Question

Protein mutation rates Random mutations lead to amino acid substitutions in proteins that are described by the Poisson probability distribution \(p_{s}(t) .\) Namely, the probability that \(s\) substitutions at a given amino acid position in a protein occur over an evolutionary time \(t\) is \\[p_{s}(t)=\frac{e^{-\lambda t}(\lambda t)^{s}}{s !}\\] where \(\lambda\) is the rate of amino acid substitutions per site per unit time. For example, some proteins like fibrinopeptides evolve rapidily, and \(\lambda_{F}=9\) substitutions per site per \(10^{9}\) years. Histones, on the other hand, evolve slowly, with \(\lambda_{H}=0.01\) substitutions per site per \(10^{9}\) years. (a) What is the probability that a fibrinopeptide has no mutations at a given site in 1 billion years? What is this probability for a histone? (b) We want to compute the average number of mutations \((s)\) over time \(t\) \\[ \langle s\rangle=\sum_{s=0}^{\infty} s p_{s}(t) \\] First, using the fact that probabilities must sum to 1 compute the sum \(\sigma=\sum_{s=0}^{\infty}(\lambda t)^{s} / s !\). Then, write an expression for \((s),\) making use of the identity \\[\sum_{s=0}^{\infty} s \frac{(\lambda t)^{s}}{s !}=(\lambda t) \sum_{s=1}^{\infty} \frac{(\lambda t)^{s-1}}{(s-1) !}=\lambda t \sigma\\] (c) Using your answer in (b), determine the ratio of the expected number of mutations in a fibrinopeptide to that of a histone, \((s)_{F} /(s)_{H}\) (Adapted from Problem 1.16 of \(\mathrm{K}\). Dill and S. Bromberg. Molecular Driving Forces, 2nd ed. Garland Science, 2011.)

Short answer

In short, firstly the Poisson distribution is used to find the no mutations probabilities for fibrinopeptides and histones. Afterwards, an infinite series is evaluated using the formula (given in the problem) and the average number of mutations is expressed. Finally, the expected number of mutations in a fibrinopeptide is found to be a certain ratio times more than that in a histone over a billion years.

Step-by-step solution

- Understanding the Notation and Formulas

First, make sure to understand the symbols and notation. In this case, \(s\) is the number of substitutions that occur at a given amino acid position at a time \(t\). The symbol \(\lambda\) refers to the rate of amino acid substitutions per site per unit time. The formula \[p_{s}(t)=\frac{e^{-\lambda t}(\lambda t)^{s}}{s!}\] shows the Poisson distribution. It gives probability that \(s\) substitutions occur at an amino acid position in a protein over an evolutionary time \(t\).

- Calculate No Mutations Probability for Fibrinopeptide

This step involves substituting the values into the Poisson distribution formula to find the mutation probability. For fibrinopeptides we know \(\lambda_{F}=9\) and \(t = 10^9\). Since we're looking for no mutations, \(s = 0\). So substituting these, we get \[p_{0}(t)=\frac{e^{-9 \times 10^{9}}(9 \times 10^{9})^{0}}{0!}\] After evaluation this expression, we get the required probability.

- Calculate No Mutations Probability for Histone

Following the same approach as in step 2, we substitute the given values for histones into the Poisson distribution formula. We know \(\lambda_{H} = 0.01\) and \(t = 10^9\), and we're also looking for no mutations (\(s = 0\)). So, the expression becomes \[p_{0}(t)=\frac{e^{-0.01 \times 10^{9}}(0.01 \times 10^{9})^{0}}{0!}\]. After evaluating this expression, we get the required probability.

- Evaluate the Infinite Series

We evaluate the summation \(\sigma = \sum_{s=0}^{\infty} (\lambda t)^s / s!\). Using the formula for the sum of an infinite series of a geometric sequence, we see that the sum actually evaluates to \(e^{\lambda t}\).

- Write Expression for Average Number of Mutations

Using the given identity, we write an expression for the average number of mutations \(\langle s\rangle\) as follows: \[\langle s\rangle = \lambda t \sigma\] Substituting the value of sigma from Step 4, we get \(\langle s\rangle = \lambda t e^{\lambda t}\).

- Compute Ratio of Expected Number of Mutations

Finally, we substitute the calculated values for fibrinopeptides and histones to \(\langle s \rangle\), and compute the ratio as follows: \[(s)_F/ (s)_H = (\lambda_F t e^{\lambda_F t}) / (\lambda_H t e^{\lambda_H t})\]. Upon simplification, we get the ratio of the expected number of mutations in a fibrinopeptide to that of a histone over the given period of time.

Key concepts

Amino Acid Substitutions

In the study of molecular biology and evolutionary biology, amino acid substitutions are a core focus. These are changes in the protein sequence that occur when one amino acid in a protein is replaced by another. This process of change is pivotal as proteins are the workhorse molecules in cells, taking on a vast array of functions from catalyzing metabolic reactions to DNA replication.

Substitutions can happen due to errors in DNA replication or as a result of environmental damage to the DNA. Some substitutions may have little to no effect on the function of a protein, while others can be detrimental or, less commonly, advantageous, potentially leading to evolutionary change. The rate at which these substitutions occur is central to understanding both the mechanisms of protein function and the forces that drive evolutionary processes.

Considering the exercise provided, we focused on calculating the likelihood of these substitutions over an extended period—1 billion years—to illuminate how often we might expect these changes to occur in two types of proteins with widely differing mutation rates.

Poisson Probability Distribution

The Poisson probability distribution is a discrete probability distribution that expresses the likelihood of a given number of events occurring in a fixed interval of time or space, provided these events happen with a known constant mean rate and independently of the time since the last event. In our scenario, it is used to model the probability of a certain number of amino acid substitutions occurring within a given period.

The formula \(p_{s}(t) = \frac{e^{-\lambda t}(\lambda t)^{s}}{s!}\) succinctly captures the essence of this distribution. It tells us the probability that \(s\) substitutions will occur in time \(t\), with \(\lambda\) being the substitution rate. In our example, the exercise asks to apply the Poisson distribution to calculate the probability of no mutations occurring in both fast-evolving proteins like fibrinopeptides and slow-evolving proteins like histones.

Understanding the implications of the Poisson distribution enriches students' knowledge of statistical models and their critical applications in biological systems, like measuring mutation rates in evolutionary studies.

Evolutionary Biology

At the crux of evolutionary biology lies the study of how organisms change over time. Mutations, including amino acid substitutions in proteins, are the raw material of genetic variation which, in turn, is essential for evolution by natural selection. Evolutionary biologists investigate the rates at which mutations occur and how these rates contribute to the diversity and adaptability of life forms.

In our exercise, the emphasis is on contrasting the mutation rates of fibrinopeptides and histones, exemplifying the diverse evolutionary pressures and constraints acting on different proteins. The probabilities drawn through the Poisson distribution reflect the evolutionary dynamics of proteins: fast-evolving fibrinopeptides can adapt quickly, whereas histones, being stable and slow to change, preserve the structural integrity of cellular components over time. Such insights are fundamental for students to comprehend how molecular changes are quantified and their implications for the broader evolutionary landscape.

Molecular Biology

Molecular biology delves into the molecular underpinnings of the processes that govern life, including how genetic information is replicated, expressed, and regulated. Within this discipline, understanding protein mutation rates is crucial as it impacts on gene expression and function. The exercise in question marries molecular biology with mathematics by using the Poisson probability distribution to predict mutations.

This integrative approach helps students grasp how quantitative analysis can provide insights into biological mechanisms—such as the expected number of amino acid substitutions over time. When students calculate mutation rates, they are engaging with one of the pillars of molecular biology, the mutation process, which can translate into changes in phenotype and affect organism functions and survival. Hence, mastering these calculations is not merely an academic exercise, but a window into the molecular forces that shape life.