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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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Page 5

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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TABLE I. Met-enkephalin reference configurations. The

columns GEMmin and Bmin correspond to configuration 1 and

configuration 2, respectively.

Residue

1

1

1

1

2

2

3

3

4

4

4

4

5

5

5

5

5

5

5

Angle

χ1

χ2

χ6

φ

ψ

φ

ψ

φ

ψ

χ1

χ2

φ

ψ

χ1

χ2

χ3

χ4

φ

ψt

GEM [21]

−179.9

−111.3

+145.3

− 86.4

+153.7

−161.6

+ 71.2

+ 64.1

− 93.5

+179.8

+ 80.0

− 81.7

− 29.2

− 65.1

−179.2

−179.3

− 60.0

− 80.8

+143.9

GEMmin

−179.8

−111.4

+145.3

− 86.3

+153.7

−161.5

+ 71.1

+ 64.1

− 93.5

+179.8

+ 80.0

− 81.7

− 29.2

− 65.1

−179.2

−179.3

− 59.9

− 80.7

+143.5

B [19]

−179

− 95

+169

+111

+157

− 71

+ 78

159

− 37

+ 59

+ 87

−154

+151

− 68

+177

−179

+ 60

−140

− 29

Bmin

+179.4

− 94.3

−179.9

+ 55.7

+157.6

− 70.7

+ 78.0

+156.5

− 35.7

+ 55.3

+ 86.8

−155.7

+151.6

− 69.4

−176.3

−179.7

+ 59.9

−140.0

− 30.6

hydrogen bonds between Gly-2 and Met-5, and configu-

ration 2 a β-turn with a hydrogen bond between Tyr-1

and Phe-4 [22].

For our present work the two reference configurations

were obtained by minimizing the GEM and the second

lowest energy state of previous literature with respect

to the ECEPP/2 energy function. The minimization was

performed with the SMMP minimizer [16] and by quench-

ing. Both methods gave identical final energies. In table I

we list the variable dihedral angles of the configurations

before and after this minimization. The initial dihedral

angles for the GEM are taken from table 1 of Ref. [21]

and the initial dihedral angles for the second lowest en-

ergy state B are from table 1 of Ref. [19]. In table I we

give the angles in degrees, while for the MC simulations

radians were used as in equations (1) and (2) for the

overlap. Our labeling of the residues follows the SMMP

convention and deviates from those of Refs. [21,19].

The distance between the two minimized configura-

tions is d = 6.62 (q = 0.652) and their energies are given

in table II.

TABLE II. Energies (in kcal/mol) of the Met-enkephalin

reference configurations 1 and 2.

Total

−10.72

−8.42

Coulomb

+21.41

+22.59

Lennard-Jones

−27.10

−26.38

H-Bond

−6.21

−4.85

Torsion

+1.19

+0.23

1

2

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0123456789

ln[wq(d1)]

d1

FIG. 3. Weight estimates from simulations with reference

configuration 1. From up to down the weight functions cor-

respond to the following temperatures: 230K, 300K, 400K,

700K, 2,000K, 10,000K, 100,000K and infinity (β = 0).

B. Simulations with one reference configuration

Each of our multi-overlap simulations at fixed temper-

ature relies on a statistics of 16,777,216 sweeps for which

data are recorded in a time series of 524,288 events, i.e.,

with a stepsize of 32 sweeps. We started most of our sim-

ulations with the GEM configuration, but some random

starts were also performed and no noticeable differences

were encountered.

Starting with the analytical result (28), valid at β = 0,

the weights are calculated by increasing β (i.e., decreas-

ing the temperature) between simulations slowly so that

the RW of each simulation still covers the desired overlap

range when using the weight estimates from the previous

temperature. Discretization errors due to histograming

can be severe and instead of weights which are piecewise

constant within each one histogram interval, we used the

interpolation of Ref. [6]:

lnw(d) = (1 − α)lnw(di) + αlnw(di+1) , for di≤ d < di+1,

(42)

where

α =

d − di

di+1− di

.(43)

Figure 3 depicts the thus obtained weight function es-

timates from simulations with reference configuration 1.

After five simulations we arrive at the physical tempera-

ture T = 300K. The same iteration works with reference

configuration 2.

For the values dmin = 0.025n and dmax = 0.495n,

where n = 19 is the number of angels in (2), we list

in table III the number of RWCs (7) achieved at each

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Page 7

temperature. We also list the CPU time ratios for the

1-step versus the 2-step updating procedures, which we

discussed in the previous section. Especially at high tem-

peratures, which are needed in our approach, the 2-step

updating turns out to be more efficient than the 1-step

updating and all of our production runs were done with

it.

TABLE III. Number of random walk cycles in the simula-

tions with our two reference configurations. The last column

lists the CPU time ratios for 1-step versus 2-step updating.

T

Configuration 1 Configuration 2 1-step/2-step

9,4589,514

3,1223,149

2,8932,741

2,1692,227

1,342 1,693

462 610

46

100,000K

10,000K

2,000K

700K

400K

300K

230K

3.0

1.8

1.6

1.5

1.3

1.2

1.2 41

We next rely on the peaked distribution function [26]

to visualize some of the data kept in the time series of

our simulations. The peaked distribution function of a

probability density f(x) is defined by

Fpeaked(x) =

?F(x) for x ≤ 0.5,

1 − F(x) for x > 0.5,

(44)

where

F(x) =

?x

−∞

dx′f(x′) (45)

is the usual cumulative distribution function (see for in-

stance [28]).

To visualize how the canonical energy distribution

moves when we lower the temperature, we plot in fig-

ure 4 the peaked energy distributions as obtained by re-

weighting some of the multi-overlap simulations of fig-

ure 3 to the canonical ensemble of their simulation tem-

perature. Due to the re-weighting the distributions look

precisely as one expects for energies from canonical MC

simulations. In contrast to conventional canonical simu-

lations, the raw data feature a considerably larger num-

ber of events at low energies. This is illustrated in fig-

ure 5, where we plot the 300K and 400K peaked dis-

tribution functions of figure 4 together with their raw

multi-overlap peaked distributions

In figure 6 we give an example of the probability den-

sity of the distance. For the 400K simulation with refer-

ence configuration 1 we plot the probability density of d1

as obtained from the multi-overlap simulation together

with its canonically re-weighted probability density. The

simulation itself is run with the multi-overlap weights

from the 700K simulations and the multi-overlap his-

togram shown is re-weighted to the multi-overlap 400K

weights. As expected, we have a flat distribution between

0

0.1

0.2

0.3

0.4

0.5

-10 -505 10152025 30

Fp

E (kcal/mol)

T=230K

T=300K

T=400K

T=700K

FIG. 4. Canonical, peaked energy distributions obtained

by re-weighting multi-overlap simulations. From left to right

the temperatures used are: 230K, 300K, 400K, and 700K.

0 and n/2 = 9.5. Moreover, there is a good coverage of

configurations close to the GEM, which are highly sup-

pressed in the 400K canonical ensemble. The maximum

ratio of the multi-overlap density divided by the canoni-

cal density is 6 × 1016in this plot.

For the same simulation figure 7 depicts separately the

peaked distribution function of the forward and back-

ward RWCs (7). A considerable asymmetry is noticeable

and it turns out that the weights of the 1/k ensemble [25]

lead to more RWCs than the flat distribution of figure 6.

In connection with our simulations this is a lucky cir-

cumstance, because the 1/k distribution of weights is in

essence the distribution at a somewhat higher temper-

ature than that of the simulation. This increases the

flexibility when estimating good weights at a lower tem-

perature from the already existing simulation results at

a higher temperature.

For multi-overlap simulations the re-weighting towards

low temperatures can work much better than for canon-

ical simulations. This is due to the fact that the low-

energy configurations close to low-energy reference con-

figuration are already in the ensemble.

trated in figure 8, where we re-weight the data from a

multi-overlap simulation with reference configuration 1 at

T = 300K and compare with a conventional multicanon-

ical simulation based on the SMMP package [16]. The

specific heat CV and the derivative of the overlap with

respect to the temperature are shown. From 200K to

400K the deviations of the results are of the order of the

statistical errors, which are not shown for clarity of the

figure. Below 200K deviations of the re-weighted overlap

simulation from the correct behavior become visible, first

in

dTthen in CV. Such deviations are expected as the

low-energy attractor does not lead to a uniform cover-

age of all low-energy states. The successful re-weighting

This is illus-

dq1

7

Page 8

0

0.1

0.2

0.3

0.4

0.5

-10-50

E (kcal/mol)

5 10 15

Fp

T=300K

T=400K

FIG. 5. Peaked multi-overlap (left-shifted) and canonical

energy distributions at T = 300K and T = 400K.

from high simulation temperatures to lower temperatures

is an improvement, because the Metropolis dynamics at

high temperatures is faster. But the re-weighting of a

multi-overlap simulation to a lower temperature will fail

at some point, because the reference configuration in-

troduces a bias towards particular low-energy configura-

tions.

The temperature at which CV and −dq1

ues correspond to the coil-globule transition temperature

Tθand the folding temperature Tf[24]. From figure 8 we

read off the following approximate values:

dTtake peak val-

Tθ= 280K and Tf= 245K .(46)

C. Simulations with two reference configurations

At 300K we combine the weights from the runs with

reference configurations 1 and 2 to one weight function

according to our equation (34). We record now three

different RWCs:

1. With respect to reference configuration 1 from dmin

to dmaxand back, found 315 times.

2. With respect to reference configuration 2 from dmin

to dmaxand back, found 545 times.

3. From dmin of reference configuration 1 to dmin

of reference configuration 2 and back, found 196

times.

In figure 9 we show the probability densities of this sim-

ulation with respect to the distances from our reference

configurations. They are no longer flat, but a satisfactory

coverage in the variables d1and d2is still achieved. Note

that both probability densities have peaks at d = 6.62,

0

0.005

0.01

0.015

0.02

0.025

02468 1012

Probability Density

d1

FIG. 6.

multi-overlap simulation at T = 400K (flat) and its canoni-

cally re-weighted probability density (peaked).

Probability density of the distance from a

which is the distance between configurations 1 and 2.

This implies that both reference configurations have been

visited with high probability.

D. Physics results

We would like to analyze the transitions between our

two reference configurations in some detail. For this pur-

pose we use the rms distance, which is defined by

drms= min

?

?

?

?1

N

N

?

i=1

(? xi− ? xj

i)2

, (47)

where N is the number of atoms, {? xj

nates of the reference configuration j, and the minimiza-

tion is over the translations and rotations of the coordi-

nates of the configuration {? xi}.

The distance (2) and the rms distance (47) are quite

distinct. The reason is that a change of a single dihedral

angle in the central parts of the molecule can cause a large

deviation in the rms distance. Although the two config-

urations are then close-by from the point of view of the

MC algorithm, physically they are rather far apart, as the

similarity of the three-dimensional structures is governed

by the rms distance. Therefore, the rms distance distri-

bution deviates considerably from the dihedral distance

distribution. We illustrate this by plotting in figure 10

the rms probability density of the 400K simulation for

which the dihedral distance probability density is shown

in figure 6.

We now analyze the free-energylandscape [29] from the

results of our simulation with combined weights at 300K

in some detail. We study the landscape with respect to

i} are the coordi-

8

Page 9

0

0.1

0.2

0.3

0.4

0.5

0 1000020000

MC sweeps

30000 40000

Fp

d->dmax

d->dmin

FIG. 7.

(d → dmax) and backward (d → dmin) parts of the random

walk cycles from a multi-overlap simulation at T = 400K.

Peaked distribution functions for the forward

some reaction coordinates (and hence it should be called

the potential of mean force). In order to study the tran-

sition states between reference configurations 1 and 2,

we first plotted the free-energy landscape with respect

to the distances d1and d2. However, we did not observe

any transition saddle point. A satisfactory analysis of the

saddle point becomes possible when the rms distance (in-

stead of the dihedral distance) is used. Figure 11 shows

contour lines of the free energy re-weighted to T = 250

K, which is close to the folding temperature (46). Here,

the free energy F(rms1,rms2) is defined by

F(rms1,rms2) = −kBT lnP(rms1,rms2) , (48)

where rms1 and rms2 are the rms distances defined in

(47) from the reference configuration 1 and the reference

configuration 2, respectively, and P(rms1,rms2) is the

(reweighted) probability at T = 250 K to find the peptide

with values rms1,rms2. The probability was calculated

from the two-dimensional histogram of bin size 0.06˚ A×

0.06˚ A. The contour lines were plotted every 2kBT (=

0.99 kcal/mol for T = 250 K).

Note that the reference configurations 1 and 2, which

are respectively located at (rms1,rms2) = (0,4.95) and

(4.95,0), are not local minima in free energy at the fi-

nite temperature (T = 250 K) because of the entropy

contributions. The corresponding local-minimum states

at A1 and B1 still have the characteristics of the refer-

ence configurations in that they have backbone hydrogen

bonds between Gly-2 and Met-5 and between Tyr-1 and

Phe-4, respectively. We remark that we observe in fig-

ure 11 another well-defined local minimum state around

(rms1,rms2) = (4.7,3.5). This state can also be consid-

ered to correspond to configuration 2 because we again

observe the backbone hydrogen bond between Tyr-1 and

Phe-4. The side-chain structures are, however, more de-

3

3.5

4

4.5

5

5.5

6

150200 250300

T

350 400 450

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

CV

dq1/dT

MUCA

MUOV

FIG. 8. Left-hand-side ordinate: Specific heat re-weighted

from a multicanonical (MUCA) and from a 300K multi-

-overlap (MUOV) simulation with reference configuration 1.

Right-hand-side ordinate:

ulations, where q1 is the overlap with reference configura-

tion 1.

dq1

dTre-weighted from the same sim-

viated from configuration 2 than B1, resulting in a larger

value of rms2.

The transition state C in figure 11 should have inter-

mediate structure between configurations 1 and 2. In

figure 12 we show a typical backbone structure of this

transition state. We see the backbone hydrogen bond

between Gly-2 and Phe-4. This is precisely the expected

intermediate structure between configurations 1 and 2,

because going from configuration 1 to configuration 2 we

can follow the backbone hydrogen-bond rearrangements:

The hydrogen bond between Gly-2 and Met-5 of config-

uration 1 is broken, Gly-2 forms a hydrogen bond with

Phe-4 (the transition state), this new hydrogen bond is

broken, and finally Phe-4 forms a hydrogen bond with

Tyr-1 (configuration 2).

It is interesting to see in figure 11 that there is only one

saddle point in the free-energy landscape that connects

configurations 1 and 2. Hence, the transition between

configurations 1 and 2 always passes through the state

C.

In Ref. [22] the low-energy conformations of Met-

enkephalin were studied in detail and they were classi-

fied into several groups of similar structures based on the

pattern of backbone hydorgen bonds. It was found there

that below T = 300 K there are two dominant groups,

which correspond to configurations 1 and 2 in the present

article. Although much less conspicuous, the third most

populated structure is indeed the group that is identified

to be the transition state in the present work.

In figures 13 and 14 we show the internal energy land-

scape and the entropy landscape at T = 250 K, respec-

tively. Here, the internal energy U is defined by the

(reweighted) average ECEPP/2 potential energy:

9

Page 10

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0246

d

8 1012

Probability Density

d1

d2

FIG. 9. Combined weight simulation at T = 300K: Proba-

bility densities with respect to the distances d1 and d2.

U(rms1,rms2) =< E(rms1,rms2) > .(49)

Here, the average was again calculated from the two-

dimensional histogram of bin size 0.06˚ A× 0.06˚ A. The

entropy S was then calculated by

S(rms1,rms2) =1

T[U(rms1,rms2) − F(rms1,rms2)] .

(50)

The landscape in figure 14 is actually −TS(rms1,rms2).

Both internal energy and entropy landscapes are more

rugged than free energy landscape (we observe much

more number of contour lines in figures 13 and 14 than

in figure 11). The internal energy has clear local min-

ima at the points (rms1,rms2) = (0,4.95) and (4.95,0),

which respectively correspond to configurations 1 and 2,

while the entropy landscape has local maxima at these

points. These two terms tend to cancel each other, and

the free energy landscape is smoothed out.

In table IV we list the numerical values of the free

energy, internal energy, and entropy multiplied by tem-

perature at the two local-minimum states (A1and B1in

figure 11) and the transition state (C in figure 11). The

internal energy is just the average of the ECEPP/2 po-

tential energy (without any shift of zero point). The free

energy was normalized so that the value at A1 is zero.

The values at the coordinates of reference configurations

1 and 2, which are respectively referred to as A0and B0

in the table, are also listed.

Among the five points, A0 and B0 are unfavored in

free energy mainly due to the large entropy effects, al-

though they are energetically most favored. This means

that at this temperature the exact conformations of the

reference configurations 1 and 2 are not populated much.

The relevant states are rather A1, B1, and C. The state

A1can be considered to be “deformed” configuration 1,

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0123456

Probability Density

rms1

Multi-overlap

Canonical

FIG. 10. Probability density of the rms distance from the

multi-overlap simulation at T = 400K of figure 6, and its

canonically re-weighted probability density. The abscissa is

the rms distance (˚ A) in Eq. (47) from the reference configu-

ration 1.

TABLE IV. Free energy, internal energy, entropy multi-

plied by temperature at T = 250 K (all in kcal/mol) at the two

local-minimum states (A1 and B1) and the transition state

(C) in figure 11. The values at the coordinates of reference

configurations 1 and 2, which are respectively referred to as

A0 and B0, are also listed. The rms distances are in˚ A.

Coordinate (rms1,rms2)

A1 (1.23, 4.83)

B1 (4.17, 2.43)

C (3.09, 4.05)

A0 (0.03, 4.95)

B0 (4.95, 0.03)

F

0

1.0

2.2

15

20

U

−TS

5.4

4.5

3.0

26

28

−5.4

−3.5

−0.8

−10.5

−8.1

and B1deformed configuration 2 due to the entropy ef-

fects, whereas C is the transition state between A1and

B1. Among these three points, the free energy F and

the internal energy U are the lowest at A1, while the en-

tropy contribution −TS is the lowest at C. The free en-

ergy difference ∆F, internal energy difference ∆U, and

entropy contribution difference −T∆S are 1.0 kcal/mol,

1.9 kcal/mol, and −0.9 kcal/mol between B1 and A1,

2.2 kcal/mol, 4.6 kcal/mol, and −2.4 kcal/mol between

C and A1, and 1.2 kcal/mol, 2.7 kcal/mol, and −1.5

kcal/mol between C and B1. Hence, the internal energy

contribution and the entropy contribution to free energy

are opposite in sign and the magnitude of the former is

roughly twice as that of the latter at this temperature.

10

Page 11

A1

B1

C

0123456

rms 1

0

1

2

3

4

5

6

rms 2

FIG. 11. Free-energy landscape at T = 250 K with respect

to rms distances (˚ A) from the two reference configurations,

F(rms1,rms2). Contour lines are drawn every 2kBT. The

labels A1 and B1 indicate the positions for the local-minimum

states at T = 250 K that originate from the reference config-

uration 1 and the reference configuration 2, respectively. The

label C stands for the saddle point that corresponds to the

transition state.

IV. SUMMARY AND CONCLUSIONS

We have outlined an approach to perform MC simu-

lations which yield the free-energy distribution between

two reference configurations. The multi-overlap weights

for this purpose were obtained by a novel, iterative pro-

cess. The main point of this iterative process is not that

it is supposed to be more efficient than the recursion that

was used in the multi-self-overlap simulations of Ref. [12],

but that it is an entirely independent approach, which

starts from an analytically controlled limit. Recursions

like the one used in [12] are not “foolproof”. For in-

stance, while most of the spin glass replica in Ref. [12]

were well-behaved, a few did not complete their recur-

sion after more than an entire year of single processor

CPU time. Similar situations could be encountered in

all-atom simulations of larger peptides, where the normal

multicanonical weight recursion as well as similar multi-

overlap weight recursion could fail. The present method

provides then an alternative, approaching the physical

region from a different limit.

Noticeable, our multi-overlap approach is well-suited

to be combined with a recently introduced, biased

Metropolis sampling [30]. Namely, the required config-

FIG. 12. The transition state between reference configura-

tions 1 and 2. See the caption of figure 1 for details.

urations at higher temperatures are as well necessary for

our particular multi-overlap recursion, so that no extra

simulations are required in this respect.

On the physical side, we have found that entropy ef-

fects are rather important for a small peptide. The ef-

fects of entropy on the folding of real proteins in realistic

solvent have yet to be studied in detail.

We have also performed the analysis of this paper for

Met-enkephalin with variable ω angles and, in particular,

simulated with combined weights at a number of temper-

atures. The results found are quite similar to those re-

ported in this paper. In future work we intend to analyze

the transition between reference configuration for larger

systems of actual interest like β-lactoglobulin.

ACKNOWLEDGMENTS

We are grateful for the financial support from the Joint

Studies Program of the Institute for Molecular Science

(IMS). One of the authors (B.B.) would like to thank

the IMS faculty and staff for their kind hospitality dur-

ing his stay in spring 2002. In part, this work was sup-

ported by grants from the US Department of Energy un-

der contract DE-FG02-97ER40608 (for B.B.), from the

Research Fellowships of the Japan Society for the Promo-

tion of Science for Young Scientists (for H.N.) and from

the Research for the Future Program of the Japan Soci-

ety for the Promotion of Science (JSPS-RFTF98P01101)

(for Y.O.).

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11

Page 12

0123456

rms 1

0

1

2

3

4

5

6

rms 2

FIG. 13. Internal energy landscape at T = 250 K with re-

spect to rms distances (˚ A) from the two reference configura-

tions, U(rms1,rms2). Contour lines are drawn every 2kBT.

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0123456

rms 1

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rms 2

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12