Simulating dynamics of rare events using discretised relaxation path sampling

Simulating dynamics of rare events
using discretised relaxation path
sampling
Boris Fačkovec
Wales group
Department of Chemistry
University of Cambridge
Energy landscapes workshop
Durham, 18th August 2014
Discretised relaxation path sampling (DRPS) 1 /19 bf269@cam.ac.uk

Rare events in nature

• ratio of the largest and lowest relevant timescales 1
molecular vibration
(~1 fs)
elongation of
protein chain (~40 ms)
folding
(~1 ms)
(another definition: computer time patience)

Towards dynamics of rare events
• coarse-graining / rigidification
• biasing the potential (metadynamics, …)
• efficient sampling in the trajectory space (TPS)
• space partitioning (milestoning, TIS, FFS, MSM,…)
choose
appropriate
space
partitioning
perform
constrained
simulations
group!
passage times
into final
rate constants
process data
from simulation
(passage times
between cells)
• discrete path sampling (DPS)

Cell-to-cell rate constant
A
B
C
D A
B
C
AC
D AB BC
BD
• soft cells were developed to allow complete space
partitioning using surface-surface rate constants
• inconvenient for observables, incompatible with DPS
A A
incomplete
partitioning
soft
cells
CD
hard
cells
A
B
B B

Methods using hard cells
• boxed molecular dynamics
(BXD) - sampling enhanced
by space partitioning
B
position
kBXD
AB =
lag time
time
log (probability)
• hyperdynamics (hyperMD) -
sampling enhanced by
biasing the potential
• dividing surface defined by
a certain plane involving the
saddle point
biased potential
position
potential
A
C
1
h⌧lagi

Limits of TST
• transition state theory - rate constant = equilibrium flux /
equilibrium population
kTST
AB =
flux
population
= hP(x) [n(x) · v]i@A
hP(x)iA
=
@tpA(0)
pA(0)
• the rate constant is very sensitive to definition of the dividing
time
population in A
A A
time
population in A
surface

Rate constants from trajectories
• (if we have to simulate trajectories) CAN WE DO BETTER?
• for system of 2 states, over damped dynamics, rate constant
from mean exit times (formula proven for 1D):
• relaxation involves both exit and penetration - best approach
Discretised relaxation path sampling (DRPS) 7
/19 bf269@cam.ac.uk
position
potential
probability
density
position
relaxation
kex
AB =
peq
B
peq
A ⌧ ex
B + peq
B ⌧ ex
A

No-flux boundary conditions
C
A B
D
E
G
F
A B
• isolation of A and B from the rest by placing hard walls
• straightforward for overdamped dynamics
• impossible for Hamiltonian (deterministic) dynamics

Equilibrium boundary conditions
• placing an equilibrated neighbour at the boundary
• exiting particle is replaced by a particle from the equilibrium
distribution
• the equilibrium distribution is not known a priori
• TRICK: independence of exiting and entering particles ➣
reduction of dynamics to propagation of response functions

Insight into the response functions
• exit response function Fi|k(t) is
the distribution of first passage
times
• response to supply at the
boundary Lij|k(t) is the
trajectory lengths
• analytical solution for 1
trajectory averaged over
trajectories
• RIGHT: response function for a
system with 2 cells (A and B)
B A
trajectories from
equilibrium
distribution
trajectories from
the boundaries
FB|A(t) = @tpA(t) ,
LBB|A(t) =
@tFB|A(t)
FB|A(0)
Discretised relaxation path sampling (DRPS) 10/19 bf269@cam.ac.uk

Optimised simulation protocol
• different for over damped dynamics and inertial dynamics
with large friction
1. partitioning of space into cells
2. constrained Monte Carlo sampling
3. molecular dynamics from random initial points terminated
at cell boundaries
4. propagation of the response functions - fit of the relaxation
5. graph transformation / lumping

Limits of the method
• decorrelation of exiting and entering trajectories at the
boundaries is assumed
• internal barriers invalidate Markovian assumption
• some types of breakdowns can be identified from the
simulation
A
B
A
C B

Application of estimator:
Sinai billiards
• non-interacting particles with
equal velocities elastically
reflect from walls of the billiard
and the circle inside
• circle forms a bottleneck -
transition through the dividing
surface is a rare event
• small change in definition of
the dividing surface should not
cause large change in the rate
constants
A
B

Application of estimator:
Sinai billiards
• RXN - “true” rate constant -
from relaxation
• TST rate constant - analytical
• BXD constant from simulated
equilibrium flux
• exit rate constant from mean
exit times
• DRPS - our estimator
• results for 2 dividing surfaces
kRXN
AB
kTST
AB
kBXD
AB
kex
AB
kDRPS
AB
position of the
dividing surface
good poor
11.1
12.11
12.11
5.64
10.62
10.8
31.58
31.6
7.96
10.56
rate constants
(in arbitrary units)

Application of DRPS:
Cluster of Lennard-Jones disks
-10.77
-11.40 -11.48
-11.04
-12.54
-11.50
4
3
2
1
0.2 0.6 1.0
distance from min 1 along
the minimum energy path
potential
-11.2
-12.0
-12.8
• search for rotation-permutation isomers
• 10-50 cells placed along the collective coordinate - root
mean square distance from minimum 2

Application of DRPS:
Cluster of Lennard-Jones disks
0.6
simul
0.2
k fit
k equil
-0.6
1.00
0.95
0.90
0.85
0.80
0.75
0.70
-1.0
0.9
0.8
0.0 0.4 0.8 0.2 0.6
0.65
in A
free energy
/19 bf269@cam.ac.uk
Rate constants
box 2 box 5 box 10 TPS DPS
ln(k12/⌧0) -28.9 ± 0.3 -29.6 ± 0.5 -30.5 ± 1.0 -28.7 -26.9
ln(k21/⌧0) -9.3 ± 0.2 -9.4 ± 0.3 -9.7 ± 0.7 -7.2 -9.2
0.60
0.0 0.2 0.4 0.6 0.8 1.0
population
time / [τ0]
population time distance from min 2
1.0
0.7
-0.2
0.2 0.6 1.0 0.4 0.8 1.0
fit
propagation
equilibrium value
TST
MC
DRPS small cell
DRPS large cell

New application:
Polymer reversal in a narrow pore
• polymer confined in a pore is a simple model of DNA or
peptide in a synthetic or a biological pore
• dynamics of an LJ polymer can be directly studied by DPS
• at higher temperatures / truncated potentials ➣ DRPS
for polyLJ all minima can be easily found 2D
13
/19 bf269@cam.ac.uk
distance from
pore axis
potential
pore
radius

Future applications
polymer reversal polymer translocation peptide dynamics
1#FIG. where the unnecessary constant terms are ignored. kB is the
Boltzmann constant and T is the absolute temperature. ! 1!
and ! 2! are the values of !! in regions I and II, respectively.
Fm exhibits a free energy barrier as a function of m. The
nature of the free energy barrier depends on i# conforma-tional
statistics on both regions, and ii# value of the chemi-cal
potential mismatch $% relative to the entropic part first
two terms on the right-hand side of Eq. 2#'. The results are
illustrated in Figs. 2 and 3. When $%!0, the barrier is sym-metric
Native
10 20 30 40 50
in f!m/N if the size exponent is the same in both
Stationary Point
regions. For ( 1)( 2 , the barrier becomes asymmetric. The
free energy minima are downhill from I to II if the chain is
more compact in II than in I. This is illustrated in Fig. 2 by
taking ! 1!!0.9, ! 2!!0.5, and N$%/kBT!3.0. On the
other hand, the free energy minima are uphill if the chain is
*
* t
Wm t #!km1Wm1 t ##km#1 ! Wm#1 where Wm is the probability in region II. km is the of a nucleus with m#1 segments segments. km !
is the rate constant with m segments into a nucleus detailed balance to express km! m to be a continuous variable, *
30
20
10
#!!
*
1. Polymer escape in transition.
FIG. 3. Plot of Fm in units of kBT represent: solid line, N$%/kBT!0; dashed line, N$%/kBT!1.
biomolecular folding
0
Energy Difference (kcal/mol)
Unfolded
Unfolded
• surface reactions
• (with rigidification) conformational
transitions of large proteins
• collaboration? 

Acknowledgements

Simulating dynamics of rare events using discretised relaxation path sampling

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