Biophysics Lecture2

8/13/2019 Biophysics Lecture2

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24 MACROMOLECULES

2.1.2 The Gaussian Chain Model

Important Concepts: fluctuating bond, Hamiltonian for the

Gaussian Chain Model

We consider a chain made up of orientationally uncorrelated (freely-jointed) links

where the length of any link vector is no longer constant but has a probability

distribution

G(r) =

3

2b2

3/2exp

3r2

2b2

(2.36)

with the expectation for the link length being

< r2 >=b2 . (2.37)

The probability distribution for the end-to-end vector is then

P(Re) = P({rn}) (2.38)

=N

n=1 3

2b2

3/2

exp3r2n

2b2 (2.39)

=

3

2b2

3/2exp

Nn=1

3(Rn Rn1)2

2b2

(2.40)

and hence for the entropy

S = ln P =Nn=1

ln P(rn) (2.41)

= const 32b2

Nn=1

r2n . (2.42)

From this we obtain the free energy

F({rn}) =E+3T

2b2

Nn=1

r2n (2.43)

with the internal energyEbeing independent of{rn}.

Prof. Heermann, Heidelberg University


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2.1 GENERAL PROPERTIES OF MACROMOLECULES 25

Hence we obtain the same equilibrium distribution as for the freely-jointed chain.

Eq (2.40) also results if we start off with the Hamiltonian for a chain of springs

H=3

2

kBT

b2

Nn=1

(Rn Rn1)2 (2.44)

and we also obtain the scaling of the end-to-end distance

< R2e > N . (2.45)

Further Reading 2.1.4 (Continuous scales)

For the later development we note the generalization to all scales for the Gaussianmodel at fixed contour lengthL

P(R) =

r(L)=R

r(0)=0

D[r(s)] exp

3L

2b2

L0

ds

r(s)

s

2 , (2.46)

wherer(s)denotes the contour or conformation of the chain. This is solved by

P(R) =

3

2b2

3/2exp

3R2

2b2

(2.47)

and is the proper continuum limit of the FJC, where the limits N andb 0were taken, such thatN b2 stayed finite.

We further generalize the effective Hamiltonian

H=3

2

kBT

b2

L0

dsr2(s) (2.48)

where the derivation is to be taken along the contour.

2.1.3 Worm-like Chain Model

Important Concepts: bending rigidity and the relation with the

persistence length

A short-coming of the above models (besides they being phantom chains, i.e.

no self-avoidance) is that there is no intrinsic stiffness. Intuitively, we expect a

bending of the chain to cost energy. A model that provides this is the worm-like

chain model(WLC)



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26 MACROMOLECULES

R(s)

r(s)

rn

rn+1

Figure 2.10: Definition of the coordinates used to describe the chain as a con-

tinuous curve

H= N1n=1

rn rn+1 (2.49)

which is simply the one-dimensional Heisenberg model for ferromagnets. Here

|rn| =b. This model can be treated in the continuum limit whereN ,b 0and with

/N= constant , (2.50)

keeping the contour length also constant. Using

rn+1 rn=1

2[(rn rn+1)

2 2b2] (2.51)

we have

H= limb0,,N

b

2

N1n=1

b

rn rn+1

b

2. (2.52)

To cross over to the continuum limit we use the tangent vector with the arc length

s

r(s)

s = lim

b0

rn+1 rn

b

(2.53)



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andN1n=1 b L0 dsto findH=

2

L0

ds

r(s)

s

2=

2

L0

ds

2R(s)

s2

2(2.54)

with thebending modulus = b.

Thus the partition function is given by

Z=

D[r(s)](|r(s)| 1)exp(H[r(s)]) . (2.55)

The bending modulus must have a relation with the persistence length. To find

this relation we need to calculate the correlation function

< r(s)r(s)> exp(|s s|/p) . (2.56)

We can now calculate the mean squared end-to-end-distance and the mean squared

radius of gyration

< R2e > = (2.57)

= L0

ds L0

ds

< r(s) r(s

)> (2.58)

= 22p

L

p 1 +eL/p

(2.59)

= L2fD

L

p

, (2.60)

wherefD(x) = 2(x 1 +ex)/x2 being the Debye-function (see figure 2.11).

Further Reading 2.1.5 (Lattice Chains)

We can also go about calculating the average end-to-end distance staying discrete

(Kratky-Porod model [36] ). For this we write

H= N1n=1

rn rn+1= Nn=1

cos n . (2.61)

The partition function is given by

ZN=Nn=1

dne

cos n =QNl (2.62)



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28 MACROMOLECULES

3010

x

6040 50

fd(x)

1

0,8

0,6

20

0,4

0,2

0

TheDebyefunction

Figure 2.11: The Debye-function

with

Ql = 2 1

1

d(cos)e cos = 4sinh

. (2.63)

With this we can calculate the distance

< R2e >=b2 =b2n,m

< rn rm> (2.64)

again as the correlation between the directions of bonds. For the nearest neighbour

correlation we have

< rn rn+1> = (2.65)

=

dcos e cos

de cos (2.66)

= d ln Ql

d() (2.67)

= coth() 1/ (2.68)

= fL() , (2.69)

wherefLis theLangevin function. In general the correlation is given by [37]



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< rn rn+1>=eNln(1/c) , (2.70)

which can be written in terms of the persistence lengthp

< rn rn+1 >=eNb/p (2.71)

with

p = b

ln(1/c) (2.72)

If the macromolecule is stiff >>1 and

c 1 kBT / , (2.73)

then

ln(1/c) kBT / (2.74)

and thus

p = b . (2.75)

Let us rewrite the end-to-end distance in terms ofc

1

b2 < R2e >=

Ni=1

i1j=1

cij + 1 +N

j=i+1

cji

, (2.76)

then for mosti the two sums in the braces can be replaced by

k=1 = c1c

and

with this we obtain

< R2e > Nb2

1 +

2c

1 c

= N b2

1 +c

1 c . (2.77)

1+c1c

is the Flory factor. The Kuhn segment lengthlKis then given by

lK=1 +c

1 cb . (2.78)



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30 MACROMOLECULES

P

b

t n

b

t

n

P`

Figure 2.12: Motion of the Fernet triad along a curve

Since we are dealing in some sense with mathematical curves in three dimen-

sional space recall that a curve in space is mathematicallydetermined by the two

parameters:curvature andtorsion ( the basic triad {tk})

dbds

= n , dnds

= t+b , tds

=n (2.79)

Choosing = = 0 we obtain a straight line, r = 1/ and = 0 for a circle

with radiusr.

Consider the case of a helix (see figure 2.13) with

r(s) =

r cos

qs

(qr)2 + 1

, r sin

qs

(qr)2 + 1

,

s

(qr)2 + 1

(2.80)

and

(s) = rq2

1 + (qr)2 (2.81)

(s) = q

1 + (qr)2 (2.82)

Clearly asr 0we must approach a straight line. The curvature vanishes indeed

in this limit but the torsion does not.



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Pitch p

Radius r

Figure 2.13: Definition of the parameters for a helix

We can generalize the Fernet-frame introducing a third parameter

dti(s)

ds =

j,k

ijkj(s)tk(s) (2.83)

where

1 = cos , 2= sin , 3 = +d

ds (2.84)

This could be used to define an elastic energy with spontaneous curvature and

torsion {0k} (the deviation from equilibrium is denoted byk=k 0k

U=1

2

k

bk

L0

2kds (2.85)

The parametersbk define the rigidity. Defining



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32 MACROMOLECULES

A=

0

0

0

(2.86)with

Aij =k

ijkk (2.87)

we can develop also a numerical procedure to generate curves that have a pre-

scribed spontaneous curvature with fluctuations. For this let vx = (tx1 , tx2 , tx3 , )etc.then we write the equation 2.83 as

dvi

ds =Av i (2.88)

which can be discretized as

vi(s+ds) =Ovi(s) (2.89)

with

O= (1 +ds

2A)(1

ds

2A)1 (2.90)

Note that O is an orthogonal matrix guaranteeing that we consistently generate

orthogonal triads.

In figure is depicted the case for a curve where the spontaneous curvature is set

such that a circle is preferred.

2.1.4 Self-Avoiding Chains

The above models all lack the excluded volume interaction between the monomers.

Indeed, the ubiquitous polymer model is a self-avoiding random walk.

Let U(RnRm)be a monomer-monomer interaction potential assumed to handle

the excluded volume. We treat the polymer as gas of disconnected monomers

confined within the same volumeVas the polymer coil. IfRdenotes the average

extent of the chain then the chain occupies a volume V = R3. LetFcbe the term

that arises from the chemical work done by initially preparing the monomers in


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