Question

Binding polynomials and polymerization Many cellular processes involve polymerization, where a bunch of monomers bind together to form a polymer. Examples include transcription, translation, construction 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: 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 $$(n)=\frac{d \ln Q}{d K x}$$ where \(Q=1+K x+(K x)^{2}+(K x)^{3}+\dots\) is the binding polynomial. (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.)

Short answer

The average polymer size \((n)\), which can be written as \(d(lnQ)/d(Kx)\), tends to infinity as the product of the equilibrium constant K and the monomer concentration x approaches 1 from the below. This suggests a state of complete polymerization.

Step-by-step solution

Calculating the Probability

For \(n\) monomers to associate and form a polymer \(X_n\), the equilibrium constant \(K\) is to the power of \(n-1\), and for the \(n^{th}\) monomer, it is multiplied by \(x\). Hence, the probability \(P_n\) that a polymer is \(n\) monomers in length is given by \(P_n = K^{n-1} x^n / Q\), where \(Q\) is the sum of probabilities of all different polymer sizes.

Calculating the Average Polymer Size

The average size of the polymer \((n)\) can be obtained by summing up the product of the size and its probability over all sizes, which is \((n) = \sum (n.P_n) = \sum (n.K^{n-1} x^n / Q) = x.d(Q/Kx) / dq\).

Expressing \((n)\) in terms of K*x

As per the definition of the binding polynomial \(Q = 1+Kx+(Kx)^2+(Kx)^3+...\), the derivative of ln(Q) with respect to Kx can be written as dlnQ/d(Kx) = 1/(1-Kx). Therefore, the average polymer size \((n) = x. d[Q/Kx].dq = d(lnQ)/d(Kx)\).

Plotting \((n)\) as a Function of Kx

The function \((n) = d(lnQ)/d(Kx) = 1/(1-Kx)\) is a hyperbola, and it tends asymptotic to infinity as Kx approaches 1 from below.

Interpreting the Divergence

The divergence of \((n)\) suggests that as the product of the equilibrium binding constant and the monomer concentration nears 1, the average polymer size tends to become infinitely large. Meaning, almost all monomers are getting incorporated into the polymer. In real physical scenarios, this suggests a state of complete polymerization where practically no free monomers remain. However, it is improbably to form an infinitely large polymer.

Key concepts

Equilibrium Binding Constant

The equilibrium binding constant, often denoted by the symbol \(K\), plays a vital role in understanding the behavior of polymers during cellular processes such as transcription and cytoskeleton formation. In essence, \(K\) provides a measure of the strength with which each monomer binds to a growing polymer chain.

When considering polymerization, if the value of \(K\) is high, it implies that monomers have a strong affinity for binding, thus forming larger polymers preferentially. Conversely, a low \(K\) indicates weaker binding affinity, resulting in smaller polymer formations or more monomers remaining unbound.

The calculation involving the constant \(K\) is foundational to other concepts like the average polymer size and the use of binding polynomials in quantifying polymerization dynamics. The probability that a polymer is \(n\) monomers in length, denoted as \(P_n\), factors in this constant. It's calculated using the expression \(P_n = \frac{K^{n-1} x^n}{Q}\), where \(x\) represents the concentration of free monomers and \(Q\) is the sum of all possible polymer sizes expressed through the binding polynomial.

Average Polymer Size

Understanding the average polymer size provides insight into the general state of polymerization within a cellular environment. It's an average representation of polymer lengths weighted by their respective probabilities.

To calculate this average, one must consider the potential sizes of polymers and their associated probabilities. The formula for average polymer size \((n)\) involves summing the products of each polymer size \(n\) and its probability \(P_n\), mathematically expressed as \((n) = \sum (n \cdot P_n)\). Simplifying the expression using the definition of the binding polynomial \(Q\) yields \((n) = \frac{x \cdot d(Q/Kx)}{dq}\), which is pivotal for determining the polymer distribution in a system.

Subsequently, one can also express \((n)\) in terms of the derivative of the logarithm of \(Q\) with respect to \(Kx\). This derivative representation showcases the relationship between the average polymer size and the equilibrium binding of the monomers in a clear and concise manner, facilitating better interpretation and prediction of polymer behavior.

Binding Polynomials

Binding polynomials are mathematical tools used to encapsulate the complexities of polymer sizes and their formation probabilities in polymerization reactions. Represented by \(Q\), the polynomial sums the probabilities of finding a polymer at all possible lengths, providing a comprehensive overview of the polymer distribution in a system.

Formally expressed, the binding polynomial is given by \(Q = 1 + Kx + (Kx)^2 + (Kx)^3 + ...\), with each term representing the probability of finding a polymer of a specific size. This series continues until it encompasses all practical polymer lengths that can be formed in the given conditions.

Significantly, the derivative of the natural logarithm of \(Q\) with respect to the equilibrium binding constant times the monomer concentration (\(Kx\)) provides the average polymer size \((n)\). This relationship reveals the sensitivity of the average polymer size to changes in the concentration of monomers and the strength of their binding, which is fundamental to understanding the polymerization process.