Kinetics and Markov State Models: Computational Approaches

1. Statistical Thermodynamics Lecture 17: Kinetics and Markov State Models Dr. Ronald M. Levy ronlevy@temple.edu

2. Computational approaches to kinetics Advanced conformational sampling methods from past several lectures have primarily focused on thermodynamics (ensembles, averages, PMFs) Now we turn our interest to kinetics by differentiating microstates and macrostates There is a vast theoretical literatures on the nonequilibrium statistical mechanical aspects of kinetics which is beyond the scope of this lecture. These two references can provide you with some starting points: R. Zwanzig. Nonequilibrium Statistical Mechanics. 2001. Oxford University Press. Hänggi, Talkner & Borkovec, Rev. Mod. Phys. 62:251-341 (1990)

3. A Regions of space B D C Discrete states A free energy B reaction coordinate

4. A Regions of space B D C A Discrete states D B C

5. Kinetics between macrostates as a stochastic process with discrete states stochastic process – a random function of time and past history Markov process – a random function of time and the current (macro)state A D D C B B C A time

6. Any given realization of a path among the macrostates is unpredictable, but we can still write down equations that describe the time-evolution of probabilities, e.g. P(state=D, time=t | state=A, time=0) In general, a master equation describes the time-evolution of probabilities as follows, Zwanzig, J. Stat. Phys. 30: 255 (1983)

7. Any given realization of a path among the macrostates is unpredictable, but we can still write down equations that describe the time-evolution of probabilities, e.g. P(state=D, time=t | state=A, time=0) In matrix form,

8. k1 A B Two-state Markov State Model k2

9. columns of U are eigenvectors of K eigenvalues of K

10. The eigenvalues of K give the characteristic rates of the system • One eigenvalue is always 0. This represents the system in equilibrium, and the eigenvector corresponding to the 0 eigenvalue is proportional to the probabilities of the macrostates at equilibrium. • In general, the decay to equilibrium from any non-equilibrium starting point will consist of a superposition of (N-1) exponentials, where N is the number of macrostates. λi<0 depend only on rates ai depend on rates and starting condition and can be positive or negative

11. k1 Two-state Markov State Model A B k2 if P(A,0) is 0 etc…

12. Simulating jump Markov processes How do we construct a “move set” over the kinetic network so that the statistics satisfy ? “Gillespie algorithm”: the amount of time spent in the current state should be an exponential random variable with rate parameter equal to the sum of the rates exiting the current state, and the next state should be chosen with probability proportional to the rate corresponding to that edge

13. A 1 µs-1 10 µs-1 D B C 5 µs-1 The amount of time t spent in B is a random variable with distribution where kt = 1 + 5 + 10 µs-1, i.e. the mean lifetime in state B is 1/16 µs-1 = 62.5 ps The probabilities of next jumping to states A, C or D are 1/16, 5/16 and 10/16=5/8, respectively.

14. A 1 µs-1 1 fs-1 D B C 1 ns-1 The amount of time t spent in B is a random variable with distribution where kt = 1 µs-1 + 1 ns-1 + 1 fs-1 ≈ 1 fs-1, i.e. the mean lifetime in state B is approximately 1 fs. The probabilities of next jumping to states A, C or D are approximately 10-9, 10-6, and (1-10-9-10-6), respectively.

15. How do we construct MSM from Molecular Dynamics simulation automatically? MSMBuilder3 from V. S. Pande: https://github.com/rmcgibbo/msmbuilder3 PyEmma from F. Noe: https://github.com/markovmodel/pyemma Or you can write your own C/C++ codes.

16. Introduction to Kinetic Lab • Goals: • Understand first passage time • Estimate rate constants between two macrostates, folded and unfolded states • Obtain an Arrhenius and anti-Arrhenius plot Zheng, Andrec, Gallicchio, Levy, J. Phys. Chem. B, 2009. DOI: 10.1021/jp900445t.

17. Introduction to Kinetic Lab PMF along x at three temperatures Rate constants of the 2-D potential

18. The end! Thank you!