1 Biopolymers

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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. To begin, we first need to introduce the idea of thermal fluctuations. Imagine a small particle floating in fluid. The particle will undergo Brownian motion, or random movements as it is randomly pushed around by the molecules around it. The motion of the particle depends on the temperature: if the temperature is increased, the motion of the particle will also be increased. Like Brownian motion, polymers are subject to random molecular forces that cause them to ”wiggle”. These wig- gles are called thermal fluctuations. As temperature is increased, the polymers wiggle 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 of a polymer. Both are induced by random forces induced by the surrounding molecules.

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

Ψ = W − TS. (1.1.1)

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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 free energy by lowering its energy W, or increasing its entropy S. Lets look at each of these quantities individually.

Entropy What is entropy? You may have heard the vague explanation that entropy is related to disorder. Many people have heard the ”messy room” analogy, in that a messy 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 as the number of microstates for a given macrostate corresponding to some macroscopic quantity. What is meant by this? Consider a model polymer consisting of n rigid links of size b. The links are connected to each other by freely rotating hinges. 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 length of the polymer if it was completely stretched out. In this case, since there are n links of size b, then L = nb. The second length is the end-to-end length R. This is simply the length from one end of the polymer to the other. In this case, R must always be less than or equal to L. Remember that Ω is defined as the number of microstates for a given macroscopic quantity. In our example, the different microstates are the different polymer configurations, and the macroscopic quantity of interest is the end-to-end length R. Thus, finding Ω boils down to counting the number 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 only one polymer conformation in which this can occur, and so Ω(R = L) = 1. Now suppose that the polymer is not completely stretched out, or R < L. In this case, there are multiple polymer conformations that have the same end-to-end length. Thus, Ω(R < L) > 1. The entropy is proportional to ln Ω, so the higher the value of Ω, the higher the entropy. Therefore, the unstretched polymer has higher entropy than the stretched out polymer. To decrease its free energy, the polymer wants to increase its entropy, and so it wants to be kinked. Thus, we say that entropically, the polymer wants to be kinked (i.e., it does not want to be stretched out).

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1.1 Introduction to statistical mechanics

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

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

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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 required to 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, then we can see that the temperature T will determine whether the entropy or energy dominates the free energy. At lower temperature, W dominates, and so the polymer will be straighter. At higher temperature, S dominates, and so the polymer will be wigglier. Thus, we now see that the free energy, energy, and entropy give a mathematical description as to why the polymer becomes wigglier at higher temperature!

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 the foundation of several of these models, so we investigate this here. Random walks are used to understand a wide variety of phenomena, such as Brownian motion and diffusion. What is an example of a random walk? Consider a soccer player standing at midfield of a soccer field, facing one of the goals. We define midfield to be at location r = 0. The player takes out a coin, and flips it. If it comes up heads, the player steps forward towards one goal, if it comes up tails, the player steps backwards towards the opposite goal. This ”flip and step” process is repeated a certain number of times. We will now calculate the probability of the player being a certain location after a given number of steps. In particular, after n random, unit-sized steps, we will calculate the probability that the player will end up at location r (for more detail, see Rubinstein and Colby’s [9] analysis of a drunk staggering up and down a narrow alley, from which the approach here was

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1.2 Introduction to polymer physics: models for biopolymers followed). Note that if the player has taken more backwards steps than forward steps, 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, the number M of those paths in which only n+ of the steps are forward can be calculated using the binomial coefﬁcient l!/(m!(l − m)!), which gives the number of ways in which a group of l items can be chosen from a set of m. Using this relation, M can be calculated as

n! M = . (1.2.1) n+!(n − n+)!

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

n! M(n, r) = n+r n−r , (1.2.4) 2 ! 2 ! which gives the total number of different paths the player can take to end up at the location r if he/she takes n steps. The probability P(n, r) the player will end up at the location r after taking n steps can be found by dividing W(n, r) by the total number of paths possible after n steps, or

M(n, r) 1 n! P(n, r) = n = n n+r n−r . (1.2.5) 2 2 2 ! 2 !

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

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

µ ¶ 1 r2 P1d(n, r) = √ exp − (1.2.6) 2πn 2n

Derivation of the Gaussian approximation of the random walk We 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

µ ¶ µ ¶ n + r n − r ln(P(n, r)) = −nln(2) + ln(n!) − ln ! − ln ! . (1.2.7) 2 2

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

a + b a b/c ³ a ´ ! = ! ∏ + s c c s=1 c

and ((a − b)/c)! can be written as

a − b a ! ! = c c b/c ³ a ´ ∏ + 1 − s s=1 c

Thus, the third term in 1.2.7 can be written as

µ ¶ n + r ³ n ´ r/2 ³ n ´ ln ! = ln ! ∏ + s 2 2 s=1 2

³ n ´ r/2 ³ n ´ = ln ! + ∑ ln + s . (1.2.8) 2 s=1 2

Similarly, the fourth term can be written as

µ ¶ n − r ³ n ´ r/2 ³ n ´ ln ! = ln ! − ∑ ln + 1 − s . (1.2.9) 2 2 s=1 2

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

³ n ´ r/2 ³ n ´ r/2 ³ n ´ ln(P(n, r)) = −nln(2) + ln(n!) − 2ln ! − ∑ ln + s + ∑ ln + 1 − s 2 s=1 2 s=1 2 ³ ´ r/2 µ n ¶ n 2 + s = −nln(2) + ln(n!) − 2ln ! − ∑ ln n 2 s=1 2 + 1 − s Ã ! ³ n ´ r/2 1 + 2s = −nln(2) + ln(n!) − 2ln ! − ln n , (1.2.10) 2 ∑ 2s 2 s=1 1 − n + n

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

continued on next page

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1.2 Introduction to polymer physics: models for biopolymers

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

Ã ! µ ¶ µ ¶ 1 + 2s 2s 2s 2 ln n = ln 1 + − ln 1 − + 2s 2 n n n 1 − n + n 4s 2 =∼ − . (1.2.11) n n

a a Using 1.2.11, and the identities ∑ s = a(a + 1)/2 and ∑ 1 = a, then s=1 s=1

µ ¶ ³ n ´ r/2 4s 2 ln(P(n, r)) =∼ −nln(2) + ln(n!) − 2ln ! − ∑ − 2 s=1 n n

³ n ´ 4 r/2 2 r/2 =∼ −nln(2) + ln(n!) − 2 ln ! − ∑ s + ∑ 1 2 n s=1 n s=1 ³ n ´ 4 ( r )( r + 1) r =∼ −nln(2) + ln(n!) − 2 ln ! − 2 2 + 2 n 2 n ³ n ´ r2 =∼ −nln(2) + ln(n!) − 2 ln ! − . (1.2.12) 2 2n

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

Ã ! 2 ∼ 1 n! r P(n, r) = n n n exp − 2 2 ! 2 ! 2n Ã ! r2 =∼ C exp − (1.2.13) 2n

where

1 n! C = n n n . (1.2.14) 2 2 ! 2 !

In principle, we could simplify C using algebraic manipulations, as in [9]. However, we can also obtain the same simpliﬁed expression for C by observing R ∞ R ∞ √ that 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 hr2i:

Z ∞ 2 2 hr i = r P1d(n, r)dr −∞ Z µ ¶ 1 ∞ r2 = √ r2 exp − dr 2πn −∞ 2n = n. (1.2.15)

Thus, combining 1.2.6 and 1.2.15, we have

µ 2 ¶ 2 1 r P1d(hr i, r) = p exp − , (1.2.16) 2πhr2i 2hr2i

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1 Biopolymers which gives the Gaussian probability distribution function for a one-dimensional random walk in terms of the location r and the mean square end-to-end distance hr2i.

1.2.3 Gaussian chain

b ri

Figure 1.5: Model polymer consisting of rigid links of length b connected to each other by freely rotating 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 Gaussian chain. Consider a chain of N + 1 backbone atoms Ai. Let ri be the bond vector from atom Ai to atom Ai+1. There are N bond vectors, and all of the bond vectors are assumed to have the same length b. Thus, the contour length of the chain, or the length of the chain if it were completely stretched out, is Nb. In general, the chain will not be fully stretched out. The degree to which it is stretched can be described by the end-to-end vector R, which is the sum of all bond vectors in the chain:

N R = ∑ ri. (1.2.17) i=1

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

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1.2 Introduction to polymer physics: models for biopolymers

hR2i = hR· Ri *Ã ! Ã !+ N N = ∑ ri · ∑ ri i=1 i=1 N N = ∑ ∑hri· rii i=1 i=1 N N = ∑ ∑hri· rii. (1.2.18) i=1 i=1

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

N N 2 2 hR i = ∑ ∑ b hcosθi ji (1.2.19) i=1 i=1

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

N N 2 2 hR i = ∑ ∑ b δi j i=1 i=1 = Nb2. (1.2.20)

This is the mean square end-to-end length of a three dimensional chain composed of N segments of size b. The quantity hR2i scales linearly with N. To obtain the probability distribution function for this chain, we combine this result 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 the one-dimensional mean square end-to-end length, but we can extend it to three dimensions quite easily. We know that for a three dimensional random walk, if R = Rxex + Ryey + Rzez (in Cartesian coordinates), the mean square end-to-end distance is the sum of the mean square end-to-end distances along each of the three coordinate directions, or

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2 2 2 2 hR i = hRx + Ry + Rzi. 2 2 2 = hRxi + hRyi + hRzi. (1.2.21)

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

hR2i hR2i = hR2i = hR2i = x y z 3 Nb2 = . (1.2.22) 3 Since the three components of a three-dimensional random walk along the three coordinates directions are independent of each other, the three-dimensional probability distribution function for a random walk can be computed as the product of three one-dimensional distribution functions in each of the three directions. More precisely,

2 2 2 P3d = P1d(hRxi, Rx)P1d(hRyi, Ry)P1d(hRzi, Rz). (1.2.23)

Combining Equations 1.2.16, 1.2.22, and 1.2.23,

Ã ! µ ¶3/2 2 2 2 3 3(Rx + Ry + Rz) P (N, R) = exp − 3d 2π Nb2 2Nb2 µ ¶ µ ¶ 3 3/2 3R2 = exp − , (1.2.24) 2π Nb2 2Nb2 which is the three-dimensional probability distribution for a Gaussian chain consisting 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 compute its free energy. In our model, the polymer has zero energy regardless of its conformation. Thus, Ψ = −TS. We learned earlier that entropy can be calculated from Ω, which in this case, is the number of conﬁgurations in which the polymer can have an end-to-end vector of R. However, we know that for a Gaussian

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

Ω(N, R) P3d(N, R) = (1.2.25) Ωtotal where Ωtotal is the total number of possible conﬁgurations. We can now compute the entropy as

S(N, R) = k ln(P (N, R)) + k ln(Ω ) 3d µ total¶ 3 R2 3 3 = − k + k ln + k ln(Ω ). (1.2.26) 2 Nb2 2 2π Nb2 total The last two terms in 1.2.26 do not depend on R, so for convenience we lump them into a single term denoted as S0:

3 R S(N, R) = − k + S (1.2.27) 2 Nb2 0 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) 3 R2 = kT + Ψ (1.2.28) 2 Nb2 0 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 reality, there will be energetic interactions within polymer backbone atoms, due to changes in bond angles and distances, for example. The backbone atoms can also interact with eachother from a distance to due electrostatic repulsion. So is the Gaussian chain, which has zero energy regardless of the conformation, a good representation of any polymers in reality? The answer is yes! The atomic structure of the polymer dictates its behavior at small length scales. However, at large length scales, the correlation between each of the segments vanishes. The result is that the polymer behaves as if is composed of many independently ﬂuctuating chain segments. Thus, the behavior of all polymers will tend towards that of a Gaussian chain, as long as its countour length is long enough.

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Gaussian chain as an entropic spring

Now that we have an expression for the free energy as function of the end-to-end vector 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

∂Ψ(N, R) 3kT Fx = = 2 Rx ∂Rx Nb ∂Ψ(N, R) 3kT Fy = = 2 Ry ∂Ry Nb ∂Ψ(N, R) 3kT Fz = = 2 Rz, (1.2.29) ∂Rz Nb and

3kT F = R. (1.2.30) Nb2 Equation 1.2.30 gives the force necessary to stretch a Gaussian chain such that its end-to-end vector is R. A couple of interesting things can be seen here. First, the force 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 is because the most number of polymer conformations can occur with a zero end-to- end distance, and thus is the most favorable entropically. Stretching the polymer out reduces the entropy, and thus takes force. Second, the force is linear in R, so the pulling the ends apart with a distance R produces the same force as an elastic spring with spring constant (3kT)/(Nb2). Thus, it is said that 1.2.30 gives the behavior of an entropic spring. Finally, we know that most engineering materials become less stiff with temperature. However, we can see that the stiffness of the spring 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, this model 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 of the Gaussian chain does not depend on how much it is extended. Therefore, the model predicts that the stiffness will be constant even as the chain is extended

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1.2 Introduction to polymer physics: models for biopolymers 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-end distance is much less than the contour length. We now present two models, the freely 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 are connected 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 constrained such that it can not be longer than the contour length. The average end-to-end extension in the z direction hRzi is related to the force Fz applied in the z direction as

µ ¶ 1 hRzi = Nb coth τz − (1.2.31) τz where τz = (Fzb)/(kT) (for those interested, the derivation of is presented at the end of the section). Note that the hyperbolic cotangent of x coth x is deﬁned as

ex + e−x 1 coth x = − . (1.2.32) ex − e−x x In the freely jointed chain, the polymer is modeled as discrete, rigid segments. However, in reality, polymers resemble continuous curves much more than strands of freely rotating links. The wormlike chain model addresses this by modeling polymers as continuous space curves. However, before we introduce the wormlike 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 modulus E and moment of intertia I. The persistence length lp is deﬁned as the quotient of the bending stiffness and kT, or

EI l ≡ . (1.2.33) p kT 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

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Free end Polymer undergoing (s = L) thermal fluctuations θ(s) Fixed end (s = 0)

Figure 1.6: Schematic depicting relevant quantities when defining persistence length. The quantity s runs from 0 to L. The angle the polymer makes with an imaginary horizontal line at each point s is given by θ(s). at temperature T, the orientations at two points more than lp apart are uncorrelated. Lets investigate this a little further. Consider a continuous polymer of contour length L undergoing thermal fluctuations, as in Figure 1.2.5. We define a 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 the orientation at each point θ(s) as the angle the polymer makes with an imaginary horizontal 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. If we were to draw the distribution for short s (i.e., near the fixed end), then we would expect that the chances for the polymer to be have an orientation different from θ = 0 would be very small, and thus the distribution would be very sharp with a peak at θ = 0. In this case, if we were to find the average cosine of the angle, then hcosθi ≈ hcos 0i ≈ 1. In contrast, at longer s, we would expect a much higher chance for the polymer to have a different orientation than θ = 0, so the distribution would be wider. The distribution becomes wider and wider for longer and longer s until for some point, the distribution becomes uniform. At this point s, the polymer orientation is completely random (i.e., the orientation becomes uncorrelated from the fixed end), and so hcosθi ≈ 0. Thus, hcosθ(s)i (or hcos(θ(s) − θ(0))i if θ(0) 6= 0) decreases from 1 to 0 as s gets larger and larger. It can be shown through statistical mechanics that this decrease with s is exponential such that

µ ¶ −s hcos ∆θ(s)i = exp (1.2.34) dlp where ∆θ(s) = θ(s) −θ(0), d = 1 if the polymer is in three dimensions, and d = 2 if the polymer is in two dimensions. The term hcos ∆θ(s)i is called the orientation correlation function. Thus, we say that the persistence length gives a character-

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1.2 Introduction to polymer physics: models for biopolymers istic length scale over which the orientations of a thermally fluctuating polymer become uncorrelated. The length at which this occurs can vary dramatically for different biopolymers. For example, the persistence length is 50 nm for DNA, 15 µm for F-actin, and 6 mm for microtubules. The difference in persistence lengths for these three biopolymers spans more than five orders of magnitude! Besides giving us a geometric interpretation of persistence length, there are several practical uses for 1.2.34. One is to measure the effective Youngs modulus of a 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, and then find the average values at each point, which gives hcos ∆θ(s)i. We can then fit these points to an exponential using 1.2.34 to find lp. Finally, assuming we know the moment of intertia of the polymer and the temperature at which the measurements were made, we can find the Youngs modulus using 1.2.33.

Short s Short s (sharp distribution)

Probability Long s

Long s (wider distribution) Very long s Very long s (uniform) 0 -π 0 π Orientation of free end (radians)

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

1.2.6 Wormlike chain Now that we know what the persistence length is, we can now introduce the wormlike chain. As mentioned before, in the wormlike chain, polymers are modeled as continuous space curves. The conﬁguration of these curves is given by r(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) is a vector from the origin and ends at point s (see Figure 1.8). The chain opposes

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s = L

s = 0 R

r(s)

Figure 1.8: Relevant quantities for the wormlike chain. Figure adapted from [11]. bending deformation through increases in energy with curvature. The bending energy is given by

Z µ 2 ¶2 kTlp L ∂ r(s) Wbend = 2 ds (1.2.35) 2 0 ∂s

2 where l is the persistence length, and ∂ r(s) is a measure of curvature. For exam- p ∂s2 2 ple, for a circle of radius of R, ∂ r(s) is constant, and equal to 1 R. The wormlike ∂s2 / chain is constrained to be inextensible, which is enforced by setting the tangent vector at each point equal to one:

∂r(s) = 1. (1.2.36) ∂s

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 computational models, and the results were fit to the following equation [1]:

" # µ ¶−2 kT 1 hRzi 1 hRzi Fz = 1 − − + (1.2.37) lp 4 L 4 L

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

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1.0

Magnetic bead 0.8 L / z

R 0.6

DNA 0.4 Rz Extension

0.2

0.0 −2 −1 0 1 10 10 10 10 Force Fz (pN)

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 variety of experiments. Figure 1.9 shows experimental data for the force-extension behavior of a segment of DNA. The measurements were made by attaching one end of the DNA to a glass surface, and the other end to a magnetic bead. The bead 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 you can 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 to the contour length. However, the experimental data is better fit by the wormlike chain, especially at forces greater than 10pN (the wormlike chain is stiffer than the freely jointed chain, as it predicts less extension for a given amount of force in this range). Perhaps this is not unexpected, since a strand of DNA resembles a continuous 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 us interpret some very interesting experiments investigating the interaction between a protein called lac repressor and DNA. Within a bacteria called Escherichia Coil (E. coli for short), three enzymes have been 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

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Binding sites for lac repressor

DNA

Distance between Genes encoding for RNA polymerase binding sites lacZ, lacY, and lacA (operator distance)

DNA loop of radius R R

Lac repressor

Figure 1.10: Binding of lac repressor to DNA requires looping of DNA. When this occurs, the bound 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 and reads DNA, and in turn makes mRNA. Before the genes for lacZ, lacY, and lacA are two binding sites for a protein called lac repressor. The binding sites are separated by a fragment of DNA (the distance between binding sites is called the operator 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, preventing the transcription of lacZ, lacY, and lacA. Thus lac repressor has a very descriptive name: it represses the levels of the lac enzyems lacZ, lacY, and lacA! Experiments show that when the operator distance is modiﬁed by inserting different sized fragments of DNA between the lac repressor binding sites, the repression of the lac enzymes peaks when the operator distance is about 70 base pairs long [8], and decreases when the operator distance becomes larger or smaller than this value. Why does this occur? Remember that the binding of lac repressor to DNA requires the DNA to loop. Whether lac repressor binds is determined by the probability for this loop to occur. We look at the probability of looping for very short and very long operator distances here. What model do we choose to look at the probability of looping? One of the main differences between the wormlike chain and the Gaussian chain is that the wormlike chain contains an energy of bending. Remember that the conﬁguration of a polymer is determined by a competition between energy W and entropy S. In

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Figure 1.11: Experimentally obtained behavior of repression of lac enzymes as a function of operator 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 persistence length, the more important the energy term will be. As the contour length becomes longer and longer, the polymer begins to resemble a chain of independently fluctuating segments (each segment with length on the order of lp), and the free energy becomes dominated by the entropy. Taking this into account, for very short operator distances, we model the chain as a wormlike chain, and for very 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 of looping for a wormlike chain is

Z µ ¶2 kTlp L 1 Wloop = ds 2 0 R kTl L = p 2 R2 kTlp L = ¡ ¢ 2 L 2 2π 2 2π kTlp = (1.2.38) L where we made use of the fact that L = 2πR. We know that the higher the energy of looping, the lower the probability of looping, Ploop. Thus, the shorter the operator distance L, the higher the energy of looping, and the less likely for

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

µ ¶ 3 3/2 P = loop 2π Nb2 µ ¶ 3 3/2 = 2π(Nb)b µ ¶ 3 3/2 = (1.2.39) 2π Lb where we made use of the fact that L = Nb. Thus, Ploop → 0 as L → ∞, and the repression will decrease as as L → ∞. Our predictions that the repression will decrease (i.e., mRNA levels increase) as L → 0 and as L → ∞ qualitatively match those of the experiments!

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Derivation of the force-extension relation for the FJC We 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 an orientation given by the unit vector ui such that ri = bui. The end-to-end vector R is given by

N R = b ∑ ui . (1.2.40) i=1

In spherical coordinates,

ex· ui = sinθi cos φi

ey· ui = sinθi sin φi

ez· ui = cosθi (1.2.41)

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

Rz = ez· R N = b ∑ ez· ui i=1 N = b ∑ cosθi . (1.2.42) i=1

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

N Etension = −Fz Rz = Fzb ∑ cosθi . (1.2.43) i=1

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

µ ¶ 1 E P ({u ,... u }, N) = exp − tension FJC 1 N Q kT Ã ! N 1 Fzb = exp − ∑ cosθi Q kT i=1 Ã ! 1 N = exp −τ ∑ cosθi (1.2.44) Q i=1

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

Ã ! Z 2π Z π Z 2π Z π N Q = dφ1 sinθ1dθ1 ... dφN sinθN dθN exp τ ∑ cosθi 0 0 0 0 i=1 N Z 2π Z π = ∏ sinθi exp (τ cosθi ) dφi dθi i=1 0 0 N = ∏ q i=1 = qN (1.2.45)

To solve for q,

Z 2π Z π q = sinθi exp (τ cosθi ) dφi dθi 0 0 Z π = 2π sinθ exp (τ cosθ) dθ. (1.2.46) 0 continued on next page

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

Z 1 q = 2π exp (τρ) dρ −1 exp(τ) − exp(−τ) = 2π (1.2.47) τ

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

2 sinh τ q = 2π τ sinh τ = 4π (1.2.48) τ

We can now solve for hRzi as

Z 2π Z π Z 2π Z π hRzi = dφ1 sinθ1dθ1 ... dφN sinθN dθN Rz PFJC ({u1,... uN }, N) 0 0 0 0 Z Z Z Z Ã ! Ã ! 2π π 2π π N 1 N = dφ1 sinθ1dθ1 ... dφN sinθN dθN τ ∑ cosθi exp −τ ∑ cosθi (1.2.49) 0 0 0 0 i=1 Q i=1

Notice that

Z Z Z Z Ã ! Ã ! b ∂Q 2π π 2π π N 1 N = dφ1 sinθ1dθ1 ... dφN sinθN dθN exp τ ∑ cosθi exp −τ ∑ cosθi (1.2.50) Q ∂τ 0 0 0 0 i=1 Q i=1

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

b ∂Q hR i = z Q ∂τ ∂ log Q = b ∂τ ∂ log q = bN ∂τ ∂ = bN [log(sinh τ) − log τ + log 4π] ∂τ 1 = bN(coth τ − ) (1.2.51) τ

which is the desired result.

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

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