lecture 17: kinetics and markov state models · lecture 17: kinetics and markov state models ......

Lecture 17: Kinetics and Markov State

Models

Dr. Ronald M. Levy

[email protected]

Statistical Thermodynamics

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

B

C

D fr

ee e

nerg

y

A

B

reaction coordinate

Regions of space

Discrete states

A

B

C

D

Regions of space

Discrete states

A

B

C

D

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

B

C

D

time

A

B

C

D

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)

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,

A B k1

k2

Two-state Markov State Model

columns of U are eigenvectors of K

eigenvalues of K

𝐏 𝑡 = 𝐔 · diag(𝑒𝜆1𝑡 , 𝑒𝜆2𝑡 , … , 𝑒𝜆𝑁𝑡) · 𝐔−1 𝐏(0)

• 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.

𝐏 𝑆, 𝑡 = 𝐏 𝑆,∞ + 𝑎𝑖𝑒𝜆𝑖𝑡

𝑖

λi<0 depend only on rates

ai depend on rates and starting condition and can be

positive or negative

A B k1

k2

etc…

Two-state Markov State Model

if P(A,0) is 0

𝒖2 = (−1,1) 𝒖1 = (𝑘2𝑘1 + 𝑘2

,𝑘1𝑘1 + 𝑘2

)

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

B

C

D

1 µs-1

10 µs-1

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.

A

B

C

D

1 µs-1

1 fs-1

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.

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

Bolhuis, Dellago & Chandler, PNAS 97:5877 (2000)

free e

nerg

y

Best

Artificial re-

crossings

Bad, no

peaks


simulation data?


should we use?




Eliminating artifactual re-crossings

“buffer region”

“transition-based assignment”

path begins in transition occurs when

path crosses from

buffer into

path begins in

and ends in

without returning

to

transition occurs at the

midpoint between the

departure from and

the arrival in

Buchete & Hummer JPC-B (2008)

doi:10.1021/jp0761665


simulation data?


should we use?




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

Rate constants of the 2-D

potential

PMF along x at three

temperatures

The end!

Thank you!

lecture 17: kinetics and markov state models · lecture 17: kinetics and markov state models ......

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