Question

Predicting the rate of mutation based on the Poisson probability distribution function. The evolutionary process of amino acid substitution in proteins is sometimes described by the Poisson probability distribution function. The probability \(p_{s}(t)\) that exactly \(s\) substitutions at a given amino acid position 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 substitution per site per unit time. Fibrinopeptides evolve rapidly: \(\lambda_{F}=9.0\) substitutions per site per \(10^{9}\) years. Lysozyme is intermediate: \(\lambda_{L} \approx 1.0\). Histones evolve slowly: \(\lambda_{H}=0.010\) substitutions per site per \(10^{9}\) years. (a) What is the probability that a fibrinopeptide has no mutations at a given site in \(t=1\) billion years? (b) What is the probability that lysozyme has three mutations per site in 100 million years? (c) We want to determine the expected number of mutations \(\langle s\rangle\) that will occur in time \(t\). We will do this in two steps. First, using the fact that probabilities must sum to one, write \(\alpha=\sum_{s=0}^{\infty}(\lambda t)^{s} / s !\) in a simpler form. (d) Now write an expression for \(\langle s\rangle\). Note that $$ \sum_{s=0}^{\infty} \frac{s(\lambda t)^{s}}{s !}=(\lambda t) \sum_{s=1}^{\infty} \frac{(\lambda t)^{s-1}}{(s-1) !}=\lambda t \alpha $$ (e) Using your answer to part (d), determine the ratio of the expected number of mutations in a fibrinopeptide to the expected number of mutations in histone protein, \(\langle s\rangle_{\mathrm{fib}} /\langle s\rangle_{\mathrm{his}}[6]\).

Short answer

a) \(e^{-9}\), b) \(\frac{0.001}{6e^{0.1}}\), c) \(e^{\lambda t}\), d) \(\langle s\rangle = \lambda t\), e) 900

Step-by-step solution

- Understand the Poisson Probability Distribution Function

The Poisson probability distribution function is given by: \[ p_{s}(t)=\frac{e^{-\lambda t}(\lambda t)^{s}}{s !},\] where \( \lambda \) is the rate of substitution, \( t \) is the time, and \( s \) is the number of substitutions.

- Calculate Probability for Part (a)

For fibrinopeptides with no mutations (\( s = 0 \)) and \( t = 1 \) billion years: \( \lambda_{F} = 9.0 \). Substitute these values into the formula: \[ p_{0}(1) = \frac{e^{-9 \cdot 1}(9 \cdot 1)^{0}}{0!} \] Simplify: \[ p_{0}(1) = \frac{e^{-9} \cdot 1}{1} = e^{-9}. \]

- Calculate Probability for Part (b)

For lysozyme with three mutations (\( s = 3 \)) and \( t = 0.1 \) billion years: \( \lambda_{L} = 1.0 \). Substitute these values into the formula: \[ p_{3}(0.1) = \frac{e^{-1 \cdot 0.1}(1 \cdot 0.1)^{3}}{3!} \] Simplify: \[ p_{3}(0.1) = \frac{e^{-0.1} \cdot 0.1^3}{6} = \frac{e^{-0.1} \cdot 0.001}{6} = \frac{0.001}{6e^{0.1}} \]

- Simplify Expression for Summation

Given \[ \alpha = \sum_{s=0}^{\infty}\frac{(\lambda t)^{s}}{s !}, \] recognize it as the Maclaurin series for \( e^{\lambda t} \). Thus: \[ \alpha = e^{\lambda t}. \]

- Express the Expected Number of Mutations

Given: \[ \sum_{s=0}^{\infty} \frac{s(\lambda t)^{s}}{s !} = (\lambda t) \sum_{s=1}^{\infty} \frac{(\lambda t)^{s-1}}{(s-1) !} = \lambda t \alpha, \] substitute \( \alpha = e^{\lambda t} \): \[ \langle s\rangle =\lambda t \cdot e^{-\lambda t}. \] Noting \(e^{-\lambda t} \) cancels, \[ \langle s\rangle = \lambda t. \]

- Determine the Ratio of Expected Mutations

The expected number of mutations: \( \langle s\rangle \). For fibrinopeptides: \( \lambda_{F} = 9.0 \), and histones: \( \lambda_{H} = 0.010 \). Using \( \langle s\rangle = \lambda t \): \[ \langle s \rangle _{\mathrm{fib}} = 9.0 \times t, \] \[ \langle s \rangle _{\mathrm{his}} = 0.010 \times t. \] So \[ \frac{\langle s \rangle_{\mathrm{fib}}}{\langle s \rangle_{\mathrm{his}}} = \frac{9.0 \times t}{0.010 \times t} = \frac{9.0}{0.010} = 900. \]

Key concepts

Rate of Amino Acid Substitution

In evolutionary biology, the rate of amino acid substitution is a crucial concept. It represents how frequently one amino acid in a protein sequence is replaced by another over a given period. Specifically, this rate is denoted by the symbol \(\lambda\), and it's measured per site per unit of time.
This rate varies significantly among different proteins and organisms. For example, fibrinopeptides have a high substitution rate of \(\lambda_{F}=9.0\) substitutions per site per billion years. This means they evolve rapidly. On the other hand, histones evolve very slowly with a rate of \(\lambda_{H}=0.010\) substitutions per site per billion years.
Understanding the rate of substitution helps scientists make predictions about the evolutionary history and future development of various proteins. It is also essential in calculating the probabilities of different numbers of substitutions occurring over a specific evolutionary time.

Expected Number of Mutations

Another important concept is the expected number of mutations, denoted by \(\langle s \rangle\). This refers to the average number of times mutations (substitutions) are expected to occur at a given site over a period. To calculate this, we use the formula \(\langle s \rangle = \lambda t\), where \(\lambda\) is the substitution rate, and \(t\) is the time.
For instance, if we want to know the expected number of mutations in fibrinopeptides over one billion years, we multiply the rate \(\lambda_{F}\) by the time \(t\):
\[\langle s \rangle_{\mathrm{fib}} = 9.0 \times 1 = 9.0\] Similarly, for histones:
\[\langle s \rangle_{\mathrm{his}} = 0.010 \times 1 = 0.010\]
This simple multiplication gives us a clear idea of how many mutations we can expect over a specific period, making it easier to compare the evolutionary dynamics of different proteins.

Evolutionary Time

Evolutionary time, represented by \(t\), is another fundamental aspect in evolutionary biology. It measures the duration over which evolutionary processes like mutations, gene duplications, and natural selection occur. Typically, this time is expressed in millions or billions of years to match the slow pace of evolutionary changes.
The choice of time unit can significantly impact the calculations. For example, in the given exercise, for fibrinopeptides, we consider \(t = 1\) billion years. This time frame aligns with the slow but steady process of evolution in such proteins.
Understanding evolutionary time scales enables biologists to accurately model and predict evolutionary trends. It also helps in comparing different organisms and estimating the divergence times among species.

Calculating evolutionary time, substitution rates, and the expected number of mutations all interrelate and provide a comprehensive view of evolutionary processes and patterns, which is essential for evolutionary biology research and education.