Question

Many cellular processes involve polymerization, where a bunch of monomers bind together to form a polymer. Examples include transcription, translation, construc. tion of the cytoskeleton, etc. Here we consider a simple model of polymerization where each monomer is added to the growing chain with the same equilibrium binding constant \(K\) This situation is described by the following set of chemical equations: $$\begin{array}{ccc} x_{1}+x_{1} & \stackrel{x}{=} & x_{2} \\ x_{1}+x_{2} & \stackrel{k}{=} & x_{3} \\ \vdots & \vdots & \vdots \\ x_{1}+x_{n-1} & \stackrel{n}{=} & x_{n} \\ \vdots & \vdots & \vdots \end{array}$$ The symbol \(X_{n}\) denotes a polymer \(n\) monomers in size. In equilibrium there will be polymers of all different sizes. We use this model to compute the average polymer size. and how it depends on the concentration of monomers. (a) Find an expression for the probability that a polymer Is \(n\) monomers in length in terms of \(K\) and \(x=\left[X_{1}\right]\), the concentration of free monomers. Use this result to get an expression for the average polymer size \((n)\) (b) Show that the average polymer size can be written as $$(m)=\frac{d \ln Q}{d K x}$$where \(Q-1+K x+(K x)^{2}+(K x)^{3}+\ldots\) is the binding poly. nomial. (c) Plot \((n)\) as a function of \(K x\). Show that \((n)\) diverges as \(K x\) approaches 1 from below. What is the physical interpretation of this divergence? (In reality there is no such thing as a polymer that is infinite in size. (Adapted from Problem 28.6 of \(\mathrm{K}\). Dill and S. Bromberg. Molecular Driving Forces, New York, Garland Science, \(2003 .)\)

Short answer

The probability \(P_n\) that a polymer is \(n\) monomers in length is \((KX_1)^n\), and the average polymer size is \((m) = dQ/d(KX_1) = 1/(1-KX_1)\). This shows that as \(KX_1\) approaches \(1\), the average polymer size diverges (becomes infinitely large), indicating a phase transition leading to the formation of a giant component or macroscopic cluster.

Step-by-step solution

Derive the probability distribution for polymers of size n

We can express the probability \(P_n\) that a polymer is \(n\) monomers in length using the law of mass action. From the chemical equations, a binding event between \(n-1\) aggregated monomers and a free monomer to form an aggregate of size \(n\) happens with the rate \(k\cdot [X_{n-1}]\cdot [X_1]\), where \(k\) is the binding rate constant. In equilibrium, this is equal to the rate of the reverse reaction where an aggregate of size \(n\) dissociates into an aggregate of size \(n-1\) and a free monomer, which happens with the rate \(k_n\cdot [X_n]\). Setting these two rates equal gives us \(k\cdot [X_{n-1}]\cdot [X_1] = k_n\cdot [X_n]\), which simplifies to \([X_n] = \frac{k[X_{n-1}][X_1]}{k_n} = K[X_{n-1}]\), since \(K = \frac{k}{k_n}\) is the equilibrium constant. Recursive application of this result gives the probability \(P_n = [X_n] = K^n[X_1]^{n} = (K[X_1])^n\)

Derive the expression of average polymer size

The average size \(\langle n \rangle\) can be computed as \(\langle n \rangle = \sum_{n=1}^{\infty}n\cdot P_n = \sum_{n=1}^{\infty}n(KX_1)^n\). Now \( \sum_{n=1}^{\infty}n r^{n}=-r(\frac{d}{dr})(1-r)^{-2} \). Using this and taking \(r = Kx_1\), we get \( \langle n \rangle = \frac{\partial (Kx_1)/(1-(Kx_1))}{\partial (Kx_1)} = 1/ (1- Kx_1) \). This is the sought expression for the size of a polymer.

Link this expression to the binding polynomial

The binding polynomial \(Q\) is defined as \(Q = \sum_{n=0}^{\infty}(KX_1)^n = 1/(1-KX_1)\), as it represents the series expansion of the same. Differentiating \(Q\) with respect to \(KX_1\) gives \(dQ/d(KX_1) = 1/(1-KX_1)^2 = (m)\), showing that the average polymer size can indeed be written in terms of the derivative of the binding polynomial.

Plot and interpret the results

As derived in step 2, \(\langle n \rangle = 1/(1-Kx_1)\). As \(KX_1 \rightarrow 1\), \(\langle n \rangle \) diverges, implying the average polymer size blows up, which means that monomers tend to aggregate into larger and larger polymers. This divergence corresponds to a phase transition from a dispersion of small polymers to the formation of a giant component or macroscopic cluster.

Key concepts

Equilibrium Binding Constant

In polymerization, the equilibrium binding constant, denoted as \( K \), is a crucial factor determining how readily monomers bind to form polymers. It acts as a measure of the strength and stability of the bond formation between a monomer and an existing polymer chain at equilibrium.
The idea is simple: a higher \( K \) value indicates a stronger affinity between the monomers, implying a greater likelihood that an aggregate or polymer will form. Conversely, a lower \( K \) suggests weaker interactions, posing a challenge for polymer formation.
The equilibrium constant is characterized by the equation \( K = \frac{k}{k_n} \), where \( k \) is the rate constant for the forward reaction (monomer binding) and \( k_n \) is the rate constant for the reverse reaction (polymer dissociation).

• A higher \( K \) can lead to longer chains, increasing the chances of forming sizeable polymers.
• It acts as a balance point where the rates of formation and dissolution of polymer segments equalize, leading to a stable system.

By understanding \( K \), scientists can manipulate conditions to enhance polymer stability, crucial in biological processes like DNA transcription and protein synthesis.

Average Polymer Size

The average polymer size is a significant parameter in understanding polymerization processes. It describes the mean length of the polymer aggregates in a given system and is denoted by \( \langle n \rangle \).
To find this, we use the expression \( \langle n \rangle = \frac{1}{1-Kx_1} \), where \( K \) is the equilibrium binding constant and \( x_1 = [X_1] \) is the concentration of free monomers. This formula gives insight into how changes in \( K \) and monomer concentration affect polymer size.

• When \( Kx_1 \) approaches 1, \( \langle n \rangle \) tends to infinity, indicating the possibility of infinitely large polymers. In reality, constraints and other factors would inhibit this infinite growth.
• The relationship between \( \langle n \rangle \) and \( Kx_1 \) helps predict and control polymer length in experiments.

Understanding average polymer size helps in various fields, including materials science and biology, by aiding in the synthesis of polymers with desired properties and functions.

Probability Distribution

The probability distribution in polymerization details how likely it is that a polymer will have a specific length. The probability \( P_n \) that a polymer consists of \( n \) monomers is given by the expression \( P_n = (K [X_1])^n \). This formula arises from the law of mass action, reflecting the balance between monomer addition and dissociation.

• It shows that longer polymers are less common due to the term \( (Kx_1)^n \), which decreases as \( n \) increases, unless \( Kx_1 \) is near 1, where longer chains become increasingly probable.
• This distribution is pivotal for predicting the variety in polymer sizes in a solution at equilibrium.

Polynomial models help visualize this distribution, assisting in deriving other related functions, such as the binding polynomial \( Q \), which represents the summation of all possible polymer probabilities. Understanding this concept guides research and applications involving polymer growth in both theoretical calculations and real-world scenarios.