Problem 13 - A guide to MATLAB: For Beginners and Experienced Users

Question

| Genotype of Parental
Offspring | Genotype | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | AA-AA | AA-Aa | AA-aa | Aa-Aa | Aa-aa | aa-aa | | AA | 1 | 1//2 | 0 | 1//4 | 0 | 0 | | Aa | 0 | 1//2 | 1 | 1//2 | 1//2 | 0 | (a) Write these equations as a single matrix equation \(X_{n+1}=M X_{n}, n \geq 0\). Explain carefully what the entries of the column matrix \(X_{n}\) are, and what the coefficients of the square matrix \(M\) are. (b) Apply the matrix equation recursively to express \(X_{n}\) in terms of \(X_{0}\) and powers of \(M\). (c) Next use MATLAB to compute the eigenvalues and eigenvectors of \(M\). (d) From Problem 12 you know that \(M U=U R\), where \(R\) is the diagonal matrix of eigenvalues of \(M\). Solve that equation for \(M\). Can you see what \(R_{\infty}=\lim _{n \rightarrow \infty} R^{n}\) is? Use that and your above expression of \(M\) in terms of \(R\) to compute \(M_{\infty}=\lim _{n \rightarrow \infty} M^{n}\). (e) Describe the eventual population distribution by computing \(M_{\infty} X_{0}\). (f) Check your answer by directly computing \(M^{n}\) for large specific values of M. (Hint: MATLAB can compute the powers of a matrix \(\mathrm{M}\) by entering \(\mathrm{M}^{\wedge} 10\), for example.) (g) You can alter the fundamental presumption in this problem by assuming, alternatively, that all members of the \(n\)th generation must mate only with a parent whose genotype is purely dominant. Compute the eventual population distribution of that model. Chapters 12-14 in Rorres and Anton have other interesting models. | aa | 0 | 0 | 0 | 1//4 | 1//2 | 1 |

Step-by-Step Solution

Answer

The matrix equation is set up to represent genotype transitions. Eventually, the population reaches a steady state defined by eigenvalue methods.

Step 1: Understanding Genotype Matrix

The given table represents genotype frequencies of offspring given parental genotype combinations. We will represent these frequencies using a matrix equation to predict future generations.

Step 2: Construct Matrix Equation

We denote the state of the population at generation \( n \) by a column matrix \( X_n \), where entries correspond to genotype frequencies \( [AA, Aa, aa]^T \). The transformation to the next generation, \( n+1 \), is done using the matrix \( M \), where each entry \( M_{ij} \) represents the fraction of offspring of genotype \( j \) produced by parents of genotype \( i \).

Step 3: Define Matrix M and X

The matrix \( M \) is defined based on the given table:\[ M = \begin{bmatrix} 1 & 1/2 & 0 \ 0 & 1/2 & 1/2 \ 0 & 0 & 1/2 \end{bmatrix} \]Matrix \( X_n \) is:\[ X_n = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \]where \( x_1, x_2, x_3 \) are proportions of \( AA, Aa, \) and \( aa \) respectively.

Step 4: Recursive Expression

The recursive equation for future generations in terms of the initial generation is given by:\[ X_{n+1} = M X_n \]By repetition, we find:\[ X_n = M^n X_0 \]

Step 5: Eigenvalues and Eigenvectors in MATLAB

Compute eigenvalues and eigenvectors of \( M \) using MATLAB. This helps in expressing \( M^n \) using the diagonal form \( URU^{-1} \) where \( R \) is the diagonal matrix of eigenvalues of \( M \).

Step 6: Compute \( M^n \) Using Eigenvalues

Using \( M^n = UR^nU^{-1} \), if \( R \) is diagonal, \( R^n \) involves simply raising each eigenvalue to the power \( n \). Determine \( R_\infty \) as \( n \to \infty \).

Step 7: Calculate the Limit \( M_\infty \)

As \( n \to \infty \), derive \( M_\infty \) from \( M^n \) to find the eventual stable distribution of genotypes.

Step 8: Eventual Population Distribution

Calculate \( M_\infty X_0 \) to describe the eventual genotype distribution in the population.

Step 9: Check by Empirical Matrix Multiplication

Use MATLAB command \( M^{\wedge} \text{large number} \) to directly compute \( M^{n} \) for confirmation.

Step 10: Alternative Model with Dominant Pairing

Alter the model assuming all members of generation \( n \) mate with purely dominant genotype, revise the matrix \( M \), and repeat steps 4-9 to find the eventual population distribution in this scenario.

Key Concepts

Genotype Frequencies

Understanding genotype frequencies is an essential part of matrix population models in population genetics. These frequencies refer to the proportion of different genotypes in a population. Genotypes result from combinations of alleles inherited from parent organisms. In the matrix model, each entry denotes how frequently a particular genotype appears in the offspring.

In our exercise, we use a matrix where the parental genotypes form the rows, and the offspring genotypes form the columns. Each cell in this matrix shows the probability of resulting offspring genotypes from specific parental combinations. The sum of entries in any row of this matrix gives the parental distribution, which remains constant over time, assuming that other factors do not change.

Tracking these frequencies over generations reveals patterns of inheritance and helps predict future population compositions, a valuable tool for predicting changes due to natural selection or genetic drift.

Eigenvalues and Eigenvectors

To delve deeper into matrix population models, understanding eigenvalues and eigenvectors is crucial. These concepts help summarize the effects of repeated application of a matrix, like the one used to represent genotype frequencies.

An eigenvalue represents how a particular transformation, like that of our matrix, scales or changes a vector. Eigenvectors are vectors that do not change direction during the transformation represented by the matrix. In simpler terms, they are the 'directions' in which a transformation like population changes occur consistently over time.

By calculating these in MATLAB, we can reform our initial matrix into a more manageable form, allowing us to predict population stability or long-term genotype distributions. For large-scale models, eigenvalues can signify stability or instability in genetic makeup over successive generations.

MATLAB Programming

MATLAB is used extensively in calculations for matrix population models due to its powerful matrix computation capabilities. This programming environment allows users to perform complex linear algebra operations like finding eigenvalues and eigenvectors, which are essential for analyzing genotype frequency transitions.

In our model, we can input the square matrix representing genotype transitions and then easily compute its powers or transform it into diagonal form. MATLAB simplifies the labor-intensive process of handling large data sets, thus speeding up analyses that predict the eventual population distribution over time.

• It supports direct computation of matrix powers through simple commands.
• It computes eigenvalues and eigenvectors in a unified workflow.

This makes MATLAB an indispensable tool in theoretical population genetics.

Population Genetics

Population genetics is the study of the distribution and change of allele frequencies within populations. It combines principles of Mendelian genetics with those of Darwinian evolution, facilitating understanding of how genotypic and phenotypic traits evolve over time.

Matrix population models serve as pivotal tools in population genetics. They help depict how different genotypes spread through generations, affecting allele frequencies. By using matrices, geneticists can predict changes in these frequencies based on selection pressures, mating patterns, and genetic drift, providing insights into evolutionary processes.

These models contribute to knowledge across various fields, including

• ecology and conservation biology, where they help assess species' survival chances,
• medical genetics, for predicting the spread of genetic diseases,
• agricultural science, optimizing plant and animal breeding strategies.

Grasping this field is essential for understanding not only the existing population structure but also its future shifts.