Question

Random mutations lead to amino acid substitutions in proteins which are described by the Poisson probability distribution \(p_{3}(t)\). 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 fibrinopep. tides evolve rapidly and \(\lambda_{f}=9\) substitutions per site per \(10^{9}\) years. Histones, on the other hand, evolve slowly with \(\lambda_{n}=0.01\) 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 \(t\), $$(s)=\sum_{s=0}^{\infty} s p_{s}(t)$$. First, using the fact that probabilities must sum to \(1, \mathrm{com}\) pute the sum \(\sigma=\Sigma_{s=0}^{\infty}(\lambda t)^{x} / s t\). Then, write an expression for \(\langle s)\) making use of the identity \\[ \sum_{n=0}^{\infty} s \frac{(\lambda \cdot t)^{5}}{s !}=(\lambda t) \sum_{s=1}^{\infty} \frac{(\lambda t)^{s-1}}{(s-1) !}=\lambda t o \\] (c) Using your answer in (b), determine the ratio of the expected number of mutations in a fibrinopeptide to that of a histone, \(\langle s\rangle_{F} /\langle s\rangle \mu\) (Adapted from Problem 1.16 of \(\mathrm{K}\). Dill and \(\mathrm{S}\), Bromber 8 . Molecular Driving Forces, New York: Garland Science, 2003.)

Short answer

The probability for a fibrinopeptide to experience no changes in a site over 1 billion years is \(e^{-9}\) and for a histone, it is \(e^{-0.01}\). The average number of mutations for both over time \(t\) is equal to \(\lambda t\). Hence, the ratio of expected number of substitutions in fibrinopeptide to histone is \(900\).

Step-by-step solution

Determine the Probability of No Mutations

First, we need to use the Poisson formula provided to determine the probability of no mutations (s = 0) for both fibrinopeptides (\(\lambda_{f}=9\)) and histones (\(\lambda_{n}=0.01\)). The Poisson probability function is \(p_{s}(t)=\frac{e^{-\lambda t}(\lambda, t)^{s}}{s !}\). Plugging in the values and calculating, we get \(p_{0}(t)=\frac{e^{-\lambda t}(\lambda, t)^{0}}{0 !}\) which simplifies to \(p_{0}(t)=e^{-\lambda t}\) for both proteins.

Calculate Full Probability for No Mutations

Calculate the full probabilities for no mutations over 1 billion years in each protein using the rate of mutations for each protein given in the problem. These are \(p_{0}(t)=e^{-9}\) for the fibrinopeptide and \(p_{0}(t)=e^{-0.01}\) for the histone.

Determine the Average Number of Mutations

Use the formula, \(\langle s\rangle =\sum_{s=0}^{\infty} s p_{s}(t)\) to determine the average number of mutations. Note that the term \(s \frac{(\lambda \cdot t)^{5}}{s !}\) in the identity given simplifies to \(\lambda t\), because for each term with s > 0 in the sum appearing on the right, the s in the numerator and the denominator cancel out. Thus, \(\langle s\rangle =\lambda t\).

Compute Ratio of Expected Number of Mutations in Fibrinopeptide to That of a Histone

Lastly, substitute the rates into the formula derived in the last step to find the expected numbers of mutations for each protein, then divide these to find the ratio. This yields a ratio of \(\langle s\rangle_{F} / \langle s\rangle_{H} = 9 / 0.01 = 900.\)

Key concepts

Understanding the Poisson Probability Distribution

The Poisson probability distribution is a widely used statistical tool in evolutionary biology, particularly to study the occurrence of rare events over time, such as amino acid substitutions in proteins. It can be expressed in the general form \( p_{s}(t)=\frac{e^{-\lambda t}(\lambda t)^{s}}{s!} \), where \( s \) represents the number of events (in this case, substitutions), \( t \) is the time period considered, and \( \lambda \) is the average rate of event occurrence per unit time.

This distribution is particularly useful because it makes certain assumptions about the nature of the events it models. Notably, it assumes that the events occur independently of one another and at a constant average rate. Thanks to these properties, we can use the Poisson distribution to delve into the stochastic nature of protein evolution. It helps in quantifying the probability of a given number of substitutions occurring at a specific site in a protein over evolutionary time.

For example, when considering proteins that evolve rapidly or slowly, different \( \lambda \) values are used. A high \( \lambda \) value indicates a high substitution rate, leading to a greater chance of observing multiple substitutions, while a low \( \lambda \) value indicates a protein that evolves more slowly, with fewer expected substitutions over the same time period.

Amino Acid Substitutions in Proteins

Amino acid substitutions are the bread and butter of protein evolution. They are changes in the protein sequence that can result from random mutations in the DNA. These mutations can be silent, causing no change in the protein structure or function, or they can alter the amino acid sequence, potentially impacting the protein's properties.

In evolutionary studies, it's important to consider the frequency and impact of these substitutions. A protein might evolve rapidly, accumulating a high number of substitutions quickly, or it may change very slowly, maintaining its sequence over long periods. This frequency is what the Poisson probability distribution models, by helping to predict the likelihood of a certain number of substitutions over time.

Factors like environmental pressures and the functional importance of the protein can influence the rate of substitutions. Proteins that are crucial to survival, like histones which package and order DNA into structural units called nucleosomes, tend to evolve slowly and are thus subject to fewer amino acid substitutions to avoid detrimental effects on the organism.

Protein Evolution Explained

Protein evolution is a fundamental concept in evolutionary biology, revealing how proteins change and adapt over time through a process of random mutations and natural selection. The study of protein evolution helps scientists understand the history of life, by indicating how different organisms are related and how functions in biological molecules have diversified.

The evolution of proteins includes mechanisms like gene duplication, where an organism inadvertently makes extra copies of a gene, which can then evolve new functions. Natural selection then determines which of these innovations are beneficial, favoring their continued existence and refinement.

Protein evolution is often a balance between conserving vital functions and adapting to new or changing environments. The Poisson probability distribution provides a quantitative model for these processes, allowing biologists to estimate the rate at which proteins are changing and to compare the evolutionary rates of different proteins, such as the fast-evolving fibrinopeptides versus the slow-evolving histones. In this way, evolutionary biologists use mathematics to gain insights into the molecular processes that drive life's diversity on Earth.