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Lecture 17: Kinetics and Markov State
Models
Dr. Ronald M. Levy
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?
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
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-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
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
Rate constants of the 2-D
potential
PMF along x at three
temperatures
The end!
Thank you!