Statistical mechanics model of protein folding : short and long chains have different folding transitions

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Statistical mechanics model of protein folding : short

and long chains have different folding transitions

J.-R. Garel, T. Garel, Henri Orland

To cite this version:

J.-R. Garel, T. Garel, Henri Orland.

Statistical mechanics model of protein folding :

short

and long chains have different folding transitions. Journal de Physique, 1989, 50, pp.3067-3074.

�10.1051/jphys:0198900500200306700�. �jpa-00211125�

Statistical mechanics model of

_protein

_{folding :}

short and

_long

chains have

different

_folding

transitions

J.-R. Garel

_(1),

T. Garel

_{(2, *)}

and H. Orland

₍₂₎

(1)

Laboratoire

_{d’Enzymologie,}

C.N.R.S.,

91190

Gif-sur-Yvette,

France

(2)

Service de

_{Physique Théorique,}

Institut de Recherche

Fondamentale,

CEA-CEN

_Saclay,

91191 Gif-sur-Yvette

Cedex,

France

(Reçu

le 17 avril 1989,

accepté

sous

forme

définitive

le 30

_juin

1989)

Résumé. 2014

Nous considdrons un modèle de

_mécanique

_statistique

_{pour simuler le}

_repliement

d’une chaîne

_{polypeptidique.}

_Chaque

résidu de la cheine est _{défini par} un ensemble de caractères

independants.

Les simulations _{Monte-Carlo pour} une

_géométrie

de

_champ

_moyen

_(dimension

infinie)

montrent _que

_(i)

les chaînes courtes ne se

_replient

_pas,

_(ii)

la transition de

_repliement

des

chaînes de

longueur

intermédiaire est du type « verre de

spins »

(où

tous les caractères doivent être

_{partiellement}

_{satisfaits), (iii)}

la transition de

_repliement

des chaînes

_longues

est du _type « Mattis »

(où

un seul caractère dominant est _presque

_completement

_satisfait).

Ce modèle est

peut-être applicable

au cas de

_protéines

multidomaines.

Abstract. 2014 A statistical mechanics model of _a

polypeptide

chain is used to simulate the

_folding

process. Each residue is defined

by

a set of

_independent

characters within a

_given

_sequence.

Simulations of this model in a mean-field

_{(infinite-dimensional)}

_geometry show that

_(i)

short chains do not _fold,

_(ii)

medium chains fold

_according

to a

_{spin-glass-like}

« transition »

_(where

most characters must be

_partially

_satisfied)

and

_(iii)

long

chains fold

_according

to a Mattis-like « transition »

(where

only

one dominant character is almost

_completely

satisfied).

This

_change

in

folding

mechanism associated to chain

length

may be relevant to the existence of multidomain

proteins.

Classification

Physics

Abstracts

05.90 - 61.40D - 87.10

1. Introduction.

There is

_enough

_{information in the amino-acid sequence of} a

_protein

to direct its

_folding

into its native conformation

_[1].

_Deciphering

the stereochemical code which transforms the

one-dimensional information contained in the amino-acid sequence into a

_unique

three-dimen-sional structure has become a

_{major challenge,}

since this would allow to

_predict

the conformation of a

_protein

from its sequence. The

_folding

code appears to be

_largely

degenerate

since

_(i)

_{very different sequences} can fold into almost identical

conformations,

and

_(ii)

_only

a limited subset of all

_possible

_{sequences is able} to fold into a stable conformation

(*)

On leave of absence from

Physique

des Solides,

Orsay,

France.

3068

[2].

The

_folding

of a

polypeptide

chain is a

_{thermodynamically}

driven process in favorable

conditions : the chain minimizes its energy in

reaching

an ordered conformation. It is the stereochemical _{information coded inside the chain sequence which determines the energy}

associated with a

_{given configuration}

of the chain. This

configuration

results from the balance

between the

_(attractive

as well as

_repulsive)

interactions that the chain has with itself and its

surrounding

solvent. This balance may have

dynamical

implications

due to the

_possible

existence of

_long-lived

metastable states. In this article we refine a statistical mechanics model

of a chain that we have

_recently

studied

[3],

and

_perform

Monte Carlo simulations of its

thermodynamics,

in a

_simplified

mean-field

_geometry.

This model allows us to

_study

how the information

_present

in a linear sequence is utilized to direct the

_folding

of the chain into a

stable and

_organized

structure, and it

_predicts

that short and

_long

chains will follow different

folding

transitions.

2. The model of a

_polypeptide

chain.

The chain consists of N

_links,

with a contact interaction between links i

_{at ri}

_{and j}

at

rj

given by vij

8

(rai

-

rj)

where i

_{and j}

_specify

the

_positions

of _{the links in the sequence. This}

contact interaction is an extreme limit of the more realistic situation where links i

_{and j}

interact

_{only through}

_{short range}

interactions,

such as

steric,

screened

Coulombic,

induced

dipolar,

etc., interactions.

Furthermore,

the absence of

_{angular dependent}

terms in the interaction suppresses the existence of

secondary

structures

_(helices,

_sheets,

_...).

Since the sequence of links is

given,

the

vij

are

_quenched

variables relative to the

_geometry

of the chain. The Hamiltonian reads :

Two extreme limits can be

studied,

depending

on the

_probability

distribution of the

vli :

1)

the

_Vij

_{may be taken} as

_independent

_{random variables : this supposes that the} interactions between a

_pair

of

_{links, i}

and

_j,

is not related to the interactions between another

pair, i’ and j’,

even when these two

_pairs

share a common link or when these

_pairs

are made

by

the same

_types

of links but at different

_positions.

This case, which cannot take the nature of links i

_{and j into}

account and considers

_only

their

_positions,

leads to a

_{spin-glass-like}

transition

between unfolded and folded states

_[3a].

This

_analogy

with

_spin-glasses

is consistent

_[5a]

with

long-lived

metastable states, slow

relaxations,

etc. A more

_{phenomenological}

model

_[5b],

based on a

_{random-energy}

_model,

leads to similar

conclusions ;

2)

the

_Vij

_{may be}

_decomposed

into a sum of

_separable

_{(Mattis-like)}

interactions

_{[3b] :}

where

_{the ( §f)}

are

_independent

random variables. In the

_following,

we

take ( §f =

±

_{1 }}

with

equal probability.

Note that the

factor 1

in

_equation

₍₂₎

is

_present

to avoid a

_complete

collapse

of the

_chain,

due to the absence of hard-core

_repulsion.

This form of

_vij

_{supposes that} the interactions of a

_given

link i with the others

_depend

on some

_particular

features of this

(1)

associated with a

_given

_link,

and this set of

_{( )f)}

_corresponds

to the « nature » of link i. Links of a different chemical

_type

_(such

as the different

aminoacids)

will be described

_by

different sets of M values of the

_{( )f) .}

In this case, the nature of the link

_present

at

_position

i in the sequence is

explicitly

considered

_through

a set of M

_independent

_{characters ;}

these characters can be viewed as the Van der Waals

_volume,

_{partial charge,}

_{hydrogen-bonding}

ability, hydrophobicity,

solvatation,

etc., of link i. Note that both the attractive and

_repulsive

interactions of a

_given

link i are considered

_through

the

_signs

of the

_{corresponding}

M values

vp :

for instance the

hydrophobic

character

yields

a

_{positive vp}

_(see

the

_appendix),

whereas the

short-range

Coulombic character

_yields

a

_{negative vp}

since the interaction is

_given

qi

_{q J 8}

(rai

-

rj).

With the two

_{simplifications}

mentionned above

_(zero-range

and no

_angular

dependence

of the

_{interactions),}

the frustration of the

system

is

_clearly

decreased.

_However,

our model still has some

_important

frustration effects in its

interactions,

and can be viewed as a first

_step

towards a better

_{understanding}

of the

_folding

_process.

3. An

_{example :}

the

single

character chain

_{(M = 1 ).}

In order to illustrate the

_physical

content of this model we will

_briefly

describe a chain in which the residues are defined

_by

a

_single

_character,

i.e. M = 1. For

_instance,

this chain can

be viewed as

_composed

of

_hydrophilic

( 03BE = - 1 )

and

_hydrophobic

(g

= ₊

1 )

residues. The interactions between the chain and the solvent molecules are described

_by

a

microscopic

Hamiltonian which

leads,

through

integration

over the

_positions

of solvent molecules

(see

the

appendix),

to an intra-chain interaction which is indeed described

by equation

(1).

_Although

the solvent does not _appear

_explicitly

in the

_expression

of

_vij

_{given by equation}

_(1),

it is taken into account

_through

the

_{variables ei}

-In the

_{thermodynamic}

limit N -> oo, the «

folding

» transition of such a chain

_corresponds

to a

_simple

condensation

_{governed by}

the

_hydrophobic

_{character ;}

a

_hydrophilic

residue likes

to be close to another

_hydrophilic

residue or to a solvent

_molecule,

while an

_hydrophobic

residue

_prefers

to _{be away from} an

_hydrophilic

residue or a solvent molecule

_(this

_simple

picture

breaks down if one considers the chain with hard-core

_repulsion.

In that case, one

expects

microdomain

_formation).

This transition can be

described by

an order

_parameter

(à

la

Mattis) :

where (...)

and ... denote

respectively

thermal and disorder averages. This order

parameter

m (r)

is a measure of the correlation between the distribution of

_{the 03BEi along}

the chain

_(its

«

sequence »)

and that of

the ri

(its

«

_{conformation »).}

As

_such,

it is a measure of the

_quality

of the

_coding

of this « conformation »

by

this « _sequence ».

4. The

_physical

définition of a chain.

A

_key

_parameter

of this model is the number M of characters needed to describe the interactions of a

_given

_type

of residue in the chain. In a broader view of statistical mechanics

models,

the M

_independent

characters associated with a link bear some

_similarity

to the

patterns

defined in neural networks theories

_[4].

As M

increases,

one

_expects

the

_folding

transition of a chain of N links to evolve from a Mattis-like to a

_{spin-glass-like}

behavior

_[5].

So

(1)

This definition of M is

slightly

different from the one used in reference

[3b] ;

see the discussion in

3070

far,

the _{model has been defined in any dimension. From} now on, we will restrict ourselves to

the

_following

_{(high-dimension}

or

mean-field)

_{approximation :}

the chain is constructed on a set

(simplex)

such that the _{distance between any}

_pair

of

_points

is constant

_(a

_simplex

with three

points

(resp.

four

_points)

is an

_{equilateral triangle}

(resp.

a

_{tetrahedron), etc.).}

Note that for

M

_{= 1,}

a chain on a

_{simplex always}

folds on at most four

_points

_(due

to the chain

_constraint),

and _{different chemical sequences}

_correspond

to different

_{paths going through}

these four

points.

One could also

_study

model

₍₂₎

with M = aN in the

thermodynamic

limit,

_as in reference

_[4b].

The

_{resulting equations being}

rather

intricate,

we have turned to a different

approach.

For a

_given

value of

M,

the

_folding

of a «

long »

chain will

correspond

to a Mattis-like

_transition,

and that of a « short » chain to a

_{spin-glass-like}

transition. We have first

chosen the value M = 8 for the number of characters

using

a

_{semi-biological}

_argument,

and

we have then studied the

_{possible folding}

transitions of chains of various

_lengths

(20

N

_{100 )}

on the

_simplex.

5. 8 characters _may be sufficient to describe a

_polypeptide

chain.

Each

_type

of residue

_being

defined

_by

a set of M

_{variables (§f)}

with values ±

_1,

the r.h.s. of

equation

(2)

can take

2M

different

values,

and

therefore,

there are

2M

different

_possible

values of

_vij.

With 20 amino-acids there are 210

_possible

different

_pairs

of residues

₍₂₀

x

21/2)

and

thus 210 different interaction terms

_vj,

if we assume that the interaction between residues i

and j depends only

on the « nature » of these residues and not on their

_position

in the

sequence. To obtain the 210 interaction terms needed for the 210 different

_pairs,

we must take M so that

2M >

210. The smallest

_integer

value

_possible

is M = 8

(a

similar

_argument

in the

case of RNA or DNA chains would lead to M = 4

independent

characters for each

base).

Is M = 8 a sufficient number of characters to describe all the interactions between the 20 amino-acids ? Individual amino-acids can

_probably

be discriminated

_using

more than 8

features :

_polar

or

_apolar,

neutral or

_{charged, large}

or

_small,

helix former or

_breaker,

hydrogen-bond

donor or

_acceptor,

etc., but these features do not

_correspond

to

_truely

independent

characters : for

_instance,

_properties

such as

_{hydrogen-bonding}

abilities,

_polarity,

secondary

structure

_preference,

and sidechain bulkiness are

_likely

to be related. It is difficult

to

_specify

more

_precisely

what these 8

_independent

characters could

be,

but this value of 8 is

sufficient to

_distinguish

all interactions between

_pairs

of amino-acids. The simulations

_given

below

_correspond

to the case M =

8,

_a fixed _set of 8

characters,

and _to chains of variable

lengths

(20

N

_{100 ).}

6. Simulations.

We have

_performed

Monte Carlo simulation for chains of N links

₍₂₀

N

_{100 )}

constructed

on a

_simplex

of N

_{points. Starting}

with a random initial

_{configuration}

of the

chain,

we

performed single

link moves

(cost

in energy

AE) :

an

_elementary

Monte Carlo

_step

moves a

link i from its actual

_{position ri}

on the

_simplex

to a new

_position

_{rneW (keeping}

the rest of the chain

_unchanged),

where the new

_position

is different

from ri -

1

and ri +

1 to

satisfy

the chain

constraint. These moves are

_{accepted according}

to the

_{Metropolis algorithm}

_[6].

The outcome

of the calculation is :

(i)

the internal energy U =

(H)

with H defined in

equation

(1).

which is a measure _{of the correlation of the chain conformation with character p. It is}

technically

more convenient to use

_global

order

_{parameters :}

Since on a

_given

_chain,

the chemical sequence is

_fixed,

we have not

_{averaged equation}

₍₄₎

and

₍₅₎

over the disorder.

We have taken _vl

_{= 1, V2}

=

0,5,

v3 =

0.2,

_v4 = 0.15 and

Vi +4 Vi

(i

= 1,

..., 4 )

in

equation

(2),

and the number of Monte Carlo

_steps

_{per link} was 40 for N =

40, 60, 100

(which

turned out to be sufficient to reach thermal

_{equilibrium ;}

this was checked

_{by using}

different initial

_conditions),

and 100 for N = 20. In the last _case, it will become clear that _one has _to increase this factor to reach thermal

_{equilibrium (see below).}

The internal _{energy U} is shown in

_figure

1 as a function of the

_temperature

T. A finite size « critical

_temperature

» can be

defined for N =

40, 60, 100,

corresponding

_to the

folding

of the chain

(this

is not the _case for N =

20,

due to finite size

fluctuations).

This critical

_temperature

is defined _as the

temperature

at which some order

_parameters

_Sp

start to _{grow. The theoretical critical}

temperatures

(Ref. [3b])

are shown in

_figure

1,

for

_{comparison ;}

the

_agreement

can be considered as reasonable. The order

_{parameters Sp}

have been measured

_just

below the

_(finite

size)

transition

_(Tl,

_Ti,

_Tr)

and at low

_{(finite size)}

_temperature

_{(To, Tô, TÕ)}

for N =

40, 60, 100.

We have also studied the _case N = 20 _{at two} similar

temperatures.

In that

case, there is a

_{progressive freezing}

of the

_system,

due to _{the presence of many metastable} states

_(since

one is in a

_glassy

situation)

_{separated by}

small

(since

N is

_small)

barriers. The

results are

_displayed

in table

I,

where

_only

_{the Sp}

_larger

than

10- 2

are shown.

_According

to the

simulations,

the

_folding

« transitions » seem to be first order.

For N

_{= 100,}

it is clear that one is almost in the M = 1 case, at least close to the transition. There is a

_single

character

_showing

_up at the transition

_{( Tl ).}

As N

_decreases,

more and more

characters appears at the transition

_{(if any).}

At low

_{temperatures,}

_{the frustration grows and}

Fig. 1.

-

Internal energy U vs. _temperature T for various chain

_length

N. The stars denote the

3072

Table 1. - For each value

of

N

_(N

=

20, 40,

60,100 ),

the various

Sp

(p

= 1,

...,

8 )

are

shown at two

_different

_{temperatures.}

For N =

20,

where there is _no « transition

», the

Sp

are seen to _grow

_{simultaneously}

as the

_temperature

T is decreased. For

_(N

=

40,

60,100 ),

the

_first

_temperature

_{(Tl, Tl, Tr)}

is chosen to be close to the

_folding

« transition », while

(To, To, TÕ)

illustrates how the other characters _grow at _{low temperature.} A bar

_(-)

denotes a value

_of

_Sp

smaller than

10- 2.

other dominated characters appear. In an

_analytical

calculation such as the one mentionned in

section

_4,

one would _{say that for}

M >

_{a c,} the

_folding

transition is of the

_spin-glass

_type,

N

whereas it is of the Mattis

_type

for

M

_{a C.} Our simulations are in

_qualitative

_agreement

with

n

this

_picture.

7. Conclusions.

The

_present

model has considered chains of N links of 20 different

_types,

the « amino-acids _»,

each one defined

_by

8

_independent

characters. For very short chains

(N 20),

the chain

cannot fold into an ordered state. For medium chains

₍₂₀

N

_{60 ),}

_folding

into an ordered

state involves the condensation of several characters.

_Many

_(if

not

_all)

different interactions

between

_pairs

of links contribute to the _{decrease in internal energy associated with}

_folding.

The

_cooperative

unit over which the chain finds an

_{energetic compromise}

which minimized its frustration

_corresponds

to the entire chain itself :

_folding

resembles a

_spin-glass

transition.

For

_long

chains

_{(N > 100),}

the condensation of a

_single

dominant character is sufficient to

promote

folding

into an ordered state.

_Folding

resembles a Mattis-like

_transition,

and there is a

_strong

correlation between the « conformation » of the chain and the distribution of this

dominant character

_along

_{its « sequence} ». The

change

in the

regime

which govems the

folding

of chains of

_increasing

_lengths

may be relevant to the behavior of real

_{proteins :}

smaller

_proteins

fold into a

_single

_compact

structure _upon a

_{highly cooperative}

transition,

whereas in

_{large proteins,}

different

_segments

_along

the chain behave as

_independent

_folding

corresponds

to an

_{oversimplified image}

of a

polypeptide

chain. In real

chains,

links of the same chemical

_type

_may not have identical sets of

_{( )f)}

_{if they}

are at different

_{positions along}

the

chain ; indeed,

although

two

_glycine

residues are

_chemically

identical,

_they

_{may have} different abilities to form turns because

_they

are

_preceeded

or followed

_by

different sequences. The definition of a « character »

_assigned

to a

_given

link is open

enough

to

accomodate not

_only

intrinsic

_{physicochemical}

_properties

of individual

_amino-acids,

but also

semi-empirical

data derived from the

_analysis

of known structures

_[8].

Such

_simple

models may be

helpful

to

_adapt

the

_concepts

of statistical mechanics to the correlation between sequence and conformation of

proteins

and to elucidate the stereochemical

_folding

code

_[9].

Note added : After submission of this

work,

we have received

_{preprints by}

E. I.

Shakhnovich and A. M. Gutin

_dealing

with similar models.

Appendix.

Let the

_short-range

interaction between link i of the chain

_{(at ri)}

and a water molecule

_(at

Ra)

be 03BEi

5 (ri - Ra ).

Thus § = - 1

(resp. e =

+

₁₎

denotes an

_hydrophilic

_(resp.

hyd-rophobic)

link. The

_{grand partition}

function of the chain

_plus

water

_system

reads :

where À is the

_fugacity

of a water molecule. We have :

that is : o o

Expanding

(A3)

to second order in

/3

yields :

of the form of

_{equation (2)}

with _VI =

Nk,6

/2 (the

fact that the interaction has the _same value for two

_hydro-philic

or

_-phobic

links is due to the zero _{range of the}

interaction).

3074

References

[1]

ANFINSEN C. B., Science 181

₍₁₉₇₃₎

223-230.

[2]

CREIGHTON T. E., Proteins : Structure and Molecular

_properties,

Ed. W. H. Freeman

_(New-York)

1984

_{Chapters 3}

and 6.

[3]

GAREL T. and ORLAND H.,

(a)

Europhys.

Lett. 6

₍₁₉₈₈₎

307-310 ;

(b)

Europhys.

Lett. 6

₍₁₉₈₈₎

597-601.

[4] (a)

HOPFIELD J. J., Proc. Natl. Acad. Sci. USA 79

₍₁₉₈₂₎

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