\documentclass{article}

\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{color}
\usepackage{pstricks}
\usepackage{pst-plot}
\usepackage{multido}
\newcommand{\dt}{\Delta t}
\newcommand{\dx}{\Delta x}
\newcommand{\dy}{\Delta y}
\newcommand{\gp}[1]{{\left({#1}\right)}}
\newcommand{\px}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\mx}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\dmx}[1]{\begin{vmatrix}#1\end{vmatrix}}

\begin{document}

\title{Math 142-2, Optional Homework 1}
\author{Solutions}
\date{\today}
\maketitle

\section*{Preliminary}

All mammals are capable of sexual reproduction.  We will consider a simple view
of this, where each organism receives one set of genes from its mother and
another set from its father.  In this way, each organism contains two possibly
slightly different copies of each gene.  When this organism later reproduces, it
passes on copies of each gene at random to its offspring.  We will be concerned
with one particular gene, which comes in two versions (\textit{alleles}), $R$
and $r$.  Since each individual contains two copies, its \textit{genotype} may
be $RR$, $Rr$, or $rr$ ($Rr$ is the same as $rR$).  We will assume that it does
not matter which copy was received from each parent; the two copies will be
treated the same way.  We wish to model the change in frequency of the $r$ and
$R$ genes through a population of this particular species in the presence of
resource limitations.  To do this, we make a number of assumptions.

\begin{itemize}
\item No mutations are occurring to this gene
\item $A(t)$ is the number of individuals with genotype $RR$
\item $B(t)$ is the number of individuals with genotype $Rr$
\item $C(t)$ is the number of individuals with genotype $rr$
\item The death rates are $a$ for $RR$, $b$ for $Rr$, and $c$ for $rr$.
\item Other than their differing death rates, the three types of individuals are
  indistinguishable
\end{itemize}
In addition to these assumptions, there are some reasonable properties we would
expect such a model to exhibit.
\begin{itemize}
\item In the case that the gene is irrelevant ($a = b = c$), the total
  population $T(t) = A(t) + B(t) + C(t)$ should obey a logistic equation.
\item If only one allele is initially present ($A(0) = B(0) = 0$ or
  $B(0) = C(0) = 0$), the total population should also obey a logistic equation,
  and the allele that is initially absent should remain absent forever.
\item The model should be symmetric with respect to the identity of $r$ and $R$.
\end{itemize}

\section*{Problem 1}

\textbf{Propose a model to describe the evolution of the populations $A(t)$,
  $B(t)$, and $C(t)$ over time.  You will need to introduce additional
  parameters (eg, some characterization of the birth rate) What additional
  assumptions have you made?  Justify your model.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 2}

\textbf{Implement your model in Matlab/Octave, C++, or any other programming
  language you like.  Use this implementation to explore the following by
  running simulations with a variety of initial conditions and parameters.  In
  each case, determine what affect the parameters have on the long term behavior
  of the population.  Does the fraction of $r$ allele vs $R$ allele, $\frac{2
    A(t) + B(t)}{2 C(t) + B(t)}$, change over time?  Does it approach some fixed
  value (such as $0$ ($R$ does off), $1$ (equality), or $\infty$ ($r$ dies
  off))?}

\textbf{(a) $a = b = c$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) $a = c$ but $b < a$.  (heterozygote advantage).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) $a = c$ but $b > a$.  (heterozygote disadvantage).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) $a = b$ but $c < a$.  ($rr$ offers some advantage).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(e) $a = b$ but $c > a$.  ($rr$ offers some disadvantage).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(f) $a = b$ but $c = 1$.  ($rr$ is absolutely fatal; such individuals
  die immediately after birth and never reproduce).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(g) $c > a > b$.  This can occur with recessive traits, where the
  recessive trait $rr$ is harmful, but yet the trait offers heterozygote
  advantage.  How do the results differ qualitatively from those of case (e)?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 3}

\textbf{Sickle-cell and Cystic fibrosis are two fairly common genetic diseases.
  Both are recessive and, without significant medical intervention, both are
  effectively fatal.  Sickle-cell is known to confer heterozygote advantage
  (such individuals fare better when infected with malaria).  Cystic fibrosis
  has no known heterozygote advantage.  Yet even though no such advantage has
  been found, it is nevertheless hypothesized that such an advantage must
  exist.  Why?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}

