Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model

Markov chain of the K-Ras4B dynamics and new pertinent
Markov-State Model.
Orchidea Maria Lecian
Sapienza University of Rome,
Rome, Italy.
19-th Edition of Global Conference on Catalysis, Chemical
Engineering & Technology
CAT2024
20 September 2024,
Rome, Italy.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Tec
Markov chain of the K-Ras4B dynamic

Abstract
The finite Markov chain originating the Markov-State Model of the
conformational dynamics of the K-Ras4B proteins in the catalytic
reaction is spelled.
The corresponding Markov-Sates-Models are studied according t the
experiment described in [H. Zhang et al., Markov State Models and Molecular Dynamics Simulations
Reveal the Conformational Transition of the Intrinsically Disordered Hypervariable Region of K-Ras4B to the
Ordered Conformation, J. Chem. Inf. Model. 62, 4222 (2022)]. The study is based on the
large-scale conformational changes of the Hypervariable Region from its
intrisicaly-disordered state to the ordered state.
Crucially, the conformal substates along the transition paths are reviewed
in the path description; interactions between the HVR and the catalytic
domain are recapitulated to be possible. Two possibilities are studied
from the Markov landscape accessible to the systems as one five-states
Markov-State Model and one four-states Markov-State Model.

Summary
• A new two-states Markov-State Model is constructed, according to
the qualities of the K-Ras4B dynamics processes; the new analysis of the
transition to the final state is newly analytically studied.
• The Galerkin description’s final-state transition’s related eigenvalue’s
time evolution is newly spelled out from the new 2-states Markov State
Model.
As a result, the new tools needed in the analytical computation of the
relative error are ready:
•The relative error is newly analytically calculated. The experimental
data and the characterisation of the lag time in shaping the discretization
error are used to write new analytical formulations of the time evolution
of the eigenvalue corresponding to the final-state transition.
• The new analysis is proposed, on the discretization error’s features,
according to which the discretization error is expected to increase
monotonically with increasing lag time.
The comparison with the experimental data is exposed.

Introduction
The catalysis properties of the Ras proteins are studied.
The finite Markov chain to which there correspond the qualities of the
conformational dynamics of the K-Ras4B proteins in the catalytic
reactions is written.
The K-Ras4B proteins in catalytic environment:
the originating finite Markov chains is newly written, as
- described in MSM’s;
- a new MSM is written, which incorporates the qualities of the final
state, for which
• the MFPT are newly calculated; and
• the long-time dynamics is newly explored;
• the errors are newly analytically written.

The K-Ras4B proteins in catalytic environments
The data available from the experiment allow one to make hypotheses
about the Markov landscapes of the MSM’s fitting the process.
One instance consists of a 4 − states landscape, in which the dynamics
takes place. More in details, the states are named M1, M3, M4, M5.
The second instance is a 5 − states landscape , whose states are named
M1, M2, M3, M4, M5.
H. Zhang, D. Ni, J. Fan, Minyu Li, J. Zhang, C. Hua, R. Nussinov, S. Lu, Markov State Models and Molecular
Dynamics Simulations Reveal the Conformational Transition of the Intrinsically Disordered Hypervariable Region of
K-Ras4B to the Ordered Conformation, J. Chem. Inf. Model. 62, 4222 (2022).

The originating finite Markov chain
From
OML, Analytical Expressions of the Markov Chain of K-Ras4B Protein within the Catalytic Environment and a New
Markov-State Model, IgMin Res. 1, 170 (2023).
the fundamental matrix Q̂ of the Markov chain M pertinent to the
considered process of transitions is here newly written as
Q̂ =






−q1→2 − q1→3 q1→2 q1→3 0 0
q2→1 q2→1 − q2→3 q2→3 0 0
0 0 0 q3→4 0
0 0 0 0 q4→5
0 0 0 0 0






The diagonal entries are controlled to be non-positive.

The 5-states MSM From the probability matrix P̂ is approximated s.t.
the first-order reaction as
P̂(t) ≡ eQ̂t
and is here newly written as
P̂ =






1 − p1→2 − p1→3 p1→2 p1→3 0 0
p2→1 p2→1 − p2→3 p2→3 0 0
0 0 1 − p3→4 p3→4 0
0 0 0 1 − p4→5 p4→5
0 0 0 0 1






In the probability matrix, each row is controlled to be summed as 1.

The 4-states MSM
The probability matrix is spelled as
P̃ =




1 − p1→3 p1→3 0 0
0 1 − p3→4 p3→4 0
0 0 1 − p4→5 p4→5
0 0 0 1





The new M4-M5 model
From
OML, Analytical results from the two-states Markovv-States model and aplication to validation of molecular
dynamics, 11, 3726 (2023).
the time evolution of the eigenvalue is calculated as the Laplace kernel
λ̃τ =
Z ∞
0
e−θΛ(t+τ)
e−θδ̃Λ
dθ =
1
Λ(t + τ) + δ̃Λ
,
where the auxiliary time variable is wanted not to coincide with the exit
time (which is implicit for the final state M5).

Time population of the final state immediately after the (mean)
first passage time
It is now possible to calculate the time evolution of the M5 eigenvalue
immediately after the time at which the state M5 starts being populated
after choosing the lower integration extremum as the time immediately
after the (mean) first passage time (MFPT) of the transition M4 → M5.
This way, the time evolution of the eigenvalue λ̃5,τ is newly obtained
after specifying the expression as
λ̃τ =
Z ∞
58.55+ϵ
e−θΛ(t+τ)
e−θδ̃Λ
dθ =
1
Λ(t + τ) + δ̃Λ
e−(58.55+ϵ)(Λ(t+τ)+δ̃Λ).

From
E. Suárez, J.L. Adelman, D.M. Zuckerman, Accurate Estimation of Protein Folding and Unfolding Times: Beyond
Markov State Models J. Chem. Theory Comput. 12, 3473 (2016).
it is possible to pose that the time evolution of the eigenvalue in
protein dynamics at the time τ be a function of the eigenvalue shift
evaluated at the time t + τ only.
This way, from
OML, Analytical Expressions of the Markov Chain of K-Ras4B Protein within the Catalytic Environment and a New
Markov-State Model, IgMin Res. 1, 170 (2023).
the time evolution of the eigenvalue λ̃τ is newly obtained as
λ̃τ =
Z ∞
58.55+ϵ
e−θΛ(τ)
e−θδ̃Λ
dθ =
1
Λ(τ) + δ̃Λ
e−(58.55+ϵ)(Λ(τ)+δ̃Λ),

Discretisation: time evolution of the eigenvalue
For a discretised MSM, the time evolution of the eigenvalues in the
Garlenkin description is written as
λi,nτ =
Z ∞
0
e−θΛi (t+nτ)
e−θδnΛi
Proof After discretisation, the properties of the eigenfunctions
coincide with those of the continuous system.

Time evolution of the
discretised two-states Markov model
The time evolution of the discretised two-states Markov model is
written as
λ̃2,nτ =
Z ∞
0
e−θΛ(t+nτ)
e−θ ˜
δnΛ
dθ.

The approximation error
The approximation error of the two-states Markovv-states model in
the Galerkin description is calculated.
The approximation error of the time evolution of the eigenvalues is
defined as
| λ2(τ) − λ̂2(τ) |
λ2(τ)
≤ δ2
2.

The relative error
The relative error of the time evolution of the two-states Markov
model in the Galerkin description is analytically calculated as
Erel Gal (τ, δ) =
| λ2,τ − 1
Λ(t+τ)+δ̃Λ
|
λ2,τ
The newly-found analytical expression of the relative error of the
two-states MSM allows one to newly calculate the majorisation
parameter δ2 from the expression of the approximation error.
OML, Some new theorems of Markov Models from the originating Chain measure,
LAP, Chisinau (2024);
OML, New theorems of Laplace Kernels with Radon measures in Galerkin
Markov-State Models: about time evolution of eigenvalues and about errors, e-print
Researchgate.

The coarse-graining error
The relative error is therefore newly analytically upgraded from
M. Sarich, F. Noé, C. Schuette, On the Approximation Quality of Markov State
Models, Multiscale Modeling & Simulation 8, 1 (2010)in
OML, Some new theorems of Markov Models from the originating Chain measure,
LAP, Chisinau (2024);
OML, New theorems of Laplace Kernels with Radon measures in Galerkin
Markov-State Models: about time evolution of eigenvalues and about errors, e-print
Researchgate.
as
Ẽ = maxi=1,...,m−1 | λ̃i − λ̂i |
Proof
The new constant δ̂ is newly calculated and its estimation proven to be
improved.

The propagation error is newly calculated as
Ẽk =|| QTk
T − (QTQ)k
||
The error for coarse-grained transfer operator is newly rewritten as
Êk =|| QTk
T − Q(TQ)k
||
with
D ⊂ L2
µ operator whose corresponding algebra can be rendered unital;
Q projector onto the direction orthogonal to D;
QTQ the operator whose eigenvalue are not the dominant one;
Q is defined on the n-dimensional space of the step functions Dn, where
the latter has an orthonormal basis;
the projector P̂ on the transfer operator T and on Dn is defined as
P̂ = QTQ,
such that L2
µ → Dn ⊂ L2
µ.
The transition matrix of the MSM Markov chain coincides with the
matrix representation of the projected transfer operator P̂.
As an operator on a finite-dimensional space, P̂ admits a matrix
representation.

Outlook
The novelty of the approach here presented is aimed at recovering the
items of information about the catalytic mechanism by taking into
account the irreversible transition from the penultimate state to the final
state. For this purpose, a new two-state MSM was built, which is issued
from the Markov chain of K-RAS4B. It is therefore possible to recover
the qualities of the eigenvalue of the final transition in the Galerkin
representation.
According to this representation, it is possible to retrieve the decay
constant and the implied time scale in an analytical manner.
The relative error and the discretization error can be newly analytically
calculated according to the qualites of the eigenvalue.
The comparison with the time evolution of the discretization error is
therefore newly set.

Perspective studies
In
[F. Liang, Z. Kang, X. Sun, J. Chen, X. Duan X, H. He, J. Cheng J. Inhibition mechanism of MRTX1133 on
KRASG12D: a molecular dynamics simulation and Markov state model study, J. Comput. Aided Mol. Des. 37, 157
(2023)]
, the experimental data are scanned, to focus on the inhibition
mechanisms of K-RAS(G12D) and its analogs: computer simulations
are used to study the conformational space for the cluster analysis by
means of a twelve-states MSM. The applications are envisaged to provide
guidance for the design of new small molecule inhibitors and to project
drugs against K-RAS in general.

The analysis of
[T. Pantsar, S. Rissanen, D. Dauch, T. Laitinen, I. Vattulainen, A. Poso, Assessment of mutation probabilities of
KRAS G12 missense mutants and their long-timescale dynamics by atomistic molecular simulations and Markov
state modeling, PLoS Comput. Biol. 14, e1006458 (2018)]
demonstrates that there are two clusters within the twelve-state MSM
among which the experimental transition probability is high, i.e. one of
the transitions C10 → C12, i.e. the state C11 is considered as not
playing a strong role; according to this analysis, it is reasonable to newly
postulate a Hidden MarkovState Model which comprehends only the two
states, and to further implement the techniques developed of the
two-states MSM.

Thank You for Your attention.

Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model

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