Protein Folding, Anisotropic Collapse and Blue Phases

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Protein Folding, Anisotropic Collapse and Blue Phases

E. Pitard, T. Garel, Henri Orland

To cite this version:

E. Pitard, T. Garel, Henri Orland. Protein Folding, Anisotropic Collapse and Blue Phases. Journal

de Physique I, EDP Sciences, 1997, 7 (10), pp.1201-1210. �10.1051/jp1:1997117�. �jpa-00247392�

J.

Phys.

I FFance 7

(1997)

1201-1210

OCTOBER1997,

PAGE 1201

Protein Folding, Anisotropic Collapse and Blue Phases

E.

Pitard,

T.

Garel (*)

and H.

Orland

Service de

Physique Thdorique, CE-Saclay,

91191 Gif-sur-Yvette

Cedex,

France

(Received

28 March

1997,

received in final form 5 June 1997,

accepted

10 June

1997)

PACS.61.4i.+e Polymers, elastomers, and

plastics

PACS.87.15.Da

Physical chemistry

of solutions of biomolecules; condensed states PACS.64.70.Md Transitions in

liquid crystals

Abstract, We

study

_a

homopolymer

model of _a protein chain, where each _monomer carries

a

dipole

moment. To mimic the geometry of the

peptidic

bond, these

dipoles

_are constrained to be

locally perpendicular

to the chain. The tensorial character of the dipolar interaction leads

naturally

to _a

(tensorial) liquid crystal-like

order parameter. For _non chiral

chains,

_{a mean} field

study

of this model shows that _a classical 6

collapse

transition _occurs

first;

at lower temperature,

nematic order sets in. For chiral chains, an anisotropic

(tensorial)

collapse transition may occur

before the 6 temperature is reached: the ordered

phase

_can be described _{as a}

"compact phase

of secondary structures", and possesses great similarities with the

liquid crystal

blue phases.

1, Introduction

Proteins _are chiral

heteropolymers,

made out of

twenty species

of _monomers

(aminoacids).

Natural

proteins

have the property ^of

folding

into _an

(almost) unique compact

native

structure,

which is of

biological

interest

Ii,2].

The

compactness

of this

unique

native state is

largely

due to the existence of _an

optimal

amount of

hydrophobic

aminoacid residues

[3],

^{since these}

biological objects

_are

usually designed

to work in water. The existence of

secondary

structures [4]

plays

_a

predominant

role in the existence of the

unique

native state. From _a theoretical

point

of

view,

one may either

emphasize

the

heterogeneous

character of these

biopolymers _[5],

_or choose to focus _on the _more

crystalline-like secondary

structures

(o-helices

and

fl-sheets).

In the first

approach, folding

is identified to _a

freezing transition;

in the second

approach,

that _we follow in this paper, the

folding

transition is more like _a

gas-solid

transition. Previous studies

along

the latter

path

made _use of

simplified

lattice models [6]. We consider here _a

microscopic

continuous model of _a

homopolymer chain,

where each _monomer carries _a

dipole

moment. The rationale for this choice is clear if _one

interprets

the chemist's

hydrogen

bonds _as the interaction between electrostatic

dipoles (the secondary

structures

approximately corresponding

to the classic

dipole equilibrium positions).

The

dipolar

moment of _a CONH _{is po ~bf}

3.6D, yielding

a

typical dipolar temperature (temperature

below which

dipolar

effects become

relevant)

of

TD

=

p(/(4ireoerd~)

where _fir is the relative dielectric constant of the

medium,

and d is _a

typical

distance between

dipoles.

The electrostatics of

proteins

ⁱⁿ water is _a somewhat controversial issue

iii,

_{so we} will content ourselves with an estimate of

TD

in the native state. It is then

(*)

Author for

correspondence (e-mail: garel@spht.saclay.cea.fr)

@

Les

#ditions

_de

_Physique

₁₉₉₇

appropriate

to take _er

~w 5

Iii.

With d

~w 3.8

I Iii,

_we find

TD

C~

20001er

_ci 400

K,

_so that

dipolar

effects _may

play

_a crucial role in the

energetics

of the

folding (or unfolding)

transition. The

spatial

_range of the

dipole-dipole

interaction _can be finite _or infinite: its

angular dependence (or

tensorial

character)

is the crucial issue of this _paper. To make _a closer connexion with

proteins,

we assume that the

dipole

moments _are constrained to be

perpendicular

to the chain. In this _{way, a monomer can} be

thought

of _as

linking

two successive Cm carbon atoms, and its

dipole

moment

represents

^the

peptid

CONH

dipole.

A finite _range of

the

dipolar

interaction may further describe the finite range character of the

hydrogen bonding

interactions. The Hamiltonian of the model then reads

PI~

=

j° ~ _iir~ rj)

₊

~ _{~piG~~ir~, rj)pj. ii)}

~~j ~~j _~,~

In

equation iii, fl

= I

IT

is the inverse

temperature,

_vo is the excluded

volume,

_r~ denotes the

spatial position

of _monomer I

ii

=

1,2,... N),

and p~ its

dipole

moment. If _necessary, three

body repulsive

interactions may be

introduced,

to avoid

collapse

at infinite

density.

The

(infinite range) dipolar

tensor reads

Gn~(r, r')

= A ~

,~~

(bn~ 3vnv~ (2)

r r

with _v~

=

jr r')n fir r'j

and A is _a

prefactor containing

the dielectric constant of the medium. The

dipolar

interaction

(2)

is cut-off at small distances

(jr r'j

_<

a)

and _may also be cut-off _at

large

distances

by

_an

exponential prefactor.

The

partition

function of the model

(1)

is

given by:

z

=

In _{dr~dP~ bllr~+i}

r~l

a) bllP~l Po) blP~ lr~+i r~))

_exP

I-fill) ₁₃₎

In

equation (3),

_a denotes the Kuhn

length

of the _monomers, and _po is the

magnitude

of their

dipole

moment. The third )-function restricts the

dipole

moment to be

perpendicular

to the

chain;

this mimics the free rotation of the CONH

dipole

around the

Cn-Cn

"virtual bond". As it

stands,

this

partition

function is not easy to evaluate. In

particular

the constraint of fixed

length dipoles

is not easy ^to

implement.

We have therefore studied the constrained

dipolar

chain in

(I)

^{in the framework of the virial}

expansion,

with fixed

length dipoles (it)

ⁱⁿ

the framework of _a Landau

theory,

where _we

replace

the fixed

dipole length by

_a Gaussian distributed

length.

2. The Virial

Approach

Since the virial extension is

quite

^familiar in the field of

polymeric liquid crystals _[8],

_we will be rather

sketchy

in _our

presentation. Starting

from

equations (1-3),

_we write

i~ _~~~

~" (G'

_~J

)PI

#

fl(I

₊

_f

~~~

~

~<J °,~

~

lJ

~<j

~ +

~j _fq

_+.

~<j

"

e~~<?

^~~~"'

N°10 PROTEIN FOLDING AND BLUE PHASES 1203

where

f~

=

f(r~ rj)

= exp

(-flp~ G~ pj)

1. To lowest _non trivial order in

f,

_we

get:

z =

/ _{fl dr~bllr~+i}

r~l hi

_exP

(-fli~»oi(lr~lll

~

x exP

l~ /dP~dPjfvbllP~l ^{-Po) blP~ lr~+i r~i)bllPjl Po) blPj lrj+i}

j)))

lsi

~<J

where

fl7lpoi

=

j~~~~ b(r~ rj). Introducing

the

joint

distribution function:

glr, Ul

=

~j _{blr r~l b(u u~l 16j}

where _u~

=

(r~+i r~) la,

the

partition

^function

(5)

reads:

Z

=

DgD§

_exp

i

drdu

§g

^"°

2~

dr

_~~

du g

^~~

x exp

/ drdr'dudu'g jr, u)V(r,

_u

jr', u')g jr', u')

₊

log ((§) iii

2

~~~~~

vj~, ujr', U')

_#

/ _dpdp'i(lPl

P0) b(P ~l~llP" ~°l ~l~' ~'l~~~

_{~'~ ~~~}

~~ ~~ ~~~

~i~~~~i~ partition

function

given byl

((§)

=

/fldri b(jr~+i _r~j a)e~~i~ibl~~'~~~ (9)

In the

spirit

of _mean field _or variational

approaches _[8],

_we look for _a

homogeneous (space

inde-

pendent)

distribution function:

g(r, u)

=

p#(u),

where _p

=

N/Q

is the _monomer concentration

and the

angular

distribution function is normalized to I. The free _{energy per monomer} reads:

flF

=

/ du41u) log(PlUll

+

)P

P

/ dUdu'4lUlV(U u'l4lU'l

₊

°lP~l _11°1

where

vl~'~'l

~

/d~dPdP'6(lPl ^P0) 1(P'U)6(lP'l P0) b(P' 'U'lf(~l _(~~l

and the

O(p~)

_term in

equation (10) represents

^{the three}

body repulsive

interactions.

Using

the

Onsager

variational form [9)

~~~~ =

47r

s~h

a

~°~~(°

_Cos

6) (12

where 6 is the

polar angle

of the unit vector u, we obtain

~~ °~147rei~nho~

~

si~ha ^~~~~~~~~~~ ~

~~~ ^{~ ~t2}

_~~~

~~~°~~~

~

~~~~°~ ~~

~

~~~~~

~~~~

where t

=

TITS,

with

T9

=

(/~TD,

and

w(a)

=

if §).

We have also assumed that

d

~w a, i.e. that _an estimate of the distance between

dipoles

is

given by

the _monomer

length.

Minimizing equation (13)

with

respect

to _a and p

yields

the

following

results:

I)

a usual second order H-like

collapse

transition to a finite

density phase

_occurs at

temper-

ature t ₌ I. Since _vo

~

a~,

we have

T9

~w

TDI

iii

in the dense

phase,

_a discontinuous nematic-like transition _occurs at

temperature

tN =

TN ITS

ci

1/3,

with _a small

density discontinuity.

Since

TN

is _a variational esti-

mate,

^the true transition

temperature

^is

necessarily higher.

These results show that

dipolar

interactions _are

responsible

for both transitions. We remind the reader that

they

_are

expected

to be _{correct at} small _monomer

concentration,

since

they rely

on the virial

expansion.

^At

higher

_monomer

concentration,

_{one may} find other

phases (e.g. crystalline phases [10j ).

An extension of this

approach

to chiral chains is

possible Ill

_]. In

this _{case, one} expects to encounter _a

phase

transition towards _an

inhomogeneous phase (e.g.

cholesteric)

at _a

temperature TH higher

than

TN.

The

possibility

of _a

crossing

^between

T9

and

TH

must therefore be considered. As shown in the next

section,

the Landau

approach

to the

constrained

dipolar

chain

provides

_{a more} convenient framework _to

study

this

possibility.

3. Landau

Theory

3.I. FORMAL DEVELOPMENTS. At _a

qualitative level, experimental phase diagrams

are

usually

in

agreement

^{with the Landau}

theory

of

phase

transitions.

Roughly speaking,

this

theory,

which _{rests over} the _use of

symmetry arguments,

may be summarized _as follows

Ii)

_one looks for _an order

parameter Q(r)

that describes the ordered

phase(s)

at

point

_r;

iii)

_one builds _a free _energy

density

_{as an}

expansion

in

Q(r),

and in the first

spatial

derivatives of

Q(r)

(iii)

_one minimizes the total free _energy with

respect

to

Q(r).

This

theory clearly

_assumes that the order parameter

Q(r)

varies _on distances much

larger

than the

microscopic length

scales.

Following

this

line,

we first soften the constraint of fixed

length dipoles

in

equation (3)

and

replace

it

by

_a Gaussian constraint

(the replacement

^of fixed

length by

^Gaussian

spins

is familiar in the

theory

of

phase transitions).

We therefore have

zL

=

/ _{fl dridP~ ~i'r~+i}

r~' al ~~rjj~/~ _{e-P~/i~»~~ biPi ir~+i}

r~i)

_exP

i-fl~i i141

1

where the

subscript

L _on the

partition

function stands for Landau and the Hamiltonian 7l is

given by equation ii _). Using

the

identity

~ 3/2

~~~~

/$ ^27r~

~ ~~~~~

~~~~

we may now

perform

the

(Gaussian) integrals

_over ^the

dipole

moments p~ in

equation (14).

As _a

result,

the

problem

_now

depends only

_on the

polymeric degrees

of freedom.

Introducing

the tensorial

(order) parameter Q~p(r) by

Qn»(rl

=

~ ((U~ln(U~lp bowl 6(r r~l (161

N°10 PROTEIN FOLDING AND BLUE PHASES 1205

where the notation _{u~ =}

(r~+i r~) la

_was

_again used,

_we

get

,

zL

=

/ _{fl Donplr)}

₆

^onplri ~llu~inlu~ip _{b~pi blr} ~l)

i~,p),r ~

x

fl _{dr~ b(jr~+i r~j a)}

exp

(-fl7lpoi)

_exp

1-

^Tr

^log

^B

ⁱⁱⁱ⁾

/

2

where the matrix B is defined

by

its matrix elements

Bn~(r, r')

=

bn~b(r r')

+

flp( ~j Qap(r)Gp~ jr, r'). ₍₁₈₎

v

Using

the

identity

b

ion» da»)

₌

/ _{~))" exPi4n» ion» dn»I jig)}

one may write

zL

=

/ _fl DQn»(rlD4«» (rl

_exP

i /

_dr

_~ Q~»(non»(rl

la,v),r O<v

x exp

-fl7lpoi(Q))

_exp

1-

₂^Tr

^log

^B

^{exp(log ((t~)) (20)}

with

((4)

#

fldr~ b(jr~+i _r~j a)

_exp

-I ~j ~j t~«p(r~)((u~)n(u~)p 6np) (21)

~ n<p _~

and where _we have

expressed 7lpoi

as a function of

Q (see Eq. (23) below).

So

far,

_no

approximations

^{beside the}

assumption

^of a Gaussian distributed

length

of the

dipoles

_were made. We _stress the fact that the number of

independent

variables

4n»(r)

is

equal

to

six,

since

Q

is _a

symmetric

rank 2 tensor. This is

why

the _{sum over} the

components (a, p)

in

equations (20, 21)

is restricted _{to a} < ~.

3.2. THE PHYSICAL ORDER PARAMETERS. At this

Stage,

it may be useful to establish the

relation of the

previous

section with the virial

approach.

It is clear that the _tensor

(or matrix) Q(r)

can also be

expressed through

the

joint

distribution function of

equation (6)

_as

Qn»lr)

=

/dU ^{glr, U) llU)alU)p 6np).} 122)

FUrthermore,

_{one may} build from

Q(r)

_a

(scalar) density

Plr)

=

-)

_lk

_Qlr)

(23)

and _a traceless tensor

Q(rl

=

Qlrl ~Tr ^{Q(r) 124)}

where I is the unit tensor.

Assuming

that both

quantities

_{are space}

independent,

_we

easily

recover the results of Section 2. For space

dependent

order

parameters,

^the

Landau-Ginzburg

free energy

density FL(r)

is defined

bj,:

EL

"

/ Dp(r)Dq~p(r)e~~f~~~~l~~ (25)

It _can be evaluated

by

iii performing

_a saddle

point

method _on

4

_in

_{equation (20)}

_and

_{replacing t~}

as a function Of

Qj

iii) expanding

the

resulting

free _energy

density

in the

exponent

^of

(20)

in powers of

Q

and

performing

_a local

gradient expansion.

Note that all terms of the

expansion

^{could have} been obtained from

symmetry considerations;

(iii) expressing

^{the free} energy

density

in terms of

p(r)

and

q(r).

The

resulting

Landau free _energy

density

reads:

~~~~~ ~° ~~~~~~~~

~

~~ ~~~~~~~

~

~i

^{~~~~~ ~}

@~q~~jrjba<qn<pjrj

+

~' b~qapb~qnp

+ ~~

jT TojTr q~jrj

2 2 2

~~~~

~~~~~ ^~

~~

_{~~~~~~~ ~}

~~/~~~~~~"~""~~~ ~~~~~

_~~~~~ ~ ~~~~

where _{we use} the standard summation _over

repeated

indices. The various coefficients _a, b,_c in

(26)

_can in

principle

be calculated

using

the method described above. In

particular,

the

temperature T9

is the _{same as} that defined in Section 2 and

To

would be the nematic transition

temperature

^if there _{was no} cubic term in _q in

equation (26).

3.3. INCLUDING CHIRALITY. The observation that the backbone of

proteins

has

chirality

can be

appreciated through

the Ramachandran

plot Iii.

The

chirality

ofthe

Co

"virtual chain"

is

predominantly

associated with the

#

dihedral

angles.

It is not easy to take

chirality

into account at _a

microscopic

level

[12j,

^{while it} is _so at a Landau-like level

[13j,

since

then, only

very

general symmetry

considerations matter. In

particular,

if the system is described

by

_a

tensor

q(r),

the absence of inversion

symmetry

ensures that _{a new}

(scalar)

invariant shows up in

equation (26), namely

j~chiral(r)

_# ^D

Eapuqn&fipqu& (27)

where D is _{a measure} of the

strength

^of the

chirality

and enp~ the

antisymmetric

third rank

tensor. The full Landau free energy

density 3~°~(r)

^of the constrained

dipolar

chiral chain

then reads

f~°~(r)

~

FL (r)

+

Fchirai(r) (28)

where

FL (r)

and

Fchirai(r)

are

given respectively

in

equations (26, 27).

It is clear that

equation (28)

describes the

coupling

between _a usual 6

collapse

described

by p(r)

and blue

phase-like ordering _[14]

described

by q(r).

For _our purposes, it is

important

to note that

equation (27) implies

^that _a ^{modulated order} in

q(r)

sets in at _a temperature

T~

^{which is}

higher [14,15]

than the

temperature TN

of the uniform

(nematic) order;

_on the other hand

equation (26)

shows

that,

to lowest

order,

the 6

collapse

_occurs at the _same

D-independent temperature T9.

Before

exhibiting

the

crossing

of the two temperatures

T~

and

To

in _a

specific

case, we

briefly

summarize the

existing

state of the art for blue

phases.

N°10 PROTEIN FOLDING AND BLUE PHASES 1207

3.4. SUMMARY _OF THE BLUE PHASES FOLKLORE. In chirai

Systems,

the blue

phases (BP)

may show _up in _{a narrow}

temperature

_range

(<

I

K)

between the

simple

cholesteric

(helical)

liquid crystal phase

and the

isotropic liquid.

Their

theory

is

quite complicated [14;15j;

their

ordering

is described

by

_a

symmetric

^{rank 2} tensor which _can, in some

appropriate limits,

be linked _to the molecular orientations. So far _two

crystalline phases (BP I,

BP

II)

and _one

non-crystalline phase (BP III)

have been found. More

cimplicated phases

_may exist when _an electric field is

present.

The

crystalline phases

_are characterized

by

_a

body

centered

(BP Ii

_or

simple

cubic

(BP II)

lattice. The unit cell contains of the order of

10~ moleciilds,

_so that the Landau

expansion

_may be _a

priori trustworthy.

Note that these

crystalline

structures are threaded

by

disclinations

(line

defects of the molecular

orientations).

The _non

crystalline phase (BP III)

is of

particular

interest

[16].

^{It has the} same

macroscopic symmetry

as the

isotropic liquid,

and the

(BP III)-(isotropic liquid) phase

transition has _a critical end

point,

much like the

liquid-gas

transition.

Furthermore, attempts

have been made to describe this

phase

_as

resulting

from _an unstable localized mode

(see

the references in

[14]),

as

opposed

_{to an} extended mode for the

crystalline

BP's.

4. A

Simple Example

of

Anisotropic Collapse

We will

exhibit,

in _a

simple

_way, the

crossing

between the 6

(uniform) collapse temperature To

and the

temperature T~

below which _a modulated order sets in.

Following

references

[15, Iii,

_we

consider in

equation (28)

^the appearance of helicoidal order in the _tensor

q(r).

At this

point,

it is

perhaps

useful to recall the reader that the various

"heliies"

_{of this section}

are not the a-helices present in

proteins. Indeed,

these "helices" _are linked to the order

parameter

^of

equation (24), and,

_as in all

polymeric problems,

it is difficult _to deduce the chain

configuration(s)

from the

physical

order

paraIneter(s).

The

q(r)

helicoidal order is described

by:

where the matrices M and

N(z)

_are

given by

I

⁰ 0

M= 01 0

(30)

° °

~~

and

cos

ko

z sin

koz

0

N = sin

ko

z cos

ko

_z 0

(3 Ii

0 0 0

where

ko

#

ID /b[)ez

is the _{wave vector} of the modulation. Usual cholesteric order

ii.

_e. the

"uniaxial"

helix) corresponds

to

#

=

7r/6,

while the

purely

biaxial helix is described

by #

= 0.

We insert this trial order parameter in

equation (28)

and

get:

j~2

j

^q~

^{cos~ 4.}

¹³²¹

o

This free energy

density

is to be minimized with

respect

to q,

#

and

p(r).

Of

particular impor-

tance are the terms which

couple

the order

parameters

ⁱⁿ

(28).

With _our choice of the order

parameter q«p(z),

the first _{non zero} term

corresponds

to the _cl coefficient of

equation (26).

Moreover,

_on

physical grounds,

this coefficient must be

positive,

since _an increase of the

density

is

expected

to favor the _appearance of _a

liquid crystal type

^of

ordering. Furthermore,

the order in

p(r)

_can be shown to be uniform

(k

= 0

ordering).

Due to the

large

number of coefficients in the free _energy

density,

it is

appropriate

to work with dimensionless

quantities,

and _we shall

follow the notations of reference

[15].

^{The rescaled total free} energy per unit volume

f

reads:

with

q

~ ~3

Q

²

b(/~

vl

_b4 ^{' P}

j

i/3 ^~,

(34)

4a3

(~ ~°), t'

=

+f(T To), (35j

3

a2b4

b2

~ ~

a(~~b~~~

~

18

/2b4 (36)

~ ~ 18

k( ^~~~

_~~

=

~~~~~~~~

~3 ^' 12 ^°'

(37)

and

3 ³~3

f~~

#

? _{f. (38)}

2 b~

We

just quote

here the conclusions of _a

study

of

equation (33).

Technical details will _appear elsewhere

[18].

^The

phase diagram

can be studied in the

(temperature, chirality) plane.

For

high enough

values of the

chirality

_{parameter t~,}

~y2 '~~

~

32+f(6t~2 (To Toll

^~

^~'

^~~~~

we

get

a

contiquous

transition towards _a biaxial helix _at

a reduced

temperature t~

₌ t~~

corresponding

to a real

temperature

T~

₌

To

₊ 6t~~.

(40)

Below the

transition,

the reduced

density

_grows as c c~

Q~.

The low

temperature phase

_can

be viewed _{as a}

compact phase

with helical order. From _a

protein-oriented point

of

view,

this would

correspond

to a direct transition from _a coil state

(denatured)

to _a native state

ii.

_e.

compact

with

secondary structures).

At much lower

chirality,

when

T~

<

To,

the scenario is _more like the _one in Section 2: there is first _a 6

collapse

transition at

t[

=

0,

followed at a lower

temperature by

_a discontinuous transition towards _a "uniaxial" helix.

At intermediate

chiralities,

several

phase diagrams

_are

possible, depending

on the relative values of the

parameters.

In

particular,

the location of the

phase

transition between the

compact

^modulated

phases (biaxial

and

uniaxial)

is very model

dependent.

In

Figure I,

_we

display

a

plausible phase diagram.

N°10 PROTEIN FOLDING AND BLUE PHASES ₁₂₀₉

T

~ AG2

IG

AGI

K

Fig.

I. Schematic

phase diagram

of the model in the

(temperature, chirality) plane.

The phases are

Iabelled

by

C

(coil),

IG

(isotropic globule),

AGI and AG2

(anisotropic globules).

The C-IG transition, and the C-AG2 transition

(for large enough chirality),

_are continuous.

5. Conclusion

Using

the fact that

dipolar

interactions

imply

the existence of _a tensorial order

parameter (see Eq. (16)),

_we have shown that chiral chains _may have

isotropic (9-like)

_or

anisotropic (blue phase-like) collapse

transitions. This result _was found in _a

simple

model with helicoidal order.

The connexion with blue

phases

is

perhaps

not very

surprising

since the molecules of _some of these

compounds

have

dipoles perpendicular

to the molecular axis. It is clear that the _same

questions

that arise in the

study

of blue

phases

should

arise,

mutatis mutandis, in the

study

of chiral

dipolar

chains:

iii

For very chiral

systems,

^{is the continuous transition towards the}

compact

^biaxial

phase preempted by

a discontinuous transition to a

compact phase

with _a

(hexagonal, body-

centered

cubic,.. superposition

^of biaxial helices?

iii)

Can _one find _a

description

of low

chirality systems

in terms of double twist

cylinders?

(iii)

Can _one link the order in

q(r),

the chain

configuration(s)

and the

dipole configuration(s)?

Does _a

crystalline

order in

q(r) imply

the existence of defects in the orientation of the chain?

The

application

of the

previous

results to

(bio)polymers

_can be

roughly

summarized _as follows. Since _{we are} concerned with

collapsed phases

with uniform

density,

_{we are}

typically

in _a melt situation. The

study

of _a

single

chain

(such

_{as a}

protein)

is

presently

out of reach. In both _{cases, a more} detailed

analysis requires

_a better

experimental knowledge

of the various

parameters (chirality, .).

Nevertheless, _we think that _our

approach

is useful for the

protein

folding problem.

In

particular _[18],

_{one may} look for _a double twist

interpretation

of a-helices

and

p-sheets,

the

possible

existence of localized instabilities

(such

_as the _ones in BP III

[19])

and their

implications

for the

folding

_process.

Along

these

lines,

it is

reassuring (although

not

completely unexpected)

to note that the

secondary

structures of real

proteins

_are well characterized

by

_a

q(r)

tensorial order

parameter [20].

Acknowledgments

We thank Marc

Delarue,

3ean-Renaud

Garel,

Roland Netz and Pawel Pieranski for

pleasant

and

stimulating

discussions.

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Proteins

(W.H. Freeman,

New

York, 1984).

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