/19 [email protected] relaxation path sampling (DRPS) 1
Simulating dynamics of rare events using discretised relaxation path
sampling
Wales group Department of Chemistry University of Cambridge
Boris Fačkovec
Energy landscapes workshop Durham, 18th August 2014
/19 [email protected] relaxation path sampling (DRPS) 2
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)
�
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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)
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Cell-to-cell rate constant
incomplete partitioning
soft cells
hard cells
A
B
AA
BB
• soft cells were developed to allow complete space partitioning using surface-surface rate constants
• inconvenient for observables, incompatible with DPS
AB
CD A
B
C
D AB BC
BD
AC
CD
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Methods using hard cells
lag time
log
(pro
babi
lity)
position
pote
ntia
l biased potential
• boxed molecular dynamics (BXD) - sampling enhanced by space partitioning
• hyperdynamics (hyperMD) - sampling enhanced by biasing the potential
• dividing surface defined by a certain plane involving the saddle pointtim
e
position
A
B
C
kBXDAB =
1
h⌧lagi
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Limits of TST
timepopu
latio
n in
A
timepopu
latio
n in
AA A
• transition state theory - rate constant = equilibrium flux / equilibrium population
• the rate constant is very sensitive to definition of the dividing surface
kTSTAB =
flux
population
=
hP (x) [n(x) · v]i@AhP (x)iA
=
@tpA(0)
pA(0)
/19 [email protected] relaxation path sampling (DRPS)
• relaxation involves both exit and penetration - best approach
• (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):
7
Rate constants from trajectories
position
pote
ntia
l
prob
abilit
y de
nsity
position
relaxation
kexAB
=peqB
peqA
⌧ exB
+ peqB
⌧ exA
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No-flux boundary conditions
• isolation of A and B from the rest by placing hard walls
• straightforward for overdamped dynamics
• impossible for Hamiltonian (deterministic) dynamics
A B
C
D
EF
G
A B
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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
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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)
trajectories from equilibrium distribution
trajectories from the boundaries
FB|A(t) = @tpA(t) ,
LBB|A(t) =@tFB|A(t)
FB|A(0)
B A
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• 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
Optimised simulation protocol
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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
AB
AC B
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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
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• 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
kexAB
kDRPS
AB
11.1
12.11
12.11
5.64
10.62
10.8
31.58
31.6
7.96
10.56
position of the dividing surfacegood poor
rate constants (in arbitrary units)
Application of estimator: Sinai billiards
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Application of DRPS: Cluster of Lennard-Jones disks
-11.40 -11.48
-12.54-11.50
-11.04
43
2
1
-10.77
distance from min 1 along the minimum energy path
pote
ntia
l
-11.2
-12.8
-12.0
0.2 0.6 1.0
• search for rotation-permutation isomers
• 10-50 cells placed along the collective coordinate - root mean square distance from minimum 2
/19 [email protected] relaxation path sampling (DRPS)
-0.6
16
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
0.600.650.700.750.800.850.900.951.00
0.0 0.2 0.4 0.6 0.8 1.0
popu
latio
n
time / [τ0]
simulk fit
k equil
17 / 20 Boris Fačkovec ([email protected]) FPT-BXD Cambridge, 13th December 2013
popu
latio
n in
A
time distance from min 2
free
ener
gy
0.0 0.60.20.4 0.8-1.0
0.6
0.2
1.0
0.7
0.8
0.9
-0.2
0.2 0.6 1.0 0.4 0.8 1.0
fit
propagation
equilibrium valueTST
MC
DRPS small cellDRPS large cell
Application of DRPS: Cluster of Lennard-Jones disks
/19 [email protected] relaxation path sampling (DRPS)
for polyLJ all minima can be easily found2D13
17
New application: Polymer reversal in a narrow pore
distance from pore axis
pote
ntia
l pore radius
• 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
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Future applications
peptide dynamicspolymer reversal polymer translocation
where the unnecessary constant terms are ignored. kB is theBoltzmann 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. Thenature 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 &firsttwo terms on the right-hand side of Eq. "2#'. The results areillustrated in Figs. 2 and 3. When $%!0, the barrier is sym-metric in f!m/N if the size exponent is the same in bothregions. For (1)(2 , the barrier becomes asymmetric. Thefree energy minima are downhill from I to II if the chain ismore compact in II than in I. This is illustrated in Fig. 2 bytaking !1!!0.9, !2!!0.5, and N$%/kBT!"3.0. On theother hand, the free energy minima are uphill if the chain ismore compact in I. This is illustrated in Fig. 2 by taking!1!!0.5, !2!!0.9, and N$%/kBT!3.0. By keeping the samevalue of ( in both regions, the free energy barrier can bemade asymmetric if $%)0, as illustrated in Fig. 3 with !1!!0.69!!2! .
Following the usual arguments of the nucleationtheory,13 the transport of the chain through the barrier isdescribed by
*
*t Wm" t #!km"1Wm"1" t #"km!Wm" t #"kmWm" t #
#km#1! Wm#1" t #, "3#
where Wm is the probability of finding a nucleus of m seg-ments in region II. km is the rate constant for the formationof a nucleus with m#1 segments from a nucleus with msegments. km! is the rate constant for the decay of a nucleuswith m segments into a nucleus of m"1 segments. Using thedetailed balance to express km#1! in terms of km and adoptingm to be a continuous variable, we get
*
*t Wm" t #!!"*
*m D "1 #"m ##*2
$*m2 km"Wm" t #, "4#
where
D "1 #"m #!"km*
*mFm
kBT#
*
*m km . "5#
As mentioned above, we assume that the rate constant km isindependent of m "because we deal here with only ho-mopolymers#, and is dictated by the ratchet potential arisingfrom the details of the pore. Under this condition, valid forthe experimental problem addressed in this paper, km is takento be a nonuniversal constant k0 independent of N. ThereforeWm is given by
*
*t Wm" t #!*
*m ! k0kBT *Fm
*m Wm" t ##k0*
*m Wm" t #" , "6#
It follows14 from Eq. "6# that the mean first passage time +for the process described by Eq. "6#, i.e., the average timerequired by the chain, having already placed at least onesegment in region II, to go from region I to region II is
+!1k0#0
Ndm1 exp$ Fm1
kBT% #
0
m1dm2 exp$ "
Fm2
kBT% . "7#
The limits 0 and N are actually 1 and N"1, respectively, andN in the following formulas should be replaced by (N"2) ifone is interested in small values of N. Substituting Eq. "2# inEq. "7#, the general results are given in Fig. 4, where + "in
FIG. 1. Polymer escape in transition.
FIG. 2. Plot of Fm in units of kBT against f. Curves represent: solid line,N$%/kBT!0, !1!!0.5!!2! ; long-dashed line, N$%/kBT!"3, !1!!0.9,!2!!0.5; dashed line, N$%/kBT!"3, !1!!0.5, !2!!0.9.
FIG. 3. Plot of Fm in units of kBT against f for !1!!0.69!!2! . Curvesrepresent: solid line, N$%/kBT!0; long-dashed line, N$%/kBT!"1;dashed line, N$%/kBT!"1.
10372 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 M. Muthukumar
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:131.111.184.86 On: Thu, 19 Jun 2014 00:33:12
biomolecular folding
0 10 20 30 40 50
0
10
20
30
Stationary PointPote
ntial
Ene
rgy D
iffere
nce
(kca
l/mol)
A
Misfolded
Unfolded
Native
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0
10
20
30
Stationary PointPote
ntial
Ene
rgy D
iffere
nce
(kca
l/mol)
B
Native
Unfolded
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0
10
20
30
Stationary PointPote
ntial
Ene
rgy D
iffere
nce
(kca
l/mol)
A
Misfolded
Unfolded
Native
0 10 20 30 40 50
0
10
20
30
Stationary PointPote
ntial
Ene
rgy D
iffere
nce
(kca
l/mol)
B
Native
Unfolded
• surface reactions
• (with rigidification) conformational transitions of large proteins
• collaboration? 😊
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Acknowledgements
[email protected] relaxation path sampling (DRPS) 20
popu
latio
n in
A
time
equilibrium value
@tFj|i(s, t) = @sFj|i(s, t) + Fk|i(0, t)Lkj|i(s) + Fi|j(0, t)Ljj|i(s) ,
@tFk|i(s, t) = @sFk|i(s, t) + Fk|i(0, t)Lkk|i(s) + Fi|j(0, t)Ljk|i(s) ,
@tFi|j(s, t) = @sFi|j(s, t) + Fk|j(0, t)Lki|j(s) + Fj|i(0, t)Lii|j(s) ,
@tFk|j(s, t) = @sFk|j(s, t) + Fk|j(0, t)Lkk|j(s) + Fj|i(0, t)Lik|j(s)
• with initial conditions:Fj|i(s, 0) = Fj|i(s) ,
Fk|i(s, 0) = Fj|i(s),
Fi|j(s, 0) = 0 ,
Fi|j(s, 0) = 0 .
pi(t) =
Z 1
0Fj|i(s) + Fk|i(s)ds
Rate constants from response functions
• simulating relaxation from cell i to j with neighbours k: