1

Milestoning

Ron Elber and Anthony WestDepartment of Chemistry and Biochemistry and

Institute of Computational Sciences and Engineering (ICES)University of Texas at Austin

$ NIH

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Motivation

• Understanding Myosin II:– A protein that converts biochemical energy

(ATP) to mechanical energy in the muscles

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Myosin Stroke Cycle (Lymn-Taylor)

S. Mesentean et al., JMB, 367, 591–602 (2007)

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Movie of the reaction path for the recovery stroke in myosin

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Simulation background• Computer simulations use classical

mechanics to simulate protein dynamics

Md2X

dt2= !"U X( )

X t + #t( ) = X t( ) +V t( ) $ #t !#t 2

2M

!1 $"U X t( )( )

V (t + #t) =V t( )!#t

2M

!1 $ "U X t( )( ) +"U X t + #t( )( )%& '(

Δt about 10-15s (a femtosecond) to maintain stability of thealgorithm.We are interested in time scale of 10-3s (a millisecond). On a PC ittakes months to compute 100 ns (10-7s).

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The problem

• Twelve orders of magnitude differencebetween the basic time steps (10-15s) andthe biological time scale (10-3s).

• How to bridge the time scale gap?– Coarsening space and time while keeping

molecular details

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The concept of Milestones

R P

q•Microscopic molecular motions tend to be diffusive at long time scales

•Following in detail individual trajectories is computationally expensive

•Can we compute the trajectories in pieces? The pieces will give us acoarse grained model.••Is it correct?Is it correct? Is it computationally efficient?Is it computationally efficient?

The interfaces between which we compute the pieces of thetrajectories are called MilestonesMilestones

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Is computing trajectory piecesmore efficient than computing

it as a whole?

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Chopping trajectories iscomputationally efficient (I)

RP

L

Parallelization Pieces of trajectories are trivial to parallelize.Speed up proportional to the number of processorsSpeed up proportional to the number of processors

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Chopping trajectories iscomputationally efficient (II)

RP

L

Diffusive processes The time t to diffuse a path of length L isproportional to L2 . If we divide the path to N segments the timebecomes (L/M)2. There are M Milestones and therefore the timerequired is M*(L/M)2=L2/M. Speed up factor of M number ofMilestones.

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Chopping trajectories is computationallyefficient (III)

RP

Activated process If one of N1 trajectories that starts in Rreaches q1, and one of N2 trajectories that starts at q1 reaches q2the total number of trajectories that we need to sample q2starting from R is N1*N2. Using Milestoning we only need N1+N2Speed up exponential in the number of Milestones - MSpeed up exponential in the number of Milestones - M

L

U(q)

q1 q2

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Building a coarse grained model (which is equation-free):What do we keep from the short chopped trajectories?

R P

ii !1 i +1

1. Initialize trajectories at the Milestones from a stationary time-independent distribution assumed known (e.g. canonical).

2. Compute trajectories between any pair of Milestones (i,j) witha shared volume, estimate the first passage time distribution

Kij!( )

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The local first passage time distribution

Kij !( )" The probability density that a trajectory that starts

at Milestone i will terminate exactly after time ! at

Milestone j

Kij !( ) #d!0

$

% = pij " The probability that a trajectory that was

initiated at Milestone i will terminate at Milestone j

pijj

& = 1 -- A normalization condition on pij (all traj terminate)

This is all the microscopic information needed. No mechanical modelis required.

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Example for a typical Kij!( )

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What do we want to compute from the coarsemodel?

R P

Pi!1

t( ) Pit( ) P

i+1t( )

Pit( ) microscopically it is the probability of being somewhere

between Milestones i-1 and i+1 at time t such that the lastMilestone passed by the trajectory is Milestone i.

Provides a blurred (coarse) spatial description withoutreducing the number of degrees of freedom

PNt( )

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• Define : prob of being at i at time t

• Define : prob of transition to i at t

• Define : conditional probability of a transition fromi to j after incubation time τ.

•Then hopping dynamics are defined by:

The QK picture

Qi(t) = P 0( )

i! t " 0+( ) + Q

j(t ')K

j ,i(t " t ')#

$%&

0

t

' dt 'j

(

Pi(t) = Q

i(t ') 1" K

i, j() )

j

(#

$*

%

&+d)

0

t" t '

'#

$**

%

&++0

t

' dt '

P

i(t)

Q

i(t)

Ki, j

!( )

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With the matrixdetermined, compute long time kinetics

Kij!( )

Qi(t) = P

i0( )! t " 0+( ) + Q

j(t ')K

ji(t " t ')#

$%&

0

t

' dt '

Pi(t) = Q

i(t ') 1" K

ij(( )

j

)#

$*

%

&+d(

0

t" t '

'#

$**

%

&++0

t

' dt '

• by direct integration• by Laplace transform (Shalloway)• by trajectory statistics (Vanden Eijnden)

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Computing the average first passage time fromMilestone 1 to N

! = !12

1( )+ ...+ ! ij

n( )+ ...+ ! kN

L( )"# $%l=1,...,L

& ' p12 ' ' ' pij ' ' ' pkN( )

! ijn( ) is a random variable sampled from Kij !( )

pij = Kij !( )d!0

(

) is the transition probability from i to j

pij = 1j

& A few points:The limit is consideredThe N state is absorbing τNk=0 pNN=1

To average over τ…

L!"

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Compute <τ>

Define a random matrix T( )ij! pij" ij

the matrix P( )ij! pij of size N # N

the vector "̂L( )! "1

L( )," 2

L( ),...," NL( )( )

T

with elements " iL( )

that are the overall first passage times using L steps

to go from i to N

Then

"̂L( )= P

L$lTP

l$11l=1

L

% = PL$lT1

l=1

L

%

where 1= 1,1,...,1( )T

and P1= 1

1 2 3 4 5 6 Nij

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The last tricks to compute <τ>For L!" the trajectory is absorbed at Milestone N .

Since the time at N does not count (#NN = 0) $ #̂ L+1= #̂

L

#̂L( )= P

L% lT1

l=1

L

&

#̂L+1( )

= PL+1% l

T1l=1

L+1

& = P PL% lT1

l=1

L

& + T1 = P#̂L( )+ T1

I % P( )#̂ = T1 (remove eigenvector 1 from the set),

T1 = 0 (I - P)1 = 0, define, I ,P,T of size N %1( ) ' N %1( )).

T is a random matrix and the average first passage time

is obtained by avergaing over elements of T . T( )ij

= pij # ij

Only the first moments of # ij are required to compute #̂ .

#̂ = I % P( )%1

T 1

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Is Milestoning correct (or whatare the assumptions)?

R P

ii !1 i +1

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Assumption

Let Sj be the hypersurface of Milestone j.

Let Xj be a coordinate vector Xj !R3Nand Xj !Sj .

" Xj( ) is the distribution at Sj initiating the short trajectories.

#ij Xi( ) is the distribution obtained from first passage traj on Sj

if initiated at Si according to " Xi( )

Assumption : #ij Xi( ) = " Xj( )

Sj

Si

Xj

Xi

Implies:• Loss of memory• committers

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Committers are special surfaces with equal probability ofreaching for the first time the product and the not the

reactants

p1 p2 p3

R

P

Committer surfaces can be calculated exactly (solvingpartial differential equations) in 2-3 dimensions forBrownian dynamics and approximated at higherdimensions and other types of equations of motion.

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Or a tunnel picture

Relaxation in planes faster than transition between planes.General but heuristic.

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Toy model: 1D box simulation

• Microscopic dynamics are Brownian• Simulations run at various temperatures

and for 4, 8, and 16 milestones

!dX

dt= "#U + R

R = 0 R t( )R t '( ) = C$ t " t '( )

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1D reaction curves (5000 trajs/MLST)

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2D simulation

q

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Memory loss demonstration

0.15!"!

2

0

6.34, ( )K d! ! ! !"# $

% &' ()!

"

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Alanine Dipeptide

JCP, 126, 145104 (2007)

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Reflecting boundary conditions

Absorbing boundary condition

Preparing initial conditions by samplingq !"

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Torsion velocity auto-correlation indicates whenmilestoning assumption is being violated

~ 400fsr

!

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Average Incubation Times vs. Velocity Relaxation Time

r! !" >

01

1( )i

M

s

i

K dM

! ! ! !"

=

# $%

1090253581713051137319129375873587431144

(fs)M !

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Rate Results

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Reaction curves

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• Milestoning divides RC into fragments whose kineticscan be computed independently then “glued” together

• Provides factor of improvement in computationalefficiency on serial machines, plus exp bootstrapping

• Uses LFPTDs from microscopic dynamics:• System distribution given by simple integral

equations that can be easily solved numerically• Correct kinetics for solvated alanine dipeptide (x 9

speedup)• Predicts microsecond Scapharca rate with ~10 ns total

serial time• Sub-millisecond rate for myosin recovery stroke with

total run time (serial) - speedup : more than 10,000

Summary

M

( )sP t

( )sK !

±

200sample ! 241mlst ! 0.01ns + 0.1ns ! 241 = 501.6ns

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Milestoning papers• Ron Elber, "A milestoning study of the kinetics of an allosteric

transition: Atomically detailed simulations of deoxy Scapharcahemoglobin", Biophysical J. ,2007 92: L85-L87

• Anthony M.A. West, Ron Elber, and David Shalloway, "Extendingmolecular dynamics timescales with milestoning: Example ofcomplex kinetics in a solvated peptide", J. Chem. Phys.126,145104(2007)

• Anton K. Faradjian and Ron Elber, "Computing time scales fromreaction coordinates by milestoning", J. Chem. Phys. 120:10880-10889(2004)

• Code (moil + zmoil) available fromhttps://wiki.ices.utexas.edu/clsb/wiki