advanced sampling

8/13/2019 Advanced Sampling

1/10

M. S. Shell 2009 1/10 last modified 5/31/2012

Advanced sampling ChE210D

Today's lecture: methods for facilitating equilibration and sampling in comple !frustrated! or slo"#e$ol$ing s%stems

Difficult-to-simulate systems&racticall% spea'ing! one is al"a%s limited in the length and time scales accessible from simula#tion. (he former relates to the ma imum s%stem si)e that can be used *in terms of number ofatoms+! and the latter to the length of the run that can be performed *in terms of number ofM, integration or M- steps+.

f the intrinsic! ph%sical time scale for some s%stem of interest is be%ond that accessible fromsimulation! "e cannot in$estigate it using straightfor"ard d%namical methods li'e molecular

d%namics because it "ill be challenging to equilibrate the s%stem. o"e$er! if "e are onl%interested in thermod%namic and not 'inetic properties! then "e can construct artificial d%nam#ics in the s%stem that accelerate con$ergence to"ards equilibrium. Such methods ine$itabl%in$ol$e a Monte -arlo component.

hat ma'es a s%stem difficult to sample in an equilibrium sense (%picall%! these s%stemsha$e frustrated or rugged energy landscapes . (his means that the underl%ing potential energ%surface contains man% deep minima separated b% high energ% barriers. t lo" temperatures! itis challenging for the s%stem to mo$e bet"een the rele$ant lo"#energ% regions because it must

surmount these barriers a rare e$ent process that ta'es time. "ide range of methods ha$e been designed to accelerate equilibration in simulations of

s%stems "ith rugged energ% landscapes. ere! "e "ill onl% re$ie" a small selection of these.Some 'inds of s%stems that might fall into this categor% include4

s%stems "ith strong electrostatic interactions

fluids of strongl% orientation#dependent interactions *e.g.! dipoles! h%drogen bonds+

macromolecular and pol%meric s%stems

biomolecular s%stems *proteins! lipids+

self#organi)ing or self#structuring s%stems *micelles! bila%ers+

supercooled liquid and amorphous s%stems at $er% lo" temperature


2/10


Simulated tempering(he simulated tempering algorithm is equi$alent to a canonical Monte -arlo simulation in"hich the temperature changes randoml%. (hat is! there are t"o 'inds of mo$es4

energy fluctuation moves single#particle displacements! orientational displacements!etc. (he usual canonical acceptance criteria! at the current temperature! are used.

temperature fluctuation moves the temperature is periodicall% increased or de#creased b% fi ed amounts so that the temperature can acquire one of a fi ed set of $al#ues min,min + ,min +2 ,,max.

(hese temperature fluctuations help the s%stem equilibrate b% allo"ing it to fluctuate to highertemperatures "here it can more readil% cross energ% barriers.

n these simulations! the microstate is no" a function of both the configurations and the

temperature4

, hat is the acceptance criterion for a temperature change e "ant all temperatures to

appear "ith equal probabilit%. (hat is! if "e measured the distribution of temperature o$er thesimulation run! "e "ould "ant it to be roughl% uniform. (o do this! "e introduce a "eightingfunction in temperature4

,

(his means that the acceptance criterion for a random temperature perturbation = + is gi$en b%a = min!",# $ # $% = min!",& &%

(he distribution of temperatures is found b% integrating this 6oint probabilit% o$er configura#tions4

', ( = ) ,*, = (o achie$e a uniform distribution in temperature! "e demand that this e pression is constant.(his means that "e should choose4


3/10


- = ./ e do not 'no" the free energies at the different temperatures a priori. nstead! "e must

iterati$el% determine them o$er the course of a simulation. Man% methods can be used to dothis! including4

flat histogram methods the flat histogram coordinate is the temperature

multiple histogram-reweighting "e maintain a histogram of energies at each temper#ature and periodicall% use the re"eighting equations to determine the relati$e free en#ergies for each temperature

free energy perturbation "e use! for e ample! 7ennett8s method bet"een ad6acenttemperatures to compute free energ% differences at periodic inter$als in the simulation

o" do "e compute properties at a temperature of interest from this approach e can usethe M#based re"eighting equation applied to all temperatures and all tra6ectories. efirst compute a configurational "eight according to

01 = 3 44 4567 ere! 8 is an inde o$er simulation obser$ations. (he in$erse temperature . corresponds to an

arbitrary re"eighting temperature . ne should choose min 9 9 max for good statisticalaccurac%.

:or an arbitrar% obser$able :! "e can then compute a$erages and distributions using; :< = 3 01 :113 011 : >01?@7@1

Replica exchange

(he simulated tempering approach is useful for facilitating equilibrium because it allo"s thes%stem to e plore multiple temperatures. t has t"o dra"bac's! ho"e$er4

e need to compute the free energies / 6 at e$er% temperature in order to properl%sample all temperatures.


4/10

M. S. Shell 2009 ;/10 last modified 5/31/2012

e need to "ait for the s%stem to tra$erse the entire temperature range man% times inorder to accumulate good statistics at each. (his can be a $er% long time if "e ha$eman% temperatures.

n alternati$e approach! "hich has become the method of choice for sampling challengings%stems is the replica exchange method. t o$ercomes these limitations b% ha$ing the follo"inggeneral construction4

A simulations * replicas + of the same s%stem are performed simultaneousl% at differenttemperatures 6.


5/10


ere! B corresponds to the set of configurations in "hich and are transposed to differ#ent temperatures. e ha$e4

B = $ $

) K# #

) KC4 D4E

)65

67L

B = $ #) K# $) KC

4 D4E)6

5

67L

&lugging these probabilities into the acceptance criterion!

,FGaHa = minM",$ $ # # $ # # $N >otice that the partition functions in the denominator cancel. (his means that "e do not needto 'no" the free energies in each temperature "hen e$aluating the acceptance criterion. t isbecause "e s"ap s%stems at t"o temperatures! rather than perturb a single s%stem8s tempera#ture! that "e no longer need to 'no" the free energ% *as compared to simulated tempering+.

Simplif%ing the abo$e e pression!

,s"apacc = min!",& &% "here . = . O . ,P = PO P >otice that "e are required to compute the potential energ% difference bet"een the instanta#neous configurations in the t"o temperature replicas.

S"ap mo$es are performed li'e an% other M- mo$es4 the mo$e is proposed! the acceptancecriterion is computed! a random number is dra"n! and it is decided "hether or not to performthe mo$e. =nli'e man% other M- mo$es! s"ap mo$es are $er% ine pensi$e to perform4 the%onl% require the current energies in each temperature! "hich is t%picall% maintained through#out the simulation an%"a%s.

Considerations for acceptance ratios and the temperature schedule-onsider a s"ap bet"een t"o temperatures Q . e e pect that the configuration dra"nfrom the higher temperature "ill ha$e a higher energ%. (hat is! P Q P . (hus thequantit% .P "ill most often be negati$e! "hich "ill ma'e the acceptance probabilit% small.:or $er% large temperature differences! this quantit% becomes e$en more negati$e. (he resultis that the temperatures must be spaced close enough together in order to achie$e a good rateof accepted s"aps.


6/10

M. S. Shell 2009 ?/10 last modified 5/31/2012

e can thin' about this problem in terms of the distributions P, P bet"een neigh#boring temperatures. f there is a substantial o$erlap! "e "ill ha$e a high frequenc% of s"apmo$e acceptance4

(hese considerations pla% into the "a% in "hich "e pic' the temperatures in our replica e #change simulation 4

min # t%picall% "e pic' a minimum temperature to be the temperature of interest that isdifficult to simulate

max # one chooses a high temperature "here free energ% barriers can be crossed! butnot so high as to require man% intermediate temperatures

A # the number of temperatures is usuall% chosen so as to achie$e @50A acceptance ofs"ap mo$es bet"een ad6acent replicas

6 # for a s%stem "ith a constant heat capacit%! it can be sho"n that a constant rate ofacceptance bet"een ad6acent temperatures corresponds to an e ponential distributionin temperature! equi$alentl%! a po"er la" in replica number. (hus! one normall% pic's

6 = minRmaxminS65

Scaling with system sizes the s%stem si)e gro"s! the distribution in energ% at a gi$en temperature becomes increasing#

l% narro" "ith respect to the a$erage energ%! as "TU . ithout o$erlap bet"een the energ%fluctuations of ad6acent temperatures in the replica e change scheme! fe" s"ap mo$es "ill beaccepted. (hus! as the s%stem si)e increases! more and more intermediate temperaturereplicas are needed to achie$e a 50A acceptance ratio. (his ma'es the method challenging toappl% to $er% large s%stems.

P

P

Q

P Q

s"aps frequentl% accepted s"aps infrequentl% accepted P


7/10

M. S. Shell 2009 B/10 last modified 5/31/2012

Evaluating properties and distributionsCigorousl%! all of the configurations at each temperature *regardless of "hether the% s"appedin from other temperatures+ con$erge to a canonical distribution at that temperature. (hus! "ecould compute the a$erages of properties at each temperature 6 b% simpl% a$eraging o$er the

configurations at that temperature. trajectory is therefore considered the e$olution ofconfigurations at a given temperature .

o"e$er! "e can also compute an a$erage at an% arbitrary temperature bet"een the minimumand ma imum in our simulation b% using a re"eighting approach. hen a replica e changesimulation is performed! "e can collect histograms or histories of each potential energ% $isitedin each temperature! P16. e can then use the re"eighting equations to compute the freeenergies and configurational "eights at each temperature. ith the computed configurational"eights! "e can e press the a$erage of an% propert% as a "eighted sum o$er all temperaturetra6ectories4

016 = 243 VV V245W7

; :< = 3 3 016 :16X17567

3 3 016X17567 : >>016?@47@

X17

5

67

mplementation(o maintain the different replicas! one t%picall% uses a parallel computing scheme ! in "hich onecomputer node *or processor+ is assigned to each replica. hen s"aps are performed! allnodes stop e$ol$ing the s%stem in time and a master or head node sorts through and ma'es thes"ap mo$es. (his in$ol$es parallel communication bet"een the different nodes and the headnode.

t each inter$al "here s"aps are performed! an% number of s"aps could be attempted. (%pi#call%! one chooses the number of s"ap attempts to be the number of temperature replicas.(he s"aps can be attempted in serial order! from lo"est/highest to highest/lo"est tempera#ture! or in random order.

n a M- simulation! the e$olution of the s%stem in bet"een s"aps can be go$erned b% an%number of M- mo$es that accomplish changes in the potential energ%! such as single particledisplacements.


8/10

M. S. Shell 2009 D/10 last modified 5/31/2012

n a M, simulation! the e$olution in bet"een s"aps can be performed using short M, tra6ecto#ries. (hese mo$es can be considered h%brid M-/M, mo$es "ith good energ% conser$ationsuch that the% are al"a%s accepted. fter a round of s"ap mo$es! one pic's random $elocitiesfor each atom at e$er% temperature Y from a 7olt)mann distribution at the corresponding 6.(hese $elocities are used to start the short M, tra6ector% before another round of s"ap mo$esis performed.

n alternati$e approach in M, "ould be to rescale the velocities in the configurations to thene" temperature after a s"ap mo$e. (his approach "as first deri$ed b% Sugita and 'amotoE1999F. ere! random ne" $elocities are not pic'ed at an% time. nstead! after s"apping

configurations! the momenta are scaled b% a factor Z n[G \]^_ ! "here n[G is the ne" temper#ature into "hich the configuration "as s"apped and \] is the temperature from "hich itcame. (his approach requires a thermostat to be used during the M, tra6ectories. hile the$elocit% rescaling approach is more frequentl% used in the literature than the h%brid approach*"ith random resampling+! it is not necessaril% more efficient and either approach is $alid.

!ariantsere! "e considered a replica e change simulation "here the replicas differed in temperature.

(his is often also called parallel tempering for its similarit% to simulated tempering. o"e$er!"e are not limited to temperature. ne can ha$e replicas differ in chemical potential orpressure if the indi$idual simulations are G-M- or simualtions! respecti$el%. Two-dimensional replica exchange methods allo" replicas to differ in both temperature and chemi#cal potential or pressure.

nother "a% to facilitate sampling is to modif% the potential energ% function itself. ere! one"ants to perturb the energ% function so that the underl%ing energ% landscape is smoother andeasier to sample. :or e ample! one s%stematicall% scale the partial charges from )ero to theirfull $alues as one mo$es from replica to replica. (his approach is often called Hamiltonianexchange (he appropriate form of the acceptance criterion is deri$ed as before! but "ithdifferent potential energ% functions in each simulation P64

,s"apacc = minM",$ $ $ # # # $ $ # # # $ N n e$aluating this acceptance criterion! one must e$aluate the cross#energies P and

P bet"een t"o replicas "ith each s"ap mo$e! since the energ% function is no longerconstant bet"een them.


9/10


n all of these cases! "e can use multiple histogram re"eighting techniques to e press thea$erage or distribution of an% propert% as a "eighted a$erage o$er all temperatures Y andtra6ector% configurations 8.

Extended ensemble molecular dynamicsne "a% to enhance the e ploration of phase space in M, at lo"er temperatures is bias the

ensemble probabilities to artificiall% broad fluctuations in potential energ%. (his enables thes%stem to hop o$er higher#energ% barriers more frequentl%. =ltimatel%! the beha$ior of thes%stem under normal conditions can be reco$ered b% standard re"eighting procedures. eha$e alread% discussed this approach in the conte t of M- simulations. >o"! "e consider anM, implementation.

e add to our M, simulation a "eighting function that biases the configurational probabilities!

in the same "a% that "e did during our discussion of biased sampling4

PG = P O ` - P ere! - P is a "eighting function "ith dimensionless units! and it onl% depends on the poten#

tial energ%. (he equations of motion are deri$ed as before. :or an atom 84

b1G = O(PG

( 1 = O(P

( 1 + ` (- P

(P

(P

( 1

= b1 R" O `(- P(P S n other "ords! the force on each atom in the "eighted ensemble is scaled b% a term in$ol$ing

the deri$ati$e of the "eighting function evaluated at the energ% of the current configuration. fthe "eighting function is a constant! no scaling occurs.

n order for this approach to "or'! the "eighting function must be a continuous function. net%picall% uses splines or other mathematical constructs so that an anal%tical deri$ati$e can becomputed.

o" do "e determine the "eighting function e can fit it to estimates of the probabilit%distribution function. n the canonical ensemble! "e ha$e the relationship

PGP

f "e "ant the "eighted distribution to be appro imatel% flat! "e can choose


10/10


- P = O]nP+ \nFc >otice that "e "ill ha$e to fit the "eighting function to discrete histogram calculation! since - must be continuous. t is for this reason that it becomes difficult to e$aluate - to high statisticalaccurac% using M, techniques. Still! broadening the energ% distribution e$en a little bit ma%

speed equilibration in M, simulations.

advanced sampling

Documents