Molecular modelling

Target
Identification
&Validation
Hit
Identification
Lead
Identification
Lead
Optimisa-
tion
CD
Prenomi-
nation
Concept
Testing
Development
for launch
Launch
Phase
FDA Submission
Launch
Finding Potential
Drug Targets
Validating Therapeutic Targets
Finding Potential Drugs
Drug<>Target<>Therapeutic
Effect Association Finalized
Testing in Man
(toxicity and efficacy)
Drug Discovery is a goal of research. Methods and approaches from different science areas
can be applied to achieve the goal.
2

Hugo Kubinyi , www.kubinyi.de
3

 R&D cost per new drug is $500 to $700 millions
 To sustain growth, each of top 20 pharma company
should produce more new drugs
 Currently, total industry produces only 32 new drugs
per year.
 Current rate of NDAs far below than required for
sustained growth.
4

Atomistic Continuum
Finite Periodic
Quantum Mechanical Methodies Classical Methods
Semi-Empirical Ab Initio Deterministic Stochastic
QM/MMQuantum MC DFT
Molecular
Dynamics
Monte CarloHartree Fock
5

> 10-4 m
10-6 m
10-8 m
10-10 m
Macroscale
Microscale
Nanoscale
Atomic
Scale
Classical Mechanics
Organism
Cell
Protein, membrane
Small molecules, drug
Continuum Mechanics
• Finite Element Method
• Fluid Dynamics
Statistical Mechanics
• Molecular Mechanics
• Molecular Dynamics
• Brownian Dynamics
• Stochastic Dynamics
Quantum Mechanics
•Density Functional Theory
• Hartree-Fock Theory
• Perturbation Theory
• Structural Mechanics
6

U
n
k
n
o
w
n
K
n
o
w
n
Unknown Known
Generate 3D
structures,
HTS, Comb. Chem
Build the lock and then
find the key
Molecular
Docking
Drug receptor
interaction
2D/3D QSAR and
Pharmacophore
Infer the lock by
expecting key
De NOVO Design ,
Virtual screening
Build or find the key
that fits the lock
Receptor based drug design
Rational drug designIndirect drug design
Homology modelling
7

• Molecular modelling allow scientists to use computers
to visualize molecules means representing molecular
structures numerically and simulating their behavior
with the equations of quantum and classical physics , to
discover new lead compounds for drugs or to refine
existing drugs in silico.
• Goal
: To develop a sufficient accurate model of the
system so that physical experiment may not be
necessary .
8

• The term “ Molecular modeling “expanded over the last
decades from a tool to visualize three-dimensional
structures and to simulate , predict and analyze the
properties and the behavior of the molecules on an
atomic level to data mining and platform to organize
many compounds and their properties into database and
to perform virtual drug screening via 3D database
screening for novel drug compounds .
9

Molecular modeling starts from structure determination
Selection of calculation methods in computational chemistry
Starting geometry from
standard geometry, x-ray, etc.
Molecule
Molecular
mechanics
Quantum
mechanics
Molecular
dynamics or
Monte Carlo
Is bond formation or
breaking important?
Are many force field
parameters missing ?
Is it smaller than
100 atoms?
Are charges
of interest ?
Are there many closely
spaced conformers?
Is plenty of computer
time available?
Is the free energy
Needed ?
Is solvation
Important ?
10

• Molecular modelling or more generally computational chemistry
is the scientific field of simulation of molecular systems.
• Basically in the computational chemistry , the free energy of the
system can be used to assess many interesting aspects of the
system.
• In the drug design , the free energy may be used to assess
whether a modification to a drug increase or decrease target
binding.
• The energy of the system is a function of the type and number
of atoms and their positions.
• Molecular modelling softwares are designed to calculate this
efficiently.
12

• The energy of the molecules play important role in the
computational chemistry. If an algorithm can estimate
the energy of the system, then many important
properties may be derived from it.
• On today's computer , however energy calculation takes
days or months even for simple system. So in practice,
various approximations must be introduced that
reduce the calculations time while adding acceptably
small effect on the result.
13

• Example :
• Familiar conformation of the Butane
7
6
5
4
3
2
1
0
0 60 120 180 240 300 360
0.3
0.25
0.2
0.15
0.10
0.1
0.05
0
C
B
D
E
F
Potentialenergy
Dihedral angle
Probability
14

Quantum mechanics Molecular mechanics
Ab initio methods DFT method Semiimpirical methods
Molecular Modelling
15

• Quantum mechanics is basically the molecular orbital calculation
and offers the most detailed description of a molecule’s chemical
behavior.
• HOMO – highest energy occupied molecular orbital
• LUMO – lowest energy unoccupied molecular orbital
• Quantum methods utilize the principles of particle physics to
examine structure as a function of electron distribution.
• Geometries and properties for transition state and excited state can only be
calculated with Quantum mechanics.
• Their use can be extended to the analysis of molecules as yet
unsynthesized and chemical species which are difficult (or
impossible) to isolate.
16

• Quantum mechanics is based on Schrödinger equation
HΨ = EΨ = (U + K ) Ψ
E = energy of the system relative to one in which all atomic
particles are separated to infinite distances
H = Hamiltonian for the system .
It is an “operator” ,a mathematical construct that operates
on the molecular orbital , Ψ ,to determine the energy.
U = potential energy
K = kinetic energy
Ψ = wave function describes the electron distribution around the
molecule.
17

The Hamiltonian operator H is, in general,
Where Vi2 is the Laplacian operator acting on particle i. Particles
are both electrons and nuclei. The symbols mi and qi are the mass
and charge of particle I, and rij is the distance between particles.
The first term gives the kinetic energy of the particle within a
wave formulation.
The second term is the energy due to Coulombic attraction or
repulsion of particles . 18

• In currently available software, the Hamiltonian above is nearly
never used.
• The problem can be simplified by separating the nuclear and
electron motions.
kinetic energy
of electrons
Attraction of
electrons to
nuclei
Repulsion
between
electrons
Born-Oppenheimer approximation
19

• Thus, each electronic structure calculation is performed for a
fixed nuclear configuration, and therefore the positions of all
atoms must be specified in an input file.
• The ab initio program like MOLPRO then computes the
electronic energy by solving the electronic Schrödinger equation
for this fixed nuclear configuration.
• The electronic energy as function of the 3N-6 internal nuclear
degrees of freedom defines the potential energy surface (PES)
which is in general very complicated and can have many minima
and saddle points.
• The minima correspond to equilibrium structures of different
isomers or molecules, and saddle points to transition states
between them.
20

• The term ab initio is Latin for “from the beginning” premises of
quantum theory.
• This is an approximate quantum mechanical calculation for a
function or finding an approximate solution to a differential
equation.
• In its purest form, quantum theory uses well known physical
constants such as the velocity of light , values for the masses and
charges of nuclear particles and differential equations to directly
calculate molecular properties and geometries. This formalism is
referred to as ab initio (from first principles) quantum
mechanics.
21

HARTREE±FOCK APPROXIMATION
• The most common type of ab initio calculation in which the
primary approximation is the central field approximation means
Coulombic electron-electron repulsion is taken into account by
integratinfg the repulsion term.
• This is a variational calculation, meaning that the approximate
energies calculated are all equal to or greater than the exact
energy.
• The energies are calculated in units called Hartrees (1 Hartree .
27.2116 eV)
22

• The steps in a Hartreefock calculation start with an initial guess
for the orbital coefficients ,usually using a semiempirical method.
• This function is used to calculate an energy and a new set of
orbital coefficients, which can then be used to obtain a new set
,and so on.
• This procedure continues iteratively untill the energies and
orbital coefficient remains constant from one iteration to the
next.
• This iterative procedure is called as a Self-consistent field
procedure (SCF).
23

Advantage
• Advantages of this method is that it breaks the many-electron
Schrodinger equation into many simpler one-electron equations.
• Each one electron equation is solved to yield a single-electron
wave function, called an orbital, and an energy, called an orbital
energy.
24

• What is Density ?
How something(s) is(are) distributed/spread about a given space
Electron density tells us where the electrons are likely to exist.
Allyl Cation:
*
26

• A function depends on a set of variables.
y = f (x)
E.g., wave function depend on electron coordinates.
What is a Functional?
• A functional depends on a functions, which in turn depends
on a set of variables.
E = F [ f (x) ]
E.g., energy depends on the wave function, which depends on
electron coordinates.
1
2
3
4
1
2
3
4
F(X)=Y
27

• The electron density is the square of wave
function and integrated over electron
coordinates.
• The complexity of a wave function increases as
the number of electrons grows up, but the
electron density still depends only on 3
coordinates.
x
x
• With this theory, the properties of a many-electron
system can be determined by using functionals, i.e.
functions of another function, which in this case is
the spatially dependent electron density .
28

• There are difficulties in using density functional theory to
properly describe intermolecular interactions, especially van der
Waals forces (dispersion); charge transfer excitations; transition
states, global potential energy surfaces and some other strongly
correlated systems .
29

• Density functional theory has its conceptual roots in the
Thomas-Fermi model .
• It is developed by Thomas and Fermi in 1927.
• They used a statistical model to approximate the distribution of
electrons in an atom.
• The mathematical basis postulated that electrons are distributed
uniformly in phase space with two electrons in every h3 of
volume.
• For each element of coordinate space volume d3r we can fill out
a sphere of momentum space up to the Fermi momentum pf .
Thomas Fermi model
30

• Equating the number of electrons in coordinate space to that in
phase space gives
• Solving for pf and substituting into the classical kinetic energy
formula then leads directly to a kinetic energy represented as a
functional of the electron density:
where
31

• As such, they were able to calculate the energy of an atom using
this kinetic energy functional combined with the classical
expressions for the nuclear-electron and electron-electron
interactions (which can both also be represented in terms of the
electron density).
• The Thomas-Fermi equation's accuracy is limited because the
resulting kinetic energy functional is only approximate, and
because the method does not attempt to represent the exchange
energy of an atom as a conclusion of the Pauli principle.
• An exchange energy functional was added by Dirac in 1928
called as the Thomas-Fermi-Dirac model
• However, the Thomas-Fermi-Dirac theory remained rather
inaccurate for most applications. The largest source of error was
in the representation of the kinetic energy, followed by the errors
in the exchange energy, and due to the complete neglect of
electron correlation.
32

• DFT was originated with a theorem by Hoenburg and Kohn .
• The original H-K theorems held only for non-degenerate
ground states in the absence of a magnetic field .
• The first H-K theorem demonstrates that the ground state
properties of a many-electron system are uniquely determined by
an electron density that depends on only 3 spatial coordinates.
• It lays the groundwork for reducing the many-body problem of
N electrons with 3N spatial coordinates to only 3 spatial
coordinates, through the use of functionals of the electron
density.
Hohenberg-Kohn theorems
33

• This theorem can be extended to the time-dependent domain to
develop time-dependent density functional theory (TDDFT),
which can be used to describe excited states.
• The second H-K theorem defines an energy functional for the
system and proves that the correct ground state electron density
minimizes this energy functional.
34

• Within the framework of Kohn-Sham DFT, the intractable
many-body problem of interacting electrons in a static external
potential is reduced to a tractable problem of non-interacting
electrons moving in an effective potential.
• The effective potential includes the external potential and the
effects of the Coulomb interactions between the electrons, e.g.,
the exchange and correlation interactions.
kohn-Sham theory
EDFT[ ] = T[ ] + Ene[ ] + J[ ] + Exc[ ]
Electronic
Kinetic energy
Nuclei-electrons
Coulombic energy
electrons-electrons
Coulombic energy
electrons-electrons
Exchange energy
35

• Modeling the latter two interactions becomes the difficulty
within KS DFT.
• In this formulation, the electron density is expressed as a linear
combination of basis functions similar in mathematical form to
HF orbitals.
• A determinant is then formed from these functions, called
Kohn±Sham orbitals.
• It is the electron density from this determinant of orbitals that is
used to compute the energy.
36

• So Semiempirical methods are very fast, applicable to large
molecules, and may give qualitative accurate results when applied to
molecules that are similar to the molecules used for
parameterization.
• Because Semiempirical quantum chemistry avoid two limitations,
namely slow speed and low accuracy, of the Hartree-Fock
calculation by omitting or parameterzing certain integrals based on
experimental data, such as ionization energies of atoms, or dipole
moments of molecules.
• Rather than performing a full analysis on all electrons within the
molecule, some electron interactions are ignored .
37

• Modern semiempirical models are based on the Neglect of
Diatomic Differential Overlap (NDDO) method in which the
overlap matrix S is replaced by the unit matrix.
• This allows one to replace the Hartree-Fock secular equation
|H-ES| = 0 with a simpler equation |H-E|=0.
• Existing semiempirical models differ by the further
approximations that are made when evaluating one-and two-
electron integrals and by the parameterization philosophy.
38

• Modified Neglect of Diatomic Overlap , MNDO ( by
Michael Dewar and Walter Thiel, 1977)
• Austin Model 1, AM1 (by Dewar and co-workers)
• Parametric Method 3, PM3 (by James Stewart)
• PDDG/PM3 (by William Jorgensen and co-workers)
39

• Modified Neglect of Diatomic Overlap , by Michael Dewar and
Walter Thiel, 1977
• It is the oldest NDDO-based model that parameterizes one-
center two-electron integrals based on spectroscopic data for
isolated atoms, and evaluates other two-electron integrals using
the idea of multipole-multipole interactions from classical
electrostatics.
• A classical MNDO model uses only s and p orbital basis sets
while more recent MNDO/d adds d-orbitals that are especially
important for the description of hypervalent sulphur species and
transition metals.
40

Deficiencies
• Inability to describe the hydrogen bond due to a strong
intermolecular repulsion.
• The MNDO method is characterized by a generally poor
reliability in predicting heats of formation.
• For example: highly substituted stereoisomers are predicted to
be too unstable compared to linear isomers due to
overestimation of repulsion is sterically crowded systems.
41

• By Michel Dewar and co-workers
• Takes a similar approach to MNDO in approximating two-
electron integrals but uses a modified expression for nuclear-
nuclear core repulsion.
• The modified expression results in non-physical attractive forces
that mimic van der Waals interactions.
• AM1 predicts the heat of the energy more accurately than the
MNDO.
• The results of AM1 calculations often are used as the starting
points for parameterizations of the force fields in molecular
dynamic simulation and CoMFA QSAR.
42

Some known limitations to AM1 energies
• Predicting rotational barriers to be one-third the actual barrier
and predicting five-membered rings to be too stable.
• The predicted heat of formation tends to be inaccurate for
molecules with a large amount of charge localization.
• Geometries involving phosphorus are predicted poorly.
• There are systematic errors in alkyl group energies predicting
them to be too stable.
• Nitro groups are too positive in energy.
• The peroxide bond is too short by about 0.17 A0 .
• Hydrogen bonds are predicted to have the correct strength, but
often the wrong orientation.
• So o n average, AM1 predicts energies and geometries better
than MNDO, but not as well as PM3 . 43

• By James Stewart
• Uses a Hamiltonian that uses nearly the same equations as the AM1
method along with an improved set of parametersis.
• Limitations of PM3..
• PM3 tends to predict that the barrier to rotation around the C-N
bond in peptides is too low.
• Bonds between Si and the halide atoms are too short Proton
affinities are not accurate.
• Some polycyclic rings are not flat.
• The predicted charge on nitrogen is incorrect.
• Nonbonded distances are too short..
44

Strength
• Overall heats of formation are more accurate than with MNDO
or AM1.
• Hypervalent molecules are also predicted more accurately
• PM3 also tends to predict incorrect electronic states for
germanium compounds
• It tends to predict sp3 nitrogen as always being pyramidal.
• Hydrogen bonds are too short by about 0.1AÊ , but the
orientation is usually correct .
• On average, PM3 predicts energies and bond lengths more
accurately than AM1 or MNDO
45

• By William Jorgensen and co-workers
• The Pairwize Distance Directed Gaussian (PDDG)
• Use a functional group-specific modification of the core
repulsion function.
• Its modification provides good description of the van der Waals
attraction between atoms .
• PDDG/PM3 model very accurate for estimation of heats of
formation because of reparameterization .
• But some limitations common to NDDO methods remain in
the PDDG/PM3 model: the conformational energies are
unreliable, most activation barriers are significantly
overestimated, and description of radicals is erratic.
• So far, only C, N, O, H, S, P, Si, and halogens have been
parameterized for PDDG/PM3 46

• Some freely available computational chemistry programs that
include many semiempirical models are MOPAC 6, MOPAC 7,
and WinMopac .
47

• Computational modeling of structure-activity relationships
• Design of chemical synthesis or process scale-up
• Development and testing of new methodologies and algorithms
• Checking for gross errors in experimental thermochemical data
e.g. heat of formation
• Preliminary optimization of geomteries of unusual molecules and
transition states that cannot be optimized with molecular
mechanics methods 48

• The Process of finding the minimum of an empirical potential
energy function is called as the Molecular mechanics. (MM)
• The process produce a molecule of idealized geometry.
• Molecular mechanics is a mathematical formalism which attempts
to reproduce molecular geometries, energies and other features by
adjusting bond lengths, bond angles and torsion angles to
equilibrium values that are dependent on the hybridization of an
atom and its bonding scheme.
50

• Molecular mechanics breaks down pair wise interaction into
√ Bonded interaction ( internal coordination )
- Atoms that are connected via one to three bonds
√ Non bonded interaction .
- Electrostatic and Van der waals component
The general form of the force field equation is
E P (X) = E bonded + E nonbonded
51

•Bonded interactions
• Used to better approximate the interaction of the adjacent
atoms.
• Calculations in the molecular mechanics is similar to the
Newtonians law of classical mechanics and it will calculate
geometry as a function of steric energy
• Hooke’s law is applied here
• f = kx
• f = force on the spring needed to stretch an ideal spring is
proportional to its elongation x ,and where k is the force
constant or spring constant of the spring.
52

• Ebonded = Ebond + Eangle + Edihedral
• Bond term
Ebond = ½ kb (b – bo) 2
• Angle term
EAngle = ½ kθ (θ – θ0)
• Energy of the dihedral angles
Edihedral = ½ kΦ(1 – cos (nΦ + δ)
53

H
CC
H
H
Graphical representation of the bonded and non bonded interaction and the
corresponding energy terms.
E coulomb
Electrostatic attraction
E vdw
Van der waals
Yij
θ0
K θ
K b
K Ф
E Ф
Ф 0
E θ
E b
b0
b
Bond stretching
Dihedral rotation
Angle bending
54

• Nearly applied to all pairs of atoms
• The nonbonded interaction terms usually include electrostatic
interactions and van der waals interaction , which are expressed
as coloumbic interaction as well as
Lennard-Jones type potentials, respectively.
• All of them are a function of the distance between atom pairs ,
rij .
55
Non bonded interaction

• E Nonbonded = E van der waals + E electrostatic
• E van der waals
• E electrostatic
56
Lennard Jones potential
Coulomb's Law

• The molecular mechanics energy expression consists of a simple
algebraic equation for the energy of the compound.
• A set of the equations with their associated constants which are
the energy expression is called a force field.
• Such equations describes the various aspects of the equation like
stretching, bending, torsions, electronic interactions van der
waals forces and hydrogen bonding.
57

• Valance term. Terms in the energy expression which describes a
single aspects of the molecular shape. Eg., such as bond stretching
, angle bending , ring inversion or torsional motions.
• Cross term. Terms in the energy expression which describes how
one motion of the molecule affect the motion of the another.
Eg., Stretch-bend term which describes how equilibrium bond
length tend to shift as bond angles are changed.
• Electrostatic term. force field may or may not include this term.
Eg., Coulomb’s law.
58

• Some force fields simplify the complexity of the calculations by
omitting most of the hydrogen atoms.
• The parameters describing the each backbone atom are then
modified to describe the behavior of the atoms with the attached
hydrogens.
• Thus the calculations uses a CH2 group rather than a Sp3
carbon bonded to two hydrogens.
• These are called united atom force field or intrinsic hydrogen
methods.
• Some popular force fields are
 AMBER
 CHARMM
 CFF 59

• Assisted model building with energy refinement is the name of
both a force field and a molecular mechanics program.
• It was parameterized specifically for the protein and nucleic
acids.
• It uses only five bonding and nonbonding terms and no any
cross term.
60
amber.scripps.edu

(Harvard University)
• Chemistry at Harvard macromolecular mechanics is the name of
both a force field and program incorporating the force field.
• It was originally devised for the proteins and nucleic acids. But
now it is applied to the range of the bimolecules , molecular
dynamics, solvation , crystal packing , vibrational analysis and
QM/MM studies.
• It uses the five valance terms and one of them is an electrostatic
term.
61
www.charmm.org

• The consistent force field .
• It was developed to yield consistent accuracy of results for
conformations , vibrational spectras , strain energy and
vibrational enthalpy of proteins.
• There are several variations on this
 CVFF – consistent valence forcefield
 UBCFF – Urefi Bradley consistent forcefield
 LCFF – Lynghy consistent forcefield
• These forcefields use five to six valance terms . One of which is
electrostatic and four to six others are Cross terms.
62

• Molecular mechanics energy minimization means to finds stable,
low energy conformations by changing the geometry of a
structure or identifying a point in the configuration space at
which the net force on atom vanishes .
• In other words , it is to find the coordinates where the first
derivative of the potential energy function equals zero.
• Such a conformation represents one of the many different
conformations that a molecule might assume at a temperature of
0 k0 .
63

• The potential energy function is evaluated by a certain algorithm
or minimizer that moves the atoms in the molecule to a nearest
local minimum
• Examples ;
o Steepest Decent
o Conjugate Gradient
o Newton-Raphson procedure
64

• There are three main approaches to finding a minimum of a
function of many variables. infalliable
! Search Methods :
- Utilize only values of function
- Slow and inefficient
- Search algorithms infalliable and
always find minimum
Example :SIMPLEX
! Gradient Methods :
- Utilizes values of a function and its
gradients.
- Currently most popular
Example : The conjugated gradient
algorithm
! Newton Methods :
- Require value of function and its 1st
and 2nd derivatives.
- Hessian matrix
Example : BFGS algorithm
65

• Geometry optimization is an iterative procedure of computing
the energy of a structure and then making incremental increase
changes to reduce the energy.
• Minimization involves two steps
1 – an equation describes the energy of the system as a function
of its coordinates must be defined and evaluated for a given
conformation
2 – the conformation is adjusted to lower the value of the
potential function .
66

V
L
G
X
L
X
X (1) X (2) X (min)
L = Local minimum
G = Global minimum
Local and global minima for a function of one variable and an example of a sequence of
solution.
Algorithm for decent series minimization.
67

! In Cartesian presentation of potential energy surface , the picture
would like the lots of narrow tortuous valleys of similar
depth.
→ This is because low energy paths for individual atoms are very
narrow due to the presence of hard bond stretching and angle
bending terms.
→ The low energy paths corresponds only to the rotation of groups
or large portions of the molecule as a result of varying torsional
angles.
• In the Cartesian space the minimizer walks along the bottom of a
narrow winding channel which is frequently a dead-end .
68

• In internal coordinates presentation , the potential energy surface
looks like a valley surrounded by high mountains.
• → The high peaks corresponds to stretching and bending terms
and close Vander Waals contacts while the bottom of the valley
represents the torsional degree of freedom.
• → If you happen to start at the mountain tops in the internal
coordinates space , the minimizer sees the bottom of the valley
clearly from the above .
69

• Using the internal coordinates there is a clear separation of
variables into the hard ones ( those whose small changes
produces large changes in the function values ) and soft ones (
those whose changes do not affect the function value
substantially).
• During the function optimization in the internal coordinates, the
minimizer first minimizes the hard variables and in the
subsequent iterations cleans up the details by optimizing the soft
variables.
• While in the Cartesian spaces all variables are of the same type.
70

• The atoms and molecules are in the constant motion and
especially in the case of biological macromolecules , these
movement are concerted and may be essential for biological
function.
• And so such thermodynamic properties cannot be derived from
the harmonic approximations and molecular mechanics because
they inherently assumes the simulation methods around a
systemic minimum.
• So we use molecular Dynamic simulations.
71

• Used to compute the dynamics of the molecular system,
including time-averaged structural and energetic properties,
structural fluctuations and conformational transitions.
• The dynamics of a system may be simplified as the movements
of each of its atoms. if the velocities and the forces acting on
atoms can be quantified, then their movement may be simulated.
72

• There are two approaches in molecular dynamics for the
simulations .
Stochastic
! Called Monte Carlo simulation
! Based on exploring the energy surface by randomly
probing the geometry of the molecular system.
Deterministic
! Called Molecular dynamics
! It actually simulates the time evolution of the molecular
system and provides us with the actual trajectory of the
system
73

• Based on exploring the energy surface by randomly probing the
geometry of the molecular system.
• Steps
1 - Specify the initial coordinates of atoms
2 - Generate new coordinates by changing the initial coordinates at
random.
3 - Compute the transition probability W(0,a)
4 - Generate a uniform random number R in the range [0,1]
5 - If W(0,a) < R then take the old coordinates as the new
coordinates and go to step 2
6 – Otherwise accept the new coordinates and go to step 2. 74

The most popular of the Monte Carlo method for the molecular system
See the pamplet for description 75

• Actually time evaluation of the molecular system and the
information generated from simulation methods can be used to
fully characterized the thermodynamic state of the system.
• Here the molecular system is studied as the series of the snapshots
taken at the close time intervals. ( femtoseconds usually) .
76

• Based on the potential energy function we can find components
Fi of the force F acting on atom as
Fi = - dV/ dxi
This force in an acceleration according to Newton’s equation of
motion
F = m a
• By knowing the acceleration we can calculate the velocity of an
atom in the next time step. From atom position , velocities and
acceleration at any moment in time, we can calculate atom
positions and velocities at the next time step.
• And so integrating these infiniteimal steps yields the trajectories
of the system for any desired time range.
77

The Verlet algorithm uses positions and accelerations at time t and
the positions from time t-δt to calculate new positions at time t+δt.
r(t+δt) = 2r(t) - r(t-δt)+a(t) δt2
78

:
• – Position integration is accurate (errors on order of Δt4).
• – Single force evaluation per time step.
• – The forward/backward expansion guarantees that the path is
reversible.
:
• – Velocity has large errors (order of Δt2).
• – It is hard to scale the temperature (kinetic energy of molecule).
79

1. the velocities are first calculated at time t+1/2δt (the velocities
leapover the positions)
2. these are used to calculate the positions, r, at time t+δt. (then
the positions leapover the velocities)
r(t+δt) = r(t) + v( t + ½ δt) δt
v( t + ½ δt) = v( t - ½ δt) +a(t) δt
80

:
– High quality velocity calculation, which is important in
temperature control.
:
– Velocities are known accurately at half time steps away from
when the position is known accurately.
– Estimate of velocity at integral time step:
v(t) = [v(t-Δt)+v(t+Δt)]/2
81

1) We need an initial set of atom positions (geometry) and atom
velocities.
• The initial positions of atoms are most often accepted from the
prior geometry optimization with molecular mechanics.
•
• Formally such positions corresponds to the absolute zero
temperature.
Procedure
82

2) The velocities are assigned to each atom from the Maxwell
distribution for the temperature 20 oK .
• Random assignment does not allocate correct velocities and the
system is not at thermodynamic equilibrium.
• To approach the equilibrium the “equilibration” run is performed
and the total kinetic energy of the system is monitored until it is
constant.
• The velocities are then rescaled to correspond to some higher
temperature. i.e heating is performed.
• Then the next equilibration run follows.
83

• The absolute temperature T, and atom velocities are related
through the mean kinetic energy of the system.
N = number of the atoms in the system
m = mass of the i-th atom
k = Boltzman constant.
• And by multiplying the velocities by we can
effectively “heat “ the system and that accelerate the atoms of
the molecular system.
• These cycles are repeated until the desired temperature is
achieved and at this point a “production’ run can commence.
T =
2
3 N k i=1
N
mi Vi
2
2
Tdesired / Tcurrent
84

• Molecular dynamics for larger molecules or systems in which
solvent molecules are explicitly taken into account is a
computationally intensive task even for supercomputers.
• For such a conditions we have two approximations
 Periodic boundary conditions
 Stochastic boundary conditions
85

Here we are actually simulating a crystal comprised of boxes with ideally
correlated atom movements.
86

Reaction zone :
Portion of the
system which we
want to study
Reservoir zone
Portion of the
system which Is
inert and
uninteresting
87

• So molecular dynamics and molecular mechanics are often used
together to achieve the target conformer with lowest energy
configuration
• Visualise the 3D shape of a molecule
• Carry out a complete analysis of all possible conformations and
their relative energies
• Obtain a detailed electronic structure and the polarisibility with
take account of solvent molecules.
• Predict the binding energy for docking a small molecule i.e. a drug
candidate, with a receptor or enzyme target.
• Producing Block busting drug
• Nevertheless, molecular modelling, if used with caution, can provide
very useful information to the chemist and biologist involved in
medicinal research. 89

References
1) Cohen N. C. “Guide book on Molecular Modelling on Drug Design”
Academic press limited publication, London, 1996.
2) Young D. C. “Computational Chemistry: A Practical Guide for Applying
Techniques to Real-World Problems”. John Wiley & Sons Inc., 2001.
3) Abraham D. J. “Burger’s Medicinal Chemistry and Drug Discovery” sixth
edition, A John Wiley and Sons, Inc. Publication,1998.
90

Molecular modelling

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Molecular modelling