“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. 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 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, 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, poly- mers want to minimize their free energy. As we have seen already, the free energy is defined as ª = W ¡ TS. (1.1.1) 1 “Lectures˙05˙and˙06˙and˙07” — 2007/11/10 — 12:34 — page 2 — #2 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 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 re- lated 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 ­. (1.1.2) Here, k is the Boltzmann constant and is equal to 1.38x10¡23 J/K. ­ is defined as the number of microstates for a given macrostate corresponding to some macro- scopic 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 ­ is defined as the number of microstates for a given macroscopic quantity. In our example, the different mi- crostates are the different polymer configurations, and the macroscopic quantity of interest is the end-to-end length R. Thus, finding ­ 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 ­(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, ­(R < L) > 1. The entropy is proportional to ln ­, so the higher the value of ­, 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). 2 “Lectures˙05˙and˙06˙and˙07” — 2007/11/10 — 12:34 — page 3 — #3 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 by freely 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. 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”), 3 “Lectures˙05˙and˙06˙and˙07” — 2007/11/10 — 12:34 — page 4 — #4 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 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 poly- mer 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 4 “Lectures˙05˙and˙06˙and˙07” — 2007/11/10 — 12:34 — page 5 — #5 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 cal- culated 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 ran- dom walk.