Conformational properties of randomly flexible heteropolymers
Pallavi Debnath and Binny J. Cherayil
Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India
Random copolymers made up of subunits with arbritary degrees of flexibility are useful as models
of biomolecules with different kinds of secondary structural motifs. We show that the mean square
end-to-end distance 具 R 2 典 of a two-letter A – B random heteropolymer in which the constituent
polymeric subunits are represented as continuum wormlike chains and the randomness is described
by the two-state Markov process introduced by Fredrickson, Milner, and Leibler 关Macromolecules
25, 6341 共1992兲兴 can be obtained in closed form. The expression for 具 R 2 典 is a function of several
parameters, including the number n of subunits, the fraction f of one kind of subunit, the persistence
lengths l A and l B of the two subunits, and the degree of correlation ␭ between successive subunits.
The variation of 具 R 2 典 with each of these parameters is discussed.
I. INTRODUCTION
Many of the most important biological macromolecules—proteins and polynucleotides in particular—are
copolymers of a small number of chemically distinctive monomeric subunits whose arrangement along the backbone of
the polymer often leads to quite specific three-dimensional
geometries. The relation between sequence and structure in
such molecules is generally complex, but it can sometimes
be inferred from the study of minimal models of the molecule. Random heteropolymers of just two elementary subunits A and B have proven to be especially useful as prototypes of the more complex heteropolymers typical of
biological systems.1 But there are certain recurring structural
motifs in biomolecules that are incorporated into these minimal models only with difficulty. Helices are an example. In
isolation, helices are frequently represented as wormlike
chains using the Kratky–Porod model,2,3 or one of its many
variants.4 When helices are part of a larger complex that may
contain extended regions of complete flexibility distributed
at random, however, an analogous description of the resulting randomly semiflexible polymer is less readily developed.
To varying levels of sophistication, there do exist calculations of the combined effects of stiffness and backbone
disorder5–7 共as well as of stretching forces, in some cases8兲
on the conformational properties of polymers. But an exact
treatment 共along the lines of Saitô, Takahashi, and Yunoki’s
path integral approach to the Kratky–Porod wormlike
chain3兲, which could, potentially, exploit powerful field theoretic techniques to address questions that might otherwise
prove intractable, does not appear to have been formulated.
Such treatments, being exact, could provide valuable reference points for the development of approximate theories of
related, more complicated systems when exact solutions are
unavailable.
These considerations suggest exploring the utility of the
STY methodology in analyzing the behavior of chains made
up of randomly distributed semiflexible segments. Accordingly, in this paper, we apply the methodology to calculate
the average radial dimensions of a polymer in which A and B
‘‘prepolymers’’ of arbitrary stiffness are arranged at random
along the chain backbone. The sequence of A’s and B’s along
the chain is assumed to be governed by the statistics of the
same two-state Markov process that was used by Fredrickson, Milner, and Leibler 共FML兲7 to analyze microphase ordering in random block copolymer melts. We show that with
this choice of disorder, the STY model of the random heteropolymer so defined, also admits of an exact solution.
Moreover, we find that the results are independent of
whether the disorder is regarded as annealed or quenched.
The following section introduces the model, and sets up
the expressions needed to calculate the mean square end-toend distance of the chain. Section III uses the STY propagator to reduce these expressions to simpler analytical forms,
which are then averaged over the sequence distribution, as
discussed in Sec. IV. The final expressions obtained by this
averaging operation are extremely lengthy, so all but the
most pertinent results are relegated to the Appendix. Section
V discusses these results in terms of the various parameters
that define the model.
II. THE MODEL
We are interested in the conformational properties 共specifically the mean square end-to-end distance兲 of a chain of
length M made up of a sequence of n polymeric subunits
共‘‘prepolymers’’兲. Each prepolymer is of length N 共hence
nN⫽M 兲, and is of one of two kinds: A or B. The prepolymers are regarded as semiflexible, so in a continuum representation of the chain, they can be completely characterized
by the set of unit tangent vectors u( ␶ ) at each of the points ␶
along the backbone. Within this representation, using units in
which k B T⫽1, the Hamiltonian H is given by
n
1
␦
H⫽ ⑀ A
2 i⫽1 ␪ i,1
兺
n
冕
iN
共 i⫺1 兲 N
1
⫹ ⑀B
␦
2 i⫽1 ␪ i,⫺1
兺
冕
d ␶ u̇2 共 ␶ 兲
iN
共 i⫺1 兲 N
d ␶ u̇ 2 共 ␶ 兲 ,
共1兲
n
S 1⫽
兺 具 R 2i 典
i⫽1
兺 冕共 i⫺1 兲N d ␶ 1 冕共 i⫺1 兲N d ␶ 2 具 u共 ␶ 1 兲 •u共 ␶ 2 兲 典 ,
i⫽1
n
⫽
iN
iN
共8兲
n⫺1
S 2⫽
兺
i⫽1
n⫺1
⫽
FIG. 1. A sketch of one possible realization of an A – B heteropolymer in
which the A and B segments have different degrees of stiffness.
n⫺2
S 3⫽
where ␪ i is a discrete random variable that takes on the values ⫾1, ⫹1 when the ith prepolymer is of type A and ⫺1
when it is of type B; ⑀ A and ⑀ B are the energies of bending of
the segments A and B, respectively, and can be identified
with the persistence lengths l A and l B of these segments; and
␦ is the Kronecker delta. Figure 1 is a sketch of one possible
realization of the copolymer sequence when the A and B
prepolymers have fairly different degrees of stiffness. In the
sketch, the A prepolymers are ‘‘coil-like,’’ while the B prepolymers are ‘‘helixlike.’’ Had the A segments been completely rigid, the chain would have corresponded to the broken rod model studied by Muroga et al.5
A simple transformation9 of the Kronecker deltas allows
H to be rewritten more succinctly as
n
1
⌬
H⫽
2 i⫽1 i
兺
冕
iN
共 i⫺1 兲 N
d ␶ u̇ 2 共 ␶ 兲 ,
共2兲
where
iN
共 i⫺1 兲 N
d␶1
冕
共 i⫹1 兲 N
iN
d ␶ 2 具 u共 ␶ 1 兲 •u共 ␶ 2 兲 典 ,
共9兲
n
兺 兺
i⫽1 j⫽i⫹2
具 Ri •R j 典
兺 兺 冕共 i⫺1 兲N d ␶ 1 冕共 j⫺1 兲N d ␶ 2 具 u共 ␶ 1 兲 •u共 ␶ 2 兲 典 .
i⫽1 j⫽i⫹2
n⫺2
⫽
冕
n
iN
jN
共10兲
The angular brackets in Eqs. 共7兲–共10兲 denote an average
both over the conformational degrees of freedom of the chain
as well as the distribution of the discrete random variables
␪ i . From these expressions the calculation of 具 R2 典 is seen to
reduce essentially to the calculation of the ‘‘bond’’ correlation function 具 u( ␶ 1 )•u( ␶ 2 ) 典 .
III. EVALUATION OF THE BOND CORRELATION
FUNCTION
In general, the correlation function of the tangent vectors
can be written in the form
⌬ i ⫽ 21 共 ⑀ A ⫹ ⑀ B 兲 ⫹ 21 共 ⑀ A ⫺ ⑀ B 兲 ␪ i
⬅D 1 ⫹D 2 ␪ i .
共3兲
共4兲
The vectorial distance R from one end of the chain to the
other is the sum of the end-to-end vectors Ri of each of the
prepolymers, i.e.,
具 u共 ␶ 1 兲 •u共 ␶ 2 兲 典 ⫽
R⫽
兺 Ri ,
共5兲
i⫽1
where Ri itself is given by
Ri ⫽
冕
iN
共 i⫺1 兲 N
d ␶ u共 ␶ 兲 .
共6兲
Thus, the mean square end-to-end distance 具 R 2 典 can be written in the form
具 R 2 典 ⫽S 1 ⫹2 共 S 2 ⫹S 3 兲 ,
共7兲
1
Q
冕
D关 u共 ␶ 兲兴 u共 ␶ 1 兲 •u共 ␶ 2 兲 e ⫺H ,
共11兲
where H is given by Eq. 共1兲, Q is the partition function,
defined as
n
where
兺
i⫽1
具 Ri •Ri⫹1 典
Q⫽
冕
D关 u共 ␶ 兲兴 e ⫺H ,
共12兲
and D关 u( ␶ ) 兴 is the functional integral measure on the space
of functions u( ␶ ). From the general approach to the evaluation of functional integrals described, for example, in Ref.
10, Eq. 共11兲 can be reduced to a product of ordinary integrals
involving a Green’s function G(u,u⬘ 兩 ␶ ⫺ ␶ ⬘ ), which describes the probability density that a monomer at the point ␶ ⬘
on the chain has the bond orientation u⬘ if the monomer at ␶
has the orientation u. When these points are located on the
jth and ith prepolymers, respectively, the expression for the
bond correlation function can be shown to be given by
具 u共 ␶ 1 兲 •u共 ␶ 2 兲 典
⫽
1
Q
⫻
⫻
冕 冕 冕
冕 ⬘冕 冕
冕 ⬘
du0
du1 ¯
dui⫺1
du
du j ¯
dun⫺1
冕 冕
du
dui ¯
冕
du j⫺1
dun u•u G 共 u0 ,u1 兩 N 兲
⫻G 共 u1 ,u2 兩 N 兲 ¯G 共 ui⫺1 ,u兩 ␶ 1 ⫺ 共 i⫺1 兲 N 兲
⫻G 共 u,ui 兩 iN⫺ ␶ 1 兲 ¯G 共 u j⫺1 ,u⬘ 兩 ␶ 2
⫺ 共 j⫺1 兲 N 兲 G 共 u⬘ ,u j 兩 jN⫺ ␶ 2 兲 ¯G 共 un⫺1 ,un 兩 N 兲 , 共13兲
where, in general,
G 共 uk⫺1 ,uk 兩 ␶ 2 ⫺ ␶ 1 兲 ⫽
冕
u共 ␶ 2 兲 ⫽uk
u共 ␶ 1 兲 ⫽uk⫺1
D关 u共 ␶ 兲兴
冋 冕
1
⫻exp ⫺ ⌬ k
2
␶2
␶1
册
d ␶ u̇2 共 ␶ 兲 . 共14兲
In terms of this distribution function, the partition function Q
can be similarly written as
Q⫽
冕
n
du0
兿
i⫽1
冕
dui G 共 ui⫺1 ,u兩 iN⫺ 共 i⫺1 兲 N 兲 .
共15兲
Adopting the STY model of the semiflexible chain3 共which
ensures inextensibility of the chain through the constraint
兩 u( ␶ ) 兩 ⫽1兲, one can determine the Green’s function as an
expansion in spherical harmonics:
G 共 uk⫺1 ,uk 兩 ␶ 2 ⫺ ␶ 1 兲 ⫽
e ⫺a 兩 ␶ ⫺ ␶ 兩 2⌬
兺
m,n
n
2
1
k
* 共 uk⫺1 兲 Y m,n 共 uk 兲 ,
⫻Y m,n
共16兲
where a n ⬅n(n⫹1). These spherical harmonics satisfy the
relations
冕
冕
*
duY m
1 ,n 1
共 u兲 Y m 2 ,n 2 共 u兲 ⫽ ␦ m 1 ,m 2 ␦ n 1 ,n 2 ,
duY m,n 共 u兲 ⫽ 冑4 ␲ ␦ m,0␦ n,0 ,
Y 0,0⫽
1
冑4 ␲
,
FIG. 2. 共a兲 A schematic representation of an A – B heteropolymer of n prepolymer segments showing coordinates relevant to the calculation of the
end-to-end distance. Each straight line segment terminated by full circles is
an A or a B prepolymer of contour length N. Letters above the circles and
crosses stand for the unit tangent vectors at the contour positions indicated
by the letters below these symbols. Squiggles denote sections of the chain
that are not shown in the diagram. 共b兲 The diagrammatic representation of
the integral D 1 关Eq. 共20兲兴. 共c兲 The diagrammatic representation of the integral D 2 关Eq. 共21兲兴. 共d兲 The diagrammatic representation of the integral D 3
关Eq. 共22兲兴.
共17兲
anywhere on the backbone of the chain. Squiggles indicate
prepolymer segments that are not shown. Integrations are
understood to be carried out over the vectorial positions of
all junction points 共including those labeled by ␶ 1 and ␶ 2 兲.
When the spherical harmonic expansion of the Green’s
function is substituted into Eq. 共13兲, and the integrations carried out using Eqs. 共17兲–共19兲, 具 u( ␶ 1 )•u( ␶ 2 ) 典 in S 1 is seen to
simplify to the diagram shown in Fig. 2共b兲. If the algebraic
expression for this diagram is denoted D 1 , one can show that
D 1 ⫽4 ␲
⫻
共18兲
冕 冕
兺
n,m
du1
du2 u1 •u2
* 共 u1 兲 Y n,m 共 u2 兲 .
exp共 ⫺a n 兩 ␶ 2 ⫺ ␶ 1 兩 /2⌬ i 兲 Y n,m
共20兲
共19兲
from which it immediately follows that Q⫽4 ␲ , independent
of the random variable ␪ i . This implies that there is no distinction here between quenched and annealed disorder.
To evaluate the functions S 1 , S 2 , and S 3 that are needed
in determining 具 R 2 典 , it is helpful to refer to Fig. 2共a兲, which
is a diagrammatic representation of the bond correlation
function 具 u( ␶ 1 )•u( ␶ 2 ) 典 . The straight line segments in this
figure stand for the Green’s function 关Eq. 共14兲兴 for the section of prepolymer of type A or B that lies between the junction points at the indicated contour positions. The crosses
mark the locations of the points ␶ 1 and ␶ 2 that appear in Eq.
共11兲, defining the correlation function; these points may lie
Similarly, 具 u( ␶ 1 )•u( ␶ 2 ) 典 in S 2 , reduces to D 2 关Fig.
2共c兲兴, which is given by
D 2 ⫽4 ␲
冕 冕
du1
du2 u1 •u2
兺
n,m
exp共 ⫺a n 兩 iN⫺ ␶ 1 兩 /2⌬ i 兲
* 共 u1 兲 Y n,m 共 u2 兲 .
⫻exp共 ⫺a n 兩 ␶ 2 ⫺iN 兩 /2⌬ i⫹1 兲 Y n,m
共21兲
Likewise, but after somewhat more algebra, 具 u( ␶ 1 )
•u( ␶ 2 ) 典 in S 3 , reduces to D 3 关Fig. 2共d兲兴, which is given by
D 3 ⫽4 ␲
冕 冕
du2 u1 •u2
du1
冉
j⫺i⫺1
⫻exp ⫺a n N
兺
l⫽1
兺
n,m
1
2⌬ i⫹l
exp共 ⫺a n 兩 iN⫺ ␶ 1 兩 /2⌬ i 兲
冊
* 共 u1 兲 Y n,m 共 u2 兲 .
⫻exp共 ⫺a n 兩 ␶ 2 ⫺ 共 j⫺1 兲 N 兩 /2⌬ j 兲 Y n,m
共22兲
This expression applies specifically to the case where the jth
prepolymer satisfies j⭓i⫹2.
The calculation of the average over the conformational
degrees of freedom is finally completed by evaluating the
integrals over u1 and u2 in Eqs. 共20兲–共22兲. Noting that u1
•u2 ⫽cos ␪1 cos ␪2⫹sin ␪1 sin ␪2 cos(␾1⫺␾2), one can use
standard results from the theory of spherical harmonics to
show that11
冕 冕
du1
* 共 u1 兲 Y n,m 共 u2 兲
du2 u1 •u2 Y n,m
4␲
⫽
␦ 共␦
⫹ ␦ m,0⫹ ␦ m,1兲 .
3 n,1 m,⫺1
共23兲
When the above equation is used in Eqs. 共20兲–共22兲, and the
resulting expressions then substituted into Eq. 共7兲, we obtain
n
冋
具 R 2 典 ⫽2N 兺 ⌬ i 1⫺
i⫽1
⌬i
共 1⫺e ⫺N/⌬ i 兲
N
册
n⫺1
⫹2
兺
i⫽1
n⫺2
⫹2
⌬ i ⌬ i⫹1 共 1⫺e ⫺N/⌬ i 兲共 1⫺e ⫺N/⌬ i⫹1 兲
n
兺 兺
i⫽1 j⫽i⫹2
冋
⌬ i ⌬ j exp ⫺N
⫻ 共 1⫺e ⫺N/⌬ i 兲共 1⫺e ⫺N/⌬ j 兲 .
j⫺i⫺1
兺
l⫽1
1
⌬ i⫹1
册
共24兲
When this expression is averaged over the sequence distribution 共as discussed in the following section兲, the desired
disorder-averaged end-to-end distance is obtained.
IV. AVERAGE OVER THE SEQUENCE DISTRIBUTION
In the absence of sequence disorder, there is no distinction between A and B segments, so the model describes a
homopolymer of n semiflexible prepolymer segments whose
stiffness can be characterized by a single persistence length l,
where l⫽ ⑀ A ⫽ ⑀ B . It is easily verified that in this limit, Eq.
共24兲 for 具 R 2 典 correctly reproduces the Kratky–Porod description of the chain. In particular, when lⰇ1, 具 R 2 典 scales
as n 2 N 2 , whereas when lⰆ1, 具 R 2 典 scales as M l 2 .
When the chain is a random heteropolymer, an average
over the sequence distribution must be carried out explicitly
to produce the final expression for 具 R 2 典 . To perform this
average, we adopt the model introduced by Fredrickson, Milner, and Leibler7 to discuss phase separation in random copolymer blends. In this model, the probability that ith prepolymer in the chain is of a given type is assumed to be
determined solely by the chemical identity of the immediately preceding prepolymer and no others. Thus, the probability of realizing a given sequence of A’s and B’s is deter-
mined, in general, by a set of four conditional probabilities
p KL , K, L⫽A, B, where p KL is conditional probability of
observing K given L. If it is further assumed that this sequence is the end result of a living polymerization process
under steady state conditions, the p KL can be expressed in
terms of the mole fractions f and 1⫺ f that define the composition of A and B in the initial reaction mixture 共and in the
chain that is generated thereby.兲 These assumptions, along
with the Markov condition, establish that only two parameters need be specified to fix the overall average sequence
distribution: one is f itself, and the other is the nontrivial
eigenvalue ␭ of the matrix of conditional probabilities. In
terms of these parameters, the following relations for the p KL
can be derived:7
p AA ⫽ f 共 1⫺␭ 兲 ⫹␭,
共25兲
p BB ⫽ f 共 ␭⫺1 兲 ⫹1,
共26兲
p AB ⫽1⫺ p AA ,
共27兲
p BA ⫽1⫺ p BB .
共28兲
Physically, the parameter ␭ is a measure of the extent of
‘‘blockiness’’ 共to use FML’s phrase兲 of the chain; it can be
shown to assume values between ⫺1 and ⫹1. The limit ␭
→⫺1 describes a chain in which A and B prepolymers tend
to succeed each other in alternation, the limit ␭→⫹1 describes a chain in which A’s tend to succeed A’s, and B’s tend
to succeed B’s, and the limit ␭⫽0 describes a chain in which
the A’s and B’s follow each other entirely randomly.
To return to Eq. 共24兲, we see that the sequence average
of the end-to-end distance requires averages over quantities
like exp(⫺N/⌬i). To perform such averages, recall that ⌬ i
⫽D 1 ⫹D 2 ␪ i , where D 1 and D 2 are defined in Eq. 共4兲 in
terms of the bending energies ⑀ A and ⑀ B . It follows therefore
that
D1
D2
1
⫽
⫺
␪ .
⌬ i D 21 ⫺D 22 D 21 ⫺D 22 i
共29兲
Defining ␣ ⫽D 1 /(D 21 ⫺D 22 ) and ␤ ⫽D 2 /(D 21 ⫺D 22 ), we now
have
具 e ⫺N/⌬ i 典 ⫽e ⫺N ␣ 具 e ⫺N ␤␪ i 典 ,
共30兲
where the angular brackets now refer to the average over the
distribution of the ␪ i ’s. Since ␪ i takes the values ⫾1, it is
easy to show that
具 e ⫺N ␤␪ i 典 ⫽cosh N ␤ ⫹ 具 ␪ i 典 sinh N ␤ .
共31兲
The average 具 ␪ i 典 is obtained from the relation 具 ␪ i 典
⫽ 兺 ␪ ⫽⫾1 ␪ p s ( ␪ ), where p s (⫹1)⫽ f and p s (⫺1)⫽1⫺ f .
Clearly, 具 ␪ i 典 ⫽2 f ⫺1. From this result it is easy to show that
冋
具 S 1 典 ⫽2nN D 1 ⫹ 共 2 f ⫺1 兲 D 2 ⫺
D 21 ⫹D 22
N
⫻ 兵 1⫺e ⫺N ␣ 共 cosh N ␤ ⫹ 共 2 f ⫺1 兲 sinh N ␤ 兲 其
⫺
2D 1 D 2
兵 2 f ⫺1⫺e ⫺N ␣ 共 sinh N ␤
N
册
⫹ 共 2 f ⫺1 兲 cosh N ␤ 兲 其 .
共32兲
The sequence averaged value of S 2 is similarly calculated as
具 S 2 典 ⫽2 共 n⫺1 兲 D 21 共 1⫺e ⫺N ␣ cosh N ␤ 兲 2
⫻
冋再
1⫺
冉
冊册
AD 2
D2
⫺ 共 2 f ⫺1 兲 A⫺
D1
D1
冉
⫹4 f 共 1⫺ f 兲 ␭ A⫺
D2
D1
冊冎
2
2
共33兲
,
where
A⬅
e ⫺N ␣ sinh N ␤
.
1⫺e ⫺N ␣ cosh N ␤
共34兲
The calculation of 具 S 3 典 is much less trivial, but it can be
done analytically. Details of the calculation are provided in
Appendix A. Here we quote the result in terms of four other
averages 具 S 31典 , 具 S 32典 , 具 S 33典 , and 具 S 34典 , whose complete expressions in terms of the various parameters of the model are
given in Eqs. 共A21兲, 共A24兲, 共A27兲, and 共A30兲:
n⫺2
具 S 3 典 ⫽2 兺
n
兺
i⫽1 j⫽i⫹2
⫻
D 21 共 2⫺e ⫺N ␣ cosh N ␤ 兲 2
冋冉 冊
冉 冊
冉 冊
1⫺
AD 2
D1
2
冉
e ⫺ 共 j⫺i⫺1 兲 N ␣ 具 S 31典 ⫺ 1⫺
AD 2
D1
冊
D 2 ⫺ 共 j⫺i⫺1 兲 N ␣
⫻ A⫺
e
共 具 S 32典 ⫹ 具 S 33典 兲
D1
D2
⫹ A⫺
D1
2
e
⫺ 共 j⫺i⫺1 兲 N ␣
册
具 S 34典 .
共35兲
After substituting the equations for 具 S 31典 , 具 S 32典 , 具 S 33典 , and
具 S 34典 into the above equation, the sums become trivial, and
can be done at once.
V. DISCUSSION
Equation 共7兲, along with Eqs. 共32兲 and 共33兲, and Eq. 共35兲
共after carrying out the summations兲, is the desired expression
for the disorder averaged mean square end-to-end distance of
the given A – B copolymer, but it is far too lengthy and complicated to be particularly perspicuous on its own. It is therefore depicted graphically in a series of figures 共3– 6兲 that
highlight its behavior in terms of one or other of the parameters that 具 R 2 典 depends on. These parameters are the length
N of the prepolymer, the number n of prepolymers in the
chain as a whole, the fraction f of A-type prepolymers, the
extent of blockiness ␭, and the persistence lengths l A and l B
of the A and B prepolymers, respectively. As a matter of
convenience, N is kept constant throughout. Although f and ␭
are regarded as free parameters, the fact that the conditional
probabilities in Eqs. 共25兲–共28兲 are constrained to lie between
0 and 1 implies that f and ␭ can only be chosen within certain
limits; these limits are always respected when numerical values are assigned to the parameters.
We plot ␣ R ⬅ 具 R 2 典 /M 2 against the dimensionless inverse
persistence length 1/l * ⬅N/ ⑀ B ⬅N/l B for four different values of one other parameter, all other parameters staying the
same. A fifth curve is included for reference: this is the varia-
FIG. 3. Full curves are the variation of the parameter ␣ R ⬅ 具 R 2 典 /M 2 with the
dimensionless inverse persistence length N/l B as a function of the fraction f
of A at fixed values of ␭, N/l A and n 共0.0, 0.01, and 10, respectively.兲 The
dashed curve is the Kratky–Porod expression for ␣ R of a homopolymer of
length M derived from Eq. 共36兲. Curves 1, 2, 3, and 4 correspond, respectively, to f ⫽0.1, 0.25, 0.5, and 0.9.
tion of ␣ R as a function of M /l for a semiflexible homopolymer of contour length M as calculated with the following
Kratky–Porod expression:3
冋
具 R 2 典 ⫽M l 1⫺
册
1
共 1⫺e ⫺2M /l 兲 .
2M /l
共36兲
This expression 共which is reproduced by our model in the
homopolymer limit兲 yields the results ␣ R →1 as lⰇ1 and
␣ R →0 as lⰆ1.
Figure 3 shows the variation of ␣ R 关as determined from
Eqs. 共7兲 along with 共32兲, 共33兲 and 共35兲兴 with 1/l * 共the full
lines兲 for four different values of f 共0.10, 0.25, 0.50, and
0.90兲 at the following values of the other parameters: ␭
FIG. 4. Variation of ␣ R with N/l B as a function of n at fixed values of f
共0.5兲, ␭共0.0兲, and N/l A 共0.01兲. Curves 1, 2, 3, and 4 correspond to n⫽10, 25,
50, and 100, respectively. The dashed curve is the same Kratky–Porod result
shown in Fig. 3.
FIG. 5. Variation of ␣ R with N/l B as a function of N/l A at fixed values of f
共0.5兲, ␭ 共0.0兲, and n 共10兲. Curves 1, 2, 3, and 4 correspond to N/l A ⫽0.01,
1.0, 2.0, and 5.0, respectively. The dashed curve is the same Kratky–Porod
result shown in Fig. 3.
⫽0.0, N/l A ⫽0.01 and n⫽10. The dashed line is the Kratky–
Porod result derived from Eq. 共36兲. Flexibility increases
from left to right along the abscissa. In general, N/l A values
less than unity correspond to prepolymer segments that are
semiflexible or rigid, so the choice N/l A ⫽0.01 indicates that
A is relatively stiff. The choice ␭⫽0.0 indicates that chain is
an ideal random copolymer. The figure shows that principal
effect of increasing the proportion of A in the chain is to
render it increasingly inflexible, as one would expect. At the
smallest fractions of A, however, there is a fairly narrow
range of N/l B values over which increasing the stiffness of B
leads to a fairly sharp rise in the stiffness of the chain as a
whole. This is a trend that is repeated, to greater or less
degree, in all the other figures.
The sharpness of the change from flexible to rigid geometries is especially pronounced in Fig. 4, which shows the
variation of ␣ R with 1/l * for 4 different values of n 共10, 25,
50, and 100兲 at the following fixed values of the other parameters: f ⫽0.5, ␭⫽0.0 and N/l A ⫽0.01. Presumably, as n
→⬁, there is something akin to a genuine discontinuity between the flexible and rodlike configurations of the random
heteropolymer.
The transition can be completely suppressed for certain
ranges of parameter values, as illustrated in Fig. 5, which
shows the variation of ␣ R with 1/l * for 4 different values of
N/l A 共0.01, 1, 2, and 5兲 at the following fixed values of the
other parameters: f ⫽0.5, ␭⫽0.0 and n⫽10. At the largest
values of N/l A 共1, 2 and 5兲, corresponding to the greatest
degree of conformational flexibility, the chain never attains
more than about 40% of its full extension.
The blockiness parameter ␭ provides one final measure
of conformational control, as illustrated in Fig. 6, which
shows the variation of ␣ R with 1/l * for four different values
of this parameter 共⫺0.9, ⫺0.5, 0.5, and 0.9兲 at the following
fixed values of the other parameters: f ⫽0.5, N/l A ⫽0.01 and
n⫽10. The negative values of ␭ correspond to chains with
the tendency to alternate between A and B segments, and
such chains show the greatest degree of flexibility at N/l B
values greater than about 2. Below that number, when both A
and B segments are relatively stiff, the conformation of the
chain is increasingly insensitive to how frequently the two
segments alternate with each other.
In conclusion, we have found an exact solution for the
size of a random heteropolymer based on the STY model that
highlights the interplay between flexibility and disorder in
chain statistics. The analytical expression for 具 R 2 典 that is
derived from the model can be used to make rough estimates
of the range of allowed sizes of chains of unknown structure
but definite sequence.
APPENDIX A: CALCULATION OF Š S 3 ‹
The functions S 31 , S 32 , S 33 and S 34 that appear in S 3
关Eq. 共35兲兴 are defined as
冉
兺
S 31⫽exp N ␤
␪ i⫹l ,
l⫽1
冉
j⫺i⫺1
S 32⫽ ␪ i exp N ␤
冉
冊
j⫺i⫺1
兺
l⫽1
冉
兺
l⫽1
S 34⫽ ␪ i exp N ␤
冊
␪ i⫹l ,
冊
j⫺i⫺1
S 33⫽exp N ␤
共A1兲
␪ i⫹l ␪ j ,
j⫺i⫺1
兺
l⫽1
冊
␪ i⫹l ␪ j .
共A2兲
共A3兲
共A4兲
By definition,
具 S 31典 ⫽ 兺
FIG. 6. Variation of ␣ R with N/l B as a function of ␭ at fixed values of f
共0.5兲, N/l A 共0.01兲, and n 共10兲. Curves 1, 2, 3, and 4 correspond to ␭
⫽⫺0.9, ⫺0.5, 0.5, and 0.9, respectively. The dashed curve is the same
Kratky–Porod result shown in Fig. 3.
兺
␪ i⫹1 ␪ i⫹2
¯
兺
␪ j⫺1
p s 共 ␪ i⫹1 兲 p 共 ␪ i⫹2 兩 ␪ i⫹1 兲
⫻ p 共 ␪ i⫹3 兩 ␪ i⫹2 兲 ¯p 共 ␪ j⫺1 兩 ␪ j⫺2 兲 exp共 ␮␪ i⫹1 兲
⫻exp共 ␮␪ i⫹2 兲 ¯ exp共 ␮␪ j⫺1 兲 ,
共A5兲
where p s ( ␪ i ) is the equilibrium 共i.e., steady state兲 probability for the occurrence of the state ␪ i , p( ␪ i⫹1 兩 ␪ i ) is the conditional
probability of seeing ␪ i⫹1 given ␪ i , and ␮ is N ␤ . One may verify that Eq. 共A5兲 may be expressed in matrix form as
共A6兲
where p AA ⬅p(1 兩 1), p AB ⬅p(1 兩 ⫺1), p BA ⬅p(⫺1 兩 1) and
p AA ⬅p(⫺1 兩 ⫺1). Defining the matrix
P⫽
冉
p AA e ␮
p AB e ⫺ ␮
p BA e ␮
p BB e ⫺ ␮
冊
it is seen that
具 S 31典 ⫽ 共 e
␮
e
⫺␮
兲P
j⫺i⫺2
共A7兲
冉 冊
共A8兲
␮
⌳ 1,2⫽ 共 p AA e ⫹p BB e
⫺␮
冉
1
⫺␾*
⫺␾
␾␾ *
冊
,
共A17兲
i⫽1,2
where
The eigenvalues ⌳ 1 and ⌳ 2 of P are given by
1
2
i⫽1,2. 共A16兲
Furthermore, if we define the matrices A1 and A2 according
to
Ai ⫽
pA
.
pB
1
共 p e ␮ ⫺⌳ i 兲 2 ⬅C i ,
p AB p BA AA
yTi xi ⫽1⫹
共A9兲
兲 T 1,2 ,
where
T 1,2⫽1⫾ 冑1⫺Q,
共A10兲
␮
␮
␾*
i ⫽ 共 p AA e ⫺⌳ i 兲 /p BA e ,
i⫽1,2
共A18兲
and
␾ i ⫽ 共 p AA e ␮ ⫺⌳ i 兲 /p AB e ⫺ ␮ ,
i⫽1,2,
共A19兲
then, in general, for some integer m,12
with
4␭
.
Q⫽
共 p AA e ␮ ⫹p BB e ⫺ ␮ 兲 2
共A11兲
The upper symbol in 共A10兲 corresponds to T 1 while the
lower symbol corresponds to T 2 . Introduce a set of left
eigenvectors x1 and x2 , and a set of right eigenvectors y1
and y2 satisfying
Px1 ⫽⌳ 1 x1 ,
Px2 ⫽⌳ 2 x2
yT1 P⫽yT1 ⌳ 1 ,
yT2 P⫽yT2 ⌳ 2 .
共A12兲
Pm ⫽
1
1
A ⌳ m⫹
A ⌳m .
C1 1 1 C2 2 2
Hence,
具 S 31典 ⫽ 共 e ␮ e ⫺ ␮ 兲
⫽
⌳ 1j⫺i⫺2
C1
and
共A13兲
The right eigenvectors may be chosen as
冉
冊
1
.
x1,2⫽
⫺ 共 p AA e ␮ ⫺⌳ 1,2兲 /p AB e ⫺ ␮
册冉 冊
pA
pB
␮
⫺␮
兲
关 f ⫺ 共 1⫺ f 兲 ␾ *
1 兴共 e ⫺ ␾ 1 e
⌳ 2j⫺i⫺2
C2
1
1
A1 ⌳ 1j⫺i⫺2 ⫹
A ⌳ j⫺i⫺2
C1
C2 2 2
关 f ⫺ 共 1⫺ f 兲 ␾ 2* 兴共 e ␮ ⫺ ␾ 2 e ⫺ ␮ 兲 . 共A21兲
In the same way
共A14兲
具 S 32典 ⫽ 兺
兺
␪ i ␪ i⫹1
Similarly, the left eigenvectors may be chosen as
T
⫽ 共 1, ⫺ 共 p AA e ␮ ⫺⌳ 1,2兲 /p BA e ␮ 兲 .
y1,2
⫹
冋
共A20兲
共A15兲
For i⫽ j, one may see from Eqs. 共25兲–共28兲 that yTi x j ⫽0.
When i⫽ j, we have
¯
兺
␪ j⫺1
p s 共 ␪ i 兲 p 共 ␪ i⫹1 兩 ␪ i 兲 p 共 ␪ i⫹2 兩 ␪ i⫹1 兲
¯p 共 ␪ j⫺1 兩 ␪ j⫺2 兲 ␪ i exp共 ␮␪ i⫹1 兲
⫻exp共 ␮␪ i⫹2 兲 ¯ exp共 ␮␪ j⫺1 兲 ,
共A22兲
which in matrix notation is given by
共A23兲
so using the result of Eq. 共A20兲
具 S 32典 ⫽ 共 e
⫻
⫽
␮
冉
e
⫺␮
冋
1
1
A ⌳ j⫺i⫺2 ⫹
A ⌳ j⫺i⫺2
兲
C1 1 1
C2 2 2
p AA p A ⫺p AB p B
p BA p A ⫺p BB p B
⌳ 1j⫺i⫺2
C1
⫹
冊
册
⌳ 2j⫺i⫺2
C2
共 e ␮ ⫺ ␾ 2 e ⫺ ␮ 兲关 f p AA ⫺ 共 1⫺ f 兲 p AB
⫺ ␾ 2* 共 f p BA ⫺ 共 1⫺ f 兲 p BB 兲兴 .
共A24兲
Similarly,
具 S 33典 ⫽ 兺
兺
␪ i⫹1 ␪ i⫹2
¯
p s 共 ␪ i⫹1 兲 p 共 ␪ i⫹2 兩 ␪ i⫹1 兲
兺
␪
j
⫻ p 共 ␪ i⫹3 兩 ␪ i⫹2 兲 ¯p 共 ␪ j 兩 ␪ j⫺1 兲 exp共 ␮␪ i⫹1 兲
共 e ␮⫺ ␾ 1e ⫺␮ 兲
⫻ 关 f p AA ⫺ 共 1⫺ f 兲 p AB ⫺ ␾ 1* 共 f p BA ⫺ 共 1⫺ f 兲 p BB 兲兴
⫻exp共 ␮␪ i⫹2 兲 ¯ exp共 ␮␪ j⫺1 兲 ␪ j ,
共A25兲
which in matrix notation is given by
共A26兲
⫽
⌳ 1j⫺i⫺1
C1
关 f ⫺ 共 1⫺ f 兲 ␾ *
1 兴共 1⫹ ␾ 1 兲 ⫹
⌳ 2j⫺i⫺1
C2
关 f ⫺ 共 1⫺ f 兲 ␾ 2* 兴共 1⫹ ␾ 2 兲 .
共A27兲
Finally,
具 S 34典 ⫽ 兺
兺
␪ i ␪ i⫹1
¯
兺 兺 p s共 ␪ i 兲 p 共 ␪ i⫹1兩 ␪ i 兲 p 共 ␪ i⫹2兩 ␪ i⫹1 兲 ¯p 共 ␪ j 兩 ␪ j⫺1 兲 ␪ i
␪ j⫺1 ␪ j
⫻exp共 ␮␪ i⫹1 兲 exp共 ␮␪ i⫹2 兲 ¯ exp共 ␮␪ j⫺1 兲 ␪ j ,
共A28兲
which in matrix notation is given by
共A29兲
⫽ 共 1⫺1 兲
⫽
⌳ 1j⫺i⫺1
C1
⫹
1
冋
1
1
A1 ⌳ 1j⫺i⫺1 ⫹
A ⌳ j⫺i⫺1
C1
C2 2 2
p AA p A ⫺ p AB p B
p BA p A ⫺ p BB p B
冊
共 1⫹ ␾ 1 兲关 f p AA ⫺ 共 1⫺ f 兲 p AB ⫺ ␾ *
1 共 f p BA ⫺ 共 1⫺ f 兲 p BB 兲兴
⌳ 2j⫺i⫺1
C2
册冉
共 1⫹ ␾ 2 兲关 f p AA ⫺ 共 1⫺ f 兲 p AB ⫺ ␾ 2* 共 f p BA ⫺ 共 1⫺ f 兲 p BB 兲兴 .
V. S. Pande, A. Y. Grosberg, and T. Tanaka, Rev. Mod. Phys. 72, 259
共2000兲, and references therein.
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11