Some physical approaches to protein folding

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Submitted on 1 Jan 1993

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J. Bascle, T. Garel, Henri Orland

To cite this version:

J. Bascle, T. Garel, Henri Orland. Some physical approaches to protein folding. Journal de Physique

I, EDP Sciences, 1993, 3 (2), pp.259-275. �10.1051/jp1:1993128�. �jpa-00246718�

J. Phys. I France 3

(1993)

259-275 FEBRUARY 1993, PAGE 259

Classification Physics AbstTacts

05.90 61.40D _87.10

Some physical approaches to protein folding

J.

Bascle,

T. Garel and H. Orland

Service de Physique

Thdorique(*)

CE-Saclay, 91191 Gif-sur-Yvette Cedex, France

(Received

15 May1992, accepted 5 June

1992)

R4sum4. Le repliement des protdines eat _un probl+me qui _a ^de nombreuses implications biologiques. Dana cet article, _nous pr6sentons, de deux fa&ens difl4rentes, _un point de _vue de physicien. Nous introduisons tout d'abord des mod+les simples de m4canique statistique qui exhibent, h la limite thermodynamique, des transitions de repliement. Ces mod+les peuvent Atre divis6s _en (I) _verres de spin

(6ventuellement

k la

Mattis),

al l'on peut ^{chercher des} corr41ations entre [es interactions intrachaine _et la structure replide,

(it)

_verres, al l'on _met l'accent _sur la

comp4tition g40mdtrique entre l'ordre local uni- _au bi-dimensionnel

(qui

modble [es structures

en hdlices _{a au en} feuillets

fl),

et la contrainte

globale

de compacitd. Ces deux types de modbles sent trap simples _pour l'dtude de vraies prot4ines, mars its devraient s'appliquer dons le domaine de la transition vitreuse, des polymbres collaps4s,... La deuxibme voie d'dtude _{eat une} m4thode

Monte-Carlo, al

on fait croitre la protdine atome par atome

(au

rdsidu _par

rdsidu),

I l'aide d'une forme donnde de

l'dnergie

totale de la protdine

(CHARMM,...).

Cette m4thode pent ^dtre alors compar4e _aux autres m4thodes numdriques; _{nous comparons} ainsi _nos rdsultats _avec des calculs de dynamique moldculaire _pour le _cas des poly-alanines. Cette double approche eat _une bonne illustration des difficultds _que l'on rencontre dons le probl+me du repfiement des protdines (nombreux 4tats m4tastables,...).

Abstract. To understand how _a protein folds is _a problem which has important

biological

implications. In this article, _we would like _to present a physics-oriented point of view, which is twofold. First of all, _we introduce simple statistical mechanics models which display, in the thermodynamic limit, folding ^and related transitions. These models _can be divided into (I) crude spin glass-like models

(with

their Mattis

analogs),

where _{one may} look for possible correlations

between the chain self-interactions and the folded structure, (it) glass-like models, ^where one

emphasizes the geometrical competition between _{one- or} two-dimensional local order

(mimicking

a helix _or fl sheet

structures),

and the requirement of global compactness. Both models

are too

simple to predict the spatial organization of _a realistic protein, ^but _are ^{useful for} the physicist

and should have _some feedback in other glassy _systems

(glasses,

collapsed

polymers,...).

These remarks lead

us to the second physical approach, namely _{a new} Monte-Carlo method, where _one grows the protein atom-by-atom

(or residue-by-residue),

using _a standard form

(CHARMM,...)

for the total energy. A detailed comparison with other Monte-Carlo schemes, _or M61ecular Dynamics calculations, is then possible; _we will sketch such

a comparison for poly-alanines.

Our twofold approach illustrates _some of the difficulties _one encounters in the protein

folding

problem, in particular those associated with the existence of

a large ^number of metastable states.

(*)

Laboratoire de la Direction des Sciences de la Matibre du Commissariat I l'Energie Atomique

1 Introduction.

Proteins _are

weakly

branched

polymers,

built _out of twenty

species

of _monomers

(aminoacids).

They

have the property ^of

folding

into _an

(almost) unique

compact ^native structure, which is the

biological interesting object [I].

The compactness ^is

largely

due to the existence of the

hydrophobic

aminoacid

residues,

since these

biological objects

_are

usually designed

to work

in _water. Both the compactness ^and the chemical

heterogeneity

of _a

given protein

tend to

slow down

dynamical

_processes, and the

question periodically

^arises _as to whether the

protein folding problem

in under

thermodynamic

_or kinetic control. This

question

is not unfamiliar in

the

physics

of

glassy

systems where the _same

problem

of _{a very}

rugged phase

_space is present.

In

physical

terms, the frustration in _a

protein

_can be

naively

^described in _two different _waysi

(I)

the _energy of _a

protein

is the _sum of bonded

(geometrical)

and non-bonded

(Coulomb,

Van Der

Waals)

terms, ^which cannot be

simultaneously

satisfied.

(ii) experimentally (Cristallography, NMR),

_a folded

protein

has _a local order due to

hydrogen-bonds.

This order

is, roughly speaking,

of one-dimensional

(a helix)

_or twc-dimensional

(fl sheet)

nature and is therefore

incompatible

with the

requirement

of

global

compactness.

In this

ch8pter,

_we shall follow the

thermodynamics approach

to

folding; simple

statistical mechanics models will be studied for

points (I)

and

(it).

For the

former,

_one is

naturally

lead to draw _a

parallel

with the

spin glass problem (the quenched

disorder

being

^linked to the

primary structure),

whereas the latter is _more akin _to

glasses.

Both of these

approaches

have

interesting

outputs. ^The

spin glass

_{[2] case} and its Mattis

analogs

_[3] suggest a connection with the

physics

_[4] of neural

networks, Hopfield model,..,

for the

major

^unsolved

problem

of the

coding

of the

tertiary

structure in the

primary

structure. The

"glassy glass"

_{case [5]}

points

towards

interesting

differences between helices and

sheets,

and revives

Flory-like

models of

polymer melting,

_as well _as the Gibbs-Dimarzio

theory

of the

glass

transition

[6,7].

On a more realistic

level,

_we also wish to benefit from the

biologists' experience

with their

complicated

systems. It is therefore _necessary to go

beyond

the above

qualitative

picture and

study

well-defined entities. We have therefore devised _{a new} Monte-Carlo

(MC)

method to generate a Boltzmannian ensemble of

configurations

of _a

protein.

This method _uses

an

empirical

form for the total _energy of the

protein,

and _may be

loosely

described _{as an} atom-

by-atom growth

of the

protein,

in marked contrast to other MC methods [8] or to Molecular

Dynamics

_[9]

(MD)

calculations. This

growth procedure

_was introduced [10] ^to try to

efficiently explore

the

rugged landscape

of _a

protein phase

_space and _may be

coupled

to _more traditional

techniques (simulated annealing

_[11], minimization

procedures,...).

We have tested [12] ^the method _on

peptides

with _a small number N of atoms

(alanine dipeptide (N

₌

22),

_penta-

alanine

(N

₌

53)), by comparing

the energy minima with those obtained

by

MD simulations

(CHARMM).

A short review of the _same

comparison

_[13a] with

hepta-alanine (N

₌

73)

is

given below, together

with

preliminary

results [13b] on

twenty-alanine (N

₌

203).

At this

point

_a caveat _seems in order: all the

methods, including

_{ours, are} faced with the

problem

of the

solvent,

and _use at best _an effective _energy function

taking

in _a crude _way the effect

of the _water molecules

((or

instance in the

collapsed regime,

a one-hundred residue

protein

has

something

^{like half} of its atoms _on the

surface).

Our simulations _are

usually

done with _a dielectric constant

equal

to

unity (vacuum-type calculations).

The

layout

of the _paper is _as follows. Section 2

briefly

deals with the

spin glass-like approach

to the

folding

transition, as well _as the related Mattis models

(coding

I la

Hopfield,.. ).

The

"glassy glass"

_case is studied in section 3, ^{where the link} with the

Flory-Gibbs-Dimarzio theory

of the

glass

transition is discussed. We

emphasize,

in this context, the existence of _a disorder

point.

Section 4 describes the _new AIC

growth

method and its

application

to

poly-alanines

(Sect. 5).

N°2 SO&fE PHYSICAL APPROACIIES TO PROTEIN FOLDING 261

2.

Spin glasses

^and

folding

transitions.

We model _a

protein

_{as a} chain of N links

(residues),

,vhere link I

(at r;)

and link

j (at rj)

interact

through

_a

potential

_u;j

(r;

_rj

),

which

depends

_on the chemical _nature of links _I and

j. Physically,

_{ujj is}

expected

to be

relatively short-ranged (screened

Coulomb _{or van} der lvaals

interactions,.. ).

^{We take for}

simplicity

u;j

(ri

_{rj = u;j} b

(r;

_rj

(I)

Two types ^{of model} can be studied [2, 3].

(I)

^the

spin glass

^model: the interactions

(uj)

_are taken _as

independent

random variables

distributed,

for

instance, according

to _a Gaussian distribution

~ ~~"

~#~~~ _12~2

~~'J

°~~) (2)

In

equation (2),

_uo denotes the excluded ;olume effect

(in appropriate units).

A

"biological"

interpretation ^{of this} approach ^is the

follo,ving:

the interactions

(u)

bet,veen the _same

couples

of

residues,

but at different

places

in the

primary

_{sequcnce, are}

totally

uncorrelated because of their different environment.

(One

_may also link this

approach

to the

travelling

salesman

optimization problem [14]).

(ii)

the

separable

model: the interactions

(u;j)

_are taken _{as a sum} of AI

separable

terms

AI

~,,

~j

~ ~P~P

(3)

11 P I _j

p=i

,vhere the

(f[)

_are

taken,

for instance, _as

indepciident

mndoni,<ariables with Gaussian distri- butioii.

Apart

from thc excluded iolume

efl'ect,

thc

"charges" (f[;

_p

= I,...,

AI)

_{can represent}

the Coulomb

charge,

thc

liydropliolJicity,

^the

I;clix-forming

_or

breaking tcndency,..

The "bi-

ological" interpretation

is

opposite

to the

prcvious

_one:

here,

each residue is dcfiued

by

M

independent "ch;irges",

_or chJracters, ^ii~hicli

dcpends onl»

_on its chemical _nature and not _on its

position along

^the

primary

_sequence.

In the continuous

limit,

the

partition

^{functi~n of these models reads} [16]

Z ₌

/ _{Dr(s)exp (-}

^~

_/~

_ds ^{~~ ~}

^~ _{/~ /~}

_{ds ds'u}

_{(s, s')}

₆

_(I(s)

r

(s') )lx

2

o ds 2

o o

~ ^{S S S ~~~}

~~~~ 6 ^~~

~~' ~~" ~_{~~~~~ ~}~~'~~ ~~~~'~

~~"~~~

The last _term is included to avoid _a total

collapse

of the chain, its

usual,

the paraineters ⁱⁿ

equation (4)

_are the space dimension

d,

the in;crsc temperature

fl

_{= ~,,} and S

= Na~

(n.here

a is the _common

length

of the

links).

Introducing replicas

_[4] to

pcrform quenched averaging

ovcr the disordered intcractions

(v(s,s')),

_{we get} the

following

results

[2,3]

: there _are three

phases, namely,

_a

high

tem-

perature ^coil state, an intermediate teniperature

collapsed phase

with _a

macroscopic

entropy

(similar

to a

polymer

below tile 8

point),

and

finally

_a low temperature

collapsed

frozen

phase.

Detailing

the above

models,

_,ve have:

(I)

^the

^spin glass

model: the low temperature

phase

is _a Potts

glass

with _p

- oo states

[16],

^{at least for}

high enough

dimensions. The

(mean field)

order parameter ^{of the}

freezing

transition is:

Qnfl (r,

_r

')

=

2flv /~

^ds

(b (r ra(S))

b

(r

^'

rfl(S))) (5)

where I < _{a <}

fl

< n

(and

_{n -}

0),

and (. ^denotes a thermal average with respect to the

replicated

Hamiltonian of

equation (4). Alternatively,

the

freezing

transition _can be studied

by

the

overlaps

of two

(real) copies

of the system [17]. ^{As in}

Ising spin glasses

_{[4], one may argue}

that there _are few dominant states in the system, which could be

interpreted

in terms of _a few dominant folded structures. Numerical calculations

along

^these

lines, including dynamics,

have been

recently reported

_[18]. Note

that, by construction,

these

"protein

models" _possess ultrametric

properties

[19]. ^For a more realistic _{case, see} section 4.

To conclude _on this type of

approach,

it is

interesting

to

point

out that _a variational function introduced

by

Shakhnovich and Gutin [20] ^{in the} context of

proteins,

has been used in other disordered solid _state situations

[21].

(ii)

the

separable

model: when _one

performs

the

quenched

_{average over} the

(ff )

in

equation (4),

_some mean field order paramters _appear

naturally

in the system ^such as

mp,a

jr)

=

/~

^ds

_fp

₁₆

(r ra(S))) (6)

In

equation (6),

_a is _an

unimportant replica

index

(to

be omitted from _now

on); _(.

and denote

respectively

thermal and disorder averages. These order parameters ^{h la}

Hopfield

[22] are a measure of the correlation between the chemical nature of _a link

(characterized by

(fp Is))

^{and its}

position

_r in _space. There is _a Mattis-like

freezing phase

transition where _some

(mp (r)

_{p =} 1, 2,..

,

Mo)

condense. For these MO

characters,

the

primary

_sequence codes for the

spatial

structure of the chain. If there is

only

_one

"charge"

_or character

(e.g. hydrophc- bicity),

the

folding

transition will translate into _a

spatial separation

between

hydrophobic

and

hydropholic

links. In

general,

_a Mattis-like transition

implies

_a

single

dominant

spatial

structure, with

a

large

number of metastable states

[22(b)].

Note that when M

increases,

at fixed N, we expect _a ^smooth _crossover from the

separable

to the

spin glass

model. From

simple qualitative

arguments [23], ^it can be inferred that in real

proteins,

one should have M

= 8 relevant

(and independent)

characters for each residue. Thus for _{a =}

fl

small

(I.e, long chains),

_one deals with the

separable

_case, whereas for _a

larger (short chains),

the

glassy

model is _more

appropriate.

The critical _a is of order [22, 23] a~ ct.I

(which gives,

in this

model,

_a critical

length

of N~ _ct

80).

Similar

coding

schemes

using

Protein Data Banks have been studied

by Wolynes

and coworkers [24].

3. Glasses and

folding

transitions.

3.I THE MODEL. The

energetical

frustration described above is note

quite satisfactory,

since there is _no real disorder in

proteins.

We will

see in section 4 that _a

commonly

used form of the total energy of _a

protein

is the _sum of

a bonded

(geometric)

part and of _{a non-} bonded

(Coulomb,

Van der

Waals)

_part. In

particular,

the Coulomb part ^is

responsible,

in this

formalism,

of the formation of

hydrogen

bonds [25] ^{that tend} to

locally

stabilize _one-

dimensional

(a helix),

or two-dimensional

(fl sheets)

structures. See

figure

I. Since

we know

that the

biologically

active

protein

is compact

[I],

we are

typically

faced with the

problem

N°2 SOME PHYSICAL APPROACHES TO PROTEIN FOLDING 263

of

geometrical frustration,

where local and

global

orders _are

incompatible.

This

approach

is familiar in

glasses

where _one tries _to solve this contradiction

by

_a

mapping

onto _a curved space

[26].

^{We choose here} a

thermodynamic approach

and

model,

_{as an}

example,

the _o helix

case in the

following

_{way: we} consider _a d-dimensional

hypercubic

lattice of N ₌

L~ sites,

with

periodic boundary conditions,

and its associated Hamiltonian

paths.

We recall that _a Hamiltonian

path

visits all sites of the lattice _once and

only

_once. Hamiltonian

paths

have been often used to model

collapsed polymer globules [15]. Following Flory [6a],

_we take each link of the Hamiltonian

path

to represent a helical _turn. Since

hydrogen-bonds

have _a

tendency

to favor

long helices,

^{that is} to

align

the links of _our

model,

_we attribute _an energy

penalty

_e to the

breaking

of _an helix, that is whenever the Hamiltonian

path

makes _a turn

(corner).

This model has attracted _a lot of attention in the

theory

of

polymer melting ii, 27].

For

simplicity,

we consider closed

paths, but,

_as is well known in

polymer theory [16], boundary

conditions

play

_a role

only

in subdominant _terms of the free _energy. The

partition

function of the system,

at inverse temperature

fl

₌

,

reads

z =

~ e-P£N,jl~j

~~~

jl~j

CO

RN

co MN

NH

NH °C

NH DC

NH

(a) (bi

Fig. I. Schematic representation of hydrogen bonds in

(a)

a-helix,

(b) (antiparallel)

p-sheet-

where

(7l)

denotes the ensemble of all Hamiltonian

paths,

and

N~(7l)

denotes the number of

corners present ⁱⁿ

path

7l.

Following

^reference [28], one may rewrite Z _as

f fl$~~

_d~an

(r)

^e~~G

fl~ (£~ )~aJ (r)

+ e~fl~

£~

~~ ~an

(r)

_~a~

(r))

Z = lim

~

(8a)

"-° _n

f fl~_~

d~aa

(r)

e-AG

with

AG

=

jj _[

_~an

_{(r) (Air ,)~~}

_~an

_{(r ')} (8bi

~= m

where _~oa

(r)

is _an n-component

(n

₌

0)

real

field,

defined in each direction _{a =}

I,..., d,

attached to all

points

r of the lattice. The operator

AQ~,

^{is I if} r and r' _are nearest

neighbours

in direction _o and 0

otherwise; (AQ~,)~~

^{denotes its inverse.}

Using

Wick's theorem and

extracting, through

the _n

= 0 trick

[29],

^the contribution of all connected

paths,

it is

easily

shown that

(8a)

^and

(7)

_are

equal.

Note that in the above

description,

_one does not consider the

primary

_{sequence anymore,} in marked contrast to the

approach

of section 2. In the _non

weighted problem (e

₌

0),

the

saddle-point (SP)

method of reference [28]

yields

Zsp(e

₌

0)

₌

(~) ⁽⁹⁾

e

~

where _q

= 2d is the lattice coordination number and _{e ci} 2.71828...

Equation (9)

is in excellent

agreement

^{with numerical}

data,

in marked contrast to the "old"

Flory theory

[6a] ^which

gives

ZF(e

₌

0)

₌

(~ ⁽¹⁰⁾

e

~

3. 2 THE HIGH TEMPERATURE ISOTROPIC APPROACH. We have extended the SP

approach

to the model defined in

equations (8).

We get

§ ~~~

'~ ~~ ~~'~

l Co @(~/~~~i~(~i~)~~'

i'fl

(~)

^~~~~

At

high

temperature, ^{it is natural} to look for _a

homogeneous

and

isotropic solution,

_~OJ

(r)

_{= ~o.}

We break the

O(n)

symmetry

by choosing

_~a in _a

given "direction",

_say 0, and obtain

~a( =

~

(12a)

and

ZSP(61

₌

(M)~ (12bj

with

q(fl)

₌ 2 ₊

2(d _I)e~~~ (12c)

The "old"

Flory theory [6a]

would

yield

similar results with

ZF(e)

₌

~~~~~)~

(13a)

e

where

qF(iii

_" 1+

2(d _I)e~~~ (13bj

Both

approaches

have the

following properties:

(I)

^{there exists} a temperature TG where the

entropy

^{vanishes. This}

remark,

in the framework of the

Flory theory,

is the basis of the Gibbs-Dimarzio

theory

of the

glass

transition

[6b].

(ii)

before _one reaches

TG,

there is _a first order

freezing

^transition at

Tc,

such that

q

(flc)

_{= e}

(14)

N°2 SOME PHYSICAL APPROACHES TO PROTEIN FOLDING 265

o

-J 5

-2 0

0 2 3

I

Fig. 2. Various approximations to the free energy of the glass model of III

as a function of temper- ature. Curve

(I)

is the "old" Flory theory. Curve (2) ^is the low temperature anisotropic saddle point result

(with

the disorder point at TD Ci 2.24

e).

Curve

(3)

is the

high

temperature isotropic saddle point result. In all _cases, the transition _occurs when the free energy vanishes.

The low temperature

phase

is frozen

(Fig. 2),

since it consists of

fully

stretched

paths making

turns at the surface.

Using (12c)

and

(13b),

_we get, ^{for d} = 3

Tc[~p

ci 0.58 _e

(Isa)

for the SP

approach

to model

(8)

^and

Tc[~ ci 1.18 _e

(lsb)

for

Flory's theory.

However,

_as

pointed

out

by Gujrati

and coworkers

iii,

such _a

freezing

transition cannot be

thoroughly

correct, ^{since the free} energy may be shown to be

strictly negative

at low tem-

peratures. ^This

(slight)

correction to the

Flory freezing

scenario _comes from _one dimensional excitations that _are not well treated in _an

isotropic

SP

approach.

3.3 THE LOW TEMPERATURE ANISOTROPIC APPROACH.

Considering

the above _men-

tioned criticism of the

isotropic

SP

approach,

_we have considered [30] an

anisotropic approach

to the model described in

(8a)

and

(8b):

_we treat

exactly

_one direction of the

lattice,

_say

I,

and treat the

(d I) remaining

directions in _{a mean} field

(saddle-point) approach.

Using

the fact that the denominator of

equation (8a)

_goes to one when _n goes ^to zero, we

rewrite

(8a)

as

Z _# llDl

f ^jj

_d§2a

(~) ~~~~ fl (~ )~'? ^(~)

^{~ ~~~~}

ll

~~ ~~~ "

~~j

~~~~

n-0 n

~_~ r n o<fl

which _we

approximate by

Zi _ci lim

/

_d~ai

_(r) e'~ie~ +(~~~)~

^~

fl

(~

^{~a~~(r) + A ~ai}

^(r)

^{~l + C ~l}

~l(17)

n-0 _n 2

~

~~~~~

A~ ₌

~

_~gi

_{(r) (A)r ,)}

_~~1°1

(r') _(~~)

~

~,~'

and

A ₌

(d I)e~~~ (19)

and

C =

~~

~

~~

(l

₊

(d 2)e~~~ (20)

In

(17),

_{~l is} the

(mean-field)

value of _{~an, a}

#

I.

Integrating exactly (17) yields

_a free _energy per site

fl

ii ₌ ^{~~ ~~}~ ^~

Log (l

^{+ C~I ~ +}

((l

+ C~I ~)~ ⁴

(C _A~)

_~l

~)

~~j (21)

4 2

Equation (21) exhibits,

at T' _ci 0.68 _e

(d

₌

3),

_a first order transition

(Fig. 2)

^between _a

frozen

phase (cristal)

with ~l = 0 and

a

high

temperature

(liquid) phase

with _~l

#

0. At this

order,

the free energy is _zero in the frozen

phase,

but becomes

negative

if fluctuations

(in

_~l)

are taken into account

(30].

^In _{any case,} the corrections to

Flory's free2ing

picture _are weak.

In the

high

temperature

phase however,

_we have found [30] a disorder

point

of the second kind [31], ^{where the} nature of the correlations

along

direction I

changes.

The disorder

point

TD is

given by

C = A~

(22)

For d

= 3, we get TD t 2.24 _e; in _a

polymeric chain,

such _a disorder

point

is

likely

to have

more severe

dynamic implications

than in usual

spin

systems [32].

3.4 CONCLUSION. We have also considered [33] ^the case of

fl

sheets and found similar conclusions. The

isotropic

SP

approach

should be "better" in this _case since twc-dimensional

long

_range order _may exist _at finite temperature: we

get

_a first order

freezing

transition h la

Flory.

The results of these

geometrically

frustrated models should be relevant for other

thermody-

namic systems, ^such as

glasses [34], polyelectrolytes

in _a bad solvent

[35],

chiral

liquid crystals [36],...

For

instance,

it is rather

tempting,

in the _case of

glasses,

to

identify

the low temperature

phase

_as the

(unreachable) crystal phase,

and _to link the disorder

point

with the

glass

transi- tion. In the _case of

proteins however,

_one deals with finite systems: we thus

cautiously identify

the low temperature

phase

_as the native structure, ^{whereas the}

high

temperature

phase

looks like _a "molten

globule"

[37]. We _now consider _{a more} "realistic"

approach,

which will allow _us to benefit from the

biologists' experience

^with these rather

complicated

systems.

4. The Monte Carlo

growth

^method.

4. I INTRODUCTION. As

previously mentioned,

_one of the main difficulties of the

protein folding problem

is the existence, in

phase

_space, of _a

large

number of local minima. Traditional

single

_move MC methods _are therefore doomed to fail, _as

large

collective motions will be necessary ^to

"untrap"

the chain. One _may

improve

these methods

by using

simulated

annealing

procedures,

_{or any} other minimization scheme. We have chosen to devise _{a new} MC

method,

N°2 SOME PHYSICAL APPROACHES TO PROTEIN FOLDING 267

@

-180 J00 20 20 100 180

qidegi

@

180 -loo lo lo loo 180

Q

(degl

Fig. 3. Ramachandran's plots for the third residue of _an hepta-alanine chain

(a)

MC results,

(b)

MD results.

where _{one grows}

an ensemble of chains

atom-by-atom (or

residue

by residue), replicating

and

deleting

chains _{so as} to

generate

an ensemble that

obeys

the Boltzmann statistics.

(Note

that there _are other methods of

growing

chains atom

by

atom

[38]).

^{A central idea in this method}

is to avoid to go over

large

_energy barriers

(as

in MC methods where the chain is

completed),

but to go around them. As far _as

comparison

with MD calculations is

concerned,

_our method

does not _assume any

particular

_guess for the initial state. We will illustrate the method for

the _case of linear

polymers

_[10] and its

application

to

poly-alanines [13a,b].

4.2 DESCRIPTION FOR THE CASE OF LINEAR POLYMERS. In this section _we recall the

principles

_on which the method is based. For

simplicity,

_we shall illustrate it _on the _case of linear

polymers [10].

Our aim is _{to construct}

a Boltzmann ensemble of

chains,

that

is,

_a statistical ensemble of

M chains such that the

probability

to find _a chain of _energy E in the ensemble should be

proportional

to its Boltzmann

weight ~,

_where

_fl

=

£j

and Z is _a normalization

factor,

I-e-, ^the

partition

function of the ensemble. In other

words,

the number of chains of energy E

in the ensemble should be

M~

_Since

_M/Z

_is

a constant

independent

of

E,

_we shall say that _a chain of energy ^E should be

replicated

_a number of times

proportional

to

e~flE

_{in the}

ensemble.

To generate ^these

chains,

_{we use a} recursive

procedure.

Assume that _we have _a Boltzmann

population

of chains of size _n. In order to obtain _a Boltzmann

population

of chains of size

n +

I,

we add

one atom to each of the

previously generated

chains of size _n, and

replicate

the

new chain the number of times

proportional

to

e~flAE,

where AE is the energy ^cost of

adding

the last _atom.

To illustrate the method in _more

detail,

_{we assume} that the

partition

function of the chain is

z ₌

/ _{fl d~r;}

exP

(-

^kb

^{$ (ir;+i r;i a)~ ~}

^»

^(r;>

^r>

)1(23)

~2 =~

i#j

where _ri

=

0, (r;) being

the

position

of the I-th atom in the chain. The first term represents the elastic _energy of _a link

(of

_average

length

_a and elastic constant

kb),

and

v is _a

2-body potential acting

between the atoms.

We have

deliberately

used _a

simple

form for the _energy in

(23),

but the

generalization

to _a

peptide

chain is

easily performed

_as discussed in section 5 below.

4. 3 REPLICATION-DELETION PROCEDURE. We start with the ensemble of

Ml

atoms _{n =} I

at ri = 0. Each of these is _a seed for _a chain.

To build chains of

length

_{n =} 2, for each of the

MI seeds,

_we draw

randomly

_a

position

_r2.

The Boltzmann

weight

associated with the

configuration (ri>r2)

is

proportional

to

In order _to obtain _a

population

of chains

obeying

the Boltzmann

distribution,

_we must

repli-

cate each

(ri, r2)-chain

_a ^number _w2

(ri

_(r2 times. Since _w2 is not _an

integer,

the

replication

is

actually

done in the

following

_way:

Define _{12 =} Int

(w2)

^the

integer

part of _w2> and _r2

= w2 -12 < the rest.

Then, replicating statistically

_w2 times _means

replicating

₁₂

times, plus

_one additional time with

probability

_r2.

That is to say, one

randomly

generates a number 0 < _r < I. If _r > r2, the chain is

replicated

₁₂

times.

Otherwise,

it is

replicated

₍₁₂ + 1) ^{times. Since} w2 can be smaller than I, the

replication

can in fact amount to _a

deletion,

and the chain is _no

longer

considered in future calculations.

For this _{reason we} call this _a

replication-deletion procedure (RDP).

Once the RDP has been

applied

to each

chain,

_we obtain _a Boltzmann-distributed

population

of M2 chains of two atoms.

We _{can now} iterate the

procedure

_as follows.

Assume that _we have _a Boltzmann

population

of

Mn

chains of size _n. The number

fi4n

(ri,...,rn)

of chains

(ri, ,rn)

in the ensemble is

proportional,

within statistical _errors, to its Boltzmann

weight:

fi4n in,

,

r~)

₌ A~ _exp

(-pE~ (ri,

, rn

)) (25)

N°2 SOME PHYSICAL APPROACHES TO PROTEIN FOLDING ₂₆₉

For each chain of the

ensemble,

_we draw the

(n

+

I)-st

atom

randomly

at the

point

_rn+i.

We compute ^the

weight:

Wn+i

(rn+i in,

, rn = exp

I-

^kb

^(lrn+i

rn

a)~ fl ~

v

(r~+i, r;) (26)

~i

We

replicate

the _new chain _wn+i

(rn+i (ri,

,

rn times. Then the number of

(ri>

_rn,rn+i

)-chains

is:

fi4n+1(ri,

, rn,

rn+i)

_{= wn+i}

(rn+i (ri,

, rn

fi4n (ri, ,rn)

=

An

_exp

(-fl ^{En (ri,}

_,

^rn)

^{+ ~~}

^{([rn+i rn[ a)~}

⁺

~

2

(27)

+~ v(rn+i,r;)j)

;=1

The last _term in the

exponential

is

just

the total _energy

En+i (ri,..,rn+i)

of the chain

(ri>

, rn+i

)

We thus have:

fidn+i (ri;.,rn+i)

₌ An _exp

(-flEn+i (ri;.,rn+i))

and the _new ensemble of chains of

length (n

₊

I)

is

again

Boltzmann distributed.

By iterating

the

procedure,

_{we see} that at each

stage

^{of the} _process we construct _a Boltzmann- distributed ensemble of chains of

increasing

size. We stop when the

required length

is obtained.

The

procedure

_can be modified without alteration of the Boltzmann character of the statistics if _we allow _rn+i to be drawn several times for each chain.

Although

the method _seems

applicable

_as it

is,

_one

immediately

encounters _a

major problem, namely,

_an

exponential

increase

(or decrease)

of the

population

of chains.

Indeed,

if _we

deal,

for

example,

with _a model of _a

polymer

^with steric

repulsion,

the

potential v(r)

is

repulsive (positive)

at short distances and thus the replication

weight

_wn is smaller that

unity. Thus,

iteration of the _process will result in

a decrease in the total

population

of

chains,

and

eventually

we may end up ^at some stage ^with an empty ensemble of chains.

Conversely,

if the interaction

v is attractive

(e,g.,

_a

polymer

chain in _a bad

solvent),

the

replication weight

_wn is

larger

than

I, leading

to _an

exponential

increase of the

population.

This also _causes

computational problem,

since the available computer memory is finite.

However,

the

problem

can be

easily

handled if _one recalls that all _one needs is _a

population

in which each chain is

replicated proportionally

to its Boltzmann

weight.

4.4 POPULATION CONTROL. Instead of

replicating

each chain with

a factor

wn+i(rn+i

[ri>

>rn) (Eq.(27)),

it is

perfectly legitimate

to

replicate

it with _a factor

gn+iwn+i

(rn+i(ri;.,rn

where _gn+i is _an

arbitrary scaling

factor which _can be

adjusted

_so as to

keep

the

population

of chains under control.

Equation (27)

becomes

fi4n+1 (ri,

, rn, rn+i = gn+iwn+i

(rn+i (ri,

, rn fi4n

(ri,

, rn

(28)

The _new

population

of chains has the size of

Mtot ₌

~j _{fi4n+1 (ri,}

,

rn,

rn+i)

_{= gn+i}

~j

_wn+i

_(rn+i (ri,

, rn fi4n

(ri,

,

rn). (29)

chains chains

From this _{we see} in which _{way one} should choose _{gn so as to}

keep

the

population

under control. The iteration of

equation (29)

for _a chain of size N

yields:

fidN

(ri,

,

rN)

" gig2g3...gN eXP

~fl ~

_(~i+1

ri

a)~ fl ~

V

(~ii

_~i

lfiii

=~

i<I,j<N

(30) (where

we set gi =

I). Equation (30)

proves that the final

population

is indeed Bolt2mann-

distributed.

Note that the

product

of _g;

provides

_a

simple

^evaluation for the free _energy.

Indeed, sumlring equation (30)

_over all chains of the

ensemble,

_we obtain

N

MN _"

fl

_{g;Z MI}

₍₃₁₎

;=1

and the free energy is

given by

~~ ^N

F ₌

j (log<

⁺

Slog

_g;

(321

,=1

In

practice,

the

scaling

factors _{gn can} be determined in two ways:

for

simple problems (polymers

in

good

_or bad

solvents),

one can use gn+i " gn and

adjust (increase

_or

decrease)

_gn+i in the _case when the

population Mn+i

gets out of _some fixed _range Mmin <

Mn+i

< Mmax.

for _more

complicated problems (e.g., proteins),

it is

preferable

to make _a trial _run of

adding

the

(n

+

I)-st

atom at each stage with the

scaling

of gn+1 " 7n+1, where 7n+1 is _a property chosen

factor,

count the total

population

^{of chains}

M(+i,

and then make the actual _run with

MI g"+1 " 7n+1

~,n+1

so as to _conserve

approximately

the initial

population

MI In this

work,

_we chose _7n+1

" gn+i

Thus,

_every time _we add _an atom, _we

adjust

the

scaling

factor _{so as} to conserve the total number of chains.

4.5 THE GUIDING FIELD. Assume that the elastic constant kb in

equation (23)

is

large.

Then,

if _we distribute rn+i

uniformly,

the factor kb

((rn+i rn[ a)~

^{will be}

large,

and the

replication weight

_wn+i in

Eq.(26)

small.

Thus,

the

sampling

will be _very

inefficient,

since it will

assign

a very

large scaling

factor _gn+i to the _rare

configuration

for which

[rn+i

_rn

~ a,

leaving

_{a very} small

weight

to other

configurations.

In other

words,

if _one

configuration

is such that

[rn+i rn[

_{~ a,} then it will be

replicated

_a

large

number of

times,

while the others will be deleted from the ensemble. This results in _a deterioration of the

quality

of the ensemble

and _a

buildup

of correlations _among the

chains,

I-e- _many chains

redundantly

^follow similar

paths

in

configurational

_space.

This

difficulty

_can be avoided. The

replication weight

for the atom

(n

₊

I)

is of the form Wn+I

(rn+I lrli

_irn

# gn+I ~XP

(~fIAE (~"+l l~li

_{~n )1}

(331

N°2 SOME PHYSICAL APPROACHES TO PROTEIN FOLDING 271

where AE is the _energy cost of

adding

the atom

(n

+

I).

This

equation

_can be factorized _as follows:

wn+i

(rn+i in..

_,n~

=

p~+i ~r~+i) g~+i

_exP

(-fl§[jjjjj/jj>...>rn11)

^~~4)

where the function

Pn+i (rn+i (ri,

_,rn is _an

arbitrary probability

distribution. The _prc- cedure is _now

simple:

draw rn+i with the

probability

distribution

Pn+i>

and

replicate

it

gn+i@

_{times. In what}

_follows,

we write

Pn+i

_c~

exp(-flvn+i),

and _we call

Vn+i

the

»+i »+i

guiding

^field.

It _can be

easily

_seen that this

procedure

indeed _conserves the Boltzmann distribution. It is also clear that statistical

independence

is best achieved when the

replication

factor is close to

unity.

For

example,

for the linear

polymer

chain it _seems natural to take

Pn+i (rn+i)

_{c~ exp}

-fl~~ _([rn+i

rn

a)~ (35)

2

that is, to draw _rn+i with the _correct Gaussian distribution. Then the

(n

+

I)-st

atom is

sampled

at _a correct distance from _rn, and the residual

replication weight

will be closer to one.

The ideal choice for the

sampling

function would be

Pn+i (rn+i)

_{c~ exp}

(-fIAE (rn+i(ri>..

_>rn

)) (36)

which would lead to unit

replication factors,

and thus _a

completely

uncorrelated statistical ensemble.

However,

in the presence of

twc-body interactions,

there _{are no} known

techniques

for

sampling

distributions like

(36).

The

optimal

choice for the

sampling

function Pn is

Pn+i (rn+i)

_{CC exP}

(-flUn+i (rn+1)) (37)

where

Un+i (rn+i)

is the _mean

potential

_seen

by

the atom

(n

+

I).

But in

general,

the de- termination of this _mean field is difficult, and _one must resort to intuition in the choice of

Pn.

At this

point,

it is of interest to note that another _use of the

guiding

^field is to introduce _an extra

potential

term to bias the MC

procedure

if _one wishes _to

directly incorporate experimental (or other)

information into the search. One _may, for

instance, guide

the

sampling, using

Ramachandran's

plot

information [1].

4.6 THE RESCALING PROCEDURE. Even if the choice of Pn is

nearly optimal,

the

replica-

tion factors _are not

strictly

equal to one, and

following

the

argumentation given

in

(4.3) above,

correlations between _the chains build up in the statistical ensemble. This effect becomes _more

important

_as the chains become

longer.

The final number of uncorrelated chains in the ensemble is

proportional

to the

population

of chains.

Depending

_on the temperature, ^{the form of the}

interaction,

and the size of the

chain,

it _may be _necessary to consider _very

large

ensembles to get ^sufficient

configurational sampling.

In reference [35] ^for instance, it _was shown

that,

for the _case of

polyelectrolytes

in

good

_or bad

solvents, good

statistics _are achieved when the

population

M is of the order of10 times the

length

of the chain.

It _can be _seen that

algorithmically (not taking

into account the

possibilities

of vectorization

or

parallerization)

the

computational

time scales _as M N~.

Thus,

for short

chains,

it is

possible

to use

large populations,

whereas for

large chains,

_{one can}

only

_use small

populations