1 biopolymersbiomechanics.stanford.edu › me339 › me339_s050607.pdf1.2 introduction to polymer...

“Lectures˙05˙and˙06˙and˙07” — 2007/11/10 — 12:34 — page 1 — #1

1 Biopolymers

1.1 Introduction to statistical mechanics

1.1.1 Free energy revisited

Previously, we briefly introduced the concepts of free energy and entropy. Here,we will examine these concepts further and how they apply to biopolymers. Tobegin, we first need to introduce the idea of thermal fluctuations. Imagine a smallparticle floating in fluid. The particle will undergo Brownian motion, or randommovements as it is randomly pushed around by the molecules around it. The mo-tion of the particle depends on the temperature: if the temperature is increased,the motion of the particle will also be increased. Like Brownian motion, polymersare subject to random molecular forces that cause them to ”wiggle”. These wig-gles are called thermal fluctuations. As temperature is increased, the polymerswiggle more. Why is this?

Particle: Brownian motion Polymer: Thermal fluctuations

Figure 1.1: Schematic depicting Brownian motion of a particle in fluid and thermal fluctuations ofa polymer. Both are induced by random forces induced by the surrounding molecules.

The conformation of a polymer is determined by its free energy. Specifically, poly-mers want to minimize their free energy. As we have seen already, the free energyis defined as

Ψ = W − TS. (1.1.1)

1


1 Biopolymers

W is referred to as the strain energy, or simply as the energy. T is temperature,and S is the entropy. We can see from 1.1.1 that a polymer can lower its freeenergy by lowering its energy W, or increasing its entropy S. Lets look at each ofthese quantities individually.

Entropy

What is entropy? You may have heard the vague explanation that entropy is re-lated to disorder. Many people have heard the ”messy room” analogy, in that amessy room is more disordered than a clean room, and so has higher entropy.What the heck does that mean?! Within statistical mechanics, entropy has a pre-cise mathematical meaning. Specifically,

S = k ln Ω. (1.1.2)

Here, k is the Boltzmann constant and is equal to 1.38x10−23 J/K. Ω is defined asthe number of microstates for a given macrostate corresponding to some macro-scopic quantity. What is meant by this? Consider a model polymer consisting ofn rigid links of size b. The links are connected to each other by freely rotatinghinges. The links can be oriented only vertically or horizontally. In this model,we define two lengths. The first length is the contour length L, which is the lengthof the polymer if it was completely stretched out. In this case, since there are nlinks of size b, then L = nb. The second length is the end-to-end length R. This issimply the length from one end of the polymer to the other. In this case, R mustalways be less than or equal to L. Remember that Ω is defined as the number ofmicrostates for a given macroscopic quantity. In our example, the different mi-crostates are the different polymer configurations, and the macroscopic quantityof interest is the end-to-end length R. Thus, finding Ω boils down to counting thenumber of ways in which our polymer can have a particular end-to-end length.Suppose the polymer is completely stretched out. The polymer will have an end-to-end length equal to the contour length, or R = L. In this case, there is onlyone polymer conformation in which this can occur, and so Ω(R = L) = 1. Nowsuppose that the polymer is not completely stretched out, or R < L. In thiscase, there are multiple polymer conformations that have the same end-to-endlength. Thus, Ω(R < L) > 1. The entropy is proportional to ln Ω, so the higherthe value of Ω, the higher the entropy. Therefore, the unstretched polymer hashigher entropy than the stretched out polymer. To decrease its free energy, thepolymer wants to increase its entropy, and so it wants to be kinked. Thus, wesay that entropically, the polymer wants to be kinked (i.e., it does not want to bestretched out).

2


1.1 Introduction to statistical mechanics

b

R

Figure 1.2: Model polymer with rigid links of size b. The links are connected to each other byfreely rotating hinges.

R < L

R < L

R = L

Figure 1.3: If R = L, there is only one configuration in which this is possible. In contrast, if R < L,there are multiple configurations in which this is possible.

Energy

Now lets look at the effect of energy on the polymer configuration. Considerthe same polymer model as before, but now this time imagine that instead ofthe links being connected by freely rotating hinges, we now straighten out thepolymer and attach rotational springs between each link. In this case, the polymernaturally wants to be straight, since bending the polymer requires work. If webend the polymer, this work becomes stored in the springs as energy. Thus, if lookat the stretched out polymer (R = L) and the unstretched polymer (R < L), thenthe stretched out polymer will have zero energy (W = 0), while the unstretchedpolymer will have non-zero energy (W > 0). To decrease its free energy, thepolymer wants to be stretched, and so here we say that energetically, the polymerwants to be stretched.Now we know that entropically, the polymer wants to be kinked (or ”wiggly”),

3


1 Biopolymers

R < L

R = L

Figure 1.4: Depiction of a polymer in which the links are connected by rotational springs (swirls).No energy is stored in the springs when the polymer is straight (R = L). However, work is requiredto bend the polymer (R < L), and this work is stored in the springs as energy.

while energetically, the polymer wants to be straight. If we look at 1.1.1, thenwe can see that the temperature T will determine whether the entropy or energydominates the free energy. At lower temperature, W dominates, and so the poly-mer will be straighter. At higher temperature, S dominates, and so the polymerwill be wigglier. Thus, we now see that the free energy, energy, and entropy givea mathematical description as to why the polymer becomes wigglier at highertemperature!

1.2 Introduction to polymer physics: models for biopolymers

1.2.1 The random walk

We will not introduce some models for polymers. The random walk serves as thefoundation of several of these models, so we investigate this here. Random walksare used to understand a wide variety of phenomena, such as Brownian motionand diffusion. What is an example of a random walk? Consider a soccer playerstanding at midfield of a soccer field, facing one of the goals. We define midfieldto be at location r = 0. The player takes out a coin, and flips it. If it comesup heads, the player steps forward towards one goal, if it comes up tails, theplayer steps backwards towards the opposite goal. This ”flip and step” process isrepeated a certain number of times. We will now calculate the probability of theplayer being a certain location after a given number of steps. In particular, aftern random, unit-sized steps, we will calculate the probability that the player willend up at location r (for more detail, see Rubinstein and Colby’s [9] analysis of adrunk staggering up and down a narrow alley, from which the approach here was

4



followed). Note that if the player has taken more backwards steps than forwardsteps, r is negative, and vice versa.If the player takes n steps, and for each step he/she can only backwards or for-wards, then there are 2n different paths which can be taken. Of those paths, thenumber M of those paths in which only n+ of the steps are forward can be cal-culated using the binomial coefficient l!/(m!(l − m)!), which gives the numberof ways in which a group of l items can be chosen from a set of m. Using thisrelation, M can be calculated as

M =n!

n+!(n− n+)!. (1.2.1)

If n− is the total number of steps backwards such that

n = n+ + n−, (1.2.2)

than after n steps, the player will end up at the location

r = n+ − n−. (1.2.3)

Now combining Equations 1.2.1-1.2.3, we get that

M(n, r) =n!

n+r2 ! n−r

2 !, (1.2.4)

which gives the total number of different paths the player can take to end up atthe location r if he/she takes n steps. The probability P(n, r) the player will endup at the location r after taking n steps can be found by dividing W(n, r) by thetotal number of paths possible after n steps, or

P(n, r) =M(n, r)

2n =12n

n!n+r

2 ! n−r2 !

. (1.2.5)

1.2.2 Gaussian approximation of the random walk

Equation 1.2.5 gives the exact probability distribution for a one-dimensional ran-dom walk. However, it is not convenient mathematically to use this distributionfor large n because of the difficulty in calculating factorials for large n. All is

5


1 Biopolymers

not lost however, since as n gets larger, the distribution becomes more and moreGaussian. Thus, we can approximate 1.2.5 with a Gaussian distribution (see boxbelow for derivation if interested):

P1d(n, r) =1√2πn

exp(− r2

2n

)(1.2.6)

Derivation of the Gaussian approximation of the random walkWe derive the Gaussian approximation of the one dimensional random walk here (for further detail, see [9]). If we take the natural log of P(n, r), then

ln(P(n, r)) = −nln(2) + ln(n!)− ln(

n + r2

!)− ln

(n− r

2!)

. (1.2.7)

Now, if a, b, and c are positive integers and a≥b, ((a + b)/c)! can be written as

a + bc

! =ac

!b/c

∏s=1

( ac

+ s)

and ((a− b)/c)! can be written as

a− bc

! =ac !

b/c

∏s=1

( ac

+ 1− s)

Thus, the third term in 1.2.7 can be written as

ln(

n + r2

!)

= ln( n

2!) r/2

∏s=1

( n2

+ s)

= ln( n

2!)

+r/2

∑s=1

ln( n

2+ s

). (1.2.8)

Similarly, the fourth term can be written as

ln(

n− r2

!)

= ln( n

2!)−

r/2

∑s=1

ln( n

2+ 1− s

). (1.2.9)

Combining, Equations 1.2.7-1.2.9, ln(P(n, r)) can be written as

ln(P(n, r)) = −nln(2) + ln(n!)− 2ln( n

2!)−

r/2

∑s=1

ln( n

2+ s

)+

r/2

∑s=1

ln( n

2+ 1− s

)

= −nln(2) + ln(n!)− 2ln( n

2!)−

r/2

∑s=1

ln( n

2 + sn2 + 1− s

)

= −nln(2) + ln(n!)− 2ln( n

2!)−

r/2

∑s=1

ln

(1 + 2s

n1− 2s

n + 2n

), (1.2.10)

where we have divided the numerator and denominator by n/2 in the last term in the last line.

continued on next page

6



Derivation of the Gaussian approximation of the random walk (cont’d)In this term, since ln(1 + a) ∼= a for |a| ¿ 1, we can approximate the logarithm as

ln

(1 + 2s

n1− 2s

n + 2n

)= ln

(1 +

2sn

)− ln

(1− 2s

n+

2n

)

∼= 4sn− 2

n. (1.2.11)

Using 1.2.11, and the identitiesa∑

s=1s = a(a + 1)/2 and

a∑

s=11 = a, then

ln(P(n, r)) ∼= −nln(2) + ln(n!)− 2ln( n

2!)−

r/2

∑s=1

(4sn− 2

n

)

∼= −nln(2) + ln(n!)− 2 ln( n

2!)− 4

n

r/2

∑s=1

s +2n

r/2

∑s=1

1

∼= −nln(2) + ln(n!)− 2 ln( n

2!)− 4

n( r

2 )( r2 + 1)2

+rn

∼= −nln(2) + ln(n!)− 2 ln( n

2!)− r2

2n. (1.2.12)

Since ln(a) = b and a = exp(b) are equivalent statements, 1.2.12 can be written as

P(n, r) ∼= 12n

n!n2 ! n

2 !exp

(− r2

2n

)

∼= C exp

(− r2

2n

)(1.2.13)

where

C =1

2nn!

n2 ! n

2 !. (1.2.14)

In principle, we could simplify C using algebraic manipulations, as in [9]. However, we can also obtain the same simplified expression for C by observingthat it is a normalizing constant such that

∫ ∞−∞ P(n, r)dr = 1. In this case, we obtain C = 1/∫ ∞−∞ P(n, r)dr = 1/

√2πn. Plugging this into 1.2.13 completes the

derivation.

It is convenient to express 1.2.6 in terms of the mean-square displacement 〈r2〉:

〈r2〉 =∫ ∞

−∞r2P1d(n, r)dr

=1√2πn

∫ ∞

−∞r2 exp

(− r2

2n

)dr

= n. (1.2.15)

Thus, combining 1.2.6 and 1.2.15, we have

P1d(〈r2〉, r) =1√

2π〈r2〉 exp(− r2

2〈r2〉)

, (1.2.16)

7


1 Biopolymers

which gives the Gaussian probability distribution function for a one-dimensionalrandom walk in terms of the location r and the mean square end-to-end distance〈r2〉.

1.2.3 Gaussian chain

b

R

ri

Figure 1.5: Model polymer consisting of rigid links of length b connected to each other by freelyrotating hinges. The end-to-end vector R and bond vector ri for link i are depicted.

We now have a probability distribution to describe the behavior of a Gaussianchain. Consider a chain of N + 1 backbone atoms Ai. Let ri be the bond vectorfrom atom Ai to atom Ai+1. There are N bond vectors, and all of the bond vectorsare assumed to have the same length b. Thus, the contour length of the chain, orthe length of the chain if it were completely stretched out, is Nb. In general, thechain will not be fully stretched out. The degree to which it is stretched can bedescribed by the end-to-end vector R, which is the sum of all bond vectors in thechain:

R =N

∑i=1

ri. (1.2.17)

Since our probability distribution is in terms of the mean square end-to-end dis-tance, we calculate this quantity for our model chain here. This quantity can beexpressed as

8



〈R2〉 = 〈R·R〉

=

⟨(N

∑i=1

ri

)·(

N

∑i=1

ri

)⟩

=N

∑i=1

N

∑i=1〈ri· ri〉

=N

∑i=1

N

∑i=1〈ri· ri〉. (1.2.18)

Since |a· b|/|a||b| = cosθab, where |a| and |b| are the lengths of a and b, and θabis the angle between a and b, then 1.2.18 can be written as

〈R2〉 =N

∑i=1

N

∑i=1

b2〈cosθi j〉 (1.2.19)

The direction of each of the bond vectors is independent of all the other bond vec-tors. cosθi j ranges from−1 to 1 and will on average be 0, except when computedbetween the same bond, in which case on average will be cos 0 = 1. Mathemati-cally, an equivalent statement is 〈cosθi j〉 = 0 for i 6= j, and 〈cosθi j〉 = 1 for i = j,or 〈cosθi j〉 = δi j. Thus,

〈R2〉 =N

∑i=1

N

∑i=1

b2δi j

= Nb2. (1.2.20)

This is the mean square end-to-end length of a three dimensional chain composedof N segments of size b. The quantity 〈R2〉 scales linearly with N.To obtain the probability distribution function for this chain, we combine this re-sult with the Gaussian approximation of the random walk 1.2.16 obtained earlier.Notice that 1.2.16 is for a one-dimensional random walk and is in terms of theone-dimensional mean square end-to-end length, but we can extend it to threedimensions quite easily. We know that for a three dimensional random walk, ifR = Rxex + Ryey + Rzez (in Cartesian coordinates), the mean square end-to-enddistance is the sum of the mean square end-to-end distances along each of thethree coordinate directions, or

9


1 Biopolymers

〈R2〉 = 〈R2x + R2

y + R2z〉.

= 〈R2x〉+ 〈R2

y〉+ 〈R2z〉. (1.2.21)

Thus, the mean square end-to-end distance along one direction is simply onethird of the total mean square end-to-end distance:

〈R2x〉 = 〈R2

y〉 = 〈R2z〉 =

〈R2〉3

=Nb2

3. (1.2.22)

Since the three components of a three-dimensional random walk along the threecoordinates directions are independent of each other, the three-dimensional prob-ability distribution function for a random walk can be computed as the productof three one-dimensional distribution functions in each of the three directions.More precisely,

P3d = P1d(〈R2x〉, Rx)P1d(〈R2

y〉, Ry)P1d(〈R2z〉, Rz). (1.2.23)

Combining Equations 1.2.16, 1.2.22, and 1.2.23,

P3d(N, R) =(

32πNb2

)3/2

exp

(−3(R2

x + R2y + R2

z)2Nb2

)

=(

32πNb2

)3/2

exp(− 3R2

2Nb2

), (1.2.24)

which is the three-dimensional probability distribution for a Gaussian chain con-sisting of N bonds of length b to have an end-to-end vector of R.

Free energy of the Gaussian chain

We will now use the probability distribution for the Gaussian chain to computeits free energy. In our model, the polymer has zero energy regardless of its con-formation. Thus, Ψ = −TS. We learned earlier that entropy can be calculatedfrom Ω, which in this case, is the number of configurations in which the poly-mer can have an end-to-end vector of R. However, we know that for a Gaussian

10



chain, the probability distribution function P3d(N, R) is the number of polymerconfigurations that have an end-to-end vector R divided by the total number ofpossible configurations. Mathematically,

P3d(N, R) =Ω(N, R)

Ωtotal(1.2.25)

where Ωtotal is the total number of possible configurations. We can now computethe entropy as

S(N, R) = k ln(P3d(N, R)) + k ln(Ωtotal)

= −32

kR2

Nb2 +32

k ln(

32πNb2

)+ k ln(Ωtotal). (1.2.26)

The last two terms in 1.2.26 do not depend on R, so for convenience we lumpthem into a single term denoted as S0:

S(N, R) = −32

kR

Nb2 + S0 (1.2.27)

Now, the free energy Ψ(N, R) for a chain composed of N segments with an end-to-end vector R can be written as

Ψ(N, R) = −TS(N, R)

=32

kTR2

Nb2 + Ψ0 (1.2.28)

where Ψ0 = −TS0 is again a constant that does not depend on R.You might be asking yourself now whether this is an accurate model. In real-ity, there will be energetic interactions within polymer backbone atoms, due tochanges in bond angles and distances, for example. The backbone atoms can alsointeract with eachother from a distance to due electrostatic repulsion. So is theGaussian chain, which has zero energy regardless of the conformation, a goodrepresentation of any polymers in reality?The answer is yes! The atomic structure of the polymer dictates its behavior atsmall length scales. However, at large length scales, the correlation between eachof the segments vanishes. The result is that the polymer behaves as if is com-posed of many independently fluctuating chain segments. Thus, the behavior ofall polymers will tend towards that of a Gaussian chain, as long as its countourlength is long enough.

11


1 Biopolymers

Gaussian chain as an entropic spring

Now that we have an expression for the free energy as function of the end-to-endvector R, we can calculate the force necessary to extend the chain. In our case,force stems from the change in free energy with the end-to-end vector. Mathe-matically, we can write

Fx =∂Ψ(N, R)

∂Rx=

3kTNb2 Rx

Fy =∂Ψ(N, R)

∂Ry=

3kTNb2 Ry

Fz =∂Ψ(N, R)

∂Rz=

3kTNb2 Rz, (1.2.29)

and

F =3kTNb2 R. (1.2.30)

Equation 1.2.30 gives the force necessary to stretch a Gaussian chain such that itsend-to-end vector is R. A couple of interesting things can be seen here. First, theforce is non-zero! Even though the polymer consists of freely rotating segments(i.e., no energy), the polymer can exert a force if the ends are separated. This isbecause the most number of polymer conformations can occur with a zero end-to-end distance, and thus is the most favorable entropically. Stretching the polymerout reduces the entropy, and thus takes force. Second, the force is linear in R, sothe pulling the ends apart with a distance R produces the same force as an elasticspring with spring constant (3kT)/(Nb2). Thus, it is said that 1.2.30 gives thebehavior of an entropic spring. Finally, we know that most engineering materialsbecome less stiff with temperature. However, we can see that the stiffness of thespring is proportional to temperature, so increasing T increases the stiffness!

Limitations of the Gaussian chain

Although many polymers can be accurately modeled as a Gaussian chain, thismodel will break down in certain instances. For example, we found that the re-sponse of a Gaussian chain to tension is given by 1.2.30. Notice that the stiffness ofthe Gaussian chain does not depend on how much it is extended. Therefore, themodel predicts that the stiffness will be constant even as the chain is extended

12



past its contour length Nb! This is clearly unphysical, since the chain can in-finitely extend past its contour length. Thus, 1.2.30 is only valid if the end-to-enddistance is much less than the contour length. We now present two models, thefreely jointed chain (FJC) (see e.g., [3–6]) and the wormlike chain (WLC) (see,e.g., [2, 7]) that give more realistic behavior at long extensions.

1.2.4 Freely jointed chain

For the freely jointed chain, the model chain consists of N links of length b that areconnected to each other via freely rotating hinges, identical to the Gaussian chain(see Figure 1.5). However, in this model, the end-to-end length is constrainedsuch that it can not be longer than the contour length. The average end-to-endextension in the z direction 〈Rz〉 is related to the force Fz applied in the z directionas

〈Rz〉 = Nb(

coth τz − 1τz

)(1.2.31)

where τz = (Fzb)/(kT) (for those interested, the derivation of is presented at theend of the section). Note that the hyperbolic cotangent of x coth x is defined as

coth x =ex + e−x

ex − e−x −1x

. (1.2.32)

In the freely jointed chain, the polymer is modeled as discrete, rigid segments.However, in reality, polymers resemble continuous curves much more than strandsof freely rotating links. The wormlike chain model addresses this by modelingpolymers as continuous space curves. However, before we introduce the worm-like chain model, we first need to introduce the notion of persistence length.

1.2.5 Persistence length

Consider a polymer modeled as a continuous, curvy beam with Youngs modulusE and moment of intertia I. The persistence length lp is defined as the quotient ofthe bending stiffness and kT, or

lp ≡ EIkT

. (1.2.33)

What good is this quantity? It turns out that the persistence length gives a char-acteristic length scale such that for a polymer undergoing thermal fluctuations

13


1 Biopolymers

θ(s)

Polymer undergoing

thermal fluctuations

Fixed end

(s = 0)

Free end

(s = L)

Figure 1.6: Schematic depicting relevant quantities when defining persistence length. The quantitys runs from 0 to L. The angle the polymer makes with an imaginary horizontal line at each point sis given by θ(s).

at temperature T, the orientations at two points more than lp apart are uncor-related. Lets investigate this a little further. Consider a continuous polymer ofcontour length L undergoing thermal fluctuations, as in Figure 1.2.5. We definea quantity s that runs from zero to the contour length L and thus gives a param-eterization by which each point on the polymer can be identified. We define theorientation at each point θ(s) as the angle the polymer makes with an imaginaryhorizontal line. At s = 0, the polymer end is fixed such that θ(0) = 0. Pre-tend that we monitor θ(s) over time, and draw probability distributions for θ. Ifwe were to draw the distribution for short s (i.e., near the fixed end), then wewould expect that the chances for the polymer to be have an orientation differentfrom θ = 0 would be very small, and thus the distribution would be very sharpwith a peak at θ = 0. In this case, if we were to find the average cosine of theangle, then 〈cosθ〉 ≈ 〈cos 0〉 ≈ 1. In contrast, at longer s, we would expect amuch higher chance for the polymer to have a different orientation than θ = 0, sothe distribution would be wider. The distribution becomes wider and wider forlonger and longer s until for some point, the distribution becomes uniform. Atthis point s, the polymer orientation is completely random (i.e., the orientationbecomes uncorrelated from the fixed end), and so 〈cosθ〉 ≈ 0. Thus, 〈cosθ(s)〉 (or〈cos(θ(s)−θ(0))〉 if θ(0) 6= 0) decreases from 1 to 0 as s gets larger and larger. Itcan be shown through statistical mechanics that this decrease with s is exponen-tial such that

〈cos ∆θ(s)〉 = exp(−s

dlp

)(1.2.34)

where ∆θ(s) = θ(s)−θ(0), d = 1 if the polymer is in three dimensions, and d = 2if the polymer is in two dimensions. The term 〈cos ∆θ(s)〉 is called the orientationcorrelation function. Thus, we say that the persistence length gives a character-

14



istic length scale over which the orientations of a thermally fluctuating polymerbecome uncorrelated. The length at which this occurs can vary dramatically fordifferent biopolymers. For example, the persistence length is 50 nm for DNA, 15µm for F-actin, and 6 mm for microtubules. The difference in persistence lengthsfor these three biopolymers spans more than five orders of magnitude!Besides giving us a geometric interpretation of persistence length, there are sev-eral practical uses for 1.2.34. One is to measure the effective Youngs modulus ofa polymer. For example, we can observe the thermal fluctuations of a polymer,and calculate cos ∆θ(s) for different values of s. We do this multiple times, andthen find the average values at each point, which gives 〈cos ∆θ(s)〉. We can thenfit these points to an exponential using 1.2.34 to find lp. Finally, assuming weknow the moment of intertia of the polymer and the temperature at which themeasurements were made, we can find the Youngs modulus using 1.2.33.

-π 0 π

0

Short s (sharp distribution)

Very long s (uniform)

Long s (wider distribution)

Orientation of free end (radians)

Pro

ba

bili

ty

Short s

Long s

Very long s

Figure 1.7: For very short s, the chances for the polymer to be have an orientation different fromθ = 0 is small, and thus the distribution is very sharp. At long s, there is a much higher chancefor the polymer to have a different orientation than θ = 0, and so the distribution widens. At verylong s, the distribution is uniform.

1.2.6 Wormlike chain

Now that we know what the persistence length is, we can now introduce thewormlike chain. As mentioned before, in the wormlike chain, polymers are mod-eled as continuous space curves. The configuration of these curves is given byr(s). The quantity s runs from 0 to the contour length L and again gives a pa-rameterization by which each point on the polymer can be identified, and r(s) isa vector from the origin and ends at point s (see Figure 1.8). The chain opposes

15


1 Biopolymers

r(s)

s = 0

s = L

R

Figure 1.8: Relevant quantities for the wormlike chain. Figure adapted from [11].

bending deformation through increases in energy with curvature. The bendingenergy is given by

Wbend =kTlp

2

∫ L

0

(∂2r(s)

∂s2

)2

ds (1.2.35)

where lp is the persistence length, and ∂2r(s)∂s2 is a measure of curvature. For exam-

ple, for a circle of radius of R, ∂2r(s)∂s2 is constant, and equal to 1/R. The wormlike

chain is constrained to be inextensible, which is enforced by setting the tangentvector at each point equal to one:

∂r(s)∂s

= 1. (1.2.36)

Mathematically, the wormlike chain is difficult to work with. In fact, an analyt-ical result for the force-extension relation of the wormlike chain does not exist.However, the force-extension behavior has been quantified using computationalmodels, and the results were fit to the following equation [1]:

Fz =kTlp

[14

(1− 〈Rz〉

L

)−2

− 14

+〈Rz〉

L

](1.2.37)

The similarities and differences in the force-extension behavior of the freely jointedchain and the wormlike chain can be seen below.

16



10−2

10−1

100

101

0.0

0.2

0.4

0.6

0.8

1.0

Force F (pN)

Exte

nsio

n R

/L

z

z

Magnetic

bead

Rz

DNA

Figure 1.9: Experimentally obtained force-extension data for DNA (circles) fit to a wormlike chain(solid line) and freely jointed chain (dashed line). Figure adapted from [1,11]

1.2.7 Fitting DNA force-extension curves to the FJC and WLC

The force-extension behavior of biopolymers can be investigated through a vari-ety of experiments. Figure 1.9 shows experimental data for the force-extensionbehavior of a segment of DNA. The measurements were made by attaching oneend of the DNA to a glass surface, and the other end to a magnetic bead. Thebead is pulled with a known force Fz, and the extension Rz is measured optically.The data was fit with the freely jointed chain and the wormlike chain. As youcan see, the behavior for both models and the experimental data fit our physi-cal intuition: that at large forces, the extension asymptotes at a value equal tothe contour length. However, the experimental data is better fit by the wormlikechain, especially at forces greater than 10pN (the wormlike chain is stiffer thanthe freely jointed chain, as it predicts less extension for a given amount of forcein this range). Perhaps this is not unexpected, since a strand of DNA resembles acontinuous curve much more than a strand of discrete, freely rotating segments!

1.2.8 DNA looping and lac repressor

Now that we have learned about these different chain models, they can help usinterpret some very interesting experiments investigating the interaction betweena protein called lac repressor and DNA.Within a bacteria called Escherichia Coil (E. coli for short), three enzymes havebeen identified that have been found to be necessary for metabolizing lactose.These enzymes are called lacZ, lacY, and lacA. Remember that the first step of

17


1 Biopolymers

RNA polymerase

Lac repressor

DNA

Binding sites for

lac repressor

Distance between

binding sites

(operator distance)

Genes encoding for

lacZ, lacY, and lacA

DNA loop of

radius R

R

Figure 1.10: Binding of lac repressor to DNA requires looping of DNA. When this occurs, thebound lac repressor prevents RNA polymerase from translating the genes for lacZ, lacY, and lacA(adapted from http://www.pdb.org).

protein production is transcription, where RNA polymerase moves along andreads DNA, and in turn makes mRNA. Before the genes for lacZ, lacY, and lacAare two binding sites for a protein called lac repressor. The binding sites areseparated by a fragment of DNA (the distance between binding sites is called theoperator distance). In order for lac repressor to bind, the DNA must form a loop.Once bound, the lac repressor blocks the path of RNA polymerase, preventingthe transcription of lacZ, lacY, and lacA. Thus lac repressor has a very descriptivename: it represses the levels of the lac enzyems lacZ, lacY, and lacA!Experiments show that when the operator distance is modified by inserting dif-ferent sized fragments of DNA between the lac repressor binding sites, the repres-sion of the lac enzymes peaks when the operator distance is about 70 base pairslong [8], and decreases when the operator distance becomes larger or smaller thanthis value. Why does this occur? Remember that the binding of lac repressor toDNA requires the DNA to loop. Whether lac repressor binds is determined bythe probability for this loop to occur. We look at the probability of looping forvery short and very long operator distances here.What model do we choose to look at the probability of looping? One of the maindifferences between the wormlike chain and the Gaussian chain is that the worm-like chain contains an energy of bending. Remember that the configuration of apolymer is determined by a competition between energy W and entropy S. In

18



Figure 1.11: Experimentally obtained behavior of repression of lac enzymes as a function of op-erator distance (figure adapted from [8]). The repression peaks at approximately 70 base pairs.As the operator distance decreases or increases from this value, the repression goes to zero (i.e.,the levels of mRNA increase).

general, the smaller the polymer contour length is compared to its persistencelength, the more important the energy term will be. As the contour length be-comes longer and longer, the polymer begins to resemble a chain of indepen-dently fluctuating segments (each segment with length on the order of lp), andthe free energy becomes dominated by the entropy. Taking this into account, forvery short operator distances, we model the chain as a wormlike chain, and forvery long operator distances, we model it as a Gaussian chain.We first investigate what happens for very short operator distances [10, 11]. As-suming an operator distance of L loops into a circle of radius R, the energy oflooping for a wormlike chain is

Wloop =kTlp

2

∫ L

0

(1R

)2

ds

=kTlp

2L

R2

=kTlp

2L

( L2π

)2

=2π2kTlp

L(1.2.38)

where we made use of the fact that L = 2πR. We know that the higher theenergy of looping, the lower the probability of looping, Ploop. Thus, the shorterthe operator distance L, the higher the energy of looping, and the less likely for

19


1 Biopolymers

the DNA will loop. In other words, Ploop → 0 as L → 0, and the repression willdecrease (i.e., the levels of mRNA will increase) as L → 0.For very long operator distances, the probability distribution for a Gaussian chainis given by 1.2.24 [10, 11]. A loop occurs when R = 0, and so

Ploop =(

32πNb2

)3/2

=(

32π(Nb)b

)3/2

=(

32πLb

)3/2

(1.2.39)

where we made use of the fact that L = Nb. Thus, Ploop → 0 as L → ∞, and therepression will decrease as as L → ∞. Our predictions that the repression willdecrease (i.e., mRNA levels increase) as L → 0 and as L → ∞ qualitatively matchthose of the experiments!

20



Derivation of the force-extension relation for the FJCWe derive the force-extension relation for the FJC here, following the approach taken by [11]. Consider the chain given in Figure 1.5. The links each have anorientation given by the unit vector ui such that ri = bui . The end-to-end vector R is given by

R = bN∑i=1

ui . (1.2.40)

In spherical coordinates,

ex · ui = sinθi cosφi

ey · ui = sinθi sinφi

ez · ui = cosθi

(1.2.41)

The end-to-end distance along the z-axis is given by

Rz = ez ·R

= bN∑i=1

ez · ui

= bN∑i=1

cosθi . (1.2.42)

If the chain is subject to a tension Fz along the z direction, then the energy is given by

Etension = −Fz Rz = FzbN∑i=1

cosθi . (1.2.43)

It can be shown that the probability PFJC of a particular chain conformation with energy Etension to occur is given by the Boltzmann probability as

PFJC(u1 , . . . uN, N) =1Q

exp(− Etension

kT

)

=1Q

exp

(− Fzb

kT

N∑i=1

cosθi

)

=1Q

exp

(−τ

N∑i=1

cosθi

)(1.2.44)

where τ = Fzb/(kT). Q is the chain partition function, and can be found by integrating PFJC over each of the spherical coordinates as

Q =∫ 2π

0dφ1

∫ π

0sinθ1dθ1 . . .

∫ 2π

0dφN

∫ π

0sinθN dθN exp

(τ

N∑i=1

cosθi

)

=N∏i=1

∫ 2π

0

∫ π

0sinθi exp (τ cosθi) dφidθi

=N∏i=1

q

= qN (1.2.45)

To solve for q,

q =∫ 2π

0

∫ π

0sinθi exp (τ cosθi) dφidθi

= 2π

∫ π

0sinθ exp (τ cosθ) dθ. (1.2.46)

continued on next page

21


1 Biopolymers

Derivation of force-extension relation for the FJC (cont’d)If we let ρ = cosθ, then

q = 2π

∫ 1

−1exp (τρ) dρ

= 2πexp(τ)− exp(−τ)

τ(1.2.47)

Now, since sinh a = (1/2)(exp(a)− exp(−a)), then 1.2.47 can be written as

q = 2π2 sinh τ

τ

= 4πsinh τ

τ(1.2.48)

We can now solve for 〈Rz〉 as

〈Rz〉 =∫ 2π

0dφ1

∫ π

0sinθ1dθ1 . . .

∫ 2π

0dφN

∫ π

0sinθN dθN Rz PFJC(u1 , . . . uN, N)

=∫ 2π

0dφ1

∫ π

0sinθ1dθ1 . . .

∫ 2π

0dφN

∫ π

0sinθN dθN

(τ

N∑i=1

cosθi

)1Q

exp

(−τ

N∑i=1

cosθi

)(1.2.49)

Notice that

bQ

∂Q∂τ

=∫ 2π

0dφ1

∫ π

0sinθ1dθ1 . . .

∫ 2π

0dφN

∫ π

0sinθN dθN exp

(τ

N∑i=1

cosθi

)1Q

exp

(−τ

N∑i=1

cosθi

)(1.2.50)

which is the exact expresion we obtained in 1.2.49. Therefore, 〈Rz〉 = (b/Q)(∂Q/∂τ), and

〈Rz〉 =bQ

∂Q∂τ

= b∂ log Q

∂τ

= bN∂ log q

∂τ

= bN∂

∂τ[log(sinh τ)− log τ + log 4π ]

= bN(coth τ − 1τ

) (1.2.51)

which is the desired result.

22


Bibliography

[1] BUSTAMANTE, C., J. F. MARKO, & E. D. SIGGIA [1994]. ‘Entropic elasticity of l-phage DNA.’ Science, 265, pp. 1599–1600.

[2] BUSTAMANTE, C., Z. BRYANT & S. B. SMITH [2003]. ‘Ten years of tension: single-molecule DNA mechanics.’ Nature, 421, pp. 423–427.

[3] FLORY, P. J. [1969]. Statistical Mechanics of Chain Molecules. John Wiley & Sons,Chichester – New York.

[4] FLORY, P. J. [1976]. ‘Statistical thermodynamics of random networks.’ Proceedings ofthe Royal Society of London A, 351, pp. 351–378.

[5] KUHN, W. [1934]. ‘Uber die Gestalt fadenformiger Molekule in Losungen.’ Kolloid-Zeitschrift, 52, pp. 2–15.

[6] KUHN, W. & F. GRUN [1942]. ‘Beziehungen zwischen elastischen Konstanten undDehnungsdoppelbrechung hochelastischer Stoffe.’ Kolloid-Zeitschrift, 101, pp. 248–271.

[7] MARKO, J. F. & E. D. SIGGIA [1995]. ‘Stretching DNA.’ Macromolecules, 28, pp. 8759–8770.

[8] MULLER, J., S. OEHLER, & B. MULLER-HILL [1996]. ‘Repression of lac promoter asa function of distance, phase, and quality of an auxiliary lac operator.’ J. Mol. Biol.,267, pp. 21–29.

[9] RUBINSTEIN, M. & R. H. COLBY [2003]. Polymer Physics. Oxford University Press,Oxford.

[10] SPAKOWITZ, A. J. [2006]. ‘Wormlike chain statistics with twist and fixed ends.’Europhys. Lett., 73(5), pp. 684–690.

[11] SPAKOWITZ, A.J. [2007]. ChemEng 466 Polymer Physics Class Notes.

23

1 biopolymersbiomechanics.stanford.edu › me339 › me339_s050607.pdf1.2 introduction to polymer...

Documents