Statistical Thermodynamics: Molecules to Machines

Statistical Thermodynamics: Molecules to MachinesVenkat Viswanathan 3 November 2014Module 7: PolymersLearning Objectives: Introduce several key concepts in polymers and proteins. Analyze the response of a single polymer chain to tension.Key Concepts:Polymer, protein, copolymers, polymer architecture, block copolymer, DNA origami, central limit theorem, ideal chain models, entropic elas- ticity, Gaussian chain model, protein foldingPolymers and proteinsPolymer is a term used to describe molecules consisting of a large number of repeating units connected by covalent chemical bonds (Fig. 1).Proteins are polymers that are composed of specific sequence of amino acids that are linked by peptide bonds. Nature has exquisite control over primary structure of proteins (i.e. amino acid sequence), facilitating the manufacture of proteins that fold into precise structures with specific biological functions. There are 20 natural amino acids that are used to construct biological proteins. The diverse chemical function- ality of these amino acids provides the flexibility to create enzymes and structural proteins that engage in a wide range of biological processes (Fig. 2).Physical behavior at different length scalesThe chemical structure of a polymer dictates the atomic level physical behavior. Thermal fluctuations at small length scales result in local de- formations that are randomly distributed about a mean structure with a finite variance. At larger length scales, the correlation between the fluctuating chain segments vanishes, and the behavior is that of many independently fluctuating chain segments. The Central Limit The- orem states that any sum of many independent identically distributed random variables will tend to a normal (or Gaussian) distribution, pro- vided the sum of variables has a finite variance.Various models for the small-length-scale physical behavior of a poly- mer exist (see Fig.4). Though they differ in their small length scale be- havior, they all tend to a Gaussian distribution at large length scales. Today, we focus on a discrete random walk (or a freely jointed chain) to demonstrate the crossover to a Gaussian distribution.

Figure 4: Several ideal chain models.

Figure 1: Some examples of Poly- mer. The figure above corresponds to the chemical structure of polyethylene. The figure below is the chemical struc- ture of DNA

Figure 2: The twenty amino acids di- vided into six categories according to the chemical nature of their side chains.

Figure 3: A polyethylene molecule at different length scales. The atomic length scale exhibits a very specific physical behavior. However, the specifics of the underlying physics is washed out as we proceed to larger and larger length scales.Ideal random-walk (or freely jointed) polymer chain: Simplest idealization of a flexible polymer. Polymer does not interact with itself, phantom chain. Random walks appear in a range of seemingly different physical pro- cesses (diffusion, heat transfer, quantum mechanics, ...).Entropic elasticityConsider a walk on a one-dimensional lattice. After N steps, how far do you travel on average?NX = s1 + s2 + ... + sN = . si(1)i=1where si = 1 is the jump vector. The mean-square end displacement is:N N(X2) = . .(sisj)(2)i=1 j=1Since jumps are uncorrelated, we have:(sisj) = ij(3)hwere ij if i = j and ij = 0 if i = j. ThereforeN NN(X2) = . . ij = . 1 = N(4)i=1 j=1i=1This gives the size of a 1-dimensional polymer with discrete subunits of unit length.This is extended to 3 dimensions. Consider a chain of N bond vectorsbi (i = 1, 2, ..., N ) with length b(bi bi = b2), with end-to-endvector R = .N bi. The mean-square end-to-end distance is:N NN N(R 2) = . .(bibj) = . . b2ij = b2N(5)i=1 j=1i=1 j=1using similar arguments as the 1-dimensional case.Decomposing the end-to-end vector into its components R = Xx +Y y + Zz, we have:2X2 = Y 2 = Z2 =3

(6)since no direction is preferred.The central limit theorem (see Lecture 5) states that the limiting behavior of any statistical distribution tends to a Gaussian distributionin the limit of large sample size. In our case, the limit of large N leads to a chain probability distribution that is Gaussian. With this, we construct the polymer chain distribution function.Generally, a Gaussian distribution for a variable X with zero mean ((X) = 0) and variance (X2) is written as:1px(X, N ) = ,2 X2

exp

.X2 . 2(X2)

(7)This is the limiting behavior of a random walk in one dimension as the number of steps is very large (N 1).Assuming the probability distribution is decoupled in the 3 direc- tions, we write the 3-dimensional probability distribution asp(R , N ) = px(X, N )py (Y , N )py (Z, N ). 2Nb2 .3/2

.3R2 .=3exp

2Nb2

(8)giving the probability that a chain of length N that begins at the origin will end at position R .The chain probability gives the free energy for constraining the chain ends at a fixed position; specifically, we have:F (R ) F (0) = kBT log

. p(R , N ) . p(0, N )=

3kBT2Nb2

R 2(9)Thus, the chain behaves as a Hookean spring, i.e a Gaussian chain exhibits a linear response to tension. Pulling a Gaussian chain along the z-axis a distance Z requires a tension given by: = F (Z) = 3kBTZNb2 Z(10)This linear response governs the small-scale deformation of a polymer chain. The response is purely entropic, indicated by the linear scaling in temperature. This leads to the curious behavior for a stretch elastic band retracts when heated. The Gaussian chain model does not capture the behavior once the chain is nearly extended. Full extension reflects the microscopic details of the model; for example, the nearly inextensible covalent bonds of the polymer chain. However, many polymer properties are captured by the Gaussian chain model. This is highly advantageous because the properties tend to be governed by just a few parameters (Fig. 5).Protein FoldingThe entropic elasticity of a polymer to tension arises due to the confor- mational entropy of the polymer. Proteins fold into a unique structure

Figure 5: The response of DNA to ten- sion [?] compared to three competing theories: the Gaussian chain (dotted green curve), the freely jointed chain (dashed blue curve), and the worm- like chain (red dashed-dotted curve are asymptotic results and solid red curve are exact results from Ref.[?]). In all cases, the Kuhn length is set to b = 2lp = 106nm.that is stabilized by favorable contacts between amino acid residues at the expense of the loss of conformational entropy (6).Folding typically requires milliseconds to occur. Large-scale projects such as Folding@Home attain sufficient simulation time scales to com- putationally simulate (http://folding.stanford.edu/). Energy land- scape theory of protein folding has been an important concept in under- standing protein folding. Energy landscape theory of protein folding has been an important concept in understanding protein folding. To overcome the entropy loss associated with folding, the folded state must have a substantially low potential energy.

Figure 6:A simple lattice model of a protein elucidates the physical processes that un- derlies protein folding (see http://www2.lbl.gov/Science-Articl es/Archive/model-protein-folding.h tml for more details.

Figure 7: The folding of a protein into its native structure (N ) frequently in- volves kinetic trapping in intermedi- ate states. One way to picture this is through a potential landscape that is funnel-like with many intermediate bumps between the unfolded state and the native state [?].i=1

()()()b N

(

)