
310
330
350
Temperature
[K]
Nondirectional
0
0
0.015
0.15
Protein concentration
310
330
350
Temperature
[K]
Strongly directional
Denatured
Native
Denatured
Native
0.005
0.05
0.01
0.1
(a)
(b)
Fig. 1.5 Stability diagram for the (a) nondirectional protein and (b) strongly
directional protein in the temperature vs. protein concentration plane. Protein
concentration is defined as in Fig. 1.4. The native state is thermodynamically
favored ( fN 0.5) in the white region, while the denatured state is favored
( fN 0.5) in the shaded region. Also shown are the loci of conditions where
fclust = 0.5 (dash). To the right of the fclust = 0.5 curve, more than half of the
proteins are part of geometric “clusters.” For conditions where the denatured state
is favored, note that the location of the fclust = 0.5 curve for both panels (a) and (b)
are the same. This result is expected because the attractive strength and relative
size of the denatured proteins are the same for the protein variants studied here.
Adapted from Ref. [18].](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-32-320.jpg)
![Protein stability in concentrated and crowded solutions 15
favor the denatured state ( fN 0.5), while the white region indicates conditions
that favor the native state ( fN 0.5). The loci of temperature-dependent con-
centrations where fclust = 0.5 (dash) are also shown. Proteins form clusters
in solution for all states to the right of these curves. Note that the clustering
behavior of the nondirectional protein occurs at lower protein concentrations
for the denatured state as compared to the native state because the former has a
larger effective radius of gyration and stronger inter-protein attractions. For the
strongly directional proteins, the native-state global clustering trends are dif-
ferent. Small increases in native protein concentration promote self-assembly
of linear clusters due to the fact that the patch–patch attractive interactions are
much stronger than even the denatured–denatured interactions for this protein.
One key prediction of our model is that strong directional hydrophobic inter-
actions can help stabilize anisotropic native proteins. In other words, folded
proteins that associate through hydrophobic patches may show enhanced sta-
bility. To illustrate the generality of this result, we briefly discuss below some
experimental stability behaviors for proteins in their native states that asso-
ciate via hydrophobic patches in native or mildly denaturing conditions. We
also mention an extreme case where an unstable, unfolded peptide monomer
folds only upon specific interaction with other peptide chains.
We begin our discussion with β-lactoglobulin, which exists as a dimer in
solution. The interface between the two protein monomers contains a large
hydrophobic patch [89]. This geometry suggests that hydrophobic interac-
tions help stabilize the complex. Because the protein must first dissociate to
unfold [90], the hydrophobic attractions also aid in preventing nucleation and
growth of β-lactoglobulin fibrils [89, 90]. This stability behavior is similar to the
enhanced stability seen in the clusters of our strongly directional proteins.
Ribonuclease A is also known to form dimers (as well as trimers and higher-
order oligomers) in solution under a variety of experimental conditions [91–93]
due to specific, directional interactions at its slightly unfolded N- and C-termini
[94]. Like β-lactoglobulin, hydrophobic interactions may help to stabilize the
oligomer complex [92]. However, it also has been observed that at high tem-
peratures, the presence of the ribonuclease oligomers is significantly decreased
[92]. This oligomer stability behavior is in qualitative agreement with the tem-
perature dependent stability behavior observed for our strongly directional
protein clusters. Figure 1.5b shows that the slope of fclust is positive. There-
fore, native proteins (unshaded region) that initially favor self-assembly ( fclust
0.5) at low temperatures, do not favor self-assembly (fclust 0.5) at higher
temperatures.
Sickle-cell hemoglobin has the propensity to form ordered aggregates
because of a point mutation that converts a surface hydrophilic residue to a](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-33-320.jpg)
![16 J. K. Cheung et al.
hydrophobic residue [95]. Because wild-type hemoglobin, like our model nondi-
rectional proteins, does not aggregate at low to intermediate concentrations,
this protein is an example of how changes to native-state surface hydrophobic-
ity can result in protein polymerization (i.e., the formation of ordered protein
clusters). Interestingly, self-association of native sickle-cell hemoglobin can be
extrapolated to occur for temperatures above the folding transition of the wild-
type protein (see, e.g., Ref. [96]). This suggests that the strongly favorable native
interactions of the sickle variant could play an active role in stabilizing the
native (“clustering”) form over the denatured state.
Finally we review the peptide p53tet. Due in part to its short chain length
of 64 residues, the monomeric form of the peptide is unstable. However, when
in contact with three neighboring peptide chains, the peptide “folds” to form a
tetrameric protein. The interface between the peptides consists of mostly apo-
lar residues [97], suggesting that the hydrophobic interactions can stabilize an
inherently unstable protein [97, 98]. In this case, the p53tet tetrameric protein
is analogous to our strongly directional model protein clusters, since the clus-
ters are formed by hydrophobic attractions and composed of native proteins
(i.e., directional interactions stabilize the geometric formation and the native
state).
Does the nature of the crowding species matter?
As mentioned previously, proteins are often a component of crowded
solutions in both pharmaceutical and biological settings. For example, the
concentration of therapeutic proteins used for subcutaneous injections can
be as high as 100 mg/ml [15]. In other cases, proteins are part of a complex
mixture of both interacting proteins and weakly interacting (inert) particles.
For example, cellular cytoplasm consists of macromolecules, structural pro-
teins (e.g., actin fibrils) and nonstructural proteins [13, 14]. The estimated
concentration of the nonstructural proteins is 110 mg/ml [99]. Addition-
ally, inert polymers and macromolecules are often found in pharmaceuti-
cal protein solutions [100]. Is the equilibrium stability behavior of proteins
in crowded solutions, where there are multiple species present, similar to
the equilibrium stability behavior of native proteins in concentrated protein
solutions, where there is a single protein species present? In other words,
does the type of crowding species strongly affect the equilibrium protein
stability?
Thermodynamically, the more compact state (usually the native conforma-
tion) is favored in solutions that contain high concentrations of inert crowding
species [101–105]. Within the framework of our coarse-grained model, however,
we have shown that the stability of native proteins in concentrated protein](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-34-320.jpg)
![Protein stability in concentrated and crowded solutions 17
solution depends not only on stabilizing crowding effects but also on destabi-
lizing protein–protein attractions. The strength of these attractions depends on
some of the intrinsic properties of the protein – e.g., the sequence hydropho-
bicity, the distribution of hydrophobic residues on the native protein surface,
and the sizes of the native and denatured states – as well as on solution con-
ditions such as temperature and pressure [16, 17, 39]. For proteins that do not
exhibit directional interactions, high sequence hydrophobicity results in non-
monotonic concentration effects on stability [16] (see Fig. 1.4a) and liquid–liquid
demixing [17] of protein solutions.
To study the effects of crowders on equilibrium protein stability, we extend
our original coarse-grained model to account for the presence of multiple
species in solution. We compare the equilibrium fraction of folded proteins
for two different cases. The first solution consists of foldable proteins (i.e.,
marginally stable proteins that will fold and unfold) and an ultrastable native
crowder (i.e., proteins with interactions identical to the foldable proteins but
that remain in their native state and do not unfold). The second solution con-
sists of foldable proteins and inert, hard-sphere crowders with the same effective
diameter as the foldable species.
Figure 1.6 shows a preliminary result for a nondirectional protein of chain
length Nr = 154 and sequence hydrophobicity = 0.455. We observe that the
0.2
0.4
0.6
0.1 0.2 0.3
0.1 0.2 0.3
Foldable protein concentration
0.2
0.4
0.6
Native
fraction
of
foldable
proteins
HS crowders
Native crowders
(a) (b)
Fig. 1.6 Native fraction of foldable proteins as a function of foldable protein
concentration for the protein Nr = 154, = 0.455 at its infinite dilution folding
temperature. Protein concentration is defined as in Fig. 1.4. The arrow indicates
increasing concentration of (a) ultrastable native crowders and (b) inert hard-sphere
(HS) crowders. (See plate section for color version.)](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-35-320.jpg)
![Protein stability in concentrated and crowded solutions 19
will unfold if there is enough local free volume and the state-dependent contact
free energy of the unfolded protein with its neighbors is sufficiently favorable
to overcome any intrinsic thermodynamic stability of the native state. At high
protein concentrations, the compact, native conformation is favored over the
expanded, denatured state due to self-crowding effects. However, at low con-
centrations where crowding effects are minimal, a protein may be destabilized
due to the more favorable free energies associated with denatured protein–
protein contacts. If other factors are equal, we find that proteins with low
sequence hydrophobicity tend to be stabilized by increasing protein concen-
tration while proteins with high sequence hydrophobicity show nonmonotonic
stability trends [16, 17]. This behavior qualitatively agrees with experimentally
observed protein behavior [87, 88].
We also studied the effects of anisotropic protein–protein interactions on
protein stability and self-assembly behavior. In contrast with the nondirectional
proteins, the strongly directional proteins we studied were stabilized against
denaturation at even low protein concentrations by forming highly ordered
chains. This behavior is similar to the oligomerization and polymerization of
proteins in solution [90, 92, 95].
We are currently examining how different crowding species in solution
affect the equilibrium fraction of folded proteins. Our initial results show
that attractions play a nontrivial role in determining protein stability. While
inert crowders only stabilize marginally stable proteins, attractive crowders
can destabilize the native state at low foldable protein concentrations. Here
our findings may lead to some insights on the stability of proteins in environ-
ments that contain a broad mixture of macromolecules like cellular cytoplasm
or concentrated pharmaceutical biological solutions.
Protein solutions show a variety of rich behaviors and open questions remain
as to how environmental conditions affect their equilibrium stability trends.
For example, one can ask whether the self-assembly of strongly directional
proteins will change in the presence of other crowding species. This type of
heterogeneous protein solution may more accurately represent in vivo biolog-
ical environments. One can also ask whether protein solution stability (e.g.,
the demixing transition observed in Ref. [17]) is affected by protein and inert
crowding species. If the equilibrium demixing transition can be prevented by
the addition of stabilizing cosolute crowders, protein drug shelf-life may be
improved. The general versatility of the coarse-grained framework reviewed
here provides the flexibility to investigate many different types of protein solu-
tions commonly encountered in biological and pharmaceutical environments,
which should allow it to provide insights into some of these open questions.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-37-320.jpg)
![Observations on the mechanics of a molecular bond under force 27
probable time for bond separation, the maximum force that can be sustained by
the bond, and the sensitivity of bond separation behavior to the rate of loading
and bond well features, for example.
The second issue considered is the stability of a molecular bond by which
a membrane is attached to a substrate. Again, the bond is represented by an
energy well in the landscape representing the dependence of bond energy on
its configuration. In this case, the system is in thermal equilibrium with its
environment, so the membrane position fluctuates continuously as a result. In
the course of random fluctuations, the membrane exerts a fluctuating force
on the bond which is equal but opposite to the constraint force acting on the
membrane. The particular question posed concerns the largest edge-to-edge
span that the membrane can have without overwhelming the bond through
thermal fluctuations. The phenomenon of interest is a feature of steady behavior
and, consequently, it is studied by means of classical statistical mechanics. The
result of the analysis is of potential interest in understanding the spacing of
integrins within a focal adhesion zone as it is formed in the course of adhesion
of a biological cell.
The two issues addressed are fundamentally different aspects of the same
physical system. Although each can be understood in a fairly general way, there
are few results of a general nature that can be inferred about the transient
behavior of the full system of an interacting bond and the membrane to which
one of the bound molecules is attached. However, the results presented here
are seen as steps in the direction of achieving that objective.
Forced separation of a molecular bond
In this section, we examine the situation of a molecular bond being
acted upon by a loading apparatus that tends to pull the bond apart. The bond
itself is described by means of an expression of bond energy versus a reaction
coordinate, commonly known as the energy landscape of the bond. The loading
apparatus is described by means of a deformable element linking the reac-
tion coordinate to a point at which a time-dependent constraint is externally
imposed. The force on the bond is understood to be the force needed to main-
tain this constraint. The reaction coordinate is stochastic and can be described
only in terms of a time-dependent probability distribution whereas the imposed
constraint is deterministic.
That such a process can be observed has been demonstrated by Evans et al.
[1], who have reported measurements of the dependence of the maximum force
required to overcome the bond on the rate of application of that force. An
intriguing question concerns the way in which the behavior observed is related](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-45-320.jpg)
![28 L. B. Freund
Force at rupture
No.
of
rupture
events
Fixed
loading
rate
Fig. 2.1 A generic diagram illustrating the way in which data were reported by
Evans and Ritchie [2] on the observed behavior of isolated molecular bonds under
the action of steadily increasing force. The response of a large number of individual
bonds under nominally identical loading conditions was represented as a
histogram in this form.
to the properties of the bond being overcome. Equilibrium statistical mechanics
is too restrictive for addressing such questions, so the process is viewed more
generally as being a stochastic transport phenomenon.
The general character of the data on bond rupture strength reported by Evans
and Ritchie [2] is as follows. Suppose a large number (several hundred) of rupture
events are observed for nominally identical bond pairs and nominally identical
rates of loading up to rupture. If the results are presented in the form of a
histogram of number of rupture events that occurred within each of numerous
finite increments of applied force, the results take the form illustrated in Fig. 2.1.
The spread of the data along the force axis suggests that the process is statistical
in nature. The main influence of an increase in the rate of loading was found to
be a stretch of the distribution in the direction along the force axis. In particular,
the peak in the distribution moves to a larger force value as the rate of loading
increases. The purpose in this section is to discuss a theoretical framework for
interpreting such observations.
A model of bond breaking
In selecting features to be incorporated into a model of the bond rup-
ture process, we follow the lead of Evans and Ritchie [2] in several respects.
The bond itself is idealized as a one-dimensional energy landscape, as depicted
in Fig. 2.2. The abscissa x in this diagram is the so-called reaction coordi-
nate, a variable that determines the configuration of the bond. While the
process of bond rupture might require extensive conformational changes of
the molecules, such as twisting or unfolding, the energy input that effects these
changes is assumed to be delivered by a force working through the translational
coordinate x.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-46-320.jpg)
![30 L. B. Freund
substrate
bound molecules
loading
apparatus
x y (t)
f (t )
Fig. 2.3 The diagram is a schematic of the entire system. The configuration of the
bond is represented by a single time-dependent reaction coordinate x, which varies
stochastically, and a time-dependent coordinate y which is specified a priori. The
essential properties of the bond are Cb and δ, and the properties of the loading
apparatus are also specified. The applied force f (t) is calculated to be the force
necessary to maintain the specified motion y(t).
loading apparatus. As an illustration, suppose that the former is described by
Ub(x)
kT
= ub(ξ) =
−Cb + 2Cbξ2 , ξ 0.5
−2Cb(ξ − 1)2 , 0.5 ≤ ξ ≤ 1
0 , 1 ξ
(2.1)
where ξ = x/δ is the nondimensional variable introduced in Fig. 2.2 and Cb is
the depth of the well. The latter is a linear spring described by
Us(x, y)
kT
= us(ξ, η) =
1
2
κ(η − ξ)2 (2.2)
where η = y/δ and the nondimensional parameter κ is the spring stiffness
normalized by kT/δ2. The energy landscape is then the surface U(x, y) = Ub(x) +
Us(x, y) over the x, y-plane or, in normalized parameters, u(ξ, η) = ub(ξ)+us(ξ, η)
over the ξ, η-plane. The features of this surface can be visualized in several
ways. Here, we do so by plotting cross-sections of the surface for several val-
ues of η. This is illustrated in Fig. 2.4 for Cb = 10 and the stiffness value
κ = 1. The entire surface near ξ = 0 translates in the direction of the energy axis
(that is, “upward” in the figure) as η increases, so the quantity plotted is actu-
ally u(ξ, η) − 1
2 κη2 to compensate for this translation. In their original report,
Evans and Ritchie [2] assumed that the external force was applied directly to
the reaction coordinate, whereas a spring to represent loading compliance was
adopted in the qualitative discussion of force spectroscopy given by Evans and
Calderwood [3].
The main features of the evolving surface are evident in Fig. 2.4. The surface
has a zero energy trough along the line y = x. This is the state with the spring
relaxed and the molecules fully separated. This is where the system ends up on
the energy surface after complete bond separation.
Initially, with y = 0, the bond well is the expected location or state of the
system. As y increases, the well is elevated to higher energy levels relative to the](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-48-320.jpg)
![Observations on the mechanics of a molecular bond under force 31
–1 1 2 3 4 5
–15
–10
–5
5
10
h = 0
20
40
60
j
0
u (j,h) –
1
2
kh2
Fig. 2.4 The two-dimensional energy landscape of the system u(ξ, η) is illustrated
by means of several cross-sections with η =constant in terms of nondimensional
coordinates ξ = x/δ and η = y/δ. The level of the landscape rises quite rapidly with
increasing η because energy is continually added by the applied force.
Consequently, a time-dependent translation is subtracted for each section so as to
represent all sections in the same coordinate system.
zero energy trough, and more so at the left end than at the right end, thereby
giving the impression that the local landscape is rotated in a clockwise sense.
This distortion promotes the flux of states from within the well over the barrier
and in the direction of the zero energy trough. This is more or less the same
viewpoint introduced by Evans and Ritchie [2], except that they had the applied
force working directly through the reaction coordinate. Some differences will
emerge in the analysis of the model below, however.
It should be recognized that this representation of the energy landscape of the
molecular bond in terms of one or two degrees of freedom is extremely coarse,
in light of the structural complexity of the molecules whose interactions form
the bond. However, at this point in the development of the topic, there is little
basis for choosing a more elaborate representation, although numerical studies
of molecular interactions such as those reported in [4] should begin to provide
guidance for doing so in the future. In an interesting discussion of breaking of
so-called catch bonds between certain types of molecules due to force, Thomas
[5] has suggested that such structures may account for the unusual behavior
exhibited by these molecular pairs.
With this picture of the energy landscape in place, the question to be
addressed is the following. If the state of the system initially resides in the
bond interaction well when y = 0 and if the coordinate y is specified to be some
increasing function of time thereafter, what is the time history of the force
acting on the bond and of the probability distribution of bond configurations?](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-49-320.jpg)
![32 L. B. Freund
The state of the bond
In view of the stochastic nature of bond behavior, the state of the bond
is represented by a time-dependent probability distribution which is a function
of the reaction coordinate x, say ρ(x, t). If a large number of identical systems are
observed under nominally identical constraints, the corresponding distribution
in values of the reaction coordinate at a certain instant of time provides a density
of states for the system at that time. If this density of states is normalized so
as to represent the fractional distribution of states, it can be interpreted as a
probability distribution for the current value of reaction coordinate in a particular
system. In the discussion that follows, the terms density of states and probability
distribution are used interchangeably with the understanding that they refer
to one and the same thing. To ensure that all states are taken into account over
time, this probability distribution ρ(x, t) on the finite interval x0 x x1 is
constrained by the conservation condition
x1
x0
ρ(x, t) dx −
t
0
j0(s) ds +
t
0
j1(s) ds = 1 (2.3)
identically in time, where j0 and j1 represent fluxes of states inward at x = x0
and outward at x = x1.
In the thermal environment of a solvent, the molecules are continually bom-
barded by impulses, each of which delivers an energy on the order of kT. This
energy is dissipated back into the environment through the viscous resistance
of the solvent in the course of motion of the molecules over a small distance in
a short time. If the depth of the interaction energy well in the energy landscape
is significantly greater than kT, then the likelihood of escape from the well (that
is, debonding) on the timescale of individual excitations is very small and the
probability distribution evolves diffusively, rather than ballistically. In other
words, transition of a configuration out of the well is an improbable event. In
this setting, the diffusion coefficient D is expressible in terms of the viscosity
γ of the solvent and the thermal energy unit kT through the Einstein relation
D = kT/γ [6]. This is the point of view of chemical kinetics that was developed
through the work of Smoluchowski and Kramers, among others, as described
in systematic detail by Risken [7] and Hanngi [8]. This viewpoint provides the
basis for extracting a partial differential equation governing the probability
distribution.
The local flux of states in the direction of increasing x, here denoted by j(x, t),
has the form
j(x, t) = −
kT
γ
∂ρ
∂x
(x, t) + ρ(x, t)
∂u
∂x
(x, t)
(2.4)
where u(x, t) = U(x, y(t))/kT. The first term represents the diffusive transport
of states in a force-free environment due to a thermally activated random walk](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-50-320.jpg)
![Observations on the mechanics of a molecular bond under force 33
whereas the second term represents directed transport in the downhill direction
on the local energy landscape. The requirement of local conservation of states,
∂ρ
∂t
(x, t) +
∂j
∂x
(x, t) = 0 , (2.5)
is equivalent to (2.3). Combining (2.4) with the conservation condition leads
to the Smoluchowski partial differential equation [7] governing the probability
density,
∂ρ
∂t
(x, t) =
kT
γ
∂
∂x
∂ρ
∂x
(x, t) + ρ(x, t)
∂u(x, t)
∂x
. (2.6)
This equation must be augmented by an initial condition and suitable boundary
conditions in order to obtain a solution. Boundary conditions will be specified
in connection with a particular case of interest.
As an initial condition at time t = 0, we assume that the distribution of states
is confined to the bond well and that the flux of states is zero. This implies that
the initial condition is determined by setting the quantity enclosed in square
brackets on the right side of (2.6) equal to zero. The initial probability distri-
bution is then determined as the solution of the ordinary differential equation
∂xρ(x, 0) + ρ(x, 0)∂xu(x, 0) = 0 and the normality condition (2.3) to be
ρ(x, 0) =
e−u(x,0)
L e−u(x,0) dx
(2.7)
where L denotes the spatial range of the reaction coordinate over the landscape.
We again denote the half-width of the bond well by the length δ and normal-
ize all distances by this characteristic dimension. Suppose we also normalize
time by the diffusion time over the distance δ, which is γ δ2/kT. If a nondi-
mensional reaction coordinate ξ and a nondimensional time τ, as defined by
ξ = x/δ τ = tkT/γ δ2 (2.8)
are introduced, then the Smoluchowski equation takes on the parameter-free
form
∂ρ
∂τ
(ξ, τ) =
∂
∂ξ
∂ρ
∂ξ
(ξ, τ) + ρ(ξ, τ)
∂u
∂ξ
(ξ, τ)
. (2.9)
This equation is to be enforced over the spatial range L of the landscape, which
is −1 ξ 3 in Fig. 2.4, and over all time τ 0. When augmented by the ini-
tial condition (2.7) and suitable boundary conditions, this partial differential
equation can be solved approximately by any number of numerical strate-
gies. However, before proceeding in that direction it is worthwhile to examine
the range of the dimensionless independent variables that are relevant to the
phenomenon being considered.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-51-320.jpg)
![34 L. B. Freund
The important range for ξ is set by the range of the physical coordinate x
required to span the interaction energy well and its surroundings, which is a
distance equal to several times the length δ. Consequently, the range of ξ of
interest is roughly −1 ξ 3. Similarly, the important range for τ is set by the
elapsed time from the instant a force is first applied to the time of bond sepa-
ration, which is typically on the order of 10 ms or greater. The corresponding
range of τ can be estimated from (2.8). Assuming the factor kT/γ to have a value
equal to the diffusion coefficient for water, or roughly 10−10 m2/s, and δ to be
about 5 nm, then the range of τ corresponding to elapsed time of 10 ms is
roughly 102. This implies that the probability distribution ρ(ξ, τ) varies slowly
on the landscape in the τ direction compared to its variation in the ξ direction
for comparable increments in ξ and τ. This observation suggests, in turn, that
the rate of change of ρ on the left side of (2.9) is very small compared to the
other contributions. It is assumed to be negligibly small for present purposes
so that the governing equation reduces to
∂
∂ξ
∂ρ
∂ξ
(ξ, τ) + ρ(ξ, τ)
∂u
∂ξ
(ξ, τ)
≈ 0 (2.10)
with τ viewed as a parameter. In this way, the partial differential equation has
been reduced to an ordinary differential equation to be solved for all values of
time τ.
A formulation similar to the present development was proposed by Heymann
and Grubmuller [9] but with one significant difference. No specific landscape
profile was presumed at the outset. Instead, conclusions were drawn by intro-
ducing reaction rate coefficients early in the analysis and, subsequently, the
shape of the landscape profile was extracted from observational data by means
of the results obtained.
The probability distribution ρ(ξ, τ)
In an important contribution to physical chemistry, Kramers (see
Ref. [7]) showed how the “on” and “off” rate coefficients of elementary rate reac-
tion theory can be determined from any presumed time-independent energy
landscape for a reaction occurring under diffusion-dominated circumstances.
Through this result, it became possible to determine the influence of distance
between an equilibrium state and the transition state in terms of reaction coor-
dinate, the influence of attempt frequency on rate, the influence of curvature
of the energy surface at the transition state, and so on. As introduced originally
in elementary reaction rate theory, the coefficients depend on the energies at
the equilibrium states and the transition state, but not on the shape of the
landscape between these energy levels. By superimposing a constant uniform
flux of probability density from one equilibrium state to another, the govern-
ing Smoluchowski partial differential equation could be reduced to an ordinary](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-52-320.jpg)
![flexure of the stand under oscillations of the pendulum. At the Stuttgart
conference of the geodetic association in 1877, Hervé Faye proposed to solve the
problem of flexure by swinging two similar pendulums from the same support with
equal amplitudes and in opposite phases. Peirce, in 1879, demonstrated
theoretically the soundness of the method and presented a design for its
application, but the “double pendulum” was rejected at that time. Peirce also
designed and had constructed four examples of a new type of invariable,
reversible pendulum of cylindrical form which made possible the experimental
study of Stokes’ theory of the resistance to motion of a pendulum in a viscous
fluid. Commandant Defforges, of France, also designed and used cylindrical
reversible pendulums, but of different length so that the effect of flexure was
eliminated in the reduction of observations. Maj. Robert von Sterneck, of Austria-
Hungary, initiated a new era in gravity research by the invention of an apparatus
with a short pendulum for relative determinations of gravity. Stands were then
constructed in Europe on which two or four pendulums were hung at the same
time. Finally, early in the present century, Vening Meinesz found that the Faye-
Peirce method of swinging pendulums hung on a Stückrath four-pendulum stand
solved the problem of instability due to the mobility of the soil in Holland.
The 20th century has witnessed increasing activity in the determination of
absolute and relative values of gravity. Gravimeters have been perfected and have
been widely used for rapid relative determinations, but the compound pendulums
remain as indispensable instruments. Mendenhall’s replacement of knives by
planes attached to nonreversible pendulums has been used also for reversible
ones. The Geodetic Institute at Potsdam is presently applying the Faye-Peirce
method to the reversible pendulum. [115] Pendulums have been constructed of
new materials, such as invar, fused silica, and fused quartz. Minimum pendulums
for precise relative determinations have been constructed and used. Reversible
pendulums have been made with “I” cross sections for better stiffness. With all
these modifications, however, the foundations of the present designs of compound
pendulum apparatus were created in the 19th century.
FOOTNOTES
[1] The basic historical documents have been collected, with a bibliography of works and
memoirs published from 1629 to the end of 1885, in Collection de mémoires relatifs a la
physique, publiés par la Société française de Physique [hereinafter referred to as
Collection de mémoires]: vol. 4, Mémoires sur le pendule, précédés d’une bibliographie
(Paris: Gauthier-Villars, 1889); and vol. 5, Mémoires sur le pendule, part 2 (Paris:
Gauthier-Villars, 1891). Important secondary sources are: C. Wolf, “Introduction
historique,” pp. 1-42 in vol. 4, above; and George Biddell Airy, “Figure of the Earth,” pp.
165-240 in vol. 5 of Encyclopaedia metropolitana (London, 1845).](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-56-320.jpg)
![[2] Galileo Galilei’s principal statements concerning the pendulum occur in his Discourses
Concerning Two New Sciences, transl. from Italian and Latin into English by Henry Crew
and Alfonso de Salvio (Evanston: Northwestern University Press, 1939), pp. 95-97, 170-
172.
[3] P. Marin Mersenne, Cogitata physico-mathematica (Paris, 1644), p. 44.
[4] Christiaan Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia
adaptato demonstrationes geometricae (Paris, 1673), proposition 20.
[5] The historical events reported in the present section are from Airy, “Figure of the Earth.”
[6] Abbé Jean Picard, La Mesure de la terre (Paris, 1671). John W. Olmsted, “The ‘Application’ of
Telescopes to Astronomical Instruments, 1667-1669,” Isis (1949), vol. 40, p. 213.
[7] The toise as a unit of length was 6 Paris feet or about 1,949 millimeters.
[8] Jean Richer, Observations astronomiques et physiques faites en l’isle de Caïenne (Paris,
1679). John W. Olmsted, “The Expedition of Jean Richer to Cayenne 1672-1673,” Isis
(1942), vol. 34, pp. 117-128.
[9] The Paris foot was 1.066 English feet, and there were 12 lines to the inch.
[10] Christiaan Huygens, “De la cause de la pesanteur,” Divers ouvrages de mathematiques et
de physique par MM. de l’Académie Royale des Sciences (Paris, 1693), p. 305.
[11] Isaac Newton, Philosophiae naturalis principia mathematica (London, 1687), vol. 3,
propositions 18-20.
[12] Pierre Bouguer, La figure de la terre, déterminée par les observations de Messieurs
Bouguer et de La Condamine, envoyés par ordre du Roy au Pérou, pour observer aux
environs de l’equateur (Paris, 1749).
[13] P. L. Moreau de Maupertuis, La figure de la terre déterminée par les observations de
Messieurs de Maupertuis, Clairaut, Camus, Le Monnier, l’Abbé Outhier et Celsius, faites
par ordre du Roy au cercle polaire (Paris, 1738).
[14] Paris, 1743.
[15] George Gabriel Stokes, “On Attraction and on Clairaut’s Theorem,” Cambridge and Dublin
Mathematical Journal (1849), vol. 4, p. 194.
[16] See Collection de mémoires, vol. 4, p. B-34, and J. H. Poynting and Sir J. J. Thomson,
Properties of Matter (London, 1927), p. 24.
[17] Poynting and Thomson, ibid., p. 22.
[18]
Charles M. de la Condamine, “De la mesure du pendule à Saint Domingue,” Collection de
mémoires, vol. 4, pp. 3-16.
[19]
Père R. J. Boscovich, Opera pertinentia ad Opticam et Astronomiam (Bassani, 1785), vol.
5, no. 3.
[20]
J. C. Borda and J. D. Cassini de Thury, “Expériences pour connaître la longueur du pendule
qui bat les secondes à Paris,” Collection de mémoires, vol. 4, pp. 17-64.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-57-320.jpg)
![[21] F. W. Bessel, “Untersuchungen über die Länge des einfachen Secundenpendels,”
Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, 1826 (Berlin,
1828).
[22] Bessel used as a standard of length a toise which had been made by Fortin in Paris and
had been compared with the original of the “toise de Peru” by Arago.
[23] L. G. du Buat, Principes d’hydraulique (Paris, 1786). See excerpts in Collection de
mémoires, pp. B-64 to B-67.
[24] Capt. Henry Kater, “An Account of Experiments for Determining the Length of the
Pendulum Vibrating Seconds in the Latitude of London,” Philosophical Transactions of
the Royal Society of London (1818), vol. 108, p. 33. [Hereinafter abbreviated Phil.
Trans.]
[25] M. G. de Prony, “Méthode pour déterminer la longueur du pendule simple qui bat les
secondes,” Collection de mémoires, vol. 4, pp. 65-76.
[26] Collection de mémoires, vol. 4, p. B-74.
[27] Phil. Trans. (1819), vol. 109, p. 337.
[28] John Herschel, “Notes for a History of the Use of Invariable Pendulums,” The Great
Trigonometrical Survey of India (Calcutta, 1879), vol. 5.
[29] Capt. Edward Sabine, “An Account of Experiments to Determine the Figure of the Earth,”
Phil. Trans. (1828), vol. 118, p. 76.
[30] John Goldingham, “Observations for Ascertaining the Length of the Pendulum at Madras in
the East Indies,” Phil. Trans. (1822), vol. 112, p. 127.
[31] Basil Hall, “Letter to Captain Kater Communicating the Details of Experiments made by
him and Mr. Henry Foster with an Invariable Pendulum,” Phil. Trans. (1823), vol. 113, p.
211.
[32] See Collection de mémoires, vol. 4, p. B-103.
[33] Ibid., p. B-88.
[34] Ibid., p. B-94.
[35] Francis Baily, “On the Correction of a Pendulum for the Reduction to a Vacuum, Together
with Remarks on Some Anomalies Observed in Pendulum Experiments,” Phil. Trans.
(1832), vol. 122, pp. 399-492. See also Collection de mémoires, vol. 4, pp. B-105, B-
112, B-115, B-116, and B-117.
[36] One was of case brass and the other of rolled iron, 68 in. long, 2 in. wide, and 1
/2 in.
thick. Triangular knife edges 2 in. long were inserted through triangular apertures 19.7
in. from the center towards each end. These pendulums seem not to have survived.
There is, however, in the collection of the U.S. National Museum, a similar brass
pendulum, 375
/8 in. long (fig. 15) stamped with the name of Edward Kübel (1820-96),
who maintained an instrument business in Washington, D.C., from about 1849. The
history of this instrument is unknown.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-58-320.jpg)
![[37] See Baily’s remarks in the Monthly Notices of the Royal Astronomical Society (1839), vol.
4, pp. 141-143. See also letters mentioned in footnote 38.
[38] This document, together with certain manuscript notes on the pendulum experiments and
six letters between Wilkes and Baily, is in the U.S. National Archives, Navy Records Gp.
37. These were the source materials for the information presented here on the
Expedition. We are indebted to Miss Doris Ann Esch and Mr. Joseph Rudmann of the
staff of the U.S. National Museum for calling our attention to this early American
pendulum work.
[39] G. B. Airy, “Account of Experiments Undertaken in the Harton Colliery, for the Purpose of
Determining the Mean Density of the Earth,” Phil. Trans. (1856), vol. 146, p. 297.
[40] T. C. Mendenhall, “Measurements of the Force of Gravity at Tokyo, and on the Summit of
Fujiyama,” Memoirs of the Science Department, University of Tokyo (1881), no. 5.
[41] J. T. Walker, Account of Operations of The Great Trigonometrical Survey of India
(Calcutta, 1879), vol. 5, app. no. 2.
[42] Bessel, op. cit. (footnote 21), article 31.
[43] C. A. F. Peters, Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (Altona,
Germany, 1860), Band 2, p. 3. The correction required if the times of swing are not
exactly the same is said to have been given also by Bohnenberger.
[44] F. W. Bessel, “Construction eines symmetrisch geformten Pendels mit reciproken Axen, von
Bessel,” Astronomische Nachrichten (1849), vol. 30, p. 1.
[45] E. Plantamour, “Expériences faites à Genève avec le pendule à réversion,” Mémoires de la
Société de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, p.
309.
[46] Ibid., pp. 309-416.
[47] C. Cellérier, “Note sur la Mesure de la Pesanteur par le Pendule,” Mémoires de la Société
de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, pp. 197-
218.
[48] A. Sawitsch, “Les variations de la pesanteur dans les provinces occidentales de l’Empire
russe,” Memoirs of the Royal Astronomical Society (1872), vol. 39, p. 19.
[49] J. J. Baeyer, Über die Grösse und Figur der Erde (Berlin, 1861).
[50] Comptes-rendus de la Conférence Géodésique Internationale réunie à Berlin du 15-22
Octobre 1864 (Neuchâtel, 1865).
[51] Ibid., part III, subpart E.
[52] Bericht über die Verhandlungen der vom 30 September bis 7 October 1867 zu Berlin
abgehaltenen allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1868).
See report of fourth session, October 3, 1867.
[53]
C. Bruhns and Albrecht, “Bestimmung der Länge des Secundenpendels in Bonn, Leiden
und Mannheim,” Astronomisch-Geodätische Arbeiten im Jahre 1870 (Leipzig:
Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1871).](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-59-320.jpg)
![[54] Bericht über die Verhandlungen der vom 23 bis 28 September 1874 in Dresden
abgehaltenen vierten allgemeinen Conferenz der Europäischen Gradmessung (Berlin,
1875). See report of second session, September 24, 1874.
[55] Carolyn Eisele, “Charles S. Peirce—Nineteenth-Century Man of Science,” Scripta
Mathematica (1959), vol 24, p. 305. For the account of the work of Peirce, the authors
are greatly indebted to this pioneer paper on Peirce’s work on gravity. It is worth noting
that the history of pendulum work in North America goes back to the celebrated Mason
and Dixon, who made observations of “the going rate of a clock” at “the forks of the
river Brandiwine in Pennsylvania,” in 1766-67. These observations were published in
Phil. Trans. (1768), vol. 58, pp. 329-335.
[56] The pendulums with conical bobs are described and illustrated in E. D. Preston,
“Determinations of Gravity and the Magnetic Elements in Connection with the United
States Scientific Expedition to the West Coast of Africa, 1889-90,” Report of the
Superintendent of the Coast and Geodetic Survey for 1889-90 (Washington, 1891), app.
no. 12.
[57] Eisele, op. cit. (footnote 55), p. 311.
[58] The record of Peirce’s observations in Europe during 1875-76 is given in C. S. Peirce,
“Measurements of Gravity at Initial Stations in America and Europe,” Report of the
Superintendent of the Coast Survey for 1875-76 (Washington, 1879), pp. 202-337 and
410-416. Peirce’s report is dated December 13, 1878, by which time the name of the
Survey had been changed to U.S. Coast and Geodetic Survey.
[59] Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten Permanenten
Commission der Europäischen Gradmessung (Berlin, 1876).
[60] Ibid. See report for fifth session, September 25, 1875.
[61] The experiments at the Stevens Institute, Hoboken, were reported by Peirce to the
Permanent Commission which met in Hamburg, September 4-8, 1878, and his report
was published in the general Bericht for 1878 in the Verhandlungen der vom 4 bis 8
September 1878 in Hamburg Vereinigten Permanenten Commission der Europäischen
Gradmessung (Berlin, 1879), pp. 116-120. Assistant J. E. Hilgard attended for the U.S.
Coast and Geodetic Survey. The experiments are described in detail in C. S. Peirce, “On
the Flexure of Pendulum Supports,” Report of the Superintendent of the U.S. Coast and
Geodetic Survey for 1880-81 (Washington, 1883), app. no. 14, pp. 359-441.
[62] Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten Permanenten
Commission der Europäischen Gradmessung (Berlin, 1877). See report of third session,
October 7, 1876.
[63] Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart abgehaltenen
fünften allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1878).
[64] Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten Permanenten
Commission der Europäischen Gradmessung (Berlin, 1880).
[65] Assistants’ Reports, U.S. Coast and Geodetic Survey, 1879-80. Peirce’s paper was
published in the American Journal of Science (1879), vol. 18, p. 112.
[66] Comptes-rendus de l’Académie des Sciences (Paris, 1879), vol. 89, p. 462.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-60-320.jpg)
![[67] Verhandlungen der vom 13 bis 16 September 1880 zu München abgehaltenen sechsten
allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1881).
[68] Ibid., app. 2.
[69] Ibid., app. 2a.
[70] Verhandlungen der vom 11 bis zum 15 September 1882 im Haag Vereinigten
Permanenten Commission der Europäischen Gradmessung (Berlin, 1883).
[71] Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen siebenten
allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1884). Gen. Cutts
attended for the U.S. Coast and Geodetic Survey.
[72] Ibid., app. 6. See also, Zeitschrift für Instrumentenkunde (1884), vol. 4, pp. 303 and 379.
[73] Op. cit. (footnote 67).
[74] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81
(Washington, 1883), p. 26.
[75] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1889-90
(Washington, 1891), app. no. 12.
[76] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82
(Washington, 1883).
[77] Transactions of the Cambridge Philosophical Society (1856), vol. 9, part 2, p. 8. Also
published in Mathematical and Physical Papers (Cambridge, 1901), vol. 3, p. 1.
[78] Peirce’s comparison of theory and experiment is discussed in a report on the Peirce
memoir by William Ferrel, dated October 19, 1890, Martinsburg, West Virginia. U.S.
Coast and Geodetic Survey, Special Reports, 1887-1891 (MS, National Archives,
Washington).
[79] The stations at which observations were conducted with the Peirce pendulums are
recorded in the reports of the Superintendent of the U.S. Coast and Geodetic Survey
from 1881 to 1890.
[80] Comptes-rendus de l’Académie des Sciences (Paris, 1880), vol. 90, p. 1401. Hervé Faye’s
report, dated June 21, 1880, is in the same Comptes-rendus, p. 1463.
[81] Commandant C. Defforges, “Sur l’Intensité absolue de la pesanteur,” Journal de Physique
(1888), vol. 17, pp. 239, 347, 455. See also, Defforges, “Observations du pendule,”
Mémorial du Dépôt général de la Guerre (Paris, 1894), vol. 15. In the latter work,
Defforges described a pendulum “reversible inversable,” which he declared to be truly
invariable and therefore appropriate for relative determinations. The knives remained
fixed to the pendulums, and the effect of interchanging knives was obtained by
interchanging weights within the pendulum tube.
[82] Papers by Maj. von Sterneck in Mitteilungen des K. u. K. Militär-geographischen Instituts,
Wien, 1882-87; see, in particular, vol. 7 (1887).
[83]
T. C. Mendenhall, “Determinations of Gravity with the New Half-Second Pendulum …,”
Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1890-91
(Washington, 1892), part 2, pp. 503-564.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-61-320.jpg)
![[84] W. H. Burger, “The Measurement of the Flexure of Pendulum Supports with the
Interferometer,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for
1909-10 (Washington, 1911), app. no. 6.
[85] E. J. Brown, A Determination of the Relative Values of Gravity at Potsdam and Washington
(Special Publication No. 204, U.S. Coast and Geodetic Survey; Washington, 1936).
[86] M. Haid, “Neues Pendelstativ,” Zeitschrift für Instrumentenkunde (July 1896), vol. 16, p.
193.
[87] Dr. R. Schumann, “Über eine Methode, das Mitschwingen bei relativen Schweremessungen
zu bestimmen,” Zeitschrift für Instrumentenkunde (January 1897), vol. 17, p. 7. The
design for the stand is similar to that of Peirce’s of 1879.
[88] Dr. R. Schumann, “Über die Verwendung zweier Pendel auf gemeinsamer Unterlage zur
Bestimmung der Mitschwingung,” Zeitschrift für Mathematik und Physik (1899), vol. 44,
p. 44.
[89] P. Furtwängler, “Über die Schwingungen zweier Pendel mit annähernd gleicher
Schwingungsdauer auf gemeinsamer Unterlage,” Sitzungsberichte der Königlicher
Preussischen Akademie der Wissenschaften zu Berlin (Berlin, 1902) pp. 245-253. Peirce
investigated the plan of swinging two pendulums on the same stand (Report of the
Superintendent of the U.S. Coast and Geodetic Survey for 1880-81, Washington, 1883,
p. 26; also in Charles Sanders Peirce, Collected Papers, 6.273). At a conference on
gravity held in Washington during May 1882, Peirce again advanced the method of
eliminating flexure by hanging two pendulums on one support and oscillating them in
antiphase (“Report of a conference on gravity determinations held in Washington, D.C.,
in May, 1882,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for
1881-82, Washington, 1883, app. no. 22, pp. 503-516).
[90] F. A. Vening Meinesz, Observations de pendule dans les Pays-Bas (Delft, 1923).
[91] A. Berroth, “Schweremessungen mit zwei und vier gleichzeitig auf demselben Stativ
schwingenden Pendeln,” Zeitschrift für Geophysik, vol. 1 (1924-25), no. 3, p. 93.
[92] “Pendulum Apparatus for Gravity Determinations,” Engineering (1926), vol. 122, pp. 271-
272.
[93] Malcolm W. Gay, “Relative Gravity Measurements Using Precision Pendulum Equipment,”
Geophysics (1940), vol. 5, pp. 176-191.
[94] L. G. D. Thompson, “An Improved Bronze Pendulum Apparatus for Relative Gravity
Determinations,” [published by] Dominion Observatory (Ottawa, 1959), vol. 21, no. 3,
pp. 145-176.
[95] W. A. Heiskanen and F. A. Vening Meinesz, The Earth and its Gravity Field (McGraw: New
York, 1958).
[96] F. Kühnen and P. Furtwängler, Bestimmung der Absoluten Grösze der Schwerkraft zu
Potsdam mit Reversionspendeln (Berlin: Veröffentlichungen des Königlichen
Preussischen Geodätischen Instituts, 1906), new ser., no. 27.
[97] Reported by Dr. F. Kühnen to the fifth session, October 9, 1895, of the Eleventh General
Conference, Die Internationale Erdmessung, held in Berlin from September 25 to
October 12, 1895. A footnote states that Assistant O. H. Tittmann, who represented the](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-62-320.jpg)
![United States, subsequently reported Peirce’s prior discovery of the influence of the
flexure of the pendulum itself upon the period (Report of the Superintendent of the U.S.
Coast and Geodetic Survey for 1883-84, Washington, 1885, app. 16, pp. 483-485).
[98] Assistants’ Reports, U.S. Coast and Geodetic Survey, 1883-84 (MS, National Archives,
Washington).
[99]
C. S. Peirce, “Effect of the Flexure of a Pendulum Upon its Period of Oscillation,” Report of
the Superintendent of the U.S. Coast and Geodetic Survey for 1883-84 (Washington,
1885), app. no. 16.
[100]
F. R. Helmert, Beiträge zur Theorie des Reversionspendels (Potsdam: Veröffentlichungen
des Königlichen Preussischen Geodätischen Instituts, 1898).
[101]
J. A. Duerksen, Pendulum Gravity Data in the United States (Special Publication No. 244,
U.S. Coast and Geodetic Survey; Washington, 1949).
[102]
Ibid., p. 2. See also, E. J. Brown, loc. cit. (footnote 85).
[103]
Paul R. Heyl and Guy S. Cook, “The Value of Gravity at Washington,” Journal of Research,
National Bureau of Standards (1936), vol. 17, p. 805.
[104]
Sir Harold Jeffreys, “The Absolute Value of Gravity,” Monthly Notices of the Royal
Astronomical Society, Geophysical Supplement (London, 1949), vol. 5, p. 398.
[105]
J. S. Clark, “The Acceleration Due to Gravity,” Phil. Trans. (1939), vol. 238, p. 65.
[106]
Hugh L. Dryden, “A Reexamination of the Potsdam Absolute Determination of Gravity,”
Journal of Research, National Bureau of Standards (1942), vol. 29, p. 303; and A.
Berroth, “Das Fundamentalsystem der Schwere im Lichte neuer
Reversionspendelmessungen,” Bulletin Géodésique (1949), no. 12, pp. 183-204.
[107]
T. C. Mendenhall, op. cit. (footnote 83), p. 522.
[108]
A. H. Cook, “Recent Developments in the Absolute Measurement of Gravity,” Bulletin
Géodésique (June 1, 1957), no. 44, pp. 34-59.
[109]
See footnote 89.
[110]
C. S. Peirce, “On the Deduction of the Ellipticity of the Earth, from Pendulum
Experiments,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for
1880-81 (Washington, 1883), app. no. 15, pp. 442-456.
[111]
Heiskanen and Vening Meinesz, op. cit. (footnote 95), p. 74.
[112]
Ibid., p. 76.
[113] Ibid., p. 309.
[114] Ibid., p. 310.
[115] K. Reicheneder, “Method of the New Measurements at Potsdam by Means of the
Reversible Pendulum,” Bulletin Géodésique (March 1, 1959), no. 51, p.72.](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-63-320.jpg)
![‘Astronomische’; ‘Veröffentlichungen des
Königlichen’—had ‘Königliche.’
Footnote 55 ‘(1768), vol. 58, pp. 329-335.’—had ‘329-
235.’
Footnote 66 ‘Comptes-rendus de l’Académie’—had
‘L’Académie.’
Footnote 81 ‘Sur l’Intensité absolue’—had ‘l’Intensite.’
Footnote 89 ‘Sitzungsberichte der Königlicher’—had
‘Königliche.’
Footnote 100 ‘Veröffentlichungen des Königlichen’ had
‘Veröffentlichungen Königliche.’
Capitalisation of ‘Von’/‘von’ has been regulaized to ‘von’ for
all personal names, except at the beginning of a sentence,
and when referring to the Von Sterneck pendulum.
Index
Quick link to each letter:
[A] [B] [C] [D] [E] [F] [G] [H] [I] [J] [K] [L] [M]
[N] [O] [P] [Q] [R] [S] [T] [U] [V] [W] [Y] [Z]
A](https://image.slidesharecdn.com/1024442-250522192513-bfe4fc52/85/Statistical-Mechanics-Of-Cellular-Systems-And-Processes-Zaman-Mh-Ed-66-320.jpg)
Statistical Mechanics Of Cellular Systems And Processes Zaman Mh Ed
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- 7. Statistical Mechanics of Cellular Systems and Processes Cells are complex objects, representing a multitude of structures and processes. In order to understand the organization, interaction and hierarchy of these structures and processes, a quantitative understanding is absolutely critical. Traditionally, statistical mechanics based treatment of biological systems have focused on the molecular level, with larger systems being ignored. This book integrates understanding from the molecular to the cellular and multi-cellular level in a quantitative framework that will benefit a wide audience engaged in biological, biochemical, biophysical and clinical research. It will build new bridges of quantitative understanding that link fundamental physical principles governing cellular structure and function with implications in clinical and biomedical contexts. muhammad h. zaman is an Assistant Professor in the Department of Biomedical Engineering and Institute for Theoretical Chemistry at the University of Texas, Austin.
- 9. Statistical Mechanics of Cellular Systems and Processes Edited by Muhammad H. Zaman University of Texas, Austin
- 10. CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-88608-6 ISBN-13 978-0-511-51709-9 © Cambridge University Press 2009 2009 Information on this title: www.cambridge.org/9780521886086 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org eBook (EBL) hardback
- 11. Contents List of contributors page vii Preface xi 1 Concentration and crowding effects on protein stability from a coarse-grained model 1 Jason K. Cheung, Vincent K. Shen, Jeffrey R. Errington, and Thomas M. Truskett 2 Observations on the mechanics of a molecular bond under force 26 L. B. Freund 3 Statistical thermodynamics of cell–matrix interactions 54 Tianyi Yang and Muhammad H. Zaman 4 Potential landscape theory of cellular networks 74 Jin Wang 5 Modeling gene regulatory networks for cell fate specification 121 Aryeh Warmflash and Aaron R. Dinner 6 Structural and dynamical properties of cellular and regulatory networks 155 R. Sinatra, J. Gómez-Gardeñes, Y. Moreno, D. Condorelli, L. M. Floría, and V. Latora v
- 12. vi Contents 7 Statistical mechanics of the immune response to vaccines 177 Jun Sun and Michael W. Deem Index 214 Thecolorplatesaretobefoundbetweenpages84and85.
- 13. Contributors Jason K. Cheung Biological and Sterile Product Development Schering–Plough Research Institute Summit, NJ, USA D. Condorelli Department of Chemical Sciences Section of Biochemistry and Molecular Biology University of Catania Catania, Italy Michael W. Deem Department of Bioengineering and Department of Physics and Astronomy Rice University Houston, TX, USA Aaron R. Dinner Department of Chemistry James Franck Institute and Institute for Biophysical Dynamics University of Chicago Chicago, IL, USA Jeffrey R. Errington Department of Chemical and Biological Engineering State University of New York at Buffalo Buffalo, NY, USA L. M. Floría Institute for Biocomputation and Physics of Complex Systems (BIFI) and Departamento de Física de la Materia Condensada vii
- 14. viii List of contributors University of Zaragoza Zaragoza, Spain L. B. Freund Division of Engineering Brown University Providence, RI, USA J. Gómez-Gardeñes Scuola Superiore di Catania Catania, Italy and Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza Zaragoza, Spain V. Latora Dipartimento di Fisica e Astronomia University of Catania Catania, Italy Y. Moreno Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza Zaragoza, Spain Vincent K. Shen Physical and Chemical Properties Division National Institute of Standards and Technology Gaithersburg, MD, USA R. Sinatra Scuola Superiore di Catania Catania, Italy Jun Sun Department of Bioengineering and Department of Physics and Astronomy Rice University Houston, TX, USA
- 15. List of contributors ix Thomas M. Truskett Department of Chemical Engineering and Institute of Theoretical Chemistry University of Texas at Austin Austin, TX, USA Jin Wang Department of Chemistry, Department of Physics and Department of Applied Mathematics State University of New York at Stony Brook Stony Brook, NY, USA and State Key Laboratory of Electroanalytical Chemistry Changchun Institute of Applied Chemistry Chinese Academy of Sciences People’s Republic of China Aryeh Warmflash Department of Physics James Franck Institute and Institute for Biophysical Dynamics University of Chicago Chicago, IL, USA Tianyi Yang Department of Physics University of Texas at Austin Austin, TX, USA Muhammad H. Zaman Department of Biomedical Engineering and Institute for Theoretical Chemistry University of Texas at Austin Austin, TX, USA
- 17. Preface While the application of statistical thermodynamics to study molecular biophysics such as proteins, nucleic acids, and membranes has been around for decades, only in the last few years, have researchers started to study hierarchi- cal and complex cellular processes with the tools of statistical mechanics. The emergence of statistical mechanics in cellular biophysics has shown incredi- ble promise for understanding a number of cellular processes such as complex structure of cytoskeleton, biological networks, cell adhesion, cell signaling, gene expression, and immunological response to pathogens. Collaboration with experimental researchers has led to results that are now emerging as promising therapeutic targets. These studies have provided a first-principle picture that is critical for developing a fundamental biological understanding of cellular pro- cesses and diseases and will be instrumental in developing the next generation of efficient therapeutics. While there has been a surge in publications on topics in cellular systems using tools of statistical mechanics, no book has appeared in the market on this emerging and intellectually fertile discipline. This book aims to fill this void. The main purpose of the book is to introduce and discuss the various approaches and applications of statistical mechanics in cellular systems, in an effort to bridge the gap between physical sciences, cell biology, and medicine. The book is organized such that the first few chapters deal with the physical aspects of cells’ structure, followed by chapters on physiologically critical pro- cesses such as signaling, and concluding with a chapter on immunology and public health. In preparation for this I am grateful to all contributors for their timely contri- bution and to all the staff at Cambridge University press, in particular Dr. Katrina Halliday and Alison Evans for their help, support, and guidance throughout this process. xi
- 18. xii Preface I hope that the readers of this book, students, researchers, and healthcare professionals will find it useful, thought-provoking, and intellectually stimulat- ing and that the sample of chapters in this book will lead to discovery of new avenues of research in the field.
- 19. 1 Concentration and crowding effects on protein stability from a coarse-grained model jason k. cheung, vincent k. shen, jeffrey r. errington, and thomas m. truskett Introduction Most of what we know about protein folding comes from experiments on polypeptides in dilute solutions [1–4] or from theoretical models of isolated proteins in either explicit or implicit solvent [5–12]. However, neither biologi- cal cells nor protein solutions encountered in biopharmaceutical development generally classify as dilute. Instead, they are concentrated or “crowded” with solutes such as proteins, sugars, salts, DNA, and fatty acids [13–15]. How does this crowding affect native-state protein stability? Are all crowding agents cre- ated equal? If not, can generic structural or chemical features forecast their effects? To investigate these and other related questions with computer simulations requires models rich enough to capture three parts of the folding problem: the intrinsic free energy of folding of a protein in solvent, the main structural features of the native and denatured states, and the connection between protein structure and effective protein–protein interactions. The model must also be simple enough to allow for the efficient simulation of hundreds to thousands of foldable protein molecules in solution, which precludes the use of atomistically detailed descriptions of either the proteins or the solvent. We recently developed a coarse-grained modeling strategy that satisfies these criteria. It is not optimized to describe any specific protein solution. Rather, it is a general tool for understanding experimental trends regarding how concen- tration or crowding impact the thermodynamic stability of globular proteins. Statistical Mechanics of Cellular Systems and Processes, ed. Muhammad H. Zaman. Published by Cambridge University Press. © Cambridge University Press 2009. 1
- 20. 2 J. K. Cheung et al. To date, the approach has been used to study how protein concentration affects the folding transition [16], how solution demixing phase transitions (e.g., liquid– liquid phase separation) couple to protein denaturation [17], and how surface anisotropy of the native proteins relates to their unfolding and self-assembly behaviors in solution [18]. In this chapter, we review the modeling strategy and some key insights it has produced. Coarse-grained modeling strategy Intrinsic protein stability A two-state protein molecule in a pure solvent has a temperature- and pressure-dependent thermodynamic preference for either its native (folded) or denatured (unfolded) form [19–24]. The free energy difference between these states G0 f quantifies the driving force for folding in the absence of protein–protein or protein–solute interactions. It also determines the equilib- rium probability (1+exp[G0 f /kBT])−1 associated with observing the native state in an infinitely dilute solution, where kB is the Boltzmann constant and T is the temperature. Interactions that influence this “intrinsic” stability of the native state include, but are not limited to, intra-protein hydrogen bonding, electrostatics, disulfide bonds, and London–van der Waals interactions, as well as effective forces due to excluded volume, chain conformational entropy, and hydrophobic hydra- tion [25]. Here, we focus exclusively on the last three, since they are relevant not only to protein folding [26–29] but also to other self-assembly processes in aqueous solutions [30–32]. Chain conformational entropy and intra-protein excluded volume interactions favor the more expanded denatured state of a protein, while the ability to bury hydrophobic residues in a largely water-free core favors the compact native fold. Intrinsic stability characterizes how these factors for a protein in the infinitely dilute limit balance at a given temperature and pressure. The coarse-grained modeling strategy we review here [16] calculates G0 f under the assumption that a foldable protein can be represented as a collapsible heteropolymer. The effective inter-segment and segment–solvent interactions of the heteropolymer are chosen to qualitatively reflect the aqueous-phase solubilities of the amino acid residues in the protein sequence [33, 34]. One advantage of heteropolymer collapse (HPC) models is that they derive from independently testable principles of polymer physics and hydration thermodynamics. A second advantage is that their behaviors can often be predicted by approximate analytical theories or elementary numerical tech- niques, which allow them to be efficiently incorporated into multiscale
- 21. Protein stability in concentrated and crowded solutions 3 simulation strategies such as the one discussed here. Although HPC theo- ries are descriptive rather than quantitative in nature, the combination of their simplicity and their ability to reproduce experimental folding trends of globular proteins [34] makes them particularly attractive for use in model calculations. HPC theories often reflect a balance of structural detail and mathematical complexity [9, 35–37]. In our preliminary studies, we use a basic, physically insightful approach introduced by Dill and co-workers [34, 38]. This theory mod- els each protein of Nr amino acid residues as a heteropolymer of Ns = Nr/1.4 hydrophobic and polar segments. As is explained in Ref. [38], the factor of 1.4 enters due to a lattice treatment of the protein in which the chain is parti- tioned into cubic polymer segments. The amino acids in a globular protein can be represented as occupying cubic volumes with an average edge length of 0.53 nm [38], whereas the separation of α-carbons in an actual protein is about 0.38 nm (0.53/0.38 ≈ 1.4). The inputs to the theory include temperature T (and, more generally, pH and ionic strength [35]), the number of residues in the protein sequence Nr, the fraction of those residues that are hydrophobic (e.g., based on their aqueous solubilities [33, 34]), and the free energy per unit kBT associated with hydrating a hydrophobic polymer segment χ. A sim- ple parameterization for χ is available that captures experimental trends for the temperature-dependent partitioning of hydrophobic molecules between a nonpolar condensed phase and liquid water at ambient pressure [16]. Although in this chapter we focus exclusively on thermal effects, we have previously introduced a statistical mechanical method for extending the parameterization for χ to also account for hydrostatic pressure [39]. To compute G0 f using this HPC model, one first constructs an imaginary two- step thermodynamic path that reversibly connects the denatured (D) and native (N) states [34]. In step 1, the denatured state with radius of gyration RD collapses into a randomly condensed configuration with radius of gyration, RN. The the- ory assumes that the fraction of solvent-exposed residues that are hydrophobic in both the denatured and the randomly condensed states is , the sequence hydrophobicity of the protein. In step 2, the native state is formed from the ran- domly condensed state via residue rearrangement at constant radius of gyration, so that the fractional surface hydrophobicity of the protein changes from to . By independently minimizing the free energies of the native and denatured states in this analysis, HPC theory predicts the values of both RD/RN and . The intrinsic free energy of folding is the sum of the contributions from the two steps along the imaginary folding path, G0 f = G0 1 +G0 2. Approximate analyt- ical solutions for this HPC theory describe cases where the hydrophobic residues have uniform [34] or patchy [18] spatial distributions on the protein surface. We
- 22. 4 J. K. Cheung et al. discuss below how these solutions can, in turn, be used to infer approximate nondirectional and directional protein–protein interactions, respectively. Non-directional protein–protein interactions While intrinsic thermodynamic stability governs whether an isolated protein favors the native or denatured state, protein–protein interactions play a role in stabilizing or destabilizing the native state at finite protein concentrations. Protein–protein interactions reflect protein structure. Since HPC theories provide only coarse information about structure, the effects we discuss here are the most basic, generally pertaining to how protein size and surface chem- istry couple to their interactions. We first examine the case where proteins display a virtually uniform spatial distribution of solvent-exposed hydropho- bic residues, so that protein–protein interactions are, to first approximation, isotropic. We also limit our discussion to systems where the driving force of proteins to desolvate their hydrophobic surface residues by burying them into hydrophobic patches on neighboring proteins dominates the attractive part of the effective protein–protein interaction. The repulsive contribution to the inter-protein potential accounts for the volume that each protein statistically excludes from the centers of mass of other protein molecules in the solution. As should be expected, the structural differences between folded and unfolded protein states translate into distinct native–native NN, native–denatured ND, and denatured–denatured DD protein–protein interactions. HPC theory correctly predicts that denatured proteins generally exclude more volume to other proteins (RD RN) as compared to their native-state counterparts as shown in Fig. 1.1a [34]. Moreover, denatured proteins exhibit a greater fractional surface hydrophobicity than folded molecules ( ). Mean-field approximations [16, 17] predict that the magnitudes of the average “contact” attractions between two isotropic proteins scale as εND = Nsχ(T)kBT 12 fe(ρ∗ s ) [1 + ρ∗ s −1/3 ]2 + fe(1) [1 + ρ∗ s 1/3 ]2 (1.1) εNN = Nsχ(T)fe(1)2kBT 24 (1.2) εDD = Nsχ(T)fe(ρ∗ s )2kBT 24 (1.3) where ρ∗ s is the effective polymer segment density, fe(ρ∗ s ) = 1 − fi(ρ∗ s ) is the fraction of residues in the denatured state that are solvent exposed, and fi(ρ∗ s ) = [1 − (4πρ∗ s /{3Ns})1/3]3 is the fraction of residues that are on the interior of the protein. A detailed derivation of the above equations can be found in Refs. [16, 18].
- 23. Protein stability in concentrated and crowded solutions 5 1 1.5 2 R D /R N ε DD / ε NN 330 340 350 360 370 Temperature [K] 3 4 5 Φ = 0.400 Φ = 0.500 Φ = 0.500 Φ = 0.400 Φ = 0.455 Φ = 0.455 (a) (b) Fig. 1.1 Comparison of the (a) radius of gyration of the denatured protein relative to the native protein, RD/RN, and (b) the effective magnitude of the DD attraction relative to the NN attraction, εDD/εNN, for proteins of Nr = 154 residues and sequence hydrophobicity = 0.400 (solid), 0.455 (dash), and 0.500 (dot). Given (1.1–1.3) and , it follows that contact attractions involving dena- tured proteins will generally be stronger than those involving the native state (Fig. 1.1b). This is why denaturation often leads to aggregation and precipitation in protein solutions. Along these lines, attractions between proteins increase in strength with the hydrophobic content of the underlying protein sequence . In our coarse-grained strategy, the interprotein exclusion diameters, σDD/σNN ≈ RD/RN and σND/σNN ≈ (RN + RD)/2RN, and the contact energies of (1.1–1.3), all of which are derived from HPC theory [34], serve as inputs into an effective protein–protein potential Vij(r) that qualitatively captures many
- 24. 6 J. K. Cheung et al. aspects of protein solution thermodynamics and phase behavior (see, e.g., Refs. [40, 41]): Vij(r) = ∞ r σij Vij(r) = ij 625 1 [( r σij )2 − 1]6 − 50 [( r σij )2 − 1]3 r ≥ σij (1.4) where ij corresponds to the type of interaction NN, ND, or DD. Directional protein–protein interactions In Dill and co-workers’ original development of this HPC theory, they assume that there are no spatial correlations between solvent-exposed hydrophobic residues in either the denatured or the native state [34]. One way to relax this assumption is to allow for segregation of hydrophobic residues on the surface of the native state. For example, consider the hypothetical scenario where two symmetric “patches” form on the surfaces of native proteins during folding. The patches are distinguishable because they have a different hydropho- bic residue composition than the rest of the solvent-exposed “body.” As shown in Fig. 1.2, the size of each patch is defined by the polar angle α. The fractional patch hydrophobicity p and body hydrophobicity b are expressed as p = fph 1 − cos α b = (1 − fph) cos α (1.5) Θ p α Θp Θb Fig. 1.2 Schematic of two anisotropic native-state proteins. The patch (shaded) and body (white) regions have different hydrophobic residue compositions. The size of the patch is defined by the angle α. The hydrophobicities of the patch p and body b are determined by (1.5). Since the dashed line connecting the protein centers passes through a patch region on each molecule, these two proteins are currently in a patch–patch alignment. Adapted from Ref. [18].
- 25. Protein stability in concentrated and crowded solutions 7 where fph quantifies the fraction of the surface hydrophobic residues that are sequestered into the patch regions on the native protein. Increasing fph increases p, which results in higher surface anisotropy and, as we see below, stronger patch–patch attractions. Knowledge of native-state structure would, in principle, allow one to formulate an approximate patch model for a given protein [42], but predicting this structure directly from sequence information using HPC theory is still not generally possible. In other words, fph, α, and patch location, along with protein sequence, are still knowledge-based inputs for the coarse-grained strategy. The directional dependencies of the contact attractions of patchy proteins are approximated as follows [18]: εND = Nsχ(T)mkBT 12 fe(ρ∗ s ) [1 + ρ∗ s −1/3 ]2 + fe(1) [1 + ρ∗ s 1/3 ]2 (1.6) εNN = Nsχ(T)fe(1)mnkBT 24 (1.7) εDD = Nsχ(T)fe(ρ∗ s )2kBT 24 . (1.8) Here, m and n denote the apparent surface hydrophobicities associated with different orientational states of interacting native molecules m and n, respec- tively. For example, to compute the value of m for molecule m of a given pair interaction, one only needs to know the orientation of molecule m relative to that of the imaginary line connecting its center of mass to that of the other par- ticipating protein. If this line passes through a patch on molecule m’s surface (see Fig. 1.2), then m = p; otherwise m = b, and so on. Equations (1.6–1.8) reduce to (1.1–1.3) for the isotropic (uniform surface hydrophobicity) case (i.e., p = b = ). As an illustration, we examine below aqueous solutions of two model pro- teins of molecular weight Ns = 110 (i.e., Nr = 154) and hydrophobic residue composition = 0.4, parameters typical for medium-sized, single-domain globular proteins [43]. The difference between the two models is that their native states display distinct surface residue distributions, which in turn lead to different protein–protein interactions: “nondirectional” (i.e., no patches) and “strongly directional” ( fph = 0.75, α = π/6). For simplicity, we refer to these models by their names shown above in quotes, rather than by the fph and α parameters that define them. As we discuss below, the behaviors of these two model systems provide insights into the mechanisms for stability in several experimental protein solutions. Figure 1.3 shows the effect of native protein surface anisotropy on the strength of protein–protein attractions. The patch–patch attractions for the
- 26. 8 J. K. Cheung et al. 330 Temperature [K] 3 6 9 12 ε N N k / B T Patch–Patch 0.1 0.2 0.3 0.4 0.5 ε N N k / B T Body–Body (b) (a) Nondirectional Strongly directional 340 350 360 370 Fig. 1.3 Comparison of the contact attraction εNN relative to kBT for (a) body–body alignment and (b) patch–patch alignment of Nr = 154, = 0.4 proteins with strongly directional interactions (dash). The contact attraction for Nr = 154, = 0.4 native proteins with nondirectional interactions (solid) is also shown. strongly directional protein are more than an order of magnitude larger than the other attractions. Pairs of directional proteins can desolvate a higher number of hydrophobic residues by self-associating (as compared to the nondirectional pro- teins), but only if they do so with their hydrophobic patches mutually aligned, which in turn imposes an entropic penalty. This balance between favorable hydrophobic interactions and unfavorable entropy yields the possibility of con- tinuous equilibrium self-assembly transitions involving the native state [18]. Given the symmetric patch geometry of the native state model studied here, the morphology of the self-assembled “clusters” would resemble linear polymeric chains. The interactions discussed above are similar in spirit to a “two-patch” description that was recently introduced to model the native–native protein interactions of the sickle cell variant of hemoglobin [44] and also to other semi-empirical anisotropic potentials developed for native proteins [42, 45–50].
- 27. Protein stability in concentrated and crowded solutions 9 However, the coarse-grained strategy described here differs significantly from these earlier models in two ways: it explicitly accounts for the possibility of protein denaturation, and it estimates the intrinsic properties of the native and denatured states using a statistical mechanical theory for heteropolymer collapse. This link to the polymeric aspect of the protein is crucial because it allows our model to be used as a tool to investigate how native-state pro- tein anisotropy affects folding equilibria, self-assembly, and the global phase behavior of protein solutions. Reducing protein stability to a classic chemical engineering problem It is worth emphasizing that the coarse-grained model described above represents an effective binary mixture of folded and unfolded proteins (the aqueous solvent only entering through χ) connected via the protein folding “reaction.” Links between the intrinsic native-state stability of the proteins, G0 f , the physical parameters defining the protein sequence (Nr, ), the native- state surface morphology (, α, fph), the interactions of hydrophobic residues with aqueous solvent χ, and the protein–protein interactions ( ij, σij) are estab- lished through the HPC model [16, 17]. As in experimental protein solutions, the fraction of proteins in the native state generally depends on both temper- ature and protein concentration. This fact, often neglected in other modeling approaches which ignore the polymeric nature of proteins or protein–protein interactions, arises because temperature affects the intrinsic stability of the native state G0 f , and both temperature and protein concentration influence the local structural and energetic environments that native and denatured proteins sample in solution. In short, the coarse-grained approach frames the stability of concentrated protein solutions in terms of a classic chemical engineering problem: mapping out the equilibrium states of a reactive, phase-separating mixture [51]. In the next section of this chapter, we discuss how advanced Monte Carlo methods designed to efficiently solve the latter problem can also be used to address the former. Simulation methods The properties of the coarse-grained protein model described above can be readily studied using transition-matrix Monte Carlo (TMMC) simulation. TMMC is a relatively new simulation technique that has emerged in recent years as a highly efficient method for investigating the thermophysical properties of fluids. It is useful for a variety of applications ranging from the precise calcula- tion of thermodynamic properties of pure and multicomponent fluids in bulk
- 28. 10 J. K. Cheung et al. and confinement [52–62], surface tension of pure fluids and mixtures [52, 63–65] and Henry’s constants [66] to the investigation of wetting transitions [67, 68] and adsorption isotherms [69, 70]. The basic goal of TMMC is to calculate the free energy of a system along some order parameter path by determining the order parameter probability distribution. To do this in a conventional simulation, a histogram is constructed by simply counting the number of times the system takes on, or visits, a given order parameter value. In a transition-matrix-based approach, the distribution is determined by accumulating the transition proba- bilities of the system moving from one order parameter value to another during the course of a Monte Carlo simulation, and subsequently applying a detailed balance condition over the explored region of order parameter space. To facili- tate sufficient and uniform sampling of order parameter space, a self-adaptive biasing scheme is often employed. The reader is referred to Refs. [52, 56–58, 63, 71–75] for further details. For this coarse-grained protein model, we use a particular TMMC implementation designed for multicomponent systems [57]. To obtain thermodynamic and structural quantities of interest, we perform TMMC simulations within the grand-canonical ensemble. Under these condi- tions (fixed chemical potentials, volume, and temperature), an appropriate choice of order parameter is the total number of molecules in the system. While the order-parameter probability distribution is unique to the chemical poten- tials used in the simulation, histogram reweighting can be used to determine the distribution at other chemical potentials [76]. Because the coarse-grained pro- tein model can be regarded as a binary mixture of native and denatured proteins where the components can undergo a unimolecular chemical reaction (folding), chemical equilibrium requires that the chemical potentials of the native and denatured proteins be identical. Thus, only a single chemical potential needs to be specified in a TMMC simulation of the protein solution. The free-energy change of the reaction, that is the intrinsic free energy of folding G0 f , enters as an activity difference between the folded and unfolded proteins, a treatment which implicitly assumes that the protein’s intermolecular and intramolecular degrees of freedom are separable. Although the main output of a TMMC simulation is the total protein num- ber (concentration) distribution, other system properties can be also calculated. This is done in a straightforward way by collecting isochoric averages during the course of a simulation. Combining these statistics with the above-mentioned histogram-reweighting technique, the fluid-phase properties of the coarse- grained protein model can be determined over a wide range of concentrations from a single TMMC simulation. In our studies, we calculate, along with other properties, the following quantities: the overall fraction of folded proteins fN, the fraction of clustered or aggregated proteins fclust, and the average fraction
- 29. Protein stability in concentrated and crowded solutions 11 of clustered proteins that are folded fNc [18]. Since this is a binary system, fN is tantamount to the composition. The midpoint unfolding transition in a protein solution occurs at fN = 0.5. The quantity fclust measures the fraction of proteins in the system that are in clusters. This has also been referred to as the extent of polymerization in other contexts [77]. Here, two proteins are considered to be in the same cluster if the magnitude of their effective pairwise attraction is greater than 80% of the interprotein potential minimum ij. Finally, the quantity fNc characterizes the overall average protein cluster composition. Besides the conventional assortment of Monte Carlo trial moves (e.g., par- ticle displacements, insertions/deletions, and identity changes), we perform so-called aggregation-volume bias trial moves [78–80]. These specialized trial moves are typically required because strong (relative to kBT) inter-protein inter- actions can otherwise result in persistent bonded configurations which prevent adequate sampling of phase space. Aggregation-volume bias moves circum- vent this difficulty by preferentially performing trial protein displacements and protein insertions or deletions in the immediate vicinity, called the “bonding region,” of a randomly chosen molecule, thereby promoting the formation and destruction of bonded configurations. Phase space sampling is further enhanced by combining this suite of moves with multiple first-bead trial insertions and configurational-bias Monte Carlo moves [80–82]. Finally, since we are interested in the liquid-state properties of the coarse-grained protein model, a constraint is imposed to prevent the system from crystallizing. This is required because, for the one-component (all native or all denatured) system, the liquid state is metastable with respect to the solid at intermediate and high concentra- tions [40]. Specifically, we apply a constraint based upon a bond-orientational metric capable of distinguishing between amorphous (liquid) and solid con- figurations [83–86]. The interested reader is referred to Ref. [17] for further details. Effects of protein concentration: uniform vs. patchy proteins In this section, we examine the predictions of our grand-canonical TMMC simulations and coarse-grained modeling strategy with a focus on eluci- dating how the surface anisotropy of the native state affects denaturation and equilibrium self-assembly behaviors of proteins in solution. In particular, we compare simulated equilibrium unfolding curves ( fN) and self-association mea- sures ( fclust and fNc) for the nondirectional and the strongly directional model proteins described above. We first explore the physics that govern the stability of the nondirectional protein in solution as a function of its concentration (Fig. 1.4a). The effects can be seen most clearly by simulating along the G0 f = 0 isotherm of the protein,
- 30. 12 J. K. Cheung et al. 0 Protein concentration 0.2 0.4 0.6 0.8 1 Strongly directional 0 0 0.2 0.4 0.6 0.8 1 Fraction Fraction Nondirectional fNc fN fclust 0.1 0.2 0.02 0.04 0.06 (a) (b) Fig. 1.4 Fraction of folded proteins fN (solid), “clustered” proteins fclust (dotted), and “clustered” proteins in the native state fNc (dashed), as a function of protein concentration for the (a) nondirectional and (b) strongly directional model proteins (Nr = 154, = 0.4) at their respective infinite-dilution midpoint folding temperatures. Protein concentration is measured as the dimensionless density (i.e., Nσ3 NN/V) where N is the total number of proteins and V is the simulation box volume. To provide a comparison to experimental concentrations, a 100 mg/ml solution of ribonuclease A would have a dimensionless density Nσ3 NN/V of approximately 0.28, assuming σNN = 4.0 nm and a molecular weight of 13.7 kDa. Adapted from Ref. [18]. i.e., its infinite dilution unfolding temperature. As we show below, the existence of patches on a protein serve to modify this baseline behavior. The minimum in the folded fraction fN curve for the nondirectional protein in Fig. 1.4a can be understood as a balance between two factors: destabilizing protein–protein attractions involving denatured species and stabilizing protein crowding effects. At finite protein concentrations, a marginally stable protein
- 31. Protein stability in concentrated and crowded solutions 13 can favorably denature in solution if it meets two criteria: (a) its current envi- ronment affords enough local free volume to accommodate the transition to the more expanded denatured state, and (b) it can form enough new inter-protein hydrophobic contacts with neighboring proteins upon denaturing so that it overcomes its intrinsic free energy penalty for unfolding. The latter destabi- lizing effect is facilitated by the fact that attractions involving the denatured state are generally much stronger than those involving native nondirectional proteins. Concentrating the protein solution decreases the probability of (a) but increases the likelihood of (b). To our knowledge, the coarse-grained strategy outlined here is the first to predict this minimum in folded fraction versus protein concentration, a nontrivial trend that is observed experimentally in, e.g., monoclonal antibody solutions [87]. Proteins with high hydrophobicity are predicted by the coarse-grained model to show more pronounced concentration-induced destabilization at low and intermedi- ate protein concentrations [16, 17], which also appears to be in agreement with available experimental trends [88]. Does clustering or self-assembly occur in solutions of nondirectional proteins? Figure 1.4a shows that the fraction of clustered nondirectional proteins fclust increases with protein concentration. Clustering of this sort can have two potential origins. On one hand, it can occur for trivial “packing” rea- sons alone at high protein concentrations, conditions for which pairs of proteins are literally forced to adopt near-contact configurations that satisfy the above geometric criteria for fclust. On the other hand, the more interesting case is when protein clustering arises due to strong inter-protein attractions, which are relevant even at low protein concentrations. For the nondirectional protein discussed above, it is known that interprotein attractions are relatively weak (Fig. 1.3). Thus, clustering for the nondirectional protein is due to geometric “packing” effects. This is also reflected in the fact that the native composition of the clusters fNc essentially tracks the fraction folded fN at higher protein concen- trations, where most of the proteins are clustered. If, instead, clustering were the result of highly favorable, native-stabilizing inter-protein attractions, then one would expect to find fNc fN, even at relatively low protein concentrations. The behaviors exhibited by the nondirectional protein discussed above are in stark contrast to that of the strongly directional protein presented in Fig. 1.4b. Notice that the relevant protein concentration range for the strongly directional protein is an order of magnitude less than that for the nondirectional proteins. Under these dilute conditions, the packing effects which drive geometric clus- tering of the proteins are absent. Increases in fclust are therefore a result of different physics: the highly energetically favorable patch–patch association.
- 32. 14 J. K. Cheung et al. The fact that the folded fraction within the clusters fNc rises above the average folded fraction fN indicates that the self-assembly behavior or “clustering” stabi- lizes the patchy native state of the directional protein relative to the denatured state. In Fig. 1.5, we plot what might be referred to as “stability diagrams” for the (a) nondirectional and (b) strongly directional proteins discussed above. The shaded regions of these diagrams indicate temperature and concentrations that 310 330 350 Temperature [K] Nondirectional 0 0 0.015 0.15 Protein concentration 310 330 350 Temperature [K] Strongly directional Denatured Native Denatured Native 0.005 0.05 0.01 0.1 (a) (b) Fig. 1.5 Stability diagram for the (a) nondirectional protein and (b) strongly directional protein in the temperature vs. protein concentration plane. Protein concentration is defined as in Fig. 1.4. The native state is thermodynamically favored ( fN 0.5) in the white region, while the denatured state is favored ( fN 0.5) in the shaded region. Also shown are the loci of conditions where fclust = 0.5 (dash). To the right of the fclust = 0.5 curve, more than half of the proteins are part of geometric “clusters.” For conditions where the denatured state is favored, note that the location of the fclust = 0.5 curve for both panels (a) and (b) are the same. This result is expected because the attractive strength and relative size of the denatured proteins are the same for the protein variants studied here. Adapted from Ref. [18].
- 33. Protein stability in concentrated and crowded solutions 15 favor the denatured state ( fN 0.5), while the white region indicates conditions that favor the native state ( fN 0.5). The loci of temperature-dependent con- centrations where fclust = 0.5 (dash) are also shown. Proteins form clusters in solution for all states to the right of these curves. Note that the clustering behavior of the nondirectional protein occurs at lower protein concentrations for the denatured state as compared to the native state because the former has a larger effective radius of gyration and stronger inter-protein attractions. For the strongly directional proteins, the native-state global clustering trends are dif- ferent. Small increases in native protein concentration promote self-assembly of linear clusters due to the fact that the patch–patch attractive interactions are much stronger than even the denatured–denatured interactions for this protein. One key prediction of our model is that strong directional hydrophobic inter- actions can help stabilize anisotropic native proteins. In other words, folded proteins that associate through hydrophobic patches may show enhanced sta- bility. To illustrate the generality of this result, we briefly discuss below some experimental stability behaviors for proteins in their native states that asso- ciate via hydrophobic patches in native or mildly denaturing conditions. We also mention an extreme case where an unstable, unfolded peptide monomer folds only upon specific interaction with other peptide chains. We begin our discussion with β-lactoglobulin, which exists as a dimer in solution. The interface between the two protein monomers contains a large hydrophobic patch [89]. This geometry suggests that hydrophobic interac- tions help stabilize the complex. Because the protein must first dissociate to unfold [90], the hydrophobic attractions also aid in preventing nucleation and growth of β-lactoglobulin fibrils [89, 90]. This stability behavior is similar to the enhanced stability seen in the clusters of our strongly directional proteins. Ribonuclease A is also known to form dimers (as well as trimers and higher- order oligomers) in solution under a variety of experimental conditions [91–93] due to specific, directional interactions at its slightly unfolded N- and C-termini [94]. Like β-lactoglobulin, hydrophobic interactions may help to stabilize the oligomer complex [92]. However, it also has been observed that at high tem- peratures, the presence of the ribonuclease oligomers is significantly decreased [92]. This oligomer stability behavior is in qualitative agreement with the tem- perature dependent stability behavior observed for our strongly directional protein clusters. Figure 1.5b shows that the slope of fclust is positive. There- fore, native proteins (unshaded region) that initially favor self-assembly ( fclust 0.5) at low temperatures, do not favor self-assembly (fclust 0.5) at higher temperatures. Sickle-cell hemoglobin has the propensity to form ordered aggregates because of a point mutation that converts a surface hydrophilic residue to a
- 34. 16 J. K. Cheung et al. hydrophobic residue [95]. Because wild-type hemoglobin, like our model nondi- rectional proteins, does not aggregate at low to intermediate concentrations, this protein is an example of how changes to native-state surface hydrophobic- ity can result in protein polymerization (i.e., the formation of ordered protein clusters). Interestingly, self-association of native sickle-cell hemoglobin can be extrapolated to occur for temperatures above the folding transition of the wild- type protein (see, e.g., Ref. [96]). This suggests that the strongly favorable native interactions of the sickle variant could play an active role in stabilizing the native (“clustering”) form over the denatured state. Finally we review the peptide p53tet. Due in part to its short chain length of 64 residues, the monomeric form of the peptide is unstable. However, when in contact with three neighboring peptide chains, the peptide “folds” to form a tetrameric protein. The interface between the peptides consists of mostly apo- lar residues [97], suggesting that the hydrophobic interactions can stabilize an inherently unstable protein [97, 98]. In this case, the p53tet tetrameric protein is analogous to our strongly directional model protein clusters, since the clus- ters are formed by hydrophobic attractions and composed of native proteins (i.e., directional interactions stabilize the geometric formation and the native state). Does the nature of the crowding species matter? As mentioned previously, proteins are often a component of crowded solutions in both pharmaceutical and biological settings. For example, the concentration of therapeutic proteins used for subcutaneous injections can be as high as 100 mg/ml [15]. In other cases, proteins are part of a complex mixture of both interacting proteins and weakly interacting (inert) particles. For example, cellular cytoplasm consists of macromolecules, structural pro- teins (e.g., actin fibrils) and nonstructural proteins [13, 14]. The estimated concentration of the nonstructural proteins is 110 mg/ml [99]. Addition- ally, inert polymers and macromolecules are often found in pharmaceuti- cal protein solutions [100]. Is the equilibrium stability behavior of proteins in crowded solutions, where there are multiple species present, similar to the equilibrium stability behavior of native proteins in concentrated protein solutions, where there is a single protein species present? In other words, does the type of crowding species strongly affect the equilibrium protein stability? Thermodynamically, the more compact state (usually the native conforma- tion) is favored in solutions that contain high concentrations of inert crowding species [101–105]. Within the framework of our coarse-grained model, however, we have shown that the stability of native proteins in concentrated protein
- 35. Protein stability in concentrated and crowded solutions 17 solution depends not only on stabilizing crowding effects but also on destabi- lizing protein–protein attractions. The strength of these attractions depends on some of the intrinsic properties of the protein – e.g., the sequence hydropho- bicity, the distribution of hydrophobic residues on the native protein surface, and the sizes of the native and denatured states – as well as on solution con- ditions such as temperature and pressure [16, 17, 39]. For proteins that do not exhibit directional interactions, high sequence hydrophobicity results in non- monotonic concentration effects on stability [16] (see Fig. 1.4a) and liquid–liquid demixing [17] of protein solutions. To study the effects of crowders on equilibrium protein stability, we extend our original coarse-grained model to account for the presence of multiple species in solution. We compare the equilibrium fraction of folded proteins for two different cases. The first solution consists of foldable proteins (i.e., marginally stable proteins that will fold and unfold) and an ultrastable native crowder (i.e., proteins with interactions identical to the foldable proteins but that remain in their native state and do not unfold). The second solution con- sists of foldable proteins and inert, hard-sphere crowders with the same effective diameter as the foldable species. Figure 1.6 shows a preliminary result for a nondirectional protein of chain length Nr = 154 and sequence hydrophobicity = 0.455. We observe that the 0.2 0.4 0.6 0.1 0.2 0.3 0.1 0.2 0.3 Foldable protein concentration 0.2 0.4 0.6 Native fraction of foldable proteins HS crowders Native crowders (a) (b) Fig. 1.6 Native fraction of foldable proteins as a function of foldable protein concentration for the protein Nr = 154, = 0.455 at its infinite dilution folding temperature. Protein concentration is defined as in Fig. 1.4. The arrow indicates increasing concentration of (a) ultrastable native crowders and (b) inert hard-sphere (HS) crowders. (See plate section for color version.)
- 36. 18 J. K. Cheung et al. ultrastable native crowders can either stabilize or destabilize the native form of the foldable proteins (Fig. 1.6a), depending on the concentration of the latter. For low foldable protein concentrations, increasing the concentration of the native crowders results in a lower fraction of folded proteins. For high foldable protein concentrations, the opposite is true. This nonmonotonic crowding behavior in solution is not observed when hard-sphere crowders are considered, which stabilize the native state at all foldable protein concentrations. The above calculation illustrates that the magnitude of the crowder–protein attraction plays a nontrivial role. These effects, however, appear to follow from our previously introduced stability criteria. A protein will unfold if (a) there is enough local free volume to accommodate the larger denatured state and (b) denaturing enables formation of sufficiently favorable inter-protein con- tacts. Inert crowders only act to stabilize the native state of the foldable protein because they decrease the likelihood of (a) and do not increase the probability of (b). However, attractive crowders increase the likelihood of (b) at low protein concentrations (where criterion (a) plays a minor role) because denatured–native (and hence denatured–crowder) attractions are larger in magnitude than native– native attractions. However, at high foldable protein concentrations (where criterion (a) dominates), both native or inert crowders reduce the total free volume and thus increase native stability of the foldable protein. Using multiscale simulations to understand the role of crowder–protein attractions on protein stability is still a relatively new area of inquiry. Progress on this problem, however, is needed as it may help to clarify some of the complicated physics present in heterogeneous protein mixtures like cellu- lar cytoplasm. It may also suggest the physico-chemical properties of new excipients to control the stability of highly concentrated protein therapeutics. Conclusions and open questions We have developed a general framework for modeling protein stability in concentrated and crowded solutions. Our approach accounts for both the intrinsic thermodynamics of folding and the general physical characteristics of the native and denatured states. Protein–protein interactions are derived using the salient physical features of the native and denatured conformations predicted by a HPC theory. Ultimately, we are able to study the effects of protein concentration and crowding on protein stability in a computationally efficient manner. First, we examined aqueous solutions of a single species of nondirectional proteins. At finite concentrations, a marginally stable protein in these systems
- 37. Protein stability in concentrated and crowded solutions 19 will unfold if there is enough local free volume and the state-dependent contact free energy of the unfolded protein with its neighbors is sufficiently favorable to overcome any intrinsic thermodynamic stability of the native state. At high protein concentrations, the compact, native conformation is favored over the expanded, denatured state due to self-crowding effects. However, at low con- centrations where crowding effects are minimal, a protein may be destabilized due to the more favorable free energies associated with denatured protein– protein contacts. If other factors are equal, we find that proteins with low sequence hydrophobicity tend to be stabilized by increasing protein concen- tration while proteins with high sequence hydrophobicity show nonmonotonic stability trends [16, 17]. This behavior qualitatively agrees with experimentally observed protein behavior [87, 88]. We also studied the effects of anisotropic protein–protein interactions on protein stability and self-assembly behavior. In contrast with the nondirectional proteins, the strongly directional proteins we studied were stabilized against denaturation at even low protein concentrations by forming highly ordered chains. This behavior is similar to the oligomerization and polymerization of proteins in solution [90, 92, 95]. We are currently examining how different crowding species in solution affect the equilibrium fraction of folded proteins. Our initial results show that attractions play a nontrivial role in determining protein stability. While inert crowders only stabilize marginally stable proteins, attractive crowders can destabilize the native state at low foldable protein concentrations. Here our findings may lead to some insights on the stability of proteins in environ- ments that contain a broad mixture of macromolecules like cellular cytoplasm or concentrated pharmaceutical biological solutions. Protein solutions show a variety of rich behaviors and open questions remain as to how environmental conditions affect their equilibrium stability trends. For example, one can ask whether the self-assembly of strongly directional proteins will change in the presence of other crowding species. This type of heterogeneous protein solution may more accurately represent in vivo biolog- ical environments. One can also ask whether protein solution stability (e.g., the demixing transition observed in Ref. [17]) is affected by protein and inert crowding species. If the equilibrium demixing transition can be prevented by the addition of stabilizing cosolute crowders, protein drug shelf-life may be improved. The general versatility of the coarse-grained framework reviewed here provides the flexibility to investigate many different types of protein solu- tions commonly encountered in biological and pharmaceutical environments, which should allow it to provide insights into some of these open questions.
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- 44. 2 Observations on the mechanics of a molecular bond under force l. b. freund Introduction In the sections that follow, two issues focusing on the mechanics of a molecular bond under the action of an applied force are addressed. This dis- cussion is motivated by the behavior of the noncovalent molecular bonds that account for adhesion of eukaryotic animal cells to extracellular matrix or to other cells, particularly bonding of transmembrane integrins to their ligands. The integrins are large, compliant molecules and the results of experiments on their behavior are reported in the literature. The discussion itself, however, is generic within the context of physical chemistry. The first issue considered is the transient response of a single molecular bond to an applied force. The configuration of the system is represented by one or more stochastic variables and by a deterministic variable, the latter account- ing for the force input. Bonding is understood to be the confinement of the configuration of the bond within an energy well, as represented by the stochas- tic variables, which exists as a feature of an overall energy landscape, and the role of the force is to distort that landscape. The system functions in a thermal environment, but the time required for bond separation is long compared to the thermal relaxation time of a single step in the Brownian motion of a small particle in that environment. The bond state is characterized by a probability distribution of possible configurations over the bond energy landscape, and the evolution of that distribution is governed by the Smoluchowski partial dif- ferential equation. The system is analyzed to extract information on the most Statistical Mechanics of Cellular Systems and Processes, ed. Muhammad H. Zaman. Published by Cambridge University Press. © Cambridge University Press 2009. 26
- 45. Observations on the mechanics of a molecular bond under force 27 probable time for bond separation, the maximum force that can be sustained by the bond, and the sensitivity of bond separation behavior to the rate of loading and bond well features, for example. The second issue considered is the stability of a molecular bond by which a membrane is attached to a substrate. Again, the bond is represented by an energy well in the landscape representing the dependence of bond energy on its configuration. In this case, the system is in thermal equilibrium with its environment, so the membrane position fluctuates continuously as a result. In the course of random fluctuations, the membrane exerts a fluctuating force on the bond which is equal but opposite to the constraint force acting on the membrane. The particular question posed concerns the largest edge-to-edge span that the membrane can have without overwhelming the bond through thermal fluctuations. The phenomenon of interest is a feature of steady behavior and, consequently, it is studied by means of classical statistical mechanics. The result of the analysis is of potential interest in understanding the spacing of integrins within a focal adhesion zone as it is formed in the course of adhesion of a biological cell. The two issues addressed are fundamentally different aspects of the same physical system. Although each can be understood in a fairly general way, there are few results of a general nature that can be inferred about the transient behavior of the full system of an interacting bond and the membrane to which one of the bound molecules is attached. However, the results presented here are seen as steps in the direction of achieving that objective. Forced separation of a molecular bond In this section, we examine the situation of a molecular bond being acted upon by a loading apparatus that tends to pull the bond apart. The bond itself is described by means of an expression of bond energy versus a reaction coordinate, commonly known as the energy landscape of the bond. The loading apparatus is described by means of a deformable element linking the reac- tion coordinate to a point at which a time-dependent constraint is externally imposed. The force on the bond is understood to be the force needed to main- tain this constraint. The reaction coordinate is stochastic and can be described only in terms of a time-dependent probability distribution whereas the imposed constraint is deterministic. That such a process can be observed has been demonstrated by Evans et al. [1], who have reported measurements of the dependence of the maximum force required to overcome the bond on the rate of application of that force. An intriguing question concerns the way in which the behavior observed is related
- 46. 28 L. B. Freund Force at rupture No. of rupture events Fixed loading rate Fig. 2.1 A generic diagram illustrating the way in which data were reported by Evans and Ritchie [2] on the observed behavior of isolated molecular bonds under the action of steadily increasing force. The response of a large number of individual bonds under nominally identical loading conditions was represented as a histogram in this form. to the properties of the bond being overcome. Equilibrium statistical mechanics is too restrictive for addressing such questions, so the process is viewed more generally as being a stochastic transport phenomenon. The general character of the data on bond rupture strength reported by Evans and Ritchie [2] is as follows. Suppose a large number (several hundred) of rupture events are observed for nominally identical bond pairs and nominally identical rates of loading up to rupture. If the results are presented in the form of a histogram of number of rupture events that occurred within each of numerous finite increments of applied force, the results take the form illustrated in Fig. 2.1. The spread of the data along the force axis suggests that the process is statistical in nature. The main influence of an increase in the rate of loading was found to be a stretch of the distribution in the direction along the force axis. In particular, the peak in the distribution moves to a larger force value as the rate of loading increases. The purpose in this section is to discuss a theoretical framework for interpreting such observations. A model of bond breaking In selecting features to be incorporated into a model of the bond rup- ture process, we follow the lead of Evans and Ritchie [2] in several respects. The bond itself is idealized as a one-dimensional energy landscape, as depicted in Fig. 2.2. The abscissa x in this diagram is the so-called reaction coordi- nate, a variable that determines the configuration of the bond. While the process of bond rupture might require extensive conformational changes of the molecules, such as twisting or unfolding, the energy input that effects these changes is assumed to be delivered by a force working through the translational coordinate x.
- 47. Observations on the mechanics of a molecular bond under force 29 1 Fig. 2.2 From the energy point of view, adhesion is represented as an energy well that is accessible in some configurations of the bond pair. The diagram illustrates the general features of such a well that are incorporated in the model. The bond energy U, which is a function of configuration as represented by a reaction coordinate x, is described mainly in terms of the depth CbkT and breadth δ of the well. The diagram shows the system landscape in the vicinity of the well in terms of normalized parameters. The quantity Ub(x) in Fig. 2.2 represents the interaction energy of the bond pair at reaction coordinate x, and this energy is normalized by the thermal unit kT where k is the Boltzmann constant and T is the absolute temperature. The reaction coordinate is normalized by the half-width δ of the well. The depth of the well Cb is the reduction in free energy of the system upon bond formation at low temperature; typically Cb ∼ 10. When the molecules are well separated, the interaction energy is zero; the molecules are fully separated at values of the reaction coordinate greater than δ. The loading apparatus is understood to be the link between the stochastic domain of the molecule and the deterministic constraint being enforced in order to separate the molecules. It is incorporated into the model by means of a spring, either linear or nonlinear, to represent the potentially significant fea- ture of loading mechanism compliance. As illustrated in the sketch in Fig. 2.3, the position of one end of the loading apparatus is the reaction coordinate x and the position y(t) of the other end is prescribed. Both x and y are mea- sured with respect to their initial locations before the force is applied. The applied force that produces the rupture of the bonds is more appropriately con- sidered as a quantity derived through its effect rather than as a fundamental variable in a thermodynamic framework and, as such, it does not appear explic- itly in the model. Instead, we determine this force as that action which produces the energy changes in the system in the course of its response to the imposed motion y(t). The energy landscape of the system is represented by a surface over the plane spanned by the reaction coordinate x and the loading coordinate y. It is the sum of two contributions, one of which represents bond behavior as depicted in Fig. 2.2 and the other of which represents the properties of the
- 48. 30 L. B. Freund substrate bound molecules loading apparatus x y (t) f (t ) Fig. 2.3 The diagram is a schematic of the entire system. The configuration of the bond is represented by a single time-dependent reaction coordinate x, which varies stochastically, and a time-dependent coordinate y which is specified a priori. The essential properties of the bond are Cb and δ, and the properties of the loading apparatus are also specified. The applied force f (t) is calculated to be the force necessary to maintain the specified motion y(t). loading apparatus. As an illustration, suppose that the former is described by Ub(x) kT = ub(ξ) = −Cb + 2Cbξ2 , ξ 0.5 −2Cb(ξ − 1)2 , 0.5 ≤ ξ ≤ 1 0 , 1 ξ (2.1) where ξ = x/δ is the nondimensional variable introduced in Fig. 2.2 and Cb is the depth of the well. The latter is a linear spring described by Us(x, y) kT = us(ξ, η) = 1 2 κ(η − ξ)2 (2.2) where η = y/δ and the nondimensional parameter κ is the spring stiffness normalized by kT/δ2. The energy landscape is then the surface U(x, y) = Ub(x) + Us(x, y) over the x, y-plane or, in normalized parameters, u(ξ, η) = ub(ξ)+us(ξ, η) over the ξ, η-plane. The features of this surface can be visualized in several ways. Here, we do so by plotting cross-sections of the surface for several val- ues of η. This is illustrated in Fig. 2.4 for Cb = 10 and the stiffness value κ = 1. The entire surface near ξ = 0 translates in the direction of the energy axis (that is, “upward” in the figure) as η increases, so the quantity plotted is actu- ally u(ξ, η) − 1 2 κη2 to compensate for this translation. In their original report, Evans and Ritchie [2] assumed that the external force was applied directly to the reaction coordinate, whereas a spring to represent loading compliance was adopted in the qualitative discussion of force spectroscopy given by Evans and Calderwood [3]. The main features of the evolving surface are evident in Fig. 2.4. The surface has a zero energy trough along the line y = x. This is the state with the spring relaxed and the molecules fully separated. This is where the system ends up on the energy surface after complete bond separation. Initially, with y = 0, the bond well is the expected location or state of the system. As y increases, the well is elevated to higher energy levels relative to the
- 49. Observations on the mechanics of a molecular bond under force 31 –1 1 2 3 4 5 –15 –10 –5 5 10 h = 0 20 40 60 j 0 u (j,h) – 1 2 kh2 Fig. 2.4 The two-dimensional energy landscape of the system u(ξ, η) is illustrated by means of several cross-sections with η =constant in terms of nondimensional coordinates ξ = x/δ and η = y/δ. The level of the landscape rises quite rapidly with increasing η because energy is continually added by the applied force. Consequently, a time-dependent translation is subtracted for each section so as to represent all sections in the same coordinate system. zero energy trough, and more so at the left end than at the right end, thereby giving the impression that the local landscape is rotated in a clockwise sense. This distortion promotes the flux of states from within the well over the barrier and in the direction of the zero energy trough. This is more or less the same viewpoint introduced by Evans and Ritchie [2], except that they had the applied force working directly through the reaction coordinate. Some differences will emerge in the analysis of the model below, however. It should be recognized that this representation of the energy landscape of the molecular bond in terms of one or two degrees of freedom is extremely coarse, in light of the structural complexity of the molecules whose interactions form the bond. However, at this point in the development of the topic, there is little basis for choosing a more elaborate representation, although numerical studies of molecular interactions such as those reported in [4] should begin to provide guidance for doing so in the future. In an interesting discussion of breaking of so-called catch bonds between certain types of molecules due to force, Thomas [5] has suggested that such structures may account for the unusual behavior exhibited by these molecular pairs. With this picture of the energy landscape in place, the question to be addressed is the following. If the state of the system initially resides in the bond interaction well when y = 0 and if the coordinate y is specified to be some increasing function of time thereafter, what is the time history of the force acting on the bond and of the probability distribution of bond configurations?
- 50. 32 L. B. Freund The state of the bond In view of the stochastic nature of bond behavior, the state of the bond is represented by a time-dependent probability distribution which is a function of the reaction coordinate x, say ρ(x, t). If a large number of identical systems are observed under nominally identical constraints, the corresponding distribution in values of the reaction coordinate at a certain instant of time provides a density of states for the system at that time. If this density of states is normalized so as to represent the fractional distribution of states, it can be interpreted as a probability distribution for the current value of reaction coordinate in a particular system. In the discussion that follows, the terms density of states and probability distribution are used interchangeably with the understanding that they refer to one and the same thing. To ensure that all states are taken into account over time, this probability distribution ρ(x, t) on the finite interval x0 x x1 is constrained by the conservation condition x1 x0 ρ(x, t) dx − t 0 j0(s) ds + t 0 j1(s) ds = 1 (2.3) identically in time, where j0 and j1 represent fluxes of states inward at x = x0 and outward at x = x1. In the thermal environment of a solvent, the molecules are continually bom- barded by impulses, each of which delivers an energy on the order of kT. This energy is dissipated back into the environment through the viscous resistance of the solvent in the course of motion of the molecules over a small distance in a short time. If the depth of the interaction energy well in the energy landscape is significantly greater than kT, then the likelihood of escape from the well (that is, debonding) on the timescale of individual excitations is very small and the probability distribution evolves diffusively, rather than ballistically. In other words, transition of a configuration out of the well is an improbable event. In this setting, the diffusion coefficient D is expressible in terms of the viscosity γ of the solvent and the thermal energy unit kT through the Einstein relation D = kT/γ [6]. This is the point of view of chemical kinetics that was developed through the work of Smoluchowski and Kramers, among others, as described in systematic detail by Risken [7] and Hanngi [8]. This viewpoint provides the basis for extracting a partial differential equation governing the probability distribution. The local flux of states in the direction of increasing x, here denoted by j(x, t), has the form j(x, t) = − kT γ ∂ρ ∂x (x, t) + ρ(x, t) ∂u ∂x (x, t) (2.4) where u(x, t) = U(x, y(t))/kT. The first term represents the diffusive transport of states in a force-free environment due to a thermally activated random walk
- 51. Observations on the mechanics of a molecular bond under force 33 whereas the second term represents directed transport in the downhill direction on the local energy landscape. The requirement of local conservation of states, ∂ρ ∂t (x, t) + ∂j ∂x (x, t) = 0 , (2.5) is equivalent to (2.3). Combining (2.4) with the conservation condition leads to the Smoluchowski partial differential equation [7] governing the probability density, ∂ρ ∂t (x, t) = kT γ ∂ ∂x ∂ρ ∂x (x, t) + ρ(x, t) ∂u(x, t) ∂x . (2.6) This equation must be augmented by an initial condition and suitable boundary conditions in order to obtain a solution. Boundary conditions will be specified in connection with a particular case of interest. As an initial condition at time t = 0, we assume that the distribution of states is confined to the bond well and that the flux of states is zero. This implies that the initial condition is determined by setting the quantity enclosed in square brackets on the right side of (2.6) equal to zero. The initial probability distri- bution is then determined as the solution of the ordinary differential equation ∂xρ(x, 0) + ρ(x, 0)∂xu(x, 0) = 0 and the normality condition (2.3) to be ρ(x, 0) = e−u(x,0) L e−u(x,0) dx (2.7) where L denotes the spatial range of the reaction coordinate over the landscape. We again denote the half-width of the bond well by the length δ and normal- ize all distances by this characteristic dimension. Suppose we also normalize time by the diffusion time over the distance δ, which is γ δ2/kT. If a nondi- mensional reaction coordinate ξ and a nondimensional time τ, as defined by ξ = x/δ τ = tkT/γ δ2 (2.8) are introduced, then the Smoluchowski equation takes on the parameter-free form ∂ρ ∂τ (ξ, τ) = ∂ ∂ξ ∂ρ ∂ξ (ξ, τ) + ρ(ξ, τ) ∂u ∂ξ (ξ, τ) . (2.9) This equation is to be enforced over the spatial range L of the landscape, which is −1 ξ 3 in Fig. 2.4, and over all time τ 0. When augmented by the ini- tial condition (2.7) and suitable boundary conditions, this partial differential equation can be solved approximately by any number of numerical strate- gies. However, before proceeding in that direction it is worthwhile to examine the range of the dimensionless independent variables that are relevant to the phenomenon being considered.
- 52. 34 L. B. Freund The important range for ξ is set by the range of the physical coordinate x required to span the interaction energy well and its surroundings, which is a distance equal to several times the length δ. Consequently, the range of ξ of interest is roughly −1 ξ 3. Similarly, the important range for τ is set by the elapsed time from the instant a force is first applied to the time of bond sepa- ration, which is typically on the order of 10 ms or greater. The corresponding range of τ can be estimated from (2.8). Assuming the factor kT/γ to have a value equal to the diffusion coefficient for water, or roughly 10−10 m2/s, and δ to be about 5 nm, then the range of τ corresponding to elapsed time of 10 ms is roughly 102. This implies that the probability distribution ρ(ξ, τ) varies slowly on the landscape in the τ direction compared to its variation in the ξ direction for comparable increments in ξ and τ. This observation suggests, in turn, that the rate of change of ρ on the left side of (2.9) is very small compared to the other contributions. It is assumed to be negligibly small for present purposes so that the governing equation reduces to ∂ ∂ξ ∂ρ ∂ξ (ξ, τ) + ρ(ξ, τ) ∂u ∂ξ (ξ, τ) ≈ 0 (2.10) with τ viewed as a parameter. In this way, the partial differential equation has been reduced to an ordinary differential equation to be solved for all values of time τ. A formulation similar to the present development was proposed by Heymann and Grubmuller [9] but with one significant difference. No specific landscape profile was presumed at the outset. Instead, conclusions were drawn by intro- ducing reaction rate coefficients early in the analysis and, subsequently, the shape of the landscape profile was extracted from observational data by means of the results obtained. The probability distribution ρ(ξ, τ) In an important contribution to physical chemistry, Kramers (see Ref. [7]) showed how the “on” and “off” rate coefficients of elementary rate reac- tion theory can be determined from any presumed time-independent energy landscape for a reaction occurring under diffusion-dominated circumstances. Through this result, it became possible to determine the influence of distance between an equilibrium state and the transition state in terms of reaction coor- dinate, the influence of attempt frequency on rate, the influence of curvature of the energy surface at the transition state, and so on. As introduced originally in elementary reaction rate theory, the coefficients depend on the energies at the equilibrium states and the transition state, but not on the shape of the landscape between these energy levels. By superimposing a constant uniform flux of probability density from one equilibrium state to another, the govern- ing Smoluchowski partial differential equation could be reduced to an ordinary
- 53. Observations on the mechanics of a molecular bond under force 35 differential equation. The reduction can be rendered exact in the case of the time-independent landscape through an insightful selection of boundary condi- tions, and this uniform flux was shown to be exactly the “on” or “off” rate coefficient. The reasoning is similar here, but the time dependence of the inter- action energy landscape and the fact that probability density is not conserved within the interval of interest necessitate adoption of a different point of view. The first integral of the ordinary differential equation (2.10) is dρ dξ (ξ, τ) + ρ(ξ, τ) du dξ (ξ, τ) = −j(τ) (2.11) with parameter of integration j(τ). While this equation must be satisfied throughout ξ0 ξ ξ1, it is not essential for each term to be continuous. We expect the probability density ρ(ξ, τ) and the gradient in the landscape ∂ξ u(ξ, τ) to be continuous on physical grounds. On the other hand, it is possible that j(τ) is only piecewise constant, in which case its discontinuities must be exactly offset by corresponding discontinuities in dρ/dξ. That this possibility exists is not a trivial observation. If j(τ) is required to be a spatially uniform flux throughout the entire interval and if we enforce the boundary condition that the flux at ξ = ξ0 is equal to zero, then j(τ) must be equal to zero throughout the interval and no solution with nonzero flux can be found. The ordinary differ- ential equation is readily solved within any interval for which j(τ) is a constant and we proceed accordingly. Several specific points in the range of ξ that covers the landscape have partic- ular significance, and these are identified in Fig. 2.5. The point ξ = ξ0 locates the extreme left end of the range of interest; this is ξ0 = −1 in the present instance. The point ξ = ξa(τ) identifies the position of the local minimum in the inter- action energy well profile; the location of this minimum is time dependent, in general. The point ξ = ξc(τ) identifies the position of the local maximum in the landscape profile, that is, the transition state; this location is also time dependent. Finally ξ = ξ1 locates the extreme right end of the range of interest; for the profile being considered here, ξ1 = 3 although the choice is arbitrary to a certain degree within ξc(τ) ξ. There are some restrictions on the definitions of the points ξa(τ) and ξc(τ) for certain ranges of system parameters. For example, in very early times, a local maximum in the energy landscape may not exist for some specified η(τ). Likewise, at late times, the local minimum and maximum points may coalesce in some cases. These circumstances are unusual and neither is a factor in the example considered below. We could proceed to formally solve the differential equation (2.11). However, it is more enlightening to construct a solution by superposition after noting the physical roles played by the homogeneous solution and the particular solution.
- 54. 36 L. B. Freund –1 1 2 3 –10 –15 –5 5 a c 1 0 Fig. 2.5 The graph illustrates a constant time cross-section of the energy landscape of the system in normalized coordinates, identifying several values of spatial coordinate ξ that identify particular points of significance in the analysis. The point ξ0 is the extreme left end of the range of interest; the flux of states is zero at this point. The point ξ1 is the extreme right end of the interval of interest; the density of states is zero at this point. The points ξa and ξc are local minimum and maximum energies in this section, respectively; the locations of these points vary with time. The quantity of primary interest is the probability that the state of the bond remains in the well. Initially, at τ = 0, this probability is one, and we expect it to decrease as τ increases. Let this probability be represented by ρa(τ), that is, ρa(τ) = ξc(τ) ξ0 ρ(ξ, τ) dξ, (2.12) where the subscript indicates that the probability is connected with the energy well at ξ = ξa(τ). The integrating factor for this differential equation is eu(ξ,τ) , and it is convenient to introduce a compact notation for representing definite integrals of this factor and its inverse. For this purpose, we introduce I±(ξ−, ξ+) = ξ+ ξ− e±u(ξ,τ) dτ , ξ+ ≥ ξ−. (2.13) The value is identically zero if ξ+ ≤ ξ−. We see that a distribution of states within the well defined by ρ(ξ, τ) homog = ρa(τ) e−u(ξ,τ) I−(ξ0, ξc+) (2.14) provides a homogeneous solution of the differential equation over ξ0 ξ ξc+, having the essential attributes we have noted. It is clear that the homogeneous solution generates no flux of probability density. The role of the particular solution is to drain those states represented by (2.14) out of the well centered at ξa and to pass them over the barrier at ξc. It was noted above that a particular solution with uniform flux throughout the range ξ0 ξ ξc is out of the question. Therefore, we adopt a particular solution with
- 55. Other documents randomly have different content
- 56. flexure of the stand under oscillations of the pendulum. At the Stuttgart conference of the geodetic association in 1877, Hervé Faye proposed to solve the problem of flexure by swinging two similar pendulums from the same support with equal amplitudes and in opposite phases. Peirce, in 1879, demonstrated theoretically the soundness of the method and presented a design for its application, but the “double pendulum” was rejected at that time. Peirce also designed and had constructed four examples of a new type of invariable, reversible pendulum of cylindrical form which made possible the experimental study of Stokes’ theory of the resistance to motion of a pendulum in a viscous fluid. Commandant Defforges, of France, also designed and used cylindrical reversible pendulums, but of different length so that the effect of flexure was eliminated in the reduction of observations. Maj. Robert von Sterneck, of Austria- Hungary, initiated a new era in gravity research by the invention of an apparatus with a short pendulum for relative determinations of gravity. Stands were then constructed in Europe on which two or four pendulums were hung at the same time. Finally, early in the present century, Vening Meinesz found that the Faye- Peirce method of swinging pendulums hung on a Stückrath four-pendulum stand solved the problem of instability due to the mobility of the soil in Holland. The 20th century has witnessed increasing activity in the determination of absolute and relative values of gravity. Gravimeters have been perfected and have been widely used for rapid relative determinations, but the compound pendulums remain as indispensable instruments. Mendenhall’s replacement of knives by planes attached to nonreversible pendulums has been used also for reversible ones. The Geodetic Institute at Potsdam is presently applying the Faye-Peirce method to the reversible pendulum. [115] Pendulums have been constructed of new materials, such as invar, fused silica, and fused quartz. Minimum pendulums for precise relative determinations have been constructed and used. Reversible pendulums have been made with “I” cross sections for better stiffness. With all these modifications, however, the foundations of the present designs of compound pendulum apparatus were created in the 19th century. FOOTNOTES [1] The basic historical documents have been collected, with a bibliography of works and memoirs published from 1629 to the end of 1885, in Collection de mémoires relatifs a la physique, publiés par la Société française de Physique [hereinafter referred to as Collection de mémoires]: vol. 4, Mémoires sur le pendule, précédés d’une bibliographie (Paris: Gauthier-Villars, 1889); and vol. 5, Mémoires sur le pendule, part 2 (Paris: Gauthier-Villars, 1891). Important secondary sources are: C. Wolf, “Introduction historique,” pp. 1-42 in vol. 4, above; and George Biddell Airy, “Figure of the Earth,” pp. 165-240 in vol. 5 of Encyclopaedia metropolitana (London, 1845).
- 57. [2] Galileo Galilei’s principal statements concerning the pendulum occur in his Discourses Concerning Two New Sciences, transl. from Italian and Latin into English by Henry Crew and Alfonso de Salvio (Evanston: Northwestern University Press, 1939), pp. 95-97, 170- 172. [3] P. Marin Mersenne, Cogitata physico-mathematica (Paris, 1644), p. 44. [4] Christiaan Huygens, Horologium oscillatorium, sive de motu pendulorum ad horologia adaptato demonstrationes geometricae (Paris, 1673), proposition 20. [5] The historical events reported in the present section are from Airy, “Figure of the Earth.” [6] Abbé Jean Picard, La Mesure de la terre (Paris, 1671). John W. Olmsted, “The ‘Application’ of Telescopes to Astronomical Instruments, 1667-1669,” Isis (1949), vol. 40, p. 213. [7] The toise as a unit of length was 6 Paris feet or about 1,949 millimeters. [8] Jean Richer, Observations astronomiques et physiques faites en l’isle de Caïenne (Paris, 1679). John W. Olmsted, “The Expedition of Jean Richer to Cayenne 1672-1673,” Isis (1942), vol. 34, pp. 117-128. [9] The Paris foot was 1.066 English feet, and there were 12 lines to the inch. [10] Christiaan Huygens, “De la cause de la pesanteur,” Divers ouvrages de mathematiques et de physique par MM. de l’Académie Royale des Sciences (Paris, 1693), p. 305. [11] Isaac Newton, Philosophiae naturalis principia mathematica (London, 1687), vol. 3, propositions 18-20. [12] Pierre Bouguer, La figure de la terre, déterminée par les observations de Messieurs Bouguer et de La Condamine, envoyés par ordre du Roy au Pérou, pour observer aux environs de l’equateur (Paris, 1749). [13] P. L. Moreau de Maupertuis, La figure de la terre déterminée par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le Monnier, l’Abbé Outhier et Celsius, faites par ordre du Roy au cercle polaire (Paris, 1738). [14] Paris, 1743. [15] George Gabriel Stokes, “On Attraction and on Clairaut’s Theorem,” Cambridge and Dublin Mathematical Journal (1849), vol. 4, p. 194. [16] See Collection de mémoires, vol. 4, p. B-34, and J. H. Poynting and Sir J. J. Thomson, Properties of Matter (London, 1927), p. 24. [17] Poynting and Thomson, ibid., p. 22. [18] Charles M. de la Condamine, “De la mesure du pendule à Saint Domingue,” Collection de mémoires, vol. 4, pp. 3-16. [19] Père R. J. Boscovich, Opera pertinentia ad Opticam et Astronomiam (Bassani, 1785), vol. 5, no. 3. [20] J. C. Borda and J. D. Cassini de Thury, “Expériences pour connaître la longueur du pendule qui bat les secondes à Paris,” Collection de mémoires, vol. 4, pp. 17-64.
- 58. [21] F. W. Bessel, “Untersuchungen über die Länge des einfachen Secundenpendels,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, 1826 (Berlin, 1828). [22] Bessel used as a standard of length a toise which had been made by Fortin in Paris and had been compared with the original of the “toise de Peru” by Arago. [23] L. G. du Buat, Principes d’hydraulique (Paris, 1786). See excerpts in Collection de mémoires, pp. B-64 to B-67. [24] Capt. Henry Kater, “An Account of Experiments for Determining the Length of the Pendulum Vibrating Seconds in the Latitude of London,” Philosophical Transactions of the Royal Society of London (1818), vol. 108, p. 33. [Hereinafter abbreviated Phil. Trans.] [25] M. G. de Prony, “Méthode pour déterminer la longueur du pendule simple qui bat les secondes,” Collection de mémoires, vol. 4, pp. 65-76. [26] Collection de mémoires, vol. 4, p. B-74. [27] Phil. Trans. (1819), vol. 109, p. 337. [28] John Herschel, “Notes for a History of the Use of Invariable Pendulums,” The Great Trigonometrical Survey of India (Calcutta, 1879), vol. 5. [29] Capt. Edward Sabine, “An Account of Experiments to Determine the Figure of the Earth,” Phil. Trans. (1828), vol. 118, p. 76. [30] John Goldingham, “Observations for Ascertaining the Length of the Pendulum at Madras in the East Indies,” Phil. Trans. (1822), vol. 112, p. 127. [31] Basil Hall, “Letter to Captain Kater Communicating the Details of Experiments made by him and Mr. Henry Foster with an Invariable Pendulum,” Phil. Trans. (1823), vol. 113, p. 211. [32] See Collection de mémoires, vol. 4, p. B-103. [33] Ibid., p. B-88. [34] Ibid., p. B-94. [35] Francis Baily, “On the Correction of a Pendulum for the Reduction to a Vacuum, Together with Remarks on Some Anomalies Observed in Pendulum Experiments,” Phil. Trans. (1832), vol. 122, pp. 399-492. See also Collection de mémoires, vol. 4, pp. B-105, B- 112, B-115, B-116, and B-117. [36] One was of case brass and the other of rolled iron, 68 in. long, 2 in. wide, and 1 /2 in. thick. Triangular knife edges 2 in. long were inserted through triangular apertures 19.7 in. from the center towards each end. These pendulums seem not to have survived. There is, however, in the collection of the U.S. National Museum, a similar brass pendulum, 375 /8 in. long (fig. 15) stamped with the name of Edward Kübel (1820-96), who maintained an instrument business in Washington, D.C., from about 1849. The history of this instrument is unknown.
- 59. [37] See Baily’s remarks in the Monthly Notices of the Royal Astronomical Society (1839), vol. 4, pp. 141-143. See also letters mentioned in footnote 38. [38] This document, together with certain manuscript notes on the pendulum experiments and six letters between Wilkes and Baily, is in the U.S. National Archives, Navy Records Gp. 37. These were the source materials for the information presented here on the Expedition. We are indebted to Miss Doris Ann Esch and Mr. Joseph Rudmann of the staff of the U.S. National Museum for calling our attention to this early American pendulum work. [39] G. B. Airy, “Account of Experiments Undertaken in the Harton Colliery, for the Purpose of Determining the Mean Density of the Earth,” Phil. Trans. (1856), vol. 146, p. 297. [40] T. C. Mendenhall, “Measurements of the Force of Gravity at Tokyo, and on the Summit of Fujiyama,” Memoirs of the Science Department, University of Tokyo (1881), no. 5. [41] J. T. Walker, Account of Operations of The Great Trigonometrical Survey of India (Calcutta, 1879), vol. 5, app. no. 2. [42] Bessel, op. cit. (footnote 21), article 31. [43] C. A. F. Peters, Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (Altona, Germany, 1860), Band 2, p. 3. The correction required if the times of swing are not exactly the same is said to have been given also by Bohnenberger. [44] F. W. Bessel, “Construction eines symmetrisch geformten Pendels mit reciproken Axen, von Bessel,” Astronomische Nachrichten (1849), vol. 30, p. 1. [45] E. Plantamour, “Expériences faites à Genève avec le pendule à réversion,” Mémoires de la Société de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, p. 309. [46] Ibid., pp. 309-416. [47] C. Cellérier, “Note sur la Mesure de la Pesanteur par le Pendule,” Mémoires de la Société de Physique et d’histoire naturelle de Genève, 1865 (Geneva, 1866), vol. 18, pp. 197- 218. [48] A. Sawitsch, “Les variations de la pesanteur dans les provinces occidentales de l’Empire russe,” Memoirs of the Royal Astronomical Society (1872), vol. 39, p. 19. [49] J. J. Baeyer, Über die Grösse und Figur der Erde (Berlin, 1861). [50] Comptes-rendus de la Conférence Géodésique Internationale réunie à Berlin du 15-22 Octobre 1864 (Neuchâtel, 1865). [51] Ibid., part III, subpart E. [52] Bericht über die Verhandlungen der vom 30 September bis 7 October 1867 zu Berlin abgehaltenen allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1868). See report of fourth session, October 3, 1867. [53] C. Bruhns and Albrecht, “Bestimmung der Länge des Secundenpendels in Bonn, Leiden und Mannheim,” Astronomisch-Geodätische Arbeiten im Jahre 1870 (Leipzig: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1871).
- 60. [54] Bericht über die Verhandlungen der vom 23 bis 28 September 1874 in Dresden abgehaltenen vierten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1875). See report of second session, September 24, 1874. [55] Carolyn Eisele, “Charles S. Peirce—Nineteenth-Century Man of Science,” Scripta Mathematica (1959), vol 24, p. 305. For the account of the work of Peirce, the authors are greatly indebted to this pioneer paper on Peirce’s work on gravity. It is worth noting that the history of pendulum work in North America goes back to the celebrated Mason and Dixon, who made observations of “the going rate of a clock” at “the forks of the river Brandiwine in Pennsylvania,” in 1766-67. These observations were published in Phil. Trans. (1768), vol. 58, pp. 329-335. [56] The pendulums with conical bobs are described and illustrated in E. D. Preston, “Determinations of Gravity and the Magnetic Elements in Connection with the United States Scientific Expedition to the West Coast of Africa, 1889-90,” Report of the Superintendent of the Coast and Geodetic Survey for 1889-90 (Washington, 1891), app. no. 12. [57] Eisele, op. cit. (footnote 55), p. 311. [58] The record of Peirce’s observations in Europe during 1875-76 is given in C. S. Peirce, “Measurements of Gravity at Initial Stations in America and Europe,” Report of the Superintendent of the Coast Survey for 1875-76 (Washington, 1879), pp. 202-337 and 410-416. Peirce’s report is dated December 13, 1878, by which time the name of the Survey had been changed to U.S. Coast and Geodetic Survey. [59] Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1876). [60] Ibid. See report for fifth session, September 25, 1875. [61] The experiments at the Stevens Institute, Hoboken, were reported by Peirce to the Permanent Commission which met in Hamburg, September 4-8, 1878, and his report was published in the general Bericht for 1878 in the Verhandlungen der vom 4 bis 8 September 1878 in Hamburg Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1879), pp. 116-120. Assistant J. E. Hilgard attended for the U.S. Coast and Geodetic Survey. The experiments are described in detail in C. S. Peirce, “On the Flexure of Pendulum Supports,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), app. no. 14, pp. 359-441. [62] Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1877). See report of third session, October 7, 1876. [63] Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart abgehaltenen fünften allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1878). [64] Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1880). [65] Assistants’ Reports, U.S. Coast and Geodetic Survey, 1879-80. Peirce’s paper was published in the American Journal of Science (1879), vol. 18, p. 112. [66] Comptes-rendus de l’Académie des Sciences (Paris, 1879), vol. 89, p. 462.
- 61. [67] Verhandlungen der vom 13 bis 16 September 1880 zu München abgehaltenen sechsten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1881). [68] Ibid., app. 2. [69] Ibid., app. 2a. [70] Verhandlungen der vom 11 bis zum 15 September 1882 im Haag Vereinigten Permanenten Commission der Europäischen Gradmessung (Berlin, 1883). [71] Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen siebenten allgemeinen Conferenz der Europäischen Gradmessung (Berlin, 1884). Gen. Cutts attended for the U.S. Coast and Geodetic Survey. [72] Ibid., app. 6. See also, Zeitschrift für Instrumentenkunde (1884), vol. 4, pp. 303 and 379. [73] Op. cit. (footnote 67). [74] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), p. 26. [75] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1889-90 (Washington, 1891), app. no. 12. [76] Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82 (Washington, 1883). [77] Transactions of the Cambridge Philosophical Society (1856), vol. 9, part 2, p. 8. Also published in Mathematical and Physical Papers (Cambridge, 1901), vol. 3, p. 1. [78] Peirce’s comparison of theory and experiment is discussed in a report on the Peirce memoir by William Ferrel, dated October 19, 1890, Martinsburg, West Virginia. U.S. Coast and Geodetic Survey, Special Reports, 1887-1891 (MS, National Archives, Washington). [79] The stations at which observations were conducted with the Peirce pendulums are recorded in the reports of the Superintendent of the U.S. Coast and Geodetic Survey from 1881 to 1890. [80] Comptes-rendus de l’Académie des Sciences (Paris, 1880), vol. 90, p. 1401. Hervé Faye’s report, dated June 21, 1880, is in the same Comptes-rendus, p. 1463. [81] Commandant C. Defforges, “Sur l’Intensité absolue de la pesanteur,” Journal de Physique (1888), vol. 17, pp. 239, 347, 455. See also, Defforges, “Observations du pendule,” Mémorial du Dépôt général de la Guerre (Paris, 1894), vol. 15. In the latter work, Defforges described a pendulum “reversible inversable,” which he declared to be truly invariable and therefore appropriate for relative determinations. The knives remained fixed to the pendulums, and the effect of interchanging knives was obtained by interchanging weights within the pendulum tube. [82] Papers by Maj. von Sterneck in Mitteilungen des K. u. K. Militär-geographischen Instituts, Wien, 1882-87; see, in particular, vol. 7 (1887). [83] T. C. Mendenhall, “Determinations of Gravity with the New Half-Second Pendulum …,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1890-91 (Washington, 1892), part 2, pp. 503-564.
- 62. [84] W. H. Burger, “The Measurement of the Flexure of Pendulum Supports with the Interferometer,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1909-10 (Washington, 1911), app. no. 6. [85] E. J. Brown, A Determination of the Relative Values of Gravity at Potsdam and Washington (Special Publication No. 204, U.S. Coast and Geodetic Survey; Washington, 1936). [86] M. Haid, “Neues Pendelstativ,” Zeitschrift für Instrumentenkunde (July 1896), vol. 16, p. 193. [87] Dr. R. Schumann, “Über eine Methode, das Mitschwingen bei relativen Schweremessungen zu bestimmen,” Zeitschrift für Instrumentenkunde (January 1897), vol. 17, p. 7. The design for the stand is similar to that of Peirce’s of 1879. [88] Dr. R. Schumann, “Über die Verwendung zweier Pendel auf gemeinsamer Unterlage zur Bestimmung der Mitschwingung,” Zeitschrift für Mathematik und Physik (1899), vol. 44, p. 44. [89] P. Furtwängler, “Über die Schwingungen zweier Pendel mit annähernd gleicher Schwingungsdauer auf gemeinsamer Unterlage,” Sitzungsberichte der Königlicher Preussischen Akademie der Wissenschaften zu Berlin (Berlin, 1902) pp. 245-253. Peirce investigated the plan of swinging two pendulums on the same stand (Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81, Washington, 1883, p. 26; also in Charles Sanders Peirce, Collected Papers, 6.273). At a conference on gravity held in Washington during May 1882, Peirce again advanced the method of eliminating flexure by hanging two pendulums on one support and oscillating them in antiphase (“Report of a conference on gravity determinations held in Washington, D.C., in May, 1882,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82, Washington, 1883, app. no. 22, pp. 503-516). [90] F. A. Vening Meinesz, Observations de pendule dans les Pays-Bas (Delft, 1923). [91] A. Berroth, “Schweremessungen mit zwei und vier gleichzeitig auf demselben Stativ schwingenden Pendeln,” Zeitschrift für Geophysik, vol. 1 (1924-25), no. 3, p. 93. [92] “Pendulum Apparatus for Gravity Determinations,” Engineering (1926), vol. 122, pp. 271- 272. [93] Malcolm W. Gay, “Relative Gravity Measurements Using Precision Pendulum Equipment,” Geophysics (1940), vol. 5, pp. 176-191. [94] L. G. D. Thompson, “An Improved Bronze Pendulum Apparatus for Relative Gravity Determinations,” [published by] Dominion Observatory (Ottawa, 1959), vol. 21, no. 3, pp. 145-176. [95] W. A. Heiskanen and F. A. Vening Meinesz, The Earth and its Gravity Field (McGraw: New York, 1958). [96] F. Kühnen and P. Furtwängler, Bestimmung der Absoluten Grösze der Schwerkraft zu Potsdam mit Reversionspendeln (Berlin: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1906), new ser., no. 27. [97] Reported by Dr. F. Kühnen to the fifth session, October 9, 1895, of the Eleventh General Conference, Die Internationale Erdmessung, held in Berlin from September 25 to October 12, 1895. A footnote states that Assistant O. H. Tittmann, who represented the
- 63. United States, subsequently reported Peirce’s prior discovery of the influence of the flexure of the pendulum itself upon the period (Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1883-84, Washington, 1885, app. 16, pp. 483-485). [98] Assistants’ Reports, U.S. Coast and Geodetic Survey, 1883-84 (MS, National Archives, Washington). [99] C. S. Peirce, “Effect of the Flexure of a Pendulum Upon its Period of Oscillation,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1883-84 (Washington, 1885), app. no. 16. [100] F. R. Helmert, Beiträge zur Theorie des Reversionspendels (Potsdam: Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts, 1898). [101] J. A. Duerksen, Pendulum Gravity Data in the United States (Special Publication No. 244, U.S. Coast and Geodetic Survey; Washington, 1949). [102] Ibid., p. 2. See also, E. J. Brown, loc. cit. (footnote 85). [103] Paul R. Heyl and Guy S. Cook, “The Value of Gravity at Washington,” Journal of Research, National Bureau of Standards (1936), vol. 17, p. 805. [104] Sir Harold Jeffreys, “The Absolute Value of Gravity,” Monthly Notices of the Royal Astronomical Society, Geophysical Supplement (London, 1949), vol. 5, p. 398. [105] J. S. Clark, “The Acceleration Due to Gravity,” Phil. Trans. (1939), vol. 238, p. 65. [106] Hugh L. Dryden, “A Reexamination of the Potsdam Absolute Determination of Gravity,” Journal of Research, National Bureau of Standards (1942), vol. 29, p. 303; and A. Berroth, “Das Fundamentalsystem der Schwere im Lichte neuer Reversionspendelmessungen,” Bulletin Géodésique (1949), no. 12, pp. 183-204. [107] T. C. Mendenhall, op. cit. (footnote 83), p. 522. [108] A. H. Cook, “Recent Developments in the Absolute Measurement of Gravity,” Bulletin Géodésique (June 1, 1957), no. 44, pp. 34-59. [109] See footnote 89. [110] C. S. Peirce, “On the Deduction of the Ellipticity of the Earth, from Pendulum Experiments,” Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81 (Washington, 1883), app. no. 15, pp. 442-456. [111] Heiskanen and Vening Meinesz, op. cit. (footnote 95), p. 74. [112] Ibid., p. 76. [113] Ibid., p. 309. [114] Ibid., p. 310. [115] K. Reicheneder, “Method of the New Measurements at Potsdam by Means of the Reversible Pendulum,” Bulletin Géodésique (March 1, 1959), no. 51, p.72.
- 64. Paper 44 - Transcriber’s Note Formatting of equations has been altered from the original to display them ‘in line,’ and brackets have been added to clarify expressions where necessary. Footnotes have been moved to the end of the paper. Illustrations and the Glossary of Gravity Terminology section have been moved to avoid breaks in paragraphs. Minor punctuation errors have been corrected without note. Typographical errors and inconsistencies have been corrected as follows: P. 320 ‘difference T1 - T2 is sufficiently’—had ‘sufficlently.’ P. 321 ‘faites à Genève avec le pendule à réversion’— had ‘reversion.’ P. 326 ‘Schwere mit Hilfe verschiedener Apparate’—had ‘verschiedene.’ P. 328 ‘between the yard and the meter.’—closing quote mark deleted. P. 334 ‘Mendenhall apparatus were part of’—‘was’ changed to ‘were.’ P. 342 ‘of the Geodetic Institute at Potsdam’—had ‘Postdam.’
- 65. P. 345 ‘The gravimetric methods of physical’—had ‘mtehods.’ Footnote 1 ‘Société française de Physique’—had ‘Française.’ Footnote 3 ‘Cogitata physico-mathematica’—had ‘physica.’ Footnote 10 ‘mathématiques et de physique par MM. de l’Académie Royale’—had ‘mathematiques,’ ‘Royal.’ Footnote 12 ‘par ordre du Roy au Pérou, pour observer’—had ‘Perou, pour observir.’ Footnote 19 ‘Opticam et Astronomiam’—had ‘Astronomian.’ Footnote 20 ‘connaître la longueur du pendule qui’— had ‘connaitre la longuer.’ Footnote 21 ‘Abhandlungen der Königlichen Akademie’—had ‘Königliche.’ Footnote 25 ‘pour déterminer la longueur du pendule’—had ‘longeur.’ Footnote 41 ‘Survey of India (Calcutta, 1879)’— had ‘Surey.’ Footnotes 45 and 47 ‘Société de Physique et d’histoire’—had ‘d’historire.’ Footnote 49 ‘Über die Grösse und Figur der Erde’—had ‘Grosse.’ Footnote 53 ‘Bestimmung der Länge’—had ‘Lange’; ‘Astronomisch-Geodätische Arbeiten’—had
- 66. ‘Astronomische’; ‘Veröffentlichungen des Königlichen’—had ‘Königliche.’ Footnote 55 ‘(1768), vol. 58, pp. 329-335.’—had ‘329- 235.’ Footnote 66 ‘Comptes-rendus de l’Académie’—had ‘L’Académie.’ Footnote 81 ‘Sur l’Intensité absolue’—had ‘l’Intensite.’ Footnote 89 ‘Sitzungsberichte der Königlicher’—had ‘Königliche.’ Footnote 100 ‘Veröffentlichungen des Königlichen’ had ‘Veröffentlichungen Königliche.’ Capitalisation of ‘Von’/‘von’ has been regulaized to ‘von’ for all personal names, except at the beginning of a sentence, and when referring to the Von Sterneck pendulum. Index Quick link to each letter: [A] [B] [C] [D] [E] [F] [G] [H] [I] [J] [K] [L] [M] [N] [O] [P] [Q] [R] [S] [T] [U] [V] [W] [Y] [Z] A
- 67. Adams, W. B., 252 Agricola, Georgius, 215, 216 Airy, G. B., 319, 324, 332 Albrecht, Karl Theodore, 322, 338 Aldini, Giovanni, 124 Al-Mamun, seventh calif of Bagdad, 306 Almansi, Emilio, 339 Ames Manufacturing Company, 5, 7 Ampère, André Marie, 127, 129 Anckerswärd, Col. Michael, 157 Angle, Edward H., 295 Arago, Dominique François Jean, 129 Aristarchus of Samos, 54 Aristotle, 179, 306 Astor, John Jacob, 141 B Baeyer, Adolf, 193 Baeyer, J. J., 321, 322, 324–327, 338, 346 Baily, Francis, 317 Baldwin, Matthias William, 264 Baltimore, Lord. See Calvert. Barlow, Peter W., 221, 227 Bartlett, Charles A., 8 Basevi, James Palladio, 345 Battison, E. A., 18 Beach, Alfred Ely, 224, 227–229, 231, 237 Bechil, Achild, 179 Bemis, Will, 20–22, 27 Bennet, Abraham, 124 Bennett, Frank M., 139, 150, 165 Benz, Carl, 6, 7 Bergh, Christian, 145 Berroth, A., 342
- 68. Berthelot, Marcellin, 189 Bertolla, Alessandro, 65 Bertolla, Bartolomeo Antonio, 31, 34, 36–41, 47, 51, 52, 57–59, 62, 63 Berzelius, Jöns Jakob, 133, 182 Bessel, Friedrich Wilhelm, 313, 314, 319, 320, 324, 325, 338, 346 Besson, Jacques, 107 Bettany, G. T., 136 Beyer, Dr. Henry Gustav, 275, 276 Biddle, James, 141 Biot, Jean Baptiste, 135, 325, 329 Black, G. V., 295 Black and Bell, plant at Stratford, 182 Blake, John B., 290, 291 Bohnenberger, Johann Gottlieb Friedrich, 315 Bollman, W., and Company, 91, 92 Bollman, Wendel, 79, 80–83, 85, 88–92, 94–97 Borda, J. C., 311, 312, 315, 325, 329, 346 Borghesi, Father Francesco, 31–59, 70, 71 Boscovitch, Père R. J., 310, 311 Boston Locomotive Works, 260 Bouguer, Pierre, 307, 309–311 327, 343, 345 Boussingault, Jean Baptiste, 185 Boyd, John C., 276 Boyle, Robert, 178, 179 Brackenridge, S. M., 145 Brahe, Tycho, 54, 306 Brand, H., 178, 179 Brewington, M. V., 155 Brown, Adam and Noah, 141, 142, 145 Brown, Alexander Crosby, 165 Brown, E. J., 334, 339 Brown, Noah, 141, 150, 151 Browne, Charles, 157 Browne, Henry, 304, 314 Browns’ yard, 142, 144
- 69. Bruhns, C., 322, 324, 338 Brunel, I. K., 217, 218 Brunel, Marc Isambard (the elder), 204, 205, 217, 218, 221, 224, 229, 231, 236 Brunner Brothers (Paris), 329 Buchner, Hans, 197, 200 Burleigh, Charles, 212, 213 Burleigh Rock Drill Company, 212 Burr, S. D. V., 236 Butzjäger, Johann Georg, 36, 37 C Calvert, George, Lord Baltimore, 156 Calvin, Melvin, 200 Canning, Stratford, 139 Carlisle, Anthony, 124 Carrel, Alexis, 291 Casciarolo, Vicenzo, 179 Cassini, Giovanni-Domenico, 306, 307 Cassini, Jacques, 306 Cassini de Thury, J. D., 311, 312, 315, 325, 329, 346 Cavallo, Tiberio, 124 Cavendish, Henry, 123 Cellérier, Charles, 320, 321, 325, 326, 329, 336 Chapman, Fredrik Henrik af, 156, 166 Charles II of England, 152, 153 Charles VI, Emperor of Austria, 32 Chevreul, Michel, 189 Clairaut, Alexis Claude, 308, 309, 343, 345 Clark, J. S., 342 Clark, John, 91 Clarke, A. R., 345 Cles, Baron of, 57, 59 Coast and Harbor Defense Company, 141
- 70. Coast Defense Society, 141, 142 Cochrane, Sir Thomas, 231, 232 Colbert, Jean Baptiste, 306 Colburn, Zerah, 259 Colden, C. D., 149 Coleman, Laurence V., 290 Colgan, P., 10 Colton, Arthur, and Company, 278 Cook, A. H., 342 Cook, Guy S., 339, 342 Copernicus, 54 Copperthwaite, William Charles, 224 Cori, Carl F., 200 Cori, Gerti T., 200 Crookes, William, 192 Cummings, James, 125, 127–129, 133–136 D Dagger, Benjamin M., 290 Danforth Cooke Co., 252 Danish Greenland Company, 150 Danish Royal Archives, 139, 150 Davy, Sir Humphry, 185 Dearborn, Henry, 141, 142 Deats, William, 9 Decatur, Stephen, 141 Defforges, C., 314, 329, 346 De Freycinet, Louis Claude de Saulses, 317 De Hevesy, George, 198, 200 De la Hire, Gabriel Philippe, 306 De la Vega, Garcilaso, 185 De Prony, M. G., 314 Deptford Yard (England), 165 De Saussure, Théodore, 185
- 71. Di Noris, Cristoforo Sizzo, 59 Dixon, William S., 276 Doane, Thomas, 210, 212, 213, 215 Dodrill, Forest D., 290 Donner, Joseph, 277 Douglas, W. B., Company, 113 Drake, Edwin L., 213 Drinker, Henry S., 224, 237 Drury, Gardner P., 260 Dryden, Hugh L., 342 Du Buat, L. G., 314 Duperry, Capt. Louis Isidore, 317 Duryea, Charles, 3-13, 15, 16, 19–21, 26, 27 Duryea, J. Frank, 3-7, 9-13, 15–23, 26, 27 Duryea Motor Corporation, 5 Duryea Motor Wagon Company, 3, 27 Duryea Power Company, 5 E Eastwick, Andrew M., 259 Eckford, Henry, 142 Einthoven, Willem, 290 Emerson, John Haven, 285 Emmet, ——, 144 Eratosthenes, 306, 308, 342 Erman, Paul, 128, 129, 132, 133 Eudoxus of Cnidus, 306 Euler-Chelpin, Hans von, 197, 200 Evans, Samuel, 141, 145 Evelyn, John, 32 F
- 72. Faraday, Michael, 125 Faye, Hervé, 325–327, 336–338, 346, 347 Ferchl, Fritz, 285 Fernel, Jean, 306 Fernelius, Jean, 179 Feulgen, Robert, 193 Fink, Albert, 79, 91 Fischelis, Robert P., 287 Fischer, Emil, 193 Fleming, Sir Alexander, 290, 295 Flint, James Milton, 273–278 Fox, Josiah, 157 Francis I, Emperor of the Holy Roman Empire, 42, 44, 52, 58 Fulton, Robert, 139, 141, 142, 144, 147, 149, 150, 157, 159, 165 Furtwängler, P., 337–339 G Gahn, Johann Gottlieb, 182 Galilei, Galileo, 304, 305, 346 Galvani, Luigi, 124 Garfield, James A., 272 Garrison, Fielding H., 277 Gauss, C. F., 320 Gautier, P., 339 Gay-Lussac, Joseph Louis, 125, 182 Gilbert, L. W., 127–129, 132 Gobley, Nicolas Théodore, 191 Godin, Louis, 307 Goldingham, John, 316, 345 Goode, G. Brown, 273 Graham, Thomas, 182, 183, 185 Gravatt, C. U., 276 Greathead, James Henry, 204, 218, 221, 224, 229, 231, 235–237 Greely, A. W., 329
- 73. Griffenhagen, George B., 290, 291 Grubenmann, Hans, 85 Grubenmann, Johann Ulrich, 85 Gulf Oil and Development Company, 338 Gurley, Ralph R. (USN), 150, 151 Gustav III of Sweden, 156, 157 Gwynn, Stuart, 210 H Hahn, Father Philipp Matthäus, 33 Haid, M., 335 Hall, Basil, 316 Hammond, William Alexander, 273 Hankwitz, Gottfried, 180 Harden, Arthur, 197, 200 Harrington, Frank, 7 Harrison, Joseph, Jr., 259 Hartford Machine Screw Company, 6 Hartmann, Immanuel Peter, 181 Haskin, DeWitt C., 201, 232, 234–236 Haupt, Herman, 96, 204, 209, 210 Hawley, C. E., 6, 11 Hawthorn, Leslie, and Company (Scotland), 166 Heaviside, W. J., 321, 345 Heiskanen, W. A., 338, 345, 346 Hellot, Jean, 180 Helmert, F. R., 338, 339 Helmholtz, Hermann von, 326 Henderson, Alfred R., 291 Henkel, Silon, 290 Henry II, King of France, 179 Herschel, John, 319, 328, 345 Heyl, Paul R., 339, 342 Hindle, Charles F., 290
- 74. Hinkley, Holmes, 252, 260, 263 Hirsch, Adolph, 322, 324–326 Hittorf, Wilhelm, 181 Hobson, Joseph, 237 Hoefer, Ferdinand, 179 Holmberg, Wilhelm, 178 Holt, L. Emmett, 276 Hoppe-Seyler, Felix, 193 Howard, George W., Company, 8 Hull, A. S., 251, 268 Humboldt, Alexander von, 185 Huygens, Christiaan, 179, 304, 305, 307, 314, 342, 346 I Ibañez, Carlos, 325 Incas, 185 J Jefferson, Thomas, 145 Jeffreys, Sir Harold, 342 Jones, Jacob, 141 Jones, Thomas, 318 Jones, William, 147 K Kater, Henry, 304, 314–320, 325, 327, 329, 345, 346 Kells, Charles E., 295 Klein, Father ——, 33 Kletwich, Johann Christopher, 179 Knight, ——, 83
- 75. Koett, Albert B., 287 Koppe, Émile, 181 Kornberg, Arthur, 200 Kossel, Albrecht, 200 Kraft, Johann Daniel, 179 Kramer, Dr. ——, 181 Kühnen, F., 338–339 Kunckel, Johann, 179 L La Condamine, Charles Marie de, 307, 310, 311, 343 Lange, W., 199 Laplace, Marquis Pierre Simon de, 309, 313, 320 Latrobe, Benjamin H., 82, 83, 85, 87–91, 208, 209 Laurie, J., 157 Lavoisier, Antoine Laurent, 181, 185 Law, Henry, 218 LaWall, Charles H., 285 Laws, John Bennet, 186 Lederle Laboratories, 290 Leibnitz, Gottfried Wilhelm von, 179 Lennox, Charles, third Duke of Richmond, 185 Leonhardi, Johann Gottfried, 179 Levine, Phoebus Aaron Theodor, 193 Lewis, Jacob, 141 Lewis, Morgan, 141 Lewton, Frederick L., 277 Liebig, Justus, 183, 185, 186 Liebreich, Oscar, 191 Lilly, Eli, and Company, 283 Lindbergh, Charles A., 291 Lipmann, Fritz, 200 Lippi, Fra Lippo, 42 London, E. S., 193
- 76. Long, Crawford W., 294 Long, Stephen H., 85 Longomontanus, Christian Severin, 54 Lorenzoni, Giuseppe, 336, 339 Lütke, Count Feodor Petrovich, 316, 345 M Macquer, Peter Joseph, 180 Marestier, Jean Baptiste, 147, 149, 159, 162 Marggraf, Andreas Sigismund, 180 Maria Theresa, Empress of Austria, 31, 41, 42, 44, 57, 58 Mariners’ Museum, 165 Markham, Erwin F., 8, 9, 15, 16, 19–22, 27 Marmion, R. A., 276 Marsh, James, 145 Marshall, Charles, 11, 16 Maudslay, Henry, 106, 113 Maupertius, P. L. Moreau de, 308, 343 Maxwell, James Clerk, 324 May, Arthur J., 139 Mayer, Jo, 285 McMurtrie, Daniel, 276 Medi, Enrico, 342 Meigs, M. C., 96 Meineke, ——, 128 Mendenhall, Thomas Corwin, 319, 331, 332, 334, 347 Merrick, C. E., 10 Mersenne, P. Marin, 305 Meton, 48 Meyerhof, Otto, 194, 200 Miescher, Johann Friedrich, 192 Miller, Patrick, 156, 157 Mitchill, Samuel L., 141, 142 Monauni, Giovanni Battista, 40, 52
- 77. Monroe, James, 145 Montgéry, M., 147, 149–152, 159 Morgan, Mr., 144 Morris, Tasker and Company, 94 Morris, Thomas, 141, 142 Morton, Arthur O., 290 Morton, William, 294 Mount Clair shops, 83, 89, 92 Mowbray, George W., 213, 215 Muspratt, James, 186 N Nagel, Oscar P., 295 Nason, Joseph, 114 National Maritime Museum (England), 147, 156, 165 Nelson, Robert J., 295 Nesbitt, Mr. and Mrs. D. H., 13 Newton, Sir Isaac, 303–305, 307, 308, 342, 343 Nicholson, William, 124 Nietzsche, Friedrich, 186, 187, 189 Nobel, Alfred B., 213 North, Simeon, arms factory, 114 Norwood, Richard, 306 O Ochoa, Severo, 200 Oersted, Hans Christian, 125–130, 132–136 Ohm, Georg Simon, 123, 135 Oken, Lorenz, 132 Olson, Carl G., 118 Oppolzer, Theodor von, 322, 324–327 Owen, H. S., 5
- 78. P Page, Irving, 4 Parke, Davis Company, 273 Parmelee, L. J., 10 Patapsco Bridge and Iron Works, 92, 95 Patrick, Mr. and Mrs. ——, 13 Patterson, Carlile Pollock, 325, 326 Peirce, Charles Sanders, 314, 322–329, 332, 336–339, 342, 345–347 Pelouze, Théophile Juste, 189 Pepys, Samuel, 155 Perry, Oliver, 141 Peters, C. A. F., 322, 324 Petty, Sir William, 152, 153, 155, 166 Pfaff, Christian Heinrich, 132 Philolaus, 54 Phoenix Iron Works, 92 Physick, Philip Syng, 294 Picard, Abbé Jean, 306, 308–311, 342 Plantamour, E., 319–321, 324–326 Poggendorf, Johann Christian, 127–129, 132–134, 136 Poissant, A. A., 10 Pope Manufacturing Company, 6, 12 Porter, David, 144 Posidonius, 306 Pratt, Thomas W., 91 Preston, E. D., 328, 329 Ptolemy, 54 Purcell, William, 147 Putnam, G. R., 339 Putnam Machine Works, 212 Pythagoras, 54, 306
- 79. Q Quare, Daniel, 32 R Raschig, Christoph Eusebius, 129 Rasmussen, Kjeld, 150 Reed, D. A., 27 Reeves, Samuel J., 92, 95 Repsold, A., and Sons (Hamburg), 320, 322, 338, 339, 346 Richer, Jean, 307, 342 Richmond, Duke of. See Lennox. Riciolus, 54 Rigsarkivet (Denmark), 147 Ritter, Johann Wilhelm, 129 Roebling, John A., 83, 90 Roentgen, Wilhelm Konrad, 290, 294 Rouelle, Guillaume François, 181 Royal Society of London, 152 Russell, John W., Sons Company, 9, 10, 18, 20 Russell, William J., 9, 10, 15, 18 Rutgers, Henry, 141, 142 S Sabine, Capt. Edward, 315, 325, 329, 345 San Cajetano, Brother David à, 33 San Daniele, Father Aurelianus à, 33 Savage Factory, 88 Savart, Felix, 135 Sawitsch, A., 321, 322 Scheele, Karl W., 182
- 80. Schieffelin and Company, 273 Schmiedeberg, Oswald, 193 Schrader, Gerhard, 199 Schrötter, Anton, 181 Schumacher, H. C., 320 Schumann, R., 335, 336 Schweigger, Johann Salomo Christoph, 127–130, 132–134, 136 Seebeck, T., 128, 135 Shanley, Walter, 212 Shanley Bros., 215 Shea, T., 10 Smith, —— (Captain, USN), 144 Smith, Alba F., 244, 246, 247, 259 Smith, Sir Sidney (RN), 155 Smith Carriage Company, 8 Snell, Willebrord, 306 Snow, ——, 8 Soemmering, S. T., 125 Sommeiller, Germain, 210 Sonnedecker, Glenn, 296 Speter, M., 127, 128 Squibb, E. R., and Sons, 285, 286 Statens Sjöhistoriska Museum (Sweden), 147 Stephenson, Robert, 90 Stephenson, Robert, Hawthorns, Ltd., 253 Sterneck, Robert von, 331, 332, 335, 338, 346 Stevens, J., Arms and Tool Company, 4 Stevens-Duryea Company, 4 Stewart, Charles, 145 Stiles, George, 144 Stokes, George Gabriel, 324, 328, 329, 345, 346 Stoklasa, Julius, 186 Storrow, Charles S., 210 Stoudinger, Charles, 144 Strecker, Adolf Friedrich, 191 Stuart, Charles B., 139, 150
- 81. Stuart, J. E. B., 249 Sully, Henry, 32 Swaine, Jack, 26 Symington, William, 157 T Tanner, Paul H., 295 Taunton Locomotive Works, 247 Taylor, Frank A., 292 Tegmeyer, John H., 91 Thames Iron-works Company (England), 165 Thenard, Louis Jacques, 125 Thomas, George S., 287 Thudichum, Ludwig, 192 Todd, Lord Alexander, 200 Tompion, Thomas, 32 Toner, Joseph Meredith, 271 Tovazzi, Giangrisostomo, 57, 58 Town, Ithiel, 85 Tromsdorff, Johann Bartholomacus, 125 Tweed, William Marcy (Boss), 229 Tyler, Daniel, 244, 253 Tyler, David B., 139 U Ulloa, Antonio de, 308 Union Works, 260 Uppercu, Inglis M., 27 V
- 82. Vander Woerd, Charles, 116, 117 Van Marum, Martin, 123 Vening Meinesz, F. A., 337, 338, 345–347 Volet, Charles, 342 Volta, Alessandro, 123, 124, 127 Vulcan Foundry, 252 W Wallace Brothers, 273 Ward, Frederick A., 120 Warrington, Samuel, 141 Warwick, George, 18 Watts, Frederick, 249 Weale, John, 218 Wernwag, Lewis, 89 Westhaeffer, Paul, 251 Wetmore, Dr. Alexander, 287 Wetschgi, Emanuel, 108 Wetschgi, Manuel, 108, 111 Whipple, Squire, 79, 83, 87, 91, 95 Whistler, George W., 83 White, C. H., 276 Whitebread, Charles, 277, 278, 281, 283, 285, 287 Whitney arms factory, 114 Wilkes, Charles, 317, 318 Wilkinson, David, 113 Williamson, Dowe, 32 Williamson, Joseph, 32 Willm, Edmond, 182 Willstätter, Richard, 191 Wilmarth, Seth, 244, 246, 247, 249, 260 Wilson, Frank E., 290 Wilstack, Paul, 155 Winans, Ross, 83
- 83. Winters, Joseph, 244 Winters, Father S. X., S. J., 42 Winz, Johann Christian, 36, 37 Wisshofer, Peter, 36, 37 Wolcott, Oliver, 141, 142 Wollaston, W. H., 125 Wright, Benjamin, 83 Wurtz, Adolphe, 185, 191 Y Youle, John, foundry, 142 Z Zamboni, Giuseppe, 132 Index - Transcriber's Notes ‘Emmet, ——, 144’—was ‘Emmett, ——, 144’.
- 84. Compiler's Notes Cover image was derived from a scan of the original cover with the title added from page I of the publication. This image is placed in the public domain. Modifications to the originally produced electronic versions include: standardization of formatting; addition of internal links to several Papers; correction of some errors (for example, one image was duplicated in Paper 39); and elimination of some Transcriber's Notes for simple typographical corrections.
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