Equilibrium distribution of shapes for linear and star macromolecules

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Equilibrium distribution of shapes for linear and star macromolecules

Joel Cannon, Joseph Aronovitz, Paul Goldbart

To cite this version:

Joel Cannon, Joseph Aronovitz, Paul Goldbart. Equilibrium distribution of shapes for linear and star macromolecules. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.629-645. �10.1051/jp1:1991159�.

�jpa-00246358�

Classification

Physics

Abstracts

05.20 02.70 61.41

Equilibrium distribution of shapes for linear and star macromolecules

Joel W. Cannon ('>^2>

*), Joseph

A. Aronovitz _('>

**)

and Paul Goldbart

(',

₂₎

(1)

Department

of

Physics, University

of Illinois at

Urbana-Champaign,

1II 0 West Green Street, Urbana, Illinois 61801, U-S-A-

f)

Materials Research

Laboratory, University

of Illinois at

Urbana-Champaign,

104 South Goodwin Avenue, Urbana, Illinois 61801, U-S-A-

(Received

3 December 1990,

accepted

5

February 1991)

Rksom6. Nous examinons la distribution I

l'bquilibre

des formes tri-dimensionnelles

prises

_par des macromolkcules isolbes de type ^linbaire et embranchfies, tant _avec que sans interactions

intramolbculaires, I l'aide de la mdthode Monte Carlo

appropride

_aux macromolbcules _en dtoile.

Nous calculons la fonction de distribution de

probabilitd conjointe

_pour deux

quantitds qui,

ensemble, caractdrisent les propridtds invariantes du tenseur de gyration normalisd

qui

est associd I la forme

~plut6t qu'i

la

taille)

des

configurations

macromolbculaires. La connaissance de _cette fonction de distribution nous permet entre autres de calculer les valeurs moyennes

(Ao)

et

($)

introduites par Aronovitz et Nelson

(J. Phys.

France 47

(1986) 1445)

_pour caractdriser le

degrd

et la nature de

l'anisotropie

des formes

typiques

extraites de l'ensemble des

configurations

macromoldculaires. Nous calculons

dgalement

_une troisidme valeur moyenne,

(30), qui

isoJe la

nature de

J'anisotropie

de _son

degrb-

Notre simulation permet de

plus

_une

comparaison

de

(do )

et

(So)

avec les

quantitds

moins natureJJes

(mars

^traitabJe

anaJytiquement)

A,

l'asphJricitJ

examinbe par Rudnick et

Gaspari

(J.

Phys.

A19

(J986)

L191) et par Aronovitz et Nelson, et S, exarninb _par Aronovitz et Nelson,

qui possddent

_une sensitivitd _{accrue envers} les

larges configurations,

et donc convoluent information de forme _avec information de taille. II

apparait

que si A et S fournissent bien _une certaine _mesure de

l'anisotropie

des macromoldcules, _ces

quantitbs

diffdrent considbrablement des _mesures naturelles

(do )

et

($).

En

particulier,

si A et S sont

regardbes

Comme des

approximations

I

(Ao)

et

(So)

alors _ces

quantitds

sous-estiment

sdvdrement

l'augmentation (ou

sur-estirnent la

diminution)

_en

degrd

et

allongement

de

l'anisotropie

due aux interactions intramoldculaires, et ce aussi bien pour les macromoldcules lindaires que embranchdes.

Abstract. We

investigate

the

equilibrium

distribution of three-dimensional

shapes adopted by

isolated linear and

star-shaped

macromolecules, both with and without intramolecular interac- tions,

using

_an

implementation

of the Monte Carlo method suitable for macromolecules with

branches. We compute the

joint probability

distribution function for two

quantifies

which

together

characterise invariant features of the normalised radius of

gyration

tensor associated

(*)

Current address:

Department

of

Physics

and

Astronomy,

Calvin

College,

Grand

Rapids,

MI 49506, U-S-A-

(**)

Curreni address MIT Lincoln

Laboratory,

244 Wood Street,

Lexington,

MA 02173, U.S.A.

with the

shape (rather

than

size)

of macromolecular

configurations. Amongst

other

things, knowledge

of this distribution function allows _us to compute ^the

expectation

values

(Ao)

and

($)

introduced

by

Aronovitz and Nelson

(J. Phys.

France 47

(1986) 1445)

to characterise the

extent and _nature of

anisotropy

of

typical shapes

drawn from the ensemble of macromolecular

configurations.

We also Compute a third

expectation

value

( lo )

which isolates the nature of the

anisotropy

from its extent. Furthermore, _our simulation

permits

_a comparison of

(Ao)

and

(6~)

with the less natural

(but analytically tractable)

alternative

quantities

_A, the

asphericity

examined

by

Rudnick and

Gaspari

(J.

Phys.

A 19

(1986) L191)

and by Aronovitz and Nelson, and S, examined

by

Aronovitz and Nelson, which have enhanced

sensitivity

to

larger Configurations

and therefore convolve

shape

information with size information. It is found that

although

A and S do

provide

some characterisation of

anisotropy, they

differ

considerably

from the natural

measures

(Ao)

and

($)-

In

particular,

if A and S _are

regarded

as

approximations

to

(Ao)

and

(6~)

then, for both linear and branched

macromolecules, they severely

underestimate the increase

(or

overestimate the

decrease)

in extent and prolateness of anisotropy due to

intramolecular interactions.

1. Introduction and overview.

Whilst many

aspects

^{of the}

equilibrium

distribution of the three-dimensional

configurations adopted by asymptotically long

macromolecules have been

thoroughly analysed

_over the last two decades

using

renormalisation _group methods

[I], significant questions

remain

concerning

the statistical distribution of the

shapes

of these

configurations and,

in

particular,

the effect of

intramolecular

interactions,

or

self-avoidance,

_on tills distribution.

Investigations

of the distribution of

shapes

have focused _on

Q;y,

^{the radius of}

gyration

tensor

computed

about the centre of _mass, defined

by

~/ j~~ j~ j~~~ j~ j~j

ⁱ

jj

_{~~ ~a} ⁱ

jj

_~~

~p

~~ ~~

iJ" I I J J

~p

i J

p

^{I J'}

a=I a,fl=1

where

r;°

^{is the} I-th cartesian

component

of the

position

vector of the a-th _monomer in _a macromolecule

containing

N _monomers, and the square brackets

[...]

represent an average

N

taken _over the N _monomers of the

macromolecule, I-e-, f]

m

(I IN) £ f(r~).

a I

Early investigations [2] typically

used Monte Carlo

techniques

to compute

quantities

such

as

(A~~~)/(A~i~),

where A~~~ and

A~i~

are the maximal and minimal

eigenvalues

of

Q~

^for a

given configuration

of the

macromolecule,

and

(. )

_represents

averaging

_{over an}

appropriate

ensemble of

configurations. Progress

_was made when it _was realised that invariant

polynomials

built from the cartesian components

Q,y provide

_an

elegant

characteris- ation of the

shape

of _a macromolecular

configuration.

As such

polynomials

_can be

computed

without

explicitly diagonalising Q,j ^they

can be treated

analytically,

and it is _upon

them,

and related characterisations of

anisotropy,

that _we shall focus in the present paper.

Very recently,

the

shapes

of star and linear macromolecules have been

investigated using

molecular

dynamics by Bishop

and Clarke

[3].

^{These authors focus} on one

particular

_measure of

anisotropy,

A

(to

be defined

below),

which is sensitive to overall macromolecular

size,

_an issue which _we discuss in detail below. Amovitz

[4]

has

investigated aspects

of the

shapes

of

linear macromolecules without self-avoidance

using

the Monte Carlo

method, obtaining

results consistent with those

presented

here.

Honeycutt

and Thirumalai [5] ^{have also}

investigated

certain

aspects

of the

shapes

^of linear macromolecules

using

the Monte Carlo

method,

also

obtaining

values consistent with _ours.

Analytical investigations

of macromolecular

shapes

have been

performed

via both _an

e(m

4 d

)-expansion by

Aronovitz and Nelson

[6] (referred

to _as

AN~

and _a

I/d-expansion by Rudnick, Beldjenna

and

Gaspari _[7] (referred

to _as

RBG),

where d is the dimension of the

space in which the macromolecules _move. The

s-expansion

results _are of

particular

interest because the

O(e)

corrections

give

_{a measure} of the effects of

self-avoidance,

whilst the

I/d-expansion explores

the

neighbourhood

of d

= co, where self-avoidance is irrelevant. In

addition,

Diehl and

Eisenriegler [25]

have calculated the _mean

asphericities

for open and closed

non-interacting

macromolecules in

arbitrary

_space dimensions. The relevant

portion

of

our numerical simulation is consistent with their

analytical

results.

In this _{paper we} will

report

^{results of} a Monte Carlo simulation from which _{we extract}

probability density

functions

describing

the statistical distribution of three-dimensional

shapes adopted by

linear and branched

macromolecules, independent

of the overall size of each

configuration.

We

emphasize

that such distribution functions

provide

_a

significantly

_more detailed characterisation of the

equilibrium

distribution of macromolecular

shapes

than do any of their

particular expectation

values. From these distribution functions _{one can} construct

equilibrium expectation

values of

quantities characterising

the

shapes

that macromolecules

adopt,

such _as the

(size-independent) anisotropy,

and its nature

(I.e.

the

typical prolateness

_or

oblateness)

_{one can} also construct

higher

moments and correlations

[8].

The information

contained in these distribution functions

provides

_a clear

representation

of the

impact

of both intramolecular interactions and the number of branches _on the distribution of macromolecular

shapes,

and allows _us to present a

qualitative

_as well _as

quantitative description

of the roles

played by

intramolecular interactions and the number of branches in

determining shape

statistics.

There _are many

related,

but not

equivalent, quantities

that _one

might

choose to

characterise the distribution of

shapes adopted by

macromolecules.

However,

certain of these

measures are less formidable _to

compute analytically

than

others,

_an issue which motivated

AN and RBG to select for

analysis

two

particular

_measures

A,

which characterises the

degree

of

anisotropy,

and

S,

which characterises its nature, both of which will be described in detail below

[9]. Through

this choice _{one can} avoid _{a severe} technical difficult associated with the evaluation of averages of

quotients

of fields

and, instead,

face the

simpler

task of

evaluating quotients

of averages of fields.

Schematically,

_{one can}

regard

this

approach

_as a method for

approximating

_purer

(but technically

_more

awkward)

_measures of

anisotropy,

of the form

(a/b), by simpler

_ones, of the form

jai lib ),

which _one then

computes,

as AN

did,

_e.g., within the

e-expansion. Alternatively,

_{one can}

regard

this

approach

_as

simply choosing

_a

(size-dependent)

distinct characterisation of the distribution of

shapes.

When

applied

to the

measures of

anisotropy

A and

S,

the

s-expansion

results suggest ^{that for linear macromol-}

ecules in three

spatial

dimensions the

impact

^of self-avoidance is

quite small,

of order _one percent. ^This conclusion is not borne out

by

_our Monte Carlo data _on

size-independent shape

characterisations of linear macromolecules _nor is it bome out for branched macromolecules.

As _we shall _see

below,

there is _a

particularly

natural way, described

by AN,

to isolate information

concerning

the distribution of

shapes adopted by

macromolecules from the distribution of sizes. This leads to the two measures

(do)

and

(So),

described

below,

defined in terms of the normalised and shifted

eigenvalues

of the radius of

gyration

tensor. If _one asserts that macromolecular

configurations

whose radius of

gyration

tensors differ

only by

_an

overall scale factor should be considered

equally anisotropic

then

do

^and

(So

are natural

(or pure)

_measures of

anisotropy,

uncontaminated

by

the distribution of macromolecular

sizes.

We shall also introduce _a further characterisation of the distribution of

shapes

(30)

which is not

only independent

of the overall size of

configurations

but also of the extent

of the

anisotropy.

Thus

(lo)

is _{a measure}

only

of the _nature of the

anisotropy, by

which _we

mean its oblateness _or

prolateness.

The

replacement

of the _measures

(do)

and

(So) by

A and S to facilitate

analysis

is _a

replacement

of _measures insensitive to the overall size of _a

configuration by

_measures sensitive to the overall size. In this _sense it is _a

replacement

of _{pure measures} of

shape by impure

_ones,

not

strictly

determined

by

the macromolecular

shapes

alone. As _a consequence of this strategy, ^the contribution to _{a measure} from each

configuration

is biased

by

its overall

size, larger configurations being preferentially weighted

_over smaller _ones.

Now consider the

impact

of intramolecular interactions

(or self-avoidance)

_on the distribution of

shapes adopted by

macromolecules in three dimensions, It is

plausible

to

assume that of two macromolecular

configurations

with the _same

shape

but different size

(I.e.

with radius of

gyration

tensors

differing only by

_an overall

scale)

the

larger

_one will

experience

less self-interaction than the smaller _one.

Hence,

if _measures of

shape

_are

preferentially

biased towards

larger configurations,

then such _measure

might

be less sensitive to the

incorporation

of intramolecular interactions.

Therefore,

if the _measures A and

S, analysed by AN,

_are

regarded

_as

approximations

to the pure measures of

anisotropy (Ao)

and

(So)

then

they

should not be

expected

to

provide precise

reflections of the

impact

of intramolecular interactions _on the distribution of macromolecular

shapes

in three dimensions _;

instead,

_one

might expect

^them to underestimate the true

strength

of the effect of self-avoidance _on the distribution of

shapes.

In

fact,

_our Monte Carlo data indicate

that,

for both linear and branched

macromolecules,

self-avoidance _{causes a} fractional

change

in the

impure

_measures which

consistenly

and

severely

underestimates the increase

(or

overesti-

mates the

decrease)

in both the extent and

prolateness

of

anisotropy, compared

with the

changes

incurred in pure measures.

We address the issue of the

impact

of self-avoidance _on the distribution of macromolecular

shapes

in three dimensions

using

the Monte Carlo method which allows _us to

compute joint

distribution functions P for _a

pair

of variables _p and _@, defined

below,

which characterise the distribution of

eigenvalues

of the norrnalised radius of

gyration

tensor

(Tr Q)~ _Q,~.

These distribution functions allow _us to

compute

natural _measures of

anisotropy, provided

_{we are}

not interested in

quantities dependent

_on the overall size of macromolecular

configurations.

By incorporating

information

conceming

^{the overall size of} macromolecular

configurations, through

the absolute

(rather

than

relative)

size of the

eigenvalues

of

Q;j,

we are also able to

investigate

the alternative _measures of

anisotropy

examined

by

AN and RBG

(I.e. quotients

of

averages) and, thus,

to estimate the extent to which these

quantifies

_may be

regarded

_as

approximations

to

size-independent

characterisations of

anisotropy.

To

probe

further the effects of

self-avoidance,

_we also examine three- and four-armed

macromolecules, I.e.,

star-

shaped polymers consisting

of _a certain number of _arms of

equal length,

all constrained at _one

end to meet at a common

(but mobile)

branch

point.

As _we shall _see, the

impact

of self-

avoidance _on macromolecular

shape depends qualitatively,

_as well _as

quantitatively,

_upon the number of _arms of the star.

2. The characterisation of macromolecular

shapes.

We

begin by reviewing

the characterisation of three-dimensional macromolecular

shapes,

_as described

by AN,

whose

goal

was to construct concise _measures

reflecting

the _nature of the

anisotropy

in the ensemble of macromolecular

configurations.

The

eigenvalues (A~)

of

Q,j ^give

^direct

^insight

^{into the}

^anisotropy

^of a

particular configuration

of _a macromolecule.

For

example,

if the

configuration

is

isotropic,

at least to within the

resolving

_power of _a second rank tensor, then the

eigenvalues

of

Q;y

are all

equal. Thus,

to

explore anisotropy

it is

useful _to define the shifted tensor

fi,~

m

Q;~ (1/3) 3,j

^Tr

Q, I-e-,

the de-traced version of

Q;y,

^with

eigenvalues (I

~ ⁼

A

~

i ),

where

I

m

(1/3)

Tr

Q

is the _mean

eigenvalue

of

Q;y

^{for a}

particular configuration.

Then _a characterisation of the extent of the

anisotropy

of _a

3

configuration

is

given by TrQ~

=

£ (A~- A)~, i-e-, (three times)

the variance of the

a= I

eigenvalues

of

Q,y.

^This measure of the extent of the

anisotropy

is made

independent

of the overall size of the macromolecule

by dividing by i~,

_thus

_producing

_the normalised variance

do

_w ^~~

~~~ ^(2.I)

(Tr Q )

The factor of

3/2

is inserted to normalise Ao so that 0 _w

do

_w I. It is also useful to define _{as a} second characterisation of the

anisotropy

the

quantity

6~ _m ²⁷

~~~

~~, (2.2)

(Tr Q)

which satisfies the bounds

I/4w Sow 2,

and

gives

_{a measure} of the

prolateness

_or

oblateness of _a

particular configuration.

To _see

this,

note that in _a

prolate configuration (for

which

hi »A~m

A~)

ii

_is

_positive

_whilst

i~

_and

i~

_are

_negative,

and

consequently

6~ ^is

positive. Conversely,

in _an oblate

configuration (for

which

hi

« A~ m A~)

ii

is

negative

whilst

i~

_and

13

_are

positive, causing

_So to be

negative.

In fact _{we can}

improve

So

by eliminating

its

sensitivity

to the _extent of

anisotropy,

_as measured

by do, arriving

at the

measure

Io

m

(~~.(j

~,~^,

(2.3)

r

Q

3

which satisfies the bounds I _w

lo

_w I

and,

in _{a sense} to be made

precise below,

is sensitive

only

to the nature of the

anisotropy (I.e.

oblateness or

prolateness)

and not to its extent.

In the _same way that the

expectation

value of the radius of

gyration

characteri~es the size of

a

macromolecule,

it is reasonable to characterise the

shape by

the _mean values of the

quotients do

and So

(or Io), averaged

_over the ensemble of

possible

macromolecular

configurations. However,

if _one

adopts

de Gennes'

m(- 0)-component magnet analogy [10]

then these

quantities

_are

represented

_as the _average of _a

quotient

of

products

of fields

and, consequently,

_{are very} difficult to calculate

analytically.

To circumvent this

problem

and obtain _an

analytic

characterisation of the distribution of

shapes corresponding

to

do

and 6~, ^{AN chose to}

approximate

the averages of the

quotients

in

equations (2.I)

and

(2.2) by

quotients

of _averages,

I-e-,

3

(Trfi~)

0wAm-

~

ml,

~

~~~~ ~~

(2.4)

(Detfi)

-~«Sm27

«2.

((TrQ))

Whilst these _are not pure measures of

anisotropy,

since

they

_are biased towards

larger configurations,

the

hope

_was that the distribution of

Q

is such that A and S would

approximate

JOURNAL DE PHYSIQUE i T I, M 5, MAT 1991 ?7

(Ao)

and

(6~) sufficiently closely

to

give

_a useful _measure of the

typical

character of macromolecular

anisotropy and,

in

particular,

its

sensitivity

to intramolecular interactions.

RBG chose to

study

the

quantity A,

which

they

termed the

asphericity, directly,

rather than

regarding

it has _some

approximation.

One

might

also compute the size- and

anisotropy- dependent prolateness,

4

(Det 4

w I

m ~,~ ^«

l

,

(2.5)

~₃ Tr

4~)

but _we have chosen not to do this.

Monte Carlo simulation of this

problem

allows _a much _more detailed

investigation

of the

question

of macromolecular

shapes.

Since

diagonalisation

of

Q

at each

step

^is not _a

difficulty,

the _mean values of

do

and 6~ can be calculated

explicitly,

rather than

relying

_on the alternative

measures A and S. Within this context, a natural first

question

to ask _concerns the

utility

of

these alternative

quantities

calculated-

by

AN. More

importantly,

_{we can} calculate detailed information about the

equilibrium

distribution of

shapes,

without bias from size.

In

particular,

we shall focus _{upon a} distribution

function, P, equivalent

to the distribution function for the

eigenvalues

of the rescaled and de-traced radius of

gyration

tensor

(Tr Q )~ 4;y.

^These

eigenvalues

can be

expressed

in _a

compact fashion, following

Alban

[I I].

As

4;~

^is

^traceless, only

two

parameters

are needed to encode its three

eigenvalues.

We represent ^the

eigenvalues of11;y using

the variables _p and which have the

geometric representation

shown in

figure

I. In this scheme the

eigenvalues of11;j

_are

represented

_as the

projections

on to the x-axis of three two-dimensional _vectors of

equal magnitude

i~

and

equal

relative

angles 2gr/3,

_so that

hi

₌

I(I

_{+ p cos}

(@)),

A~ =

I(i

_{+ p cos}

_(o

₊

_2«/3)), (2.6)

A~ ₌

I(I

_{+ p cos (@}

2ar/3)).

ip

_measures the

strength

of the

anisotropy,

whilst indicates whether the

configuration

is

prolate (0

_{~ ~}

gr/6,

I-e-

cigar-like)

_or oblate

(gr/6

_{< <}

gr/3,

I.e.

pancake-like).

As _{we are} interested in the

shape

of the

macromolecule,

rather than its absolute

size,

_we have

l~j

^P

, z _x3~

xi

,

o j ^St

Fig.

I. Geometric characterisation of the

eigenvalues

of

Q,j.

^The

eigenvalues

of the traceless matrix

(TrQ

)~^'

l~,j,

^which can be used to characterise the relative

anisotropy

of _an

object,

can be

represented by

the two parameters p and _{@. p}

gives

_{a measure} of relative

anisotropy

of the

eigenvalues

of l~,j~P = 0 _~

isotropy,

and _p

=

2 _~ extreme

anisotropy)

_; S indicates the nature of the

anisotropy

(0

_~ S

<

gr/6

_~

prolate,

and

w/6

_< S

<

w/3

_~

oblate). Equations (2.6) give

the

relationship

between

(I,

_p, S and the

eigenvalues

of l~,j.

introduced the relative

anisotropy strength

_p,

I.e., ip

norrnalised

by

the radius of

gyration I.

_{In terms of this}

parametrisation

the characterisations of

anisotropy

become

do

_{= p}

~/4

,

So

₌ p^~ cos

(@)

sin

(gr/6 )

sin

(gr/6

₊

)

,

(2.7) Io

= 4 _cos

(@)

sin

(gr/6 )

sin

(gr/6

₊

)

Notice that

Io

is

independent

of _p, whilst

do

is

independent

of

@,

I.e., Io

_measures

only

the nature of the

anisotropy,

whilst

do

measures

only

its extent. One

might

choose to characterise the distribution of

shapes by (Ao)

and

(Io),

rather than

(Ao)

and

(So),

since this would

isolate the _nature of the

anisotropy

from its _extent. These

(and

_any

other) shape-independent

characterisations of

anisotropy

_may be constructed

using

the distribution functions

given

in this paper.

The

joint probability

distribution function for macromolecular

shapes

P is defined

by

P(r, t)

m

j3 (r R(C))

3

(t T(C))j

=

jr

_sin

_(t)

_sin

_{(ar/3 t)}

_sin

_(ar/3

+

t)

x

j3 (r~/4 Ao(C))

3

(4

_cos

(t)

sin

(ar/6 t)

sin

(ar/6

₊

t) Io(C)) j

,

(2.8)

where R and T _are the

appropriate

values of _p and for the

configuration C,

do

and

Io

_are

quantities

to be evaluated from the radius of

gyration

tensor of

C, (... )

denotes _{an average over} the ensemble of

configurations,

and the factor outside the

second set of

angular

brackets is the

appropriate jacobian

determinant.

Distribution functions with the form of

P(p, )

contain all the information _necessary to

implement

_our

description

of macromolecular

shapes

and it is these which _we

compute using

the Monte Carlo method. From them _{we can}

obtain,

_e-g-, the _mean values and moments of

do

and

So,

as well _as those of _any other

quantities,

such _as

Io,

which

depend only

_on the relative amount and nature of

anisotropy

in the

eigenvalues

of

Q;y.

^{Provided we}

incorporate

some further information

conceming

the overall size of

configurations

_{we can} also

compute

the

quantities

A and S considered

by

AN and RBG.

One feature of this characterisation which should be noted is that the maximum

possi.ble

value of _p is

b-dependent.

To _see

this,

note that the

eigenvalues

of

Q;j

^must

^always

^be non-

negative

_as

_Q;y

is _a radius of

gyration

tensor about the centre of _mass.

Thus,

for _a

given

value of @, the maximum value of _{p occurs} when _one of the A

~

is _zero

(I.e.

_A~, in the _case of

Fig. I),

and thus 0 _{« p «}

p~~~(@)

=

I/cos (gr/3 @),

as shown in

figure

2.

20 40 60

6

Fig.

2. p~~~ versus S. The maximum value that _{p can} attain, for _a

given

value of S,

according

to the

characterisation

depicted

in

figure1.

3.

Description

of the

computational

model.

We have

performed

_our calculations

using

_a standard lattice model of _a macromolecule

[12, 13]

^{which has} been extended to model branched

(or cross-linked)

macromolecules

[14].

The

model consists of _a

self-interacting

chain _{on a} three-dimensional cubic

lattice,

the chain

comprising

N

nodes,

connected

by

N I bonds

(I.e. steps).

The nodes occupy sites of the

lattice,

and interact

through

_an excluded volume contact

potential v(ri, r~)

₌

it(ri _r~)

(where I(r)

_is

a Kronecker delta

function,

and _we have chosen units in which kB ^T =

I).

Figure

3 shows _an

example

of _a segment of such _a

chain,

and _a

possible change

in the

configuration (dotted line)

which

might

be made

during

_a Monte Carlo

update.

To allow efficient

implementation

of the Monte Carlo

procedure,

_we consider

only changes

in the chain

configuration

which _can be

accomplished by moving

_a

single

node. One feature of _our

implementation

is the _use of _a finite

(rather

than

infinite,

_or

hard-core)

excluded volume

interaction,

and the related _use of the heat bath

algorithm [15]

to

update

the

system.

^This allows faster

equilibration,

avoids

possible non-ergodicities

associated with the

self-avoiding

chain

[16] (f

= co

),

and makes it

possible

to simulate

efficiently

cross-linked macromolecules.

For

sufficiently large [17]

fl all models of flexible macromolecules _are believed to

belong

to the

same

universality

class

and, thus,

to exhibit similar

scaling properties [18, 19, 20].

we found that _a value of I

=

4.5

reproduced

known

scaling

results in reasonable

computation

times _; this value is used for all calculations with

self-avoiding

chains

reported

here.

We _now describe _our method for

simulating

cross-linked macromolecules. Due to the translational invariance of the lattice it is not necessary ^to simulate the motion of the _cross- link of _a branched macromolecule.

However,

since this method

provides

_an alternative to the

limited number of other methods

currently

available for

simulating

branched macromolecules

or cross-linked networks

[21]

we will present some of the details here.

We model the

branch-point

of _a macromolecule

by constraining

nodes from

(at least)

two different chains to lie at the _same

spatial

location

(referred

to _as the

cross-link).

The cross-link is then moved

by

the _same rule that is used for the

ordinary

nodes _{a move} is

possible

if the

node _can be moved without

changing

the

position

of any other _monomers. To observe how _a cross-link node

might

be

moved,

consider the

possible

_sequence of

configurations

shown in

figure

4.

Initially,

in

(a),

all sites _are

singly occupied,

and the

position

of the cross-link cannot

change by

itself. At _some later time the _{two moves,} indicated in

(a) by

the _arrows, have occurred and it is _now

possible

for the cross-fink _{to move} without

moving

_any additional nodes. This _can result in the

change,in

cross-link

position

between

(b)

and

(c). Finally,

_some

number of Monte Carlo steps

later,

the

possible

random _moves indicated in

(c)

^have

again

Fig. 3. A lattice model of _a macromolecule,

including

_a

possible change

in the

configuration

(denoted by dotted

lines)

that

might

_occur during Monte Carlo simulation. Notice that only one node is moved

during

the shown change,

making

it

simple

to

implement

computationally.

Although

this

depiction

of

the nature of the model is two-dimensional, all the results

reported

here _concem three-dimensional

simulations.

c

Fig.

4, A

possible

_sequence of

configurations by

which _a cross-fink

might

_move in _our Monte Carlo scheme which

requires

that all basic

configurational changes

involve

only

a

single

node.

Initially (Fig. a)

all sites _are

singly occupied,

and the

position

of the cross-fink _cannot

change independently.

At _some

later time, the two moves indicated

by

the arrows in

figure

_a have occurred, and it is now

possible

for the cross-fink to be moved without

moving

_any additional nodes

(Figs.

b and

c), Finally,

the _moves indicated in

figure

_c have

again

resulted in _a situation where there _are

doubly occupied

sites, and it is not

possible for the cross-fink

position

to

change independently.

To facilitate the motion of the cross-link _we

have set to zero the monomer-monomer interaction associated with those node

directly

connected to the

cross-link, thus

allowing configurations

such _as those shown in

figures

b and _c to occur more

frequently.

resulted in _a situation where there _{are no}

doubly occupied sites,

and it is not

possible

for the cross-link

position

to

change independently. Thus, through

_{a sequence} of

simple changes involving only

_one node at _a

time,

a new

configuration

can be reached in which the cross-link has _a different location.

This

procedure produces

_very slow motion of the

cross-link,

since

multiply occupied configurations,

such _as those in

figures

4b and

4c,

_are

suppressed by

the excluded volume

interactions. To

improve

this

behaviour,

_we set the excluded volume interaction between nodes

adjacent

to cross-links to _zero. With this

approach

the

configurations

shown in

figures

4b and 4c become _more

probable and, therefore,

cross-link nodes diffuse _more

freely,

whilst still

preserving

the connectedness of the chain.

Furthermore,

the chains also

respect topology,

in the _sense that

they

do _not

freely

_pass

through

each other

and,

for

large

§,

cannot pass

through

each other at all.

Topology

is not _{a concem} for

star-shaped

macromolecules,

but is _an

important

issue for the

modelling

of cross-linked networks. As

asymptotic properties

should not

depend

_on the

microscopic

details of the

model,

_our modification of the excluded volume interaction _{near to} the cross-link should not affect the

long-distance

behaviour _on which _we wish to focus.

4.

Qualitative

and

quantitative

results.

In this section _we shall describe the

qualitative

and

quantitative

results of _our Monte Carlo simulation

and,

in the _case of linear

macromolecules,

_compare them with conclusions drawn from the

analytical

^{work of} AN. As _we shall

repeatedly

be

comparing

data from simulations of macromolecules in the presence and absence of intramolecular interactions it will be

convenient _to introduce two acronyms:

(I)RW

will refer to macromolecules without

intramolecular

interactions, performing

free random walks and

(it)

SAW will refer to macromolecules with intramolecular

interactions, performing self-avoiding

random walks.

Furthermore,

_{as we} shall be

comparing

the

analytical

results of Aronovitz and Nelson with the results of Monte Carlo simulations _we shall _use the labels AN and MC to

distinguish

them.

4.I LINEAR MACROMOLECULES. In this subsection _we

analyse

_our results

concerning

the

distribution of

shapes

of linear

macromolecules,

and _compare them with the results obtained

by

AN.

Figures 5a,

5b and 5c

display

_{the p-@}

probability

distributions for RW and SAW calculated for linear macromolecules. These distributions _were calculated

using

chains of

length

N

= 32

nodes,

and each calculation consisted of 20 sets of 100000 Monte Carlo sweeps. The norrnalised distributions _were constructed

by collecting

the p-@ data in _{a two-}

a

P

30 60

(degreesl

b

P

30 60

6

(degrees)

c

P

30 60

6

(degreesl

Fig.

5. p-S distribution functions for _a linear macromolecule. The RW _case is shown in 5a, the SAW

case in 5b, and the difference

(RW

subtracted from

SAIV~

in 5c. The

shading

indicates the

region

where the RW distribution function exceeds the SAW distribution function. The figures _are generated from _a 20 _x 20 bin

histogram

of data obtained

through

Monte Carlo simulation. The contour fines

ccrrespond

to 0.0167 differences in

probability density

in 5a and

5b,

and 0.0067 in 5c. Note that the distributions favours

shapes

which _are rather

anisotropic

_~p

= 0 _~

isotropy)

and rather

prolate (S

=

0 _~ extreme

prolateness),

even in the RW case. The elTect of interactions is to shift the

weight

to even more

anisotropic

regions ^of the distribution. Note the

geometric

constraint

imposed

on

possible

values of _p and

depicted

in

figure

2.

dimensional

histogram consisting

of 20 bins in each of the _p and directions

(I.e.

_a total of 400

bins).

The

smoothing required

to

generate

the

plots

_was

performed using

Mathemati- caTM. As the

plots

_were constructed from

coarse-grained

statistical

data,

the

high frequency

components

^{should be}

ignored.

The

length

N

= 32 _was chosen because it is in the

asymptotic scaling regime

for the

present model,

which _we find to be reached

by

N

=

16,

at least for linear chains with interaction

strength [22]

§

= 4.5.

What _{can we} learn

simply by looking

at the distribution functions

themselves,

rather than

examining

moments ?

Inspection

of the distribution for RW shown in

figure

5a indicates

that,

even in the

non-self-avoiding

_{case, a} linear macromolecule is

typically

close to its maximum

anisotropy

and is

markedly prolate.

The maximum of the distribution is at

(p, @)

=

(1.55,

^4.5°

),

with _p

= 1.55

being quite

close _to

p~~(4.5°)

=

1.77, I.e.,

the maximum value for

=

4.5[ In

fact,

for all values of _@, the maximum of the distribution P

(p,

lies within

approximately

10 9b of

p~~~(@ ).

The SAW distribution shown in

figure

5b is similar to that of the RW. Its maximum _occurs at the _same

point

_as for the

RW,

to within the accuracy of _our

Monte Carlo

data,

and the

shapes

of the distributions _appear similar. The most

pronounced

difference is that the SAW distribution is narrower, and its

peak

is about 20 9b

larger.

As the effects of self-avoidance _are rather

delicate,

the

changes

become _more

apparent

in

figure 5c,

a contour

plot

of the

d@ferences

between the distributions shown in

figures

5a and 5b. Each

contour on this

plot

_{represents a} difference of 0.0067 in

probability density,

with measured

in

degrees.

For

comparison,

the contour fines of

figures

5a and 5b

correspond

to _a difference of 0.0167. The

negative (shaded) region, reflecting shapes

that _are less

probable

for the SAW

than for the

RW,

is broad and shallow

(-

0.0227 at

minimum)

and _occurs in the less

anisotropic portion

of p-@ space

(I.e.

_near smaller values of

p).

The

positive region,

where the

probability

is enhanced

by self-avoidance,

is

relatively

_narrow, confined in

p-b-space

to the

region

of

high anisotropy (see Fig. I)

and

considerably

_more

peaked (0.0733

at

maximum,

or

50 9b of the

peak

RW

probability).

In _{essence, a} small amount of

probability

has been removed from _a

large portion

of the

probability distribution,

and added to the

peak

of the

distribution.

There is _a

plausible physical explanation

for these results. The effect of the intramolecular interactions is to thin the

configuration

_space of linear

macromolecules, eliminating (or,

in the

case of finite

strength interactions, reducing

the

weight on

_any

configurations

with self-

intersection.

Interactions, being

_more

probable

for _{a monomer} surrounded

by

other

monomers, are less

likely

with the

large

surface-to-volume ratios of

anisotropic configur-

ations.

Thus,

the

probabilities

of less

anisotropic configurations

_are

suppressed by

in-

tramolecular

interactions, enhancing

the relative

probability

of

anisotropic regions.

The values of

A~'~

_and

_S~'~,

_the

shape

_parameters for linear macromolecules obtained

by

the

s-expansion

of

AN,

_are

compared

with the Monte Carlo values

A"I (Ao)"~,

S"~

_and

(So)"~

in table I. For this linear _case, chains of

length

N

=

32 nodes _were used to construct the distribution functions shown in

figure

5

however,

for the data

given

in table

I,

we used chains of

length

N

= 64

(RW~,

and N

= 100 nodes

(SAW).

The SAW Monte Carlo

values

Afl§~

"

0.543 _± 0.017 and

6j#§

= 0.879 _± 0.027 agree with AN's

A(flw

= 0.534 and

6j#w

"

0.893,

to within the

uncertainty

of _our calculation

(which

is I _« statistical

error).

In

fact,

the small

O(s)

correction to RW due to intramolecular

interactions,

calculated

by AN,

is of the _same order _{as our} numerical _error. In contrast,

however,

the Monte Carlo data

give

mean values

(Ao)((

= 0.396 and

(So)((

=

0.481,

and

(Ao)C$

=

0.447 and

(6~)C$

=

0.572,

which _are

considerably

smaller than their

size-dependent counterparts All

=

0.526 and

S##

"

0.887,

and

A(i~w

_" ^0.534 and

5j#w

=

0.893. The values of

(A)#(

and

(Sol $$

are

approximately

75 9b and 54 9b of their

counterparts

ARW ^and

S~w, respectively,

Table I.

Comparison of

A and

S,

calculated

by

AN, with A, S,

(Ao)

and

(So),

calculated

using

the Monte Carlo

method, for

the _case

of

linear macromolecules.

(do)

characterises the average

anisotropy (Ao)

₌ 0 _~

isotropy, (Ao)

= ~ extreme

anisotropy)

;

(6~)

^charac-

terizes the _average nature

of

the

anisotropy ( (So)

_< 0 _~

oblate, (6~)

_m 0 _~

prolate).

A and S _can be

regarded

_as

approximations

to

(Ao)

and

(So).

The bracketed numbers _are I-w uncertainties

of

values calculated

using

the Monte Carlo method. The

analytical

values

of

A and S

(due

to

AN)

_agree with those calculated

using

the _monte Carlo

method,

but

d@fer sign#icantly from

the Monte Carlo values

of

their

size-independent

counterparts

(Ao)

and

(So).

A~~ S~~

_A ^~~

S~~ Idol'~~ I _So

^'~~

RW 0.526 0.887 0.529

(0.006)

0.895

(0.019)

0.396

(0.005)

0.481

(0.009)

SAW 0.534 0.893 0.543

(0.017)

0.879

(0.027)

0.447

(0.011)

0.572

(0.025)

indicating

_a

positive

correlation between the overall size of _a

configuration

and the extent of its

anisotropy.

To _see

this,

notice that A

=

(i~)~ (i~ _p~/4),

and

(do)

=

(p~/4)

thus

A

lAo)

=

(i~) (i~ (i~) (p

~/4

ip _~/4) )) ^(4.1)

Perhaps

more

importantly,

A and S do _not

accurately

reflect the

significance

of the

change

in

shape

that results from intramolecular interactions. When intramolecular interactions _are

incorporated,

A

changes by

1.5 9b_; this should be

compared

with _a

change

of

approximately

13 9b in

(Ao) "~. Similarly,

S

changes by

0.6

9b,

whilst

(So)"~ _{changes by}

_{19 9b.}

Thus,

for linear macromolecules it _appears that the _measures of

anisotropy

used

by

AN do

change,

upon inclusion of

interactions,

in _a direction which coincides with the

corresponding changes

in

do

^and So

,

and thus _can be said _to reflect the

impact

of intramolecular interactions _on

the extent and

prolateness

of macromolecular

anisotropy. However,

the _measures A and S

severely

underestimate the effects of

self-avoidance, compared

with the _measures

(Ao)

and

(So ).

To

sumrnarise,

the effect of interactions is

large, causing (Ao)

and

(So)

to increase

by

13 9b and 19

9b, respectively, I-e-,

interactions _cause the

anisotropy

of linear macromolecules to

become

significantly

_greater and _more

prolate.

At least for linear

macromolecules,

it appears that the diminished

sensitivity

of A and S to intramolecular interactions is _a consequence of their enhanced

sensitivity

to

larger configurations,

for which intramolecular interactions _are less

significant.

4.2 BRANCHED MACROMOLECULES. In this subsection _we

analyse

the role

played by

the

number of branches in

determining

macromolecular

shape. Figures

6 and 7

display

the

p -@

probability

distributions calculated for 3-star and 4-star macromolecules. The calculations

were made _on macromolecules of branch _node

length

31

(I.e. containing

totals of 93 and 124

links, respectively).

Each calculation consisted of 20 sets of 100 000 Monte Carlo _sweeps.

As _one

might except,

in both RW and SAW _cases the distribution of macromolecular

shapes

is

strongly

affected

by

the number of branches in _a star macromolecule. Branched

macromolecules become less

anisotropic

and less

prolate

as the number of branches increases.

The decrease in

anisotropy

_can be _seen from the values of

(Ao) (where

0

~

isotropy,

and

~ extreme

anisotropy)

shown in table

II,

for which _we obtain values of

0.396,

0.298 and 0.240 for 2-star

(I.e. linear),

3-star and 4-star

RW,

and

0.447,

0.304 and 0.222 for the

corresponding

SAW. The decrease in

prolateness

can be _seen from the values of

(a) (a)

p p

30 60 30 60

(degrees)

₆

(degrees)

(b) (b)

P p

30 60

6

(degrees)

30 60

6

(degrees)

(cl

(cl

p

P

30 60

6

(degrees)

_{30 60}

6

(degrees)

Fig.

6.

Fig.

7.

Fig.

6. p-S distribution functions for _a 3-star macromolecule. The RW _case is shown in 6a, the SAW

case in 6b, and the difference

(RW

subtracted from

SAW~

in 6c. The

shading

indicates the

region

where the RW distribution function exceeds the SAW distribution function. The _contour fines

correspond

to 0.01 differences in

probability density

in 6a and 6b, and 0.005 in 6c. The distributions _are less

anisotropic,

less

prolate

and _more broad than in the linear _case. Intramolecular interactions result in enhancement of the less

prolate configurations (I.e.

transfer of

weight

to

higher

_S).

Fig.

7. p-S distribution functions for a 4-star macromolecule. The RW _case is shown in 7a, the SAW

case in 7b, and the difference

(RW

subtracted from

SAW~

in 7c. The

shading

indicates the

region

where the RW distribution function exceeds the SAW distribution function. The contour lines

correspond

to 0.0083 differences in

probability density

in 7a and

7b,

and 0.0042 in 7c. The distributions _are less

anisotropic,

less prolate and _more broad than for the linear and 3-star _cases. Intramolecular interactions enhance the less

anisotropic,

less prolate

configurations.

(Io) (where

_~ extreme

prolateness,

and -1 _~ extreme

oblateness),

for which _we obtain values of

0.706,

0.560 and 0.472 for

linear,

3-star and 4-star

RW,

and

0.745,

0.479 and 0.370 for the

corresponding

SAW. The _measure

(So)

^{shows identical trends. Error estimates} are not included in table II for

(Io)

_as these

quantities

_were

computed

from the _coarse-

grained

distribution

functions,

rather than

directly during

the simulation.