Lecture 17: Kinetics and Markov State Models Molecular Simulations in Computational Biology 2/4/08 Statistical Thermodynamics Lecture 17: Kinetics and Markov State Models Dr. Ronald M. Levy ronlevy@temple.edu Michael Andrec - lecturer (andrec@lutece.rutgers.edu, room 203 Wright-Rieman Labs)

Computational approaches to kinetics Advanced conformational sampling methods from last lecture 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)

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

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

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 time A B C D A B D C

P(state=D, time=t | state=A, time=0) 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)

P(state=D, time=t | state=A, time=0) 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,

Two-state Markov State Model B k1 k2

columns of U are eigenvectors of K eigenvalues of K

  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

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

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

A 1 µs-1 10 µs-1 B D 5 µs-1 C 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.

A 1 µs-1 1 fs-1 B D 1 ns-1 C 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.

How do we obtain rate constants from simulation data? How do we define the macrostates? What reduced coordinates should we use? Given a trajectory, how do we decide when a transition between macrostates has occurred? Given the transition times, how do we estimate the rates?

How do we obtain rate constants from simulation data? How do we define the macrostates? What reduced coordinates should we use? Given a trajectory, how do we decide when a transition between macrostates has occurred? Given the transition times, how do we estimate the rates?

Good order parameter vs good reaction coordinate Best Artificial re-crossings free energy Bad, no peaks Bolhuis, Dellago & Chandler, PNAS 97:5877 (2000)

How do we obtain rate constants from simulation data? How do we define the macrostates? What reduced coordinates should we use? Given a trajectory, how do we decide when a transition between macrostates has occurred? Given the transition times, how do we estimate the rates?

Eliminating artifactual re-crossings “buffer region” transition occurs when path crosses from buffer into path begins in “transition-based assignment” transition occurs at the midpoint between the departure from and the arrival in path begins in and ends in without returning to Buchete & Hummer JPC-B (2008) doi:10.1021/jp0761665

How do we obtain rate constants from simulation data? How do we define the macrostates? What reduced coordinates should we use? Given a trajectory, how do we decide when a transition between macrostates has occurred? Given the transition times, how do we estimate the rates?

Estimating rates from transition times If kinetics is approximately Markovian and there are only 2 macrostates, then the rate can be set to the inverse of the mean first passage time from state A to state B (or vice versa). If there are more than 2 macrostates, then we can estimate the rates from the lifetimes and “branching ratios” (by analogy to the Gillespie algorithm): the mean time spent in state i is the inverse of the sum of the rates exiting state i, and kij is proportional to the fraction of times an i to j transition was observed. Alternatively, one can use maximum likelihood estimation to obtain the rates, i.e. maximize with respect to the independent elements of K (where ai and bi are the starting and ending states of the i-th transition). Sriraman, Kevrekidis & Hummer, JPC-B 109:6479 (2005)

Introduction to Kinetic Lab Goal: Understand first passage time Estimate rate constants between two macrostates, folded and unfolded states Obtain an Arrhenius and anti-Arrhenius plot

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

The end! Thank you!