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

Author: Orchidea Maria Lecian
Speaker: Orchidea Maria Lecian
Title: Markov chain of the K-Ras4B dynamics and new pertinent
Markov-State Model
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.
Presentation at 19-th Edition of Global Conference on Catalysis, Chemical
Engineering & Technology
CAT2024
20 September 2024,
Rome, Italy.

1. 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 & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

2. 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.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

3. Summary
•Anew two-states Markov-State Modelis 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 evolutionis 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:
•Therelative erroris 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 thediscretization 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.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

4. 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.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

5. 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−stateslandscape, in which the dynamics
takes place. More in details, the states are namedM1,M3,M4,M5.
The second instance is a 5−stateslandscape , 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).
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

6. 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).
thefundamental matrix
ˆ
Qof the Markov chainMpertinent to the
considered process of transitions is here newly written as
ˆ
Q=






−q1→2−q1→3 q1→2 q1→30 0
q2→1 q2→1−q2→3q2→30 0
0 0 0 q3→40
0 0 0 0 q4→5
0 0 0 0 0






The diagonal entries are controlled to be non-positive.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

7. The 5-states MSMFrom theprobability matrix
ˆ
Pis approximated s.t.
the first-order reaction as
ˆP(t)≡e
ˆ
Qt
and is here newly written as
ˆ
P=






1−p1→2−p1→3 p1→2 p1→3 0 0
p2→1 p2→1−p2→3p2→3 0 0
0 0 1 −p3→4p3→4 0
0 0 0 1 −p4→5p4→5
0 0 0 0 1






In the probability matrix, each row is controlled to be summed as 1.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

8. The 4-states MSM
Theprobability matrixis spelled as
˜P=




1−p1→3p1→3 0 0
0 1 −p3→4p3→4 0
0 0 1 −p4→5p4→5
0 0 0 1




19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

9. 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).
thetime evolution of the eigenvalueis calculated as the Laplace kernel
˜
λτ=
Z
∞
0
e
−θΛ(t+τ)
e
−θ˜δΛ
dθ=
1
Λ(t+τ) +
˜
δΛ
,
where the auxiliary time variable is wanted not to coincide with the exit
time (which is implicit for the final stateM5).
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

10. Time population of the final state immediately after the (mean)
first passage time
It is now possible to calculate thetime evolution of theM5eigenvalue
immediately after the time at which the stateM5starts being populated
after choosing the lower integration extremum as the time immediately
after the (mean) first passage time (MFPT)of the transitionM4→M5.
This way, the time evolution of the eigenvalue
˜
λ5,τis newly obtained
after specifying the expression as
˜
λτ=
Z
∞
58.55+ϵ
e
−θΛ(t+τ)
e
−θ
˜
δΛ
dθ=
1
Λ(t+τ) +˜δΛ
e
−(58.55+ϵ)(Λ(t+τ)+
˜
δΛ)
.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

11. From
E. Su´arez, 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 thetime evolution of the eigenvalue in
protein dynamicsat the timeτbe a function of the eigenvalue shift
evaluated at the timet+τonly.
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).
thetime evolution of the eigenvalue
˜
λτis newly obtained as
˜
λτ=
Z
∞
58.55+ϵ
e
−θΛ(τ)
e
−θ˜δΛ
dθ=
1
Λ(τ) +
˜
δΛ
e
−(58.55+ϵ)(Λ(τ)+˜δΛ)
,
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

12. 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
ProofAfter discretisation, the properties of the eigenfunctions
coincide with those of the continuous system.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

13. Time evolution of the
discretised two-states Markov model
The time evolution of the discretised two-states Markov model is
written as
˜λ2,nτ=
Z
∞
0
e
−θΛ(t+nτ)
e
−θ
˜
δnΛ
dθ.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

14. 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)−
ˆ
λ2(τ)|
λ2(τ)
≤δ
2
2.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

15. 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)+˜δΛ
|
λ2,τ
The newly-found analytical expression of the relative error of the
two-states MSM allows one to newly calculate the majorisation
parameterδ2from 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.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

16. The coarse-graining error
The relative error is therefore newly analytically upgraded from
M. Sarich, F. No´e, 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
˜
E=maxi=1,...,m−1|
˜
λi−
ˆ
λi|
Proof
The new constant
ˆ
δis newly calculated and its estimation proven to be
improved.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

17. The propagation error is newly calculated as
˜
Ek=||QT
k
T−(QTQ)
k
||
Theerror for coarse-grained transfer operatoris newly rewritten as
ˆ
Ek=||QT
k
T−Q(TQ)
k
||
with
D⊂ L
2
µoperator whose corresponding algebra can be rendered unital;
Qprojector onto the direction orthogonal toD;
QTQthe operator whose eigenvalue are not the dominant one;
Qis defined on then-dimensional space of the step functionsDn, where
the latter has an orthonormal basis;
the projector
ˆ
Pon the transfer operatorTand onDnis defined as
ˆP=QTQ,
such thatL
2
µ→Dn⊂ L
2
µ.
The transition matrix of the MSM Markov chain coincides with the
matrix representation of the projected transfer operator
ˆ
P.
As an operator on a finite-dimensional space,ˆPadmits a matrix
representation.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

18. 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.
19-th Edition of Global Conference on Catalysis, Chemical Engineering & Technology CAT2024 20 September 2024, Rome, Italy.Markov chain of the K-Ras4B dynamics and new pertinent Markov-State Model. Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.

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

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

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