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arXiv:cond-mat/0305055v1 [cond-mat.stat-mech] 3 May 2003

Multi-Overlap Simulations for Transitions between Reference Configurations

Bernd A. Berg1,2, Hirochi Noguchi3,∗and Yuko Okamoto3,4

(E-mails: berg@csit.fsu.edu, noguchi@ims.ac.jp, okamotoy@ims.ac.jp )

1Department of Physics, Florida State University, Tallahassee, FL 32306, USA

2School of Computational Science and Information Technology

Florida State University, Tallahassee, FL 32306, USA

3Department of Theoretical Studies, Institute for Molecular Science

Okazaki, Aichi 444-8585, Japan

4Department of Functional Molecular Science, Graduate University for Advanced Studies

Okazaki, Aichi 444-8585, Japan

(printed February 2, 2008)

We introduce a new procedure to construct weight factors, which flatten the probability density of

the overlap with respect to some pre-defined reference configuration. This allows one to overcome

free energy barriers in the overlap variable. Subsequently, we generalize the approach to deal with the

overlaps with respect to two reference configurations so that transitions between them are induced.

We illustrate our approach by simulations of the brainpeptide Met-enkephalin with the ECEPP/2

energy function using the global-energy-minimum and the second lowest-energy states as reference

configurations. The free energy is obtained as functions of the dihedral and the root-mean-square

distances from these two configurations. The latter allows one to identify the transition state and

to estimate its associated free energy barrier.

PACS: 05.10.Ln, 87.53.Wz, 87.14.Ee, 87.15.Aa

I. INTRODUCTION

Markov chain Monte Carlo (MC) simulations, for in-

stance by means of the Metropolis method [1], are well

suited to simulate generalized ensembles. Generalized en-

sembles do not occur in nature, but are of relevance for

computer simulations (see [2–4] for recent reviews). They

may be designed to overcome free energy barriers, which

are encountered in Metropolis simulations of the Gibbs-

Boltzmann canonical ensemble. Generalized ensembles

do still allow for rigorous estimates of the canonical ex-

pectation values, because the ratios between their weight

factors and the canonical Gibbs-Boltzmann weights are

exactly known.

Umbrella sampling [5] was one of the earliest

generalized-ensemble algorithms. In the multicanonical

approach [6,7] one weights with a microcanonical tem-

perature, which corresponds, in a selected energy range,

to a working estimate of the inverse density of states. Ex-

pectation values of the canonical ensembles can be con-

structed for a wide temperature range, hence the name

“multicanonical”. Here, “working estimate” means that

running the updating procedure with the (fixed) multi-

canonical weight factors covers the desired energy range.

The Markov process exhibits random walk behavior and

moves in cycles from the maximum (or above) to the

minimum (or below) of the chosen energy range, and

∗Present address: Theory II, Institute of Solid State Re-

search, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany.

E-mail: hi.noguchi@fz-juelich.de.

back. A working estimate of the multicanonical weights

allows for calculations of the spectral density and all re-

lated thermodynamical observables with any desired ac-

curacy by simply increasing the MC statistics. Thus, we

have a two-step approach: The first step is to obtain

the working estimate of the weights, and the second step

is to perform a long production run with these weights.

There is no need for that estimate to converge towards

the exact inverse spectral density. Once the working es-

timate of the weights exists, MC simulations with frozen

weights converge and allow one to calculate thermody-

namical observables with, in principle, arbitrary preci-

sion. Various methods, ranging from finite-size scaling

estimates [8] in case of suitable systems to general pur-

pose recursions [9–11], are at our disposal to obtain a

working estimate of the weights.

In the present article we deal with a variant of the mul-

ticanonical approach: Instead of flattening the energy

distribution, we construct weights to flatten the proba-

bility density of the overlap with a given reference con-

figuration. This allows one to overcome energy barriers

in the overlap variable and to get accurate estimates of

thermodynamic observables at overlap values which are

rare in the canonical ensemble. A similar concept was

previously used in spin glass simulations [12], but there

is a crucial difference: In Ref. [12] the weighting was

done for the self-overlap of two replicas of the system

and a proper name would be multi-self-overlap simula-

tions, while in the present article we are dealing with the

overlap to a predefined configuration.

We next generalize our approach to deal with two ref-

erence configurations so that transitions between them

become covered and our method allows one then to esti-

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mate the transition states and its associated free energy

barrier. We have in mind situations where experimen-

talists determined the reference configurations and ob-

served transitions between them, but an understanding

of the free energy landscape between the configurations

is missing. An example would be the conversion from a

configuration with α helix structures to a native struc-

ture which is mostly in the β sheet, as it is the case for

β-lactoglobulin [13,14].

The paper is organized as follows: In the next sec-

tion we describe the algorithmic details, using first one

and then two reference configurations. In particular, a

two-step updating procedure is defined, which is typi-

cally more efficient than the conventional one-step up-

dating. Moreover, based on the sums of uniformly dis-

tributed random numbers, a method to obtain a working

estimate of the multi-overlap weights is introduced. In

section III we illustrate the method for a simulation with

the pentapeptide Met-enkephalin. Our simulations use

the all-atom energy function ECEPP/2 (Empirical Con-

formational Energy Program for Peptides [15]) and rely

on its implementation in the computer package SMMP

(Simple Molecular Mechanics for Proteins [16]). We use

as reference configurations the global energy minimum

(GEM) state, which has been determined by many au-

thors [17–21], and the second lowest-energy state, as

identified in Refs. [19,22]. While our overlap definition

relies on a distance definition in the space of the dihe-

dral angles, it turns out that for the data analysis the

use of the root-mean-square (rms) distance is crucial. It

is only in the latter variable that one obtains a clear pic-

ture of the transition saddle point in the two-dimensional

free energy diagram. In the final section a summary of

the present results and an outlook with respect to future

applications are given.

II. MULTI-OVERLAP METROPOLIS

ALGORITHM

In this section we explain the details of our multi-

overlap algorithm. The overlap of a configuration versus

a reference configuration is defined in the next subsec-

tion. In the second subsection we discuss details of the

updating. To achieve step one of the method, i.e., the

construction of a working estimate of the multi-overlap

weights, one could employ a similar recursion as the one

used in [12] or explore the approach of [11]. Instead of

doing so, we decided to test a new method: At infinite

temperature, β = 0, the overlap distributions can be cal-

culated analytically (see subsection IID). We use this

as starting point and estimate the overlap weights at the

desired temperature by increasing β in sufficiently small

steps so that the entire overlap range remains covered.

In the final subsection we define the overlap with respect

to two distinct reference configurations to cover the tran-

sition region between them.

A. Definition of the overlap

There is a considerable amount of freedom in defining

the overlap of two configurations. For instance, one may

rely on the rms distance between configurations, and in

subsection IIID we analyze some of our results in this

variable. However, the computation of the rms distance

is slow and for MC calculations it is important to rely on

a computationally fast definition. Therefore, we define

the overlap in the space of dihedral angles by, as it was

already used in [24],

q = (n − d)/n ,(1)

where n is the number of dihedral angles and d is the

distance between configurations defined by

d = ||v − v1|| =

1

π

n

?

i=1

da(vi,v1

i) .(2)

Here, viis our generic notation for the dihedral angle i,

−π < vi≤ π, and v1is the vector of dihedral angles of the

reference configuration. The distance da(vi,v′

two angles is defined by

i) between

da(vi,v′

i) = min(|vi− v′

i|,2π − |vi− v′

i|) . (3)

The symbol ||.|| defines a norm in a vector space. In

particular, the triangle inequality holds

||v1− v2|| ≤ ||v1− v|| + ||v − v2|| .(4)

For a single angle we have

0 ≤ |vi− v1

i| ≤ π ⇒ 0 ≤ d ≤ n . (5)

At β = 0 (i.e., infinite temperature)

di =

1

πda(vi,v1

i) (6)

is a uniformly distributed random variable in the range

0 ≤ di ≤ 1 and the distance d in (2) becomes the sum

of n such uniformly distributed random variables, which

allows for an exact calculation of its distribution.

B. Multi-overlap weights

We choose a reference configuration of n dihedral an-

gles v1

i, (i = 1,...,n) to define the dihedral distance (2).

We want to simulate the system with weight factors that

lead to a random walk (RW) process in the dihedral dis-

tance d,

d < dmin → d > dmax and back .(7)

Here, dmin is chosen sufficiently small so that one can

claim that the reference configuration has been reached,

e.g., a few percent of n/2, which is the average d at

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T = ∞. The value of dmax has to be sufficiently large

to introduce a considerable amount of disorder, e.g.,

dmax = n/2. In the following we call one event of the

form (7) a random walk cycle (RWC).

One possibility is to choose weight factors which give

a flat probability density in the dihedral distance range

0 ≤ d ≤ n/2, falling off for d > n/2 by keeping the d-

dependence of the weight constant for d ≥ n/2. This

is quite similar to multimagnetical simulations [8], for

which the external magnetic field takes the place of the

reference configuration. The analogy becomes obvious,

when the external field is defined via a ghost spin, which

couples to all other spins. For instance, the spins ? s of the

Heisenberg ferromagnet are three-dimensional vectors of

magnitude ? s2= 1. Their interaction with an external

magnetic field?H can be written as

?H ·

?

i

? si= H

?

i

? sH·? si= N H q ,(8)

where ? sH is the unit vector in the direction of the mag-

netic field, ? si is the Heisenberg spin at site i, N is the

number of spins, and q is the overlap of the spin config-

uration with the reference configuration ? sH:

q =

1

N

?

i

? sH·? si. (9)

Using the multi-overlap language [12], the multi-magneti-

cal [8] weight factors may then be re-written as

exp(−βE + S(q)) = wc(E)wq(q) , (10)

where

wc(E) = exp(−β E) ,(11)

and E = −?

berg ferromagnet (the sum is over nearest neighbor

spins). Here, S(q) has the meaning of a microcanonical

entropy of the overlap parameter, which has to be de-

termined so that the probability density becomes flat in

q. Weights for other than the flat distribution have also

been discussed in the literature, e.g., Ref. [25], on which

we shall comment in connection with figure 7 below.

?ij?? si·? sjis energy function of the Heisen-

C. The updating procedure

In essence, there are two ways to implement the up-

date.

1. Combine the multi-overlap and the canonical

weights to one probability, which is accepted or re-

jected in one random step.

2. Accept or reject the multi-overlap and the canoni-

cal probabilities sequentially in two random steps.

1. One-step updating

As defined in equations (10) and (11), the weight fac-

tor is a product of wc(E) and wq(d), where wc(E) is the

usual, canonical Gibbs-Boltzmann factor and wq(d) is the

multi-overlap weight factor, where we now use the dis-

tance d from the reference configuration (instead of the

overlap q) as argument. As is clear from equation (1),

the use of either q or d as argument is equivalent, while

in the presentation of results the use of either variable

can have intuitive advantages. In the one-step updating

we combine the weights to

w(E,d) = wc(E)wq(d) ,(12)

and accept or reject newly proposed configurations in

the standard Metropolis way. Notably, the calculation

of wq(d) (a simple table lookup) is very fast compared

with the calculation of wc(E). Therefore, the following

two-step procedure is of interest.

2. Two-step updating

Suppose that the present configuration is (d,E) and a

new configuration (d′,E′) is proposed:

(d,E) → (d′,E′) .(13)

We can sequentially first accept or reject with the wq(d)

probabilities and then conditionally, when the d-part is

accepted, with the wc(E) probabilities.

Proof: We show detailed balance for two subsequent

updates of the same dihedral angle with the two-step

procedure. There are four cases with probabilities of ac-

ceptance:

Pi, i = 1,2,3,4.(14)

They are listed in the following:

1. wq(d′) ≥ wq(d) and wc(E′) ≥ wc(E) :

P1= 1,

2. wq(d′) ≥ wq(d) and wc(E′) < wc(E) :

P2= wc(E′)/wc(E),

3. wq(d′) < wq(d) and wc(E′) ≥ wc(E) :

P3= wq(d′)/wq(d),

4. wq(d′) < wq(d) and wc(E′) < wc(E) :

P4= wq(d′)wc(E′)/[wq(d)wc(E)].

(15)

(16)

(17)

(18)

For the inverse move

(d′,E′) → (d,E)(19)

with probabilities of acceptance

P′

i, i = 1,2,3,4,(20)

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the cases are:

1. wq(d) ≤ wq(d′) and wc(E) ≤ wc(E′) :

P′

2. wq(d) ≤ wq(d′) and wc(E) > wc(E′) :

P′

3. wq(d) > wq(d′) and wc(E) ≤ wc(E′) :

P′

4. wq(d) > wq(d′) and wc(E) > wc(E′) :

P′

4= 1.

1= wq(d)wc(E)/[wq(d′)wc(E′)], (21)

2= wq(d)/wq(d′), (22)

3= wc(E)/wc(E′),(23)

(24)

For the ratios we find

Pi

P′

i

=

wq(d′)wc(E′)

wq(d)wc(E)

,(25)

independently of i = 1,2,3,4. Therefore, we have con-

structed a valid Metropolis updating procedure.

D. Sums of a uniformly distributed random variable

To calculate the overlap weights at infinite tempera-

ture, we consider the sum

ur= xr

1+ ... + xr

n

(26)

of the random variables xr

distributed in the interval [0,1) and derive a recursion

formula for the probability density fn(u) of this distribu-

tion. Care is taken to cast the recursion in a form which

allows for a numerically stable implementation [26] over

a reasonably large range of n.

Let us recall the probability density of the uniform

distribution:

j(j = 1,···,n), each uniformly

f1(x) =

?1, for 0 ≤ x < 1,

0, otherwise.

(27)

To derive the recursion formula for the probability den-

sity of the random variable (26), it is convenient to cast

it in the form

fn(u) =

n

?

k=1

fn,k(xk) withxk= u − k + 1, (28)

where

fn,k(x) =

n−1

?

i=0

0, otherwise.

ai

n,kxi, for 0 ≤ x < 1,

(29)

The master formula for the recursion is obtained from

the convolution

fn(u) =

?u

0

f1(u − v) fn−1(v) dv .(30)

Let now u = x+k−1 with 0 ≤ x < 1, and equations (27),

(28), and (29) imply

fn,k(x) =

?k−1+x

k−2+x

?x

0

fn−1(v) dv

=

?1

x

fn−1,k−1(y) dy +fn−1,k(y) dy . (31)

Using equation (29) and performing the integrations, we

obtain

fn,k(x) =

n−2

?

i=0

n−2

?

i=0

ai

n−1,k−1

1

i + 1

−

n−2

?

i=0

ai

n−1,k−1

xi+1

i + 1

+

ai

n−1,k

xi+1

i + 1.(32)

Expanding in powers of x and comparing (29) with (32)

allows one to calculate the coefficients ai

a numerically robust way:

n,krecursively in

a0

n,k=

n−1

?

j=0

aj

n−1,k−1

j + 1

, ai

n,k=

n−1

?

j=0

aj

n−1,k− aj

n−1,k−1

j + 1

.

(33)

Once the coefficients ai

evaluate the probability densities fn(u) and the corre-

sponding cumulative distribution functions.

The probability density (28) takes its maximum value

for u = n/2. Due to the central limit theorem the fall-off

behavior is Gaussian as long as u stays sufficiently close

to n/2. In the tails, for u → 0 or u → n, the fall-off is

much faster than Gaussian, namely an exponential of an

exponential as follows from extreme value statistics [27].

n,kare available, one can easily

E. Combination of two weights

In the following the weights with superscript j, wj

correspond to two distinct reference configurations vj,

(j = 1,2), and dj is the distance from the configura-

tion at hand to the configuration vj.

that multi-overlap simulations with respect to the two

reference configurations have been carried out and that

the weights, w1

q(d2), have been determined

so that they sample their distance distributions approxi-

mately uniformly.

We want to construct combined weights w12

which lead to a RW process between the configurations

v1and v2. Our choice is

q(dj),

Let us assume

q(d1) and w2

q(d1,d2)

w12

q(d1,d2) =

?w1

cjw2

q(d1), for d1< d2,

q(d2), for d1≥ d2.

(34)

The constant cj, with j either 1 or 2, is introduced to

allow for smooth transitions from d1 < d2 to d′

1≥ d′

2

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FIG. 1. Reference configuration 1. Only backbone struc-

ture is shown. The N-terminus is on the left-hand side and

the C-terminus on the right-hand side. The dotted lines stand

for hydrogen bonds. The figure was created with RasMol [23]

and vice versa.

of either run 1 (or run 2), which are the (one refer-

ence configuration) simulations leading to the weights

w1

q(d2)). The constant c1 is found from run

1 by scanning the time series for configuration for which

d1 ≥ d2 holds and which have a one-update transition

(d1,d2) → (d′

tions k we determine the constant c1so that

We determine cj from the analysis

q(d1) (or w2

1,d′

2) with d′

1< d′

2. From these configura-

?

k

w1

q[d1(k)] = c1

?

k

w2

q[d2(k)] (35)

holds. Similarly, run 2 may be used to get c2. It turns out

that the normalized weights almost agree in the transi-

tion region and, therefore, the patching (34) works. The

dependence of the constant on the run used for its de-

termination is small, and it appears not worthwhile to

explore more sophisticated methods.

It is straightforward to implement the Metropolis up-

dating with respect to the weights (34). For the transi-

tion

(d1,d2) → (d′

1,d′

2), (36)

one has to distinguish four more cases:

1.d1< d2 and d′

2.d1< d2 and d′

3.d1≥ d2 and d′

4.d1≥ d2 and d′

1< d′

1≥ d′

1< d′

1≥ d′

2,

2,

2,

2.

(37)

(38)

(39)

(40)

FIG. 2. Reference configuration 2. See the caption of figure

1 for details.

Alternatively to the approach outlined, one may com-

bine d1 and d2 into a new variable θd for which the

weights are then calculated as in the one-dimensional

case. A suitable choice along this line is

θd=2

πarctan

?d1

d2

?

. (41)

III. MET-ENKEPHALIN SIMULATIONS

In the following we introduce two reference configura-

tions. Subsequently, we discuss first the results for sim-

ulations with one reference configuration and then those

involving both reference configurations.

A. The reference configurations

Met-enkephalin has the amino-acid sequence Tyr-Gly-

Gly-Phe-Met. We fix the peptide-bond dihedral angles ω

to 180◦, which implies that the total number of variable

dihedral angles is n = 19. We neglect the solvent effects

as in previous works. The low-energy configurations of

Met-enkephalin in the gas phase have been classified into

severalgroups of similar structures [19,22]. Two reference

configurations, called configuration 1 and configuration 2,

are used in the following and depicted in figures 1 and 2,

respectively. Configuration 1 has a β-turn structure with

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