Random hydrophilic-hydrophobic copolymers

(1)

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Submitted on 1 Jan 1994

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T. Garel, L. Leibler, Henri Orland

To cite this version:

T. Garel, L. Leibler, Henri Orland. Random hydrophilic-hydrophobic copolymers. Journal de

Physique II, EDP Sciences, 1994, 4 (12), pp.2139-2148. �10.1051/jp2:1994252�. �jpa-00248120�

(2)

J. Phys. II Fiance 4

(1994)

2139-2148 DECEMBER1994, _PAGE 2139

Classification Physics Abstracts

81.205 64.75 75.10N

Random hydrophilic-hydrophobic copolymers

T. Garel

(~),

^L. ^Leibler

(2)

and H. 0rland (~,*)

(~) CEA, Service de Physique Th40rique

(**),

CE-Saclay, 91191 Gif-sur-Yvette Cedex, France (2 Groupe de Physico-Chimie Th40rique (***), E.S.P.C.1., 10 _rue Vauquelin, 75231 Paris Cedex

05, France

(Received

15 June 1994, accepted in final form 14 September 1994)

R4sumk Nous 4tudions

un copolymkre al6atoire amphiphile AB dans

un solvant s61ectif

(par

exemple, de

l'eau).

Nous consid4rons deux _cas. Dans le _cas du d4sordre mobile, les monom4res

hydrophiles

(A)

et hydrophobes

(B)

sont h l'4quilibre chirnique local, et la fraction de monom+res A ainsi _que leur position dans l'espace peuvent varier, alors quo dans le _cas du d6sordre geld

(qui

est rel16 _au probmme des

prot6ines),

la s4quence chimique est fix4e _par synt1Jbse. Dans les deux cas, le comportement de la chine depend de _son hydrophobicit4 _moyenne. Pour _une chine

fortement hydrophobe

(grande

fraction de

B),

_on trouve _un point d'elfondrement 6 continu

ordinaire, _avec _une grande entropie conformationnelle. Pour _une chaine faiblement hydrophobe

ou hydrophile, _on trouve une transition inhabituelle du premier ordre. En particulier, dans le

cas du d4sordre gaussien, cette transition discontinue _est pilot6e _par _un changement de signe

du troisibme coefficient du viriel. L'entropie de cette phase collaps4e est fortement r4duite par rapport ^I cello d'un point 6 ordinaire.

Abstract. We study _a single statistical amphiphilic copolymer chain AB in _a selective

solvent

(e.g. water).

Two situations

are considered. In the annealed _case, hydrophilic

(A)

and hydrophobic

(B)

_monomers _are at local chemical equilibrium and both the fraction of A

monomers and their location along the chain _can _vary, whereas in the quenched _case

(which

is relevant to

proteins),

the chemical sequence along the chain is fixed by synthesis. In both

cases, the physical behaviour depends _on the _average hydrophobicity of the polymer chain. For

a strongly hydrophobic chain

(large

fraction of B), _we find _an ordinary continuous 6 collapse,

with _a large conformational entropy in the collapsed phase. For _a weakly hydrophobic, _or _a hydrophilic chain, there is _an unusual first-order collapse transition. In particular, for the _case of Gaussian disorder, this discontinuous transition is driven by _a change of sign of the third virial coefficient. The entropy of this collapsed phase is strongly reduced with respect to the 9

collapsed phase.

(*) Also at Groupe de Physique Statistique, Universit4 de Cergy-Pontoise, 95806 Cergy-Pontoise Cedex, France.

(**) Laboratoire de la Direction des Sciences de la Matibre du Cornmissariat I

l'Energie Atomique.

(***) URA CNRS 1382.

Q Les Editions de Physique 1994

(3)

1 Introduction.

The

study

of random

heteropolymers

in solutions has been

recently developed,

the main _reason

being

_a

plausible

connection of this

problem

with the protein

folding puzzle

_{[1, 2].} Various kinds of

quenched

randomness have been considered.

Here,

_we wish _to

emphasize

the role of the solvent

(water)

in the

folding

_process.

Indeed,

it is

widely

believed that the

hydrophobic

effect [3, 4] ^is the main

driving

force for the

folding

transition: in _a native

protein,

the

hydrophobic

residues _are burried in _an inner _core, surrounded

by hydrophilic (or

less

hydrophobic)

residues.

In this _paper, _we consider _a

simple model,

where the _monomers of _a

single

chain _are

(quenched) randomly hydrophilic

_or

hydrophobic. They

interact with the solvent molecules

through

_an

effective short _range

interaction,

which is attractive _or

repulsive, depending

_on their _nature. A related model has been considered

by

Obukhov [5], ^but ^he ^did not

study

its

phase diagram.

We first consider the

corresponding

annealed case, which is

simpler

and

gives

useful

insights

into

the nature of the low temperature

phase.

It may well be of interest for

hydrophilic polymers

in

the _presence of

complexing amphiphilic

molecules. In section 2, _we describe the model. Section

3 deals with the annealed _case, and section 4 with the

quenched

_case, both for Gaussian and

binary

distributions of

hydrophilicities.

2 The model.

We consider _a chain of N _monomers which _can be either

hydrophilic (L)

_or

hydrophobic (P).

The

hydrophilicity degree

of _monomer I is measured

by I,,

and the monomer-solvent interaction is assumed to be

short-ranged (b-interaction).

More

precisely,

_we consider _a monomer-solvent

interaction of the form:

N+i fit

7ims _"

~ ~ l,b(r, Ra) (I)

i=i a=i

where the _r, _are the

positions

of the N _monomers and the

Ra

_are the

positions

of the fit solvent molecules.

Here,

the

I,

_are

independent

annealed _or

quenched

random variables. A

positive (resp. negative)

I

corresponds

to a

hydrophilic (resp. hydrophobic)

_monomer.

Using

the

incompressibility

condition of the chain-solvent system [6], ^I.e. ^the sum of the _monomer and solvent concentration is constant:

pm(r)

₊

ps(r)

= po, the solvent

degrees

of freedom _can be

eliminated,

and the full Hamiltonian reads:

fl7i(1,)

=

£

_[uo ₊

_fl(I,

₊

_I,

_)]

_b(r,

2 r, 1#>

+

£

w3

b(r, rj) b(rj rk) (2)

6 1#j#k

+

£

w4 b(r~

rj) b(rj rk) b(rk ri)

~~

'#J#k#1

In

equation (2),

^the _~o term accounts for

entropic

excluded volume interactions and represents the

simplest

form of direct monomer-monomer interactions.

(Note

that Obukhov's model _[5]

also includes _a random

two-body

term

proportional

to

I,lj). Usually,

when

considering

a

polymer collapse transition,

it is sufficient to include terms up ^to w3.

Here,

_we have included

a

repulsive four-body

term w4 which will

play

_an essential role.

(4)

N°12 RANDOM HYDROPHILIC-HYDROPHOBIC COPOLYMERS 2141

In the continuous limit

iii,

the

partition

function reads:

with

~ ~

fl~i(>(8))

"

/

_d8

/ _d8'(~O

+

fl(>(8)

₊

>(8~))) b(~(8) ~(8'))

o o

~ N N N

~ ~~

~~'~

~~'~ ~~~~~~ ~~~~~~ ~~~~~~~ ~~~~'~~

(3b)

~ N N N N

+fl /

_d8

/ _d8'/ _d8"/ d8'"b(r(8)-r(8'))

o o o o

6(r(s') r(s")) b(r(s") r(s"'))

where _we have

implicitly

assumed that the

origin

of the chain is pinned at 0 and its extremity is free. The parameter a denotes the statistical

segment (Kuhn) length,

and d denotes the

dimension of space. In the

following,

_we shall

study

the _case where the

hydrophilicities I(s)

are random

independent

Gaussian

variables,

with average lo and variance /h:

P(li)

₌

~p

exP

(- ~"~,(°~~ ⁽⁴⁾

We shall also discuss

briefly

the

technically

_more

complicated

_case where the I, _are random

independent binary variables, taking

the value lA with

probability

_p for a

hydrophilic

_monomer

and lB with

probability

_q

= 1 p for _a

hydrophobic

_monomer. This _case is _more relevant to

synthetic

random

copolymers.

The annealed _case amounts to take the average of the

partition

function Z _on the random-

ness, whereas the

quenched

_case

corresponds

to

taking

the _average of

log

Z. Since the

quenched

case is achieved

by introducing replicas

_[8] ^and

using

the

identity log

Z

=

lima-o ~j~,

we

compute

directly

the _average W for

integer

_n, and then treat

separately

the annealed

(n

₌ 1) and

quenched (n

₌

0)

_cases. For Gaussian disorder and

integer

_n, the average

yields:

~ /

j~ ^~~~~~~

^~~~

2~2 _~ _~j~ _i~

~

~~

~ ~ ~ ~

/~

~

(5a)

x exp

~

'

/

_ds

/

_ds'

/

_ds"

_£ _b(ra(s) ra(s')) b(rb(s) rb(s"))

2

0 0

~ ~=j

with

N N n

AG _"

-(~O

+

2fllO)

^ds ^ds'

£b(ra(s) ra(s'))

2

/ /

~=i

~ N N N n

+@ /

d~

/

_d~'

/

_d8"

_Lb(~a(8) ~a(8')) 6(~a(8') ~a(8"))

° ° °

a=1 ~~~~

~ N N N N

+I£ d8£ d8'£ d8"£

^d8"'

L

n

b(~a(8) ~a(8')) b(~a(8') ra(8")) b(~a(8") ~a(8"'))

a=1

(5)

whereas for

binary

^disorder:

~ It

~~~~~~

~~

lt ^~

~~

t ^~i1~ ^~l

~~~~

with

~ ~ ~

AB "

~°

/

_ds

/

_ds'

£b<ra<s> ^ra<s'»

2 0 0 ~=i

+^~°~

/~

_ds

/~

ds'

/~

ds"

fb<ra<s> ra<s'» b<ra<s'> ra(s"»

6 0 0 0 ~=i

~ N N N N

+i£ dS£ dS'£ dS"£

^dS"'

~~~,

L

~ b<~a<S> ~a<S'» b(~a<S'> ~a<S"» b<~a<S"> ~a<S"'»

a=1

/~

ds

log (p ^e~~~~ ^Ll=i I"

~~'~~~°~~~~~°~~'~~

o

~q ~~~~

Ll=I IT

~~'~(~"(~)~~~(~'))

We _now

study separately

the annealed ~nd

quenched

_cases.

3. The annealed _case.

We _set _n

= I in

equ~tions

_<5> ~nd (6>.

Denoting

the _monomer concentr~tion

by p<r>,

we

assume the

validity

of

ground

st~te domin~nce [6, 9]. ^The

ground

state wave-function

q<r)

is

normalized,

and _we

perform

_a

saddle-point approximation

of the

partition

function. We quote here

only

the

results, leaving

the details of the calculations for the

quenched

_case

<Sect.

4>.

These

approximations yield

the

following

annealed free _energy _per _monomer.

3. I GAUSSIAN DISORDER.

flfG ~[ijjjj~>jj [~j[)~jj[

^~

~

j~Tj~ ^~~

,

,~

<~~>

<

^r

p

^TW ^r o TWr

~~~~~

W~ " W3 3fl~~~ _~~~~

The _energy

Eo ~ppe~ring

in (7a> is the

Lagrange multiplier enforcing

the normalization of the wave-function _q. The equation of state is obtained

by minimizing

the free energy (7a> ^with respect to the wave-function

q(r>.

The _monomer concentration is

given by:

P(r>

=

Nw~(r>

(8>

The

phase diagram

of the system _can ^be

easily

discussed: for _any _non-zero concentration of

hydrophobic

_monomers, there is _a transition between _a swollen

phase (p

= 0> ^at

high

(6)

N°12 RANDOM HYDROPHILIC-HYDROPHOBIC COPOLYMERS ₂₁₄₃

temperature, ^and a

collapsed phase (finite

_p> at low temperature. ^The nature of the transition

depends

_on the _average

hydrophobicity

of the chain

lo

and variance I.

For strong _average

hydrophobicity _<lo

_< _0>, there is _a second order

collapse transition,

similar to an

ordinary

point ^for

polymers

in _a bad solvent [10]. ^This occurs when the coefficient of the

q~

term in <7a> becomes

negative

while the coefficient of the

q6

_term is still

positive.

This _occurs if the _average

hydrophobicity lo

satisfies the condition:

->o

>

(~ £<9a>

and the transition temperature is

given by:

To _" 210 (9b>

When condition (9a> is not

satisfied,

the

collapse

transition becomes first

order,

and is driven

by

the

change

of

sign

of

w(,

I-e-

by

_a

change

^of

sign

of the effective third virial coefficient. This is

quite

an unusual mechanism.

Such is the _case if the system is

weakly hydrophobic,

_or

hydrophilic.

The minimization of the free _energy

<7a)

^leads to _a non-linear

partial

differential

equation

for _q which _cannot be solved

analytically.

It is

possible

to restrict the minimization _to _a subset of Gaussian _wave-

functions,

and this will be discussed in the section _on

quenched

_systems _<see

Eq.

<20)

below>.

Within this Gaussian approximation, the first order transition temperature T* is

given by

the

equation:

~

~~~~ ~4

~°

~

~~

^~~~~~

^~3 $) _'~~'

3.2 BINARY DISORDER. In this _case, the free _energy _per _monomer reads:

~~~~~

G = w~

log (p

e

~~~~~~

+ q

~'~)

^~

~°~~

_~~~~~

By taking

derivatives with respect to the chemical

potentials l~,

it. is

easily

_seen that the concentrations of

hydrophilic

and

hydrophobic

_monomers _are

given by:

~+'~' ~~'~'~'p ~Nj~~~~i)~NpAB~~

~ q eNpAB~~ ^~~~~

~~'~' ~~' '~'p

_e~NpAA~~

+ q eNpAB~~

Note the Boltzman

weights

in (11>, ^which govern the chemical composition of the

polymer.

It

clearly

_appears that the

regions

of

high

_monomer concentration _are

hydrophobic,

whereas those of low concentration _are

hydrophilic.

The main conclusions of the Gaussian _case above remain

valid,

except that the transition temperatures cannot be calculated

explicitly.

We find

again

an

ordinary

second order point ^for

strongly hydrophobic chains,

and _a first-order

collapse

for

weakly hydrophobic

_or

hydrophilic

chains.

(7)

As _can be _seen

by

_a numerical

study

of this _case, the transition is

strongly first-order,

and this

implies

the existence of _a

large

latent heat _at the

transition,

and thus _a

sharp drop

of the entropy. ^This

drop

in entropy can be

interpreted

_as the formation of _a

hydrophobic

core, surrounded

by

_a

hydrophilic coating,

which is

energetically favourable,

but

entropically

unfavourable. This is confirmed

by equations

II_>, which show that

high

concentration

regions

are

hydrophobic,

whereas low concentration

regions

_are

hydrophilic. Finally,

_we _see ^from (11>

that at low temperature, the total number of

hydrophilic

_monomers _goes to zero, whereas the number of

hydrophobic

monomers goes ^to N.

4. The

quenched

case.

We have to take the _n

= 0 limit in (5> and (6>.

Introducing

the parameters

qab(r,r'>,

with

a < b, and

pa(r> by:

N

qab(r, r'>

₌

/

_ds

_b(ra(s>

_r>

_b(rb(s> _r'>

°~

(12>

pa(r>

=

/

_ds

_b(ra(s>

r>

o

we may write

equations

_(5> _as.

4. I GAUSSIAN DISORDER.

@

=

Dqab(r, r'>D#ab(r, r'>Dpa(r>D#a(r>

_exp

(G(qab, lab,

_Pa,

la>

+

log ((#ab, la» (13a>

where

#ab(r,r'>

and

#a(r>

_are the

Lagrange multipliers

associated with (12>, ^and:

G(~ab, lab,

_Pa,

la>

"

/

_dT

_L lPa(~>#a(~>

(U0 + 2fl>0>

~j~' ~Pl(~> )(Pl(~>)

~

+

/

dr

/

_dr'

_£ (iqab(r, r'>#ab(r, r'>

+

fl~l~ qab(r, r'>pa(r>pb(r'>)

a<b

(13b>

and

((4ab, ~a'

"

fl _~~a(S'

_(13C~

~

where

w(

is defined in

equation (7b>.

Using

the

identity

of equation (13c> ^with a

Feynman integral

[9,

iii,

we can write:

((@ab,

la>

"

/ _fl _d~ra

< ri

rn(e~~~"~4ab'~a~(0.

0 >

(14a>

~

(8)

where Hn is _a

"quantum-like"

_n _- 0

Hamiltonian, given by:

~n £ _V~

₊

£ _l~a(ra'

₊

£ _I#ab(ra,

rb

(14b~

a a a<b

Anticipating

_some kind of

(hydrophobically-driven> collapse,

_we _assume that _we _can _use

ground-state

dominance to evaluate (14>, ^and write,

omitting

_some non-extensive

prefactors:

j(#~~~

j~,

_Qf

e~NE0(4ab,S)

= exp

(-N

_l~(r)I^min

(< lP(Hn(il

>

-Eo (<

lP(lP >

-1>))

^~~~~

where Eo is the

ground

state energy of Hn. At this

point,

the

problem

is still

untractable,

and _we make the extra

approximation

of

saddle-point

method

(SPM>.

^The extremization with

respect to qab reads:

I#ab(r,r'>

=

-fl~ l~pa(r> pb(r'>

_(16>

This

equation

shows that

replica

symmetry is not

broken,

since

jab

is _a

product

of two

single- replica quantities

_pa. To get _more

analytic information,

_we _use the

Rayleigh-Ritz

variational

principle

to evaluate Eo. Due to the absence of

replica

symmetry

breaking (RSB>,

_we further restrict the variational wave-function _space to Hartree-like

replica-symmetric wave-functions,

and write:

n

~P(~l;

,~n>

_"

fl _w(ra>

_(17>

a=I

Because of

replica

symmetry, we can omit

replica

indices and take

easily

the _n _- 0 limit. The variational free _energy _now reads:

~fll~G(q,

4,

P,~,

_§~' _"

/dT ^lp(r~#(r~

(U0 +

2fl~0' ~j~~

~P~(~' j'(P~(~')

/

_dT

/

_dT'

_liq(r, _r'>4(r, _r'>

+ fl~>~

q(r, r'>p(r>p(r'>)

~2

~~~'

Nl/d~T

^W(r>

liv~

+ i

<(r>

W(r>

/

_~~~

/ ~'~'~"'"

~

~~°~ / _~~~~~~~' _~'

The SPM

equations

re~d:

P(r>

=

Nw~(r>

q(r, r'>

=

Nw~(r>w~(r'>

wj

~ w~ _~ _~ _~

j

~, _, _, (19>

~~~~~ ~~° ~ ~~~°~~~~~ ~

2 ~ ~~~ ~

6 ~ ~~~ ~

~

~ ~ _~

~~~'~

~~~~

14(~,

~'> "

~fl~>~P(~>P(~'>

We still have to minimize with respect to the normalized wave-function

q(r>.

This le~ds to a very

complicated

non-linear

Schr6dinger

equation, ^and _we ^shall restrict ourselves _to _a _one-

parameter

family

of Gaussian wave-functions of the form:

i d/4 ~2

~~~~

27rR2

~~~ 4R2 ~~~~

(9)

where R is the

only

variational parameter.

Using equations

(18> ^and (19>, ^the variational free _energy _per _monomer

()

"

GIN>

reads:

Note that the

only

^difference between this free _energy and the annealed _one of equation

(7a),

is the presence, in the

quenched

_case, of _an additional disorder induced term, ^which _we ^have

grouped together

with the

q6

term for _reasons that will become clear below.

Using

the Gaussian

wave function

(20>,

and the value of

w( given

in

(7b>,

the free _energy reads:

~ i

^~

2j~>~

_{~~~ ~}

^~

~~~

~~~~

~

(27r~$>~ ~ 2~>~ ~~~

^~~ ^~~~ ^~

~~

^~

~

3~7r~

~~~

~~

~

At low temperatures, one has _to

study

the

sign

^of the third _term of

equation (22>.

^This

sign

becomes

negative

at low temperature, ^due to disorder fluctuations.

It is easy ^to see that this result is not restricted to the choice (20> ^{of the} variational function

q(r>. Indeed, using

the

inequality Jd~r(q3(r>

₊

xq(r»2

_>

_0,

for all x>, one finds that the third virial coefficient of

equation

_(21> becomes

negative

at low

enough

temperatures, ^for ^all

space-dependent

normalized variational functions

q(r>.

We therefore

get

^the

following

results:

I> lo > 0: the

hydrophilic

_case. In _our

approximation,

we find that there is _a first

order transition towards _a

collapsed phase (R

r-

N~/~>

induced

by

_a

negative three-body

term

(and

stabilized

by

_a

positive four-body term>.

This transition is neither _an

ordinary point (since

the

two-body

term is

positive>,

_nor _a

freezing

point

(the replica-symmetry

is not

broken>.

Note that since _our

approach

is

variational,

the true free energy of the system is lower than the variational _one, and _we thus expect the real transition to occur at _an _even

higher

temperature. ^Since ^the transition is first

order,

_we expect

metastability

and retardation effects to be important _near the

transition;

_we also expect that

(due

to the latent heat _>, there will be

a reduction of entropy in the low temperature

phase, compared

to _an

ordinary

second order

point.

ii> lo < 0: the

hydrophobic

_case. In this _case,

defining

the two temperatures:

To _"

2(lo(/uo

~ ~

/

_(23>

/w3(1 3d/2/2d>

leads _to _two

possible

scenarii, in _a very similar way as in the annealed _case:

a>

Weakly hydrophobic

_case:

Ti

> To- The

collapse

transition is

again

driven

by

the disorder fluctuations of the

three-body

interactions. The

resulting

first-order transition is very similar to the

hydrophilic

_case _I>.

b>

Strongly hydrophobic

_case: To > Ti The

collapse

transition is _now driven

by

the strong

twc-body

lo term. The

resulting phase

transition is very similar to _an

ordinary point,

and is therefore second-order.

(10)

The

phase diagram

of the

quenched

_case _seems

quite

similar _to the annealed _case,

but,

as we shall _see _on the _case of

binary disorder,

the

physics

is

quite

different. Note also that the transition temperature ⁱⁿ the

quenched

_case is lower than in the

corresponding

annealed

case, as can be checked from (7a> ^and (21>. ^This may be viewed _as _an effect of

geometrical

frustration induced

by

the chain constraint.

4.2 BINARY DISORDER.

Using

the _same

approximations

_as in the

namely

ground

state

dominance,

_no

replica

symmetry

breaking

and Hartree variational

wave-function,

the free _energy _per _monomer reads:

~~~ '~ ~~~~~~~~~~~

~ ~

~~~~~~~~

+N~'°~ / d~rq~(r>

₊ ^'°~

^N~ / d~rq~(r>

Eo

/ d~rq~(r> _(~4,

6 24

+~~ffl~A

/

d~T§~~

(~' i~ ^~~~

^~

_1°g /

d~T§~~

(~' ~~~'~ ~~~~)

~_~ P

where I

= AA + lB.

In this _case, we see that _even the Gaussian variational form is not

tractable,

but _a numerical

study

shows that the system has

exactly

the _same phase

diagram

_as for Gaussian disorder.

However,

the interest of this model is that it allows to calculate the concentration of

hydrophilic

and

hydrophobic

_monomers. We obtain:

P+(~'

"

NW~(r>

I j§(-i,i+i (q) ^eP'lN~~

I=1 i'

i

ddT§a2eflAlN~2

m

P-(r>

=

Np2(r~ £(_i,i+i

^q

^eP'lN~2

(25>

1=1

i'~

fddT§a2eflAlN~2

It _is

easily

_seen _on _(25> that the two concentrations

satisfy

the _sum rules:

/d~TP+(4

⁼

^NP

/ d~rp-(r>

~~~,

=

Nq

and thus

(as expected

for _a

quenched chain>,

the overall chemical

composition

of the chain remains

unchanged.

5 Conclusion.

We have studied _a

simple

model of _a

randomly hydrophilic-hydrophobic

chain in water. The randomness _may be annealed _or

quenched,

and _we have considered

binary

and Gaussian distributions. Let _us first summarize _some technical aspects ^of our

study.

We have used the

ground

state dominance

approximation

_[6,

9],

^which ^is

certainly

_apprc- priate in _a

collapsed phase.

We have further used _a

saddle-point method, together

with _a

restricted variational procedure. These extra

approximations

_are more difficult to assess, but should be _correct at _a mean-field level

(see below>.

(11)

We _now consider the

physical

output ^of this calculation. Results _are very similar for annealed and

quenched

randomness. The low temperature

phase

is

always collapsed; depending

_on the

degree

of

hydrophobicity

of the

chain,

this

collapsed phase

is reached either

through

_an usual second order

point,

_or

through

_a first order transition. In both _cases, critical fluctuations _are

expected

to be weak. As

expected,

the interior of the compact

globule

is

mainly hydrophobic,

whereas its exterior is

hydrophilic (see Eqs.

(11> ^and

(25».

Note however that the detailed connection between the _monomers and their

spatial positions

is lost when _one averages over the disorder. We would like _to _stress that the first order transition,

resulting

from equation

(22>,

does not

imply

that the

hydrophilic

(A> and

hydrophobic

(B> _monomers

undergo

_a

macroscopic phase

separation. Since

q2(r>

is

proportional

to the total

polymer density,

_one _may

identify

the

coexisting phases

_as the solvent and _a

copolymer-rich phase:

the first order transition _can be

thought

of _as _a

demixing

transition, due _to the disorder fluctuations. The detailed

study

of the

spatial

structure of the

chain,

_as well _as the

properties

of the chain-solvent

interface,

_are

beyond

the scope of this work.

Finally,

the

quenched

_case _may well be relevant to proteins, since

hydrophobic

residues

play

a major ^{role in} ^the

folding

transition.

Indeed, hydrophobicity-based

codes have been

previously proposed by

Goldstein et al. [12], ^to

predict

_some folded structures. Their

approach

is however

linked to the spin

glass approach

to protein

folding,

which differs from the

approach

of the present _paper. References

ill

and _[2]

give

_a _more

complete

view of the disordered

polymer approach

to the

protein folding puzzle.

Note added in

proof:

A first order

collapse

transition for random annealed

copolymers

^has ^been obtained

by Grosberg

in Biofizika 29

(1984>

^569. We thank A.

Grosberg

for

bringing

this work to _our attention and for

stimulating correspondence.

We also benefited from

interesting

discussions with A. Gutin

and E. Shakhnovich.

References

iii

Karplus M. and Shakhnovich E-I-, Protein Folding, T-E- Creighton Ed.

(W.H.

Freeman, New

York, 1992> ^and ^references ^therein.

[2] Garel T., Orland H. and Thirumalai D., New Developments in Theoretical Studies of Proteins,

R. Elber Ed.

(World

Scientific, Singapour, 1994> and references therein.

[3] Tanford C., The hydrophobic effect

(Wiley,

New-York,

1980).

[4] Dill K-A-, Biochemistry 29 (1990> 7133.

[5] Obukhov S-P-, J. Phys. A 19 (1986> 3655.

[6] ^de ^Gennes P-G-, Scaling concepts in polymer physics

(Cornell

University Press, Ithaca, 1979).

[7] ^des ^Cloizeaux ^J. ^and ^Jannink G., Les Polymbres _en Solution

(Editions

de Physique, Les Ulis, 1987>.

[8] ^Mdzard M., Parisi G. and Virasoro M-A-, Spin glass theory and beyond

(World

Scientific, Sin- gapore, 1987).

[9] Wiegel F-W-, Introduction to path integral methods in physics and polymer science

(World

scientific, Singapore, 1986>.

[10] Flory P-J-, Principles of polymer chemistry

(Cornell

University Press, Ithaca, 1971>.

ill]

Edwards S-F-, Proc. Phys. Soc. London 85

(1965)

613.

[12] ^Goldstein R-A-, Luthey-Schulten Z-A- and Wolynes P-G., Proc. Nat]. Acad. Sci. USA 89

(1992

4918 and references therein.