% Carlo C. Maley
% This is a paper on the coevolution of mutation rates in an artificial 
% evolving system.
% It was insprired by a comment by Richard Lewontin, Oct. 1993.

\documentstyle[12pt,psfig,named]{article}

% set left and right margins
\oddsidemargin=0in
\evensidemargin=0in
\textwidth=6.5in             % US Letter paper is 8.5in wide

% set top and bottom margins
\headheight=0pt
\headsep=0pt
\topmargin=-0.2in             % better for US Letter paper
\textheight=9in             % US Letter paper is 11in high


\raggedbottom

\begin{document}

\title{The Coevolution of Mutation Rates}
\author{C.C. Maley\thanks{This work was supported in part by the
generosity of the MIT AI Lab.  I am greatful to M. Donoghue, P. Goss,
K. Rice and L. King for their comments and support.} \\
	NE43-803 \\
	Massachusetts Institute of Technology \\
	Cambridge, MA 02139 \\
	617-253-6567 \\
	email: cmaley@ai.mit.edu}

\date{30 August 1994}

\maketitle

\begin{abstract}

In order to better understand life, it is helpful to look beyond the
envelop of life as we know it.  A simple model of coevolution was
implemented with the addition of genes for longevity and mutation rate
in the individuals.  This made it possible for a lineage to evolve to be
immortal.  It also allowed the evolution of no mutation or extremely
high mutation rates.  The model shows that when the individuals interact
in a sort of zero-sum game, the lineages maintain relatively high
mutation rates.  However, when individuals engage in interactions that
have greater consequences for one individual in the interaction than the
other, lineages tend to evolve relatively low mutation rates.  This
model suggests that different genes may have evolved different mutation
rates as adaptations to the varying pressures of interactions with other genes.

\end{abstract}


\section{The Possibilities of Life}


\begin{quote}
\ldots we badly need a comparative biology.  So far, we have been able to
study only one evolving system and we cannot wait for interstellar
flight to provide us with a second.  If we want to discover
generalizations about evolving systems, we will have to look at
artificial ones.    

						- J. Maynard Smith\cite{smit92}

\end{quote}


We only have a single example of the evolution of life.  With a sample
of one, it is difficult, if not impossible, to distinguish between
historical and systematic constraints on the evolution of life.
Artificial Life represents an emerging complementary approach to
understanding biology.\cite{tayl94,lang89,lang92,lang94}  It is an
attempt to expand the size of the sample of living systems we have to
study.  Artificial Life also commonly includes the synthesis of
artificial systems designed to address specific questions in biology
that are intractable under more traditional laboratory techniques.  The
principle weakness in the Artificial Life approach is the sometimes
tenuous connection between the artificial systems and biological
systems.  This weakness stands in contrast to the high degree of
experimental control over the artificial systems.  An artificial
evolving system can be restarted under identical initial conditions,
excepting the random number generator seed, slightly different initial
conditions, or different system parameters.  Furthermore, the
experimenter has access to all of the information in the system, and so
can avoid many of the difficulties inherent in ecological research.  The
fact that Artificial Life research depends on models of life means that
the conclusions must be predicated on the acceptance of the model as a
reasonable description of life.  Even given such acceptance of the
realism of the model, the results generally can only be qualitative
rather than quantitative.  Given these limitations, Artificial Life
helps us to understand both the constraints and possibilities of
life.\cite{male94,vale86,kauf93,kauf91,boer91,coll92}

Lewontin suggested that when we think about what life might
be, rather than what we see before us, there are two dimensions that
can be considered: mortality and heredity.\cite{lewo93}  The life with which we are
most familiar is mortal and reproduces with high fidelity.  But why not
mortal organisms that pass on little information to their offspring?
For that matter, why not immortal organisms with varying degrees of
hereditary transmission of information?  This paper is an attempt to
address those questions.  Neither mortality nor
heredity need be dichotomous qualities.  Mortality might range over
varying degrees of longevity.  Heredity can be addressed through varying
rates of mutation.  At the extremes there is either no mutation and so
perfect replication, or the mutation rate is so high that the
offspring's genetic information is near random.

\subsection{A Naive Model}

Organisms described only by genes for longevity and mutation rates would
be difficult to model on a computer.  Assuming a constant rate of
reproduction, the computer's memory would fill up with near immortal
creatures.  Once a lineage struck upon a low mutation rate combined with
a high longevity, this growth would become exponential.  Of course, in
the real world, such growth would be limited by the available resources.
The computer's memory in this case is an appropriate analogy for just
such a limited resource.  So if the model contained a fixed maximum
carrying capacity, then an organism could only successfully reproduce if
another died.  Death, in this naive model, only comes from the longevity
gene.  Once an immortal organism evolved it would be there for good.
Given non-zero mutation rates and a reasonable sized population,
immortal organisms would continue to arise.  In fact, the only
selective pressure in this naive model is a pressure against mortality.
Eventually, the entire carrying capacity of the environment would be
filled with immortal organisms and evolution would come to a halt.  An
implementation of this naive model has confirmed this prediction.

\subsection{Genetic Algorithms}

Research into genetic algorithms has addressed the issue of mutation
rates.  Genetic algorithms are generally used in problems of
optimization.  See \cite{gold89} for a good introduction to genetic
algorithms.  A genetic algorithm
employs an evolving population of potential solutions to an artificial problem
in order to find a good solution to the problem.
Each individual in the population is evaluated as a solution to the
problem via the application of a ``fitness function'' to the individual.
Then some scheme is used to remove the poor solutions from the
population and replace them with variants on the better solutions in the
population.  These new variants are usually generated by minor mutations
in the successful solutions and through recombination of pieces of the
successful solutions.  Thus a highly stylized form of evolution is
employed to find a good, and sometimes optimal, solution to the problem
at hand. 
The individual solutions in the population do not directly interact with
each other.  The only form of interaction is indirect, through the
fitness function.  In general, a solution must be better than other
solutions in the population in order to survive and reproduce.

A significant amount of work on genetic algorithms has focused on
finding a mutation rate that will allow the population of solutions to
converge on the best solution in the shortest amount of
time.\cite{back93,back92,muhl92,bram91,hess90,foga89}  However, others
point out that the optimal mutation rate depends on how a solution is
encoded as a genome\cite{tate93} and the choice of fitness function to
evaluate a solution.\cite{back92}  B\"{a}ck goes as far as
to include a gene for mutation rate in each solution in the population.\cite{back92}
This allows a ``schedule'' of mutation rates over time.  Early on,
mutation rates tend to be high.  As the population approaches the
optimal solution, mutation rates descend.  Scaling down mutation rates
over time is a common improvement
technique in genetic algorithms.\cite{back93,bram91,foga89}

Traditional genetic algorithms employ an
unrealistically static concept of fitness.  The ``problem'' that the
lineages are trying to solve in a genetic algorithm does not change over
time.  On the other hand, the naive model misses the point of environmental carrying capacity in
biology.  Rather than reproduction filling a viable hole in the
population, the biological world follows the more Malthusian dynamic of
explosive reproduction curtailed by starvation, disease, and conflict.
If we allow the organisms to interact with each other, then we
open up the possibility for competition, exploitation, and even
cooperation.  With interactions, there is no longer any single, optimal
solution.\footnote{For a discussion of optimality see \cite{lewo79,smit94}.
For examples of coevolutionary models see \cite{ray92,holl92,lind94}.}
Would the evolution of mutation rates and longevity change under
coevolutionary pressures?  The following model provides a tool for
addressing that question.

\section{The Model}

Each organism in a population is explicitly represented in this
configuration model.\cite{casw92,male93,hust88}
  Time is discrete and is organized by ``generations.''  During each generation
all of the organisms are allowed to interact in parallel with the other
organisms near by in the environment.  The form of these
interactions is highly abstract, though not as simplified as other
models of ecosystems\cite{kauf91,kauf93,bak93},
 and have been inspired by the ECHO model.\cite{holl92,jone93,holl93}

\subsection{An Organism}

An organism consists of four genes: {\em offense, defense, longevity\/}
and {\em mutation.\/}  Both the {\em offense\/} and {\em defense\/}
genes consist of eight bits.  These are treated solely as bit patterns.
They determine the results of an interaction with another organism
through matching of the bit patterns as detailed below in
Section~\ref{sec:interaction}.     

The genes for {\em longevity\/} and {\em mutation\/} are integers over
the range 0 to 100, inclusive.  The gene for {\em longevity\/} codes the
percent likelihood that the organism will die in a generation.  Thus an
organism with a 0 for a {\em longevity\/} gene would not survive past
one generation.  An organism with 100 for a {\em longevity\/} gene would
be immortal.\footnote{Such an ``immortal'' organism could still fall prey
to competition, described in Sections~\ref{sec:interaction} and
\ref{sec:generation}.}

The {\em mutation\/} gene codes for the mutation rate of all the genes
in the organism.  Mutation only occurs during reproduction when the
parental genes are passed on and then modified in the offspring.
Mutation is different for the integer and bit pattern genes.  In the
case of the integer genes, {\em longevity\/} and {\em mutation\/}
itself, the genes of the offspring differ randomly from the parental
genes along a Poisson distribution centered on the parent's gene,
as defined in Equation~\ref{eq:poisson}.  

\begin{equation}
\frac{(\lambda^{x + \lambda - parental \, gene}) e^{-\lambda}}{(x +
\lambda - parental \, gene)!} \label{eq:poisson} 
\end{equation}

In Equation~\ref{eq:poisson}, $\lambda$ is the mutation gene, $x$ is a potential offspring's
gene, and $parental \, gene$ is the gene of the parent that is being
transmitted to its offspring.  This gives a Poisson distribution around
the parental gene.  The offspring's gene is truncated at 0 and 100.
When the mutation rate is low, the offspring's gene is likely to be very
close to the parent's gene.  However, when mutation rate is high, the
offspring's gene may vary widely from the parent's gene.
Figure~\ref{fig:poissons} depicts three different distributions.  When
the mutation rate gene is zero, the offspring's genes are identical to
the parent's genes.

\begin{figure}
\centerline{\psfig{figure=poissons.ps}}
\caption{For a gene with a value of 50, the genes of its offspring are
distributed around it depending on the $\lambda$ of the poisson
distribution.  This figure shows poisson distributions for $\lambda =$
5, 50 and 100.}
\label{fig:poissons}
\end{figure}

Mutation of a bit pattern gene is scaled down via
Equation~\ref{eq:scaling}.  The $\lambda$ in Equation~\ref{eq:scaling},
ranging from 0 to 8, was again used in a Poisson distribution.  Here
$\lambda$ is the average number of bits that are flipped in an 8-bit
gene during reproduction.  This scaling was chosen to give both the full
range of possibilities from 0 to 8 and to allow evolution to act on a
fine grain of resolution of different mutation rates when the rates were
low.  Figure~\ref{fig:scaling} shows the scaling curve.

\begin{equation}
	\lambda = (mutation / 50)^3  \label{eq:scaling}
\end{equation}

\begin{figure}
\centerline{\psfig{figure=mutation.ps}}
\caption{The mutation gene is scaled from a range of 0 to 100 to a range
of 0 to 8 by this curve.  This scaling has the advantage of a fine grain
resolution of different mutation rates when the mutation gene is low.}
\label{fig:scaling}
\end{figure}

\subsection{An Interaction}

\label{sec:interaction}

To model an ecosystem, interactions must at the least allow for the full
scope of coevolutionary relationships, from mutualism to competition.
More rigorously, an interaction can have a positive, negative, or
neutral effect on either of the participants in the interaction.  A
model ecosystem should minimally be able to simulate all eight of these
possible interactions.  Comparison of bit patterns is one realization of
this minimal requirement.\cite{holl92}
In this model, an interaction between two organisms, X and Y, consists
of a comparison of X's {\em offense\/} gene bit pattern versus Y's {\em
defense\/} gene bit pattern, and X's {\em defense\/} gene bit pattern
versus Y's {\em offense\/} gene bit pattern.

A comparison is scored by subtracting four from the number of bits in the
two genes that match.  This is the reverse of the hamming distance.
Subtracting 4 gives a score spread from -4 to 4.  An organism with an
{\em offense\/} gene of 11111111 would match 7 bits against an organism
with a {\em defense\/} gene of 01111111 and so score 3 points.  A full
interaction is scored by subtracting the {\em defense-punishment\/} parameter
times the comparison of X's {\em defense\/} gene and Y's {\em offense\/}
gene from the {\em offense-reward\/} parameter times the comparison of X's {\em
offense\/} gene and Y's {\em defense\/} gene.  An organism's {\em
fitness\/} score is calculated as the sum total of all the interaction
scores that organism collected during one generation.


\subsection{A Generation}
\label{sec:generation}

The population of organisms is organized along a ring\cite{bak93},
 so that every organism has exactly one neighbor on either side.  A
single generation consists of four steps.  (1) All the organisms
interact with each neighbor on either side.  An organism's {\em fitness}
score is the sum of two interaction scores.  (2) A random number from 0
to 99 is generated for each organism in the population.  If that number
exceeds the {\em longevity\/} gene number for an organism, that
organism is considered dead.  (3) The organisms with the lowest fitness
scores are then killed off until only half of the population remains.
This simulates the organisms producing twice the carrying capacity of
the environment each generation.  (4)  The remaining half of the
population is allowed to reproduce, according to their {\em mutation}
genes, to fill the empty positions on the ring.

Reproduction is asexual.  The new born offspring receive the genes of a
single parent, modified by that parent's {\em mutation\/} rate gene.
Furthermore, reproduction is fair.  Each surviving organism gets a turn
to reproduce, by order in the ring.  The position of the last organism
to reproduce in a generation is saved so that reproduction in the next
generation starts with the following organism on the ring.  Only living
organisms are allowed to reproduce.

\subsection{Missing Complexities}

The model includes many simplifying assumptions.  Interactions are
highly stylized.  The fitness payoffs for the interactions are simple,
discrete, linear functions.  The environment has a fixed carrying
capacity and the organism's reproduction rates are assumed to be both
identical and high enough to fill the environment beyond the carrying
capacity.  Organisms reproduce asexually.  Perhaps most limiting is the
absence of any life histories for the organisms.  This means that an
organism that has survived many generations has no {\em a priori\/} competitive
advantage over a new born organism.  It also means that longevity, and
its complement, senescence, cannot be examined in the context of age
dependent genetic effects.

\subsection{Running the Model}

When the model is started, parameters are input for population size, the
number of generations to simulate, the number of neighbors on each side
with which an organism will interact, the {\em offense-reward\/} and
{\em defense-punishment\/} multipliers.  All of the genes in the initial
population are generated randomly.\footnote{Knuth's pseudo-random number generator was used, with a
seed taken from the computer's clock.}  During the
course of the simulation statistics are gathered each generation on the
average fitness, minimum fitness, maximum fitness, average longevity,
average mutation rate, average age, and average number of offspring.
Finally, at the end of a simulation, the resulting population is saved
in a file for later examination.

\section{The Results}

In all of the following results the model was run with a population of 200
organisms for 2000 generations and an interaction radius of one.  The
independent variables in the experiment were the {\em offense-reward\/}
and {\em defense-punishment\/} parameters.   

\subsection{Longevity}

There are two separate selective pressures in the model, senescence and
competition.  The first depends solely on the longevity genes.  It is
perhaps not surprising then that the longevity genes of the organisms
approach 100, or immortality.   However, the population generally does
not reach fixation on the maximum.  To test if these results were
significant, a null model was implemented.  All interactions were
removed from the null model and instead of killing off the organisms
with the lowest fitness scores, half of the population was killed at
random.  The results showed that longevity of the population does not
fixate at 100 because mutation continually introduces organisms with
longevities below 100.  There was no statistically significant
difference between the evolution of longevity in the coevolutionary
model and the null model.

\subsection{Mutation Rates}

The evolution of mutation rates differed dramatically depending on the
settings of the {\em offense-reward\/} and {\em defense-punishment\/}
parameters.  The central results are detailed in Table~\ref{tab:mutations}. 
When the average mutation rates are compared for the condition where
{\em offense-reward\/} = {\em defense-punishment\/} $= 1$ against the
condition where the {\em offense-reward\/} is twice the {\em
defense-punishment\/}, the later condition results in significantly
lower mutation rates (t~=~-4.0175, df~=~9.075,
p-value~=~0.0015).\footnote{A Welch modified two-sample t-test was used
with the alternative hypothesis that the second condition resulted in
lower mutation rates.} As can be seen, the mutation rates often dropped
to fixation at a zero mutation rate.  The same is true if the {\em
defense-punishment\/} is twice the {\em offense-reward\/}.  The effect
becomes even more dramatic as one of the two parameters is weighted more
than the other.  In all the extreme cases, where the {\em
offense-reward\/} was weighted 100 times as much as the {\em
defense-punishment\/} or when one of the two parameters was 0, mutation
rates always dropped to 0.\footnote{In the two trials when one of the
two parameters was zero, the population reached fixation at a zero
mutation rate in less than 150 generations.}

\begin{table}
\begin{tabular}{|r|r|r|r|r|r|r|} \hline  
{\em trial} & {\em O=1, D=1} & {\em O=2, D=1} & {\em O=1, D=2} & {\em O=100, D=1} & {\em O=1, D=0} & {\em O=0, D=1} \\ \hline
1	& 53.525	& 61.170	& 0		& 0	& 0	& 0 \\
2	& 47.575	& 0		& 0		& 0	& 0	& 0 \\
3	& 51.310	& 0		& 50.275	& 0	& 0	& 0 \\
4	& 67.455	& 31.980	& 0		& 0	& 0	& 0 \\
5	& 66.485	& 0		& 22.805	& 0	& 0	& 0 \\
6	& 71.195	& 33.480	& 55.465	& 0	& 0	& 0 \\
7	& 67.470	& 34.890	& 24.380	& 0	& 0	& 0 \\
8	& 66.725	& 45.505	& 0		& 0	& 0	& 0 \\ \hline
\end{tabular}
\caption{These are the average mutation rates for a population of 200
after 2000 generations.  In the headings of the columns, ``O'' and ``D''
refer to the {\em offense-reward\/} and {\em defense-punishment\/}
multiplier that were used for that trial.}
\label{tab:mutations}
\end{table}




\section{Discussion}

There seems to be nothing of interest in the evolution of longevity in
this model.  The results of the null model clearly argue that the only
reason that longevity does not reach its maximum is that the mutation
rates of the organisms constantly introduce a certain amount of genetic
load.  The central issue in the evolution of longevity is the attempt to
distinguish between two theories for the phenomenon of
senescence.\cite{part93,cris88,bell84,sonn78,cutl78}  
Senescence may be caused by a life history trade off.  Genes might
persist that enhanced survival and reproduction in an organism's early
life at the expense of impairment in later life.
  The competing theory asserts that senescence is caused by deleterious
mutations that are only expressed in old age, and so are not easily
weeded out of the population.  That is, senescence is just a form of
genetic load.  Clearly, without an implementation of life histories in
the organisms, it is impossible to distinguish between these theories.



\subsection{Game Theory Analysis of Mutation Rates}

Game theory\cite{smit82,colm82}
 provides a useful tool for analyzing the evolution of mutation rates in
the model.  Why should an inequality in the {\em offense-reward\/} and {\em
defense-punishment\/} parameters lead to the evolution of low mutation
rates?  The players in these games are lineages.  Each organism in the
environment interacts with its two neighbors.  The outcome of this
interaction is fixed over the life time of the pair.  As long as they
both survive, the interaction will not change.  There is no selection of
alternate strategies within the life time of an organism.  However, when
an organism reproduces, its {\em offense\/} and {\em defense\/} genes
may mutate.  A mutation in either of these genes changes the way in
which the new organism interacts with other organisms.  So a lineage may
fix on a certain form of interaction by evolving a zero mutation rate,
or it may continually change its interactions by maintaining a non-zero
mutation rate. 

We can analyze the payoff matrices for a lineage mutating or not
mutating in interaction with other lineages.  To simplify the analysis,
it helps to consider the payoffs around the extremes of interactions,
with either a perfect match or mismatch between the {\em offense\/} and
{\em defense\/} genes.  This is not an unreasonable approach, because
these extremes are often observed to be equilibria in the model.  It
also helps to assume that the lineages are coevolving over time.  Of
course, there is no guarantee that the next generation in a lineage will
interact with the same ``player.''  However, as a small number of
lineages come to dominate the environment, the participants in the games
will stabilize.  The predictive value of a game theoretic analysis
appears to be robust to these simplifications.

In all of the following matrices, it is helpful to remember that a
perfect match scores positive (e.g., 4) fitness points and a perfect
mismatch scores negative (e.g., -4) points.  The maximum payoff for an
interaction, if {\em offense-reward\/} = {\em defense-punishment\/} $=
1$, would be $4 - (-4) = 8$, when the organism's {\em offense\/} gene
perfectly matches its neighbor's {\em defense\/} gene, and the
organism's {\em defense\/} gene perfectly mismatches the neighbor's {\em
offense\/} gene.  Let us begin with those parameter settings.  If we
consider the case where the two players perfectly match each other, in
Table~\ref{tab:p1e1match}, then a mutation in an {\em offense\/} gene
will always lower that player's payoff and a mutation in a {\em
defense\/} gene will always raise that player's payoff.  The
interactions are completely symmetric.  The first number in each
paired entry of a matrix is the score for player 1 and the second
number is the score for player 2.

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  0, 0 	& -1, 1 \\ \cline{2-3}
Mutation 		&  1, -1	& 0, 0	 \\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 1 and {\em defense-punishment} = 1.
When lineages 1 and 2 perfectly match, a mutation in the {\em defense}
gene produces these scores.  This is a zero-sum game.}
\label{tab:p1e1match}
\end{center}
\end{table}

At the opposite extreme, the two players perfectly mismatch.  Then a
mutation in a player's {\em defense} gene will always lower its score.
If we consider the mutations in the {\em offense} genes, then the exact
same payoff matrix applies.  A mutation in the player's {\em offense\/}
gene will always raise its score.  Under this condition, the benefits of
mutation in one gene is always matched by a penalty for mutation in the
other gene.  Thus, when {\em offense-reward\/} = {\em
defense-punishment\/} $= 1$ there is no selective pressure on mutation
rates.  The result is a population of lineages doing a sort of slow random walk in
their mutation genes.  The larger the population, the slower this walk.

Now consider what happens if {\em offense-reward\/}~$= 2$ and {\em
defense-punishment\/}~$= 1$.  If the players initially have a perfect
match then the payoff matrix for mutating the {\em defense} gene is
shown in Table~\ref{tab:p2e1match}.  This is a Prisoner's Dilemma payoff
matrix.\cite{axel81,axel84,nowa93,mili93,boyd89,boyd87,nowa92,wilk84}  A
lineage scores better by mutating, or ``defecting'' in the Prisoner's
Dilemma terminology, regardless of what its opponent does.  However,
mutual cooperation scores slightly better for both players than mutual
defection.  The table for a mutation in the {\em offense\/} gene
reverses each pair of scores in Table~\ref{tab:p2e1match}.  Thus there
is pressure against mutation in the {\em offense} gene and mutual
cooperation still scores better than mutual defection.  Thus, on
balance, there is pressure towards cooperation, or zero mutation rates.
A pair of well matched organisms with low mutation rates will thrive in
a sea of organisms with high mutation rates.  An historical
accident could lead to the spread of cooperation and even fixation on a
zero mutation rate.

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  4, 4 	& 2, 5 \\ \cline{2-3}
Mutation 	&  5, 2		& 3, 3	 \\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 2 and {\em defense-punishment} = 1.
When lineages 1 and 2 perfectly match, a mutation in the {\em defense}
gene produces these scores.  This is a Prisoner's Dilemma game.}
\label{tab:p2e1match}
\end{center}
\end{table}


If we consider the opposite extreme where the players are perfectly
mismatched, as in Table~\ref{tab:p2e1mismatch}, mutual defection scores
better than mutual cooperation.  As long as the lineages generally
mismatch they naturally evolve towards relatively high mutation rates.
Once the lineages tend to match each other, the
prisoner's dilemma payoff matrix emerges with its reward for mutual
cooperation.  However, the difference in payoff between mutual
cooperation and mutual defection is minimal.  It still requires an adjacent
cooperating pair of lineages to let cooperation spread.  This explains why
some of the trials fixate at zero mutation rates and others do not.
Those that do not fixate at a zero mutation rate still have lower
average mutation rates than the symmetrically weighted trials discussed
above. 

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  -4, -4 	& -5, -2 	\\ \cline{2-3}
Mutation 		&  -2, -5	& -3, -3	\\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 2 and {\em defense-punishment} = 1.
When lineages 1 and 2 perfectly mismatch, a mutation in the {\em
offense} gene produces these scores.}
\label{tab:p2e1mismatch}
\end{center}
\end{table}

Reversing the parameters does not change the analysis significantly.  If
{\em offense-reward\/} = 1 and {\em defense-punishment} = 2, mutual
benefit can be gained when the players mismatch.  The payoff matrix in
this case is identical to Table~\ref{tab:p2e1match}.  Similarly, in the
case that the genes match, the payoff matrix is identical to
Table~\ref{tab:p2e1mismatch}.  Reversing the two parameters results in
an opposite but equivalent set of pressures.  So these trials ought to
behave similarly to the trials where {\em offense-reward\/} = 2 and {\em
defense-punishment} = 1 except that cooperating organisms must mismatch
rather than match.  This is, in fact, what is observed in the
populations that fixated on zero mutation rates.

When we increase the differences between the two parameters, the
interactions become more lopsided, but the prisoner's dilemma remains.
Table~\ref{tab:p100e1match} shows the payoff matrix for {\em
offense-reward\/} = 100 and {\em defense-punishment} = 1.  It still
benefits a lineage to mutate its {\em defense} gene.  However, the
advantage of such a mutation is only a single point, whereas the
disadvantage if both lineages mutate is 99 points.  The analysis of the
mismatching case, in Table~\ref{tab:p100e1mismatch}, is also just a
caricature of the Table~\ref{tab:p2e1mismatch}.

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  396, 396 	& 296, 397\\ \cline{2-3}
Mutation 		&  397, 296	& 297, 297	 \\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 100 and {\em defense-punishment} = 1.  When lineages 1 and 2 perfectly match, a mutation in the {\em defense} gene produces these scores.}
\label{tab:p100e1match}
\end{center}
\end{table}


\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  -396, -396 	& -397, -296 	\\ \cline{2-3}
Mutation 		&  -296, -397		& -297, -297	\\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 100 and {\em defense-punishment} = 1.  When lineages 1 and 2 perfectly mismatch, a mutation in the {\em offense} gene produces these scores.}
\label{tab:p100e1mismatch}
\end{center}
\end{table}

When the differences in the parameters are exaggerated like this, there
is even greater evolutionary pressure for the lineages to reduce their
mismatches as well as to maintain cooperation.  Correspondingly, all of
the trials run under these conditions fixated on zero mutation rates. 

The limit of this trend towards asymmetric interactions is to set one of
the parameters to zero.  When {\em offense-reward\/} = 1 and {\em
defense-punishment} = 0, there is an overall bias towards cooperation.
The payoff matrix of Table~\ref{tab:p1e0match} makes it clear that there
is no benefit to mutating the {\em defense\/} gene.  When the pairs of
payoffs are reversed in the table, it becomes clear that there is a
pressure to prevent mutations in the {\em offense\/} gene.  In total,
mutual cooperation scores higher than mutual defection when the lineages
perfectly match.  All of the experimental runs under these conditions
fixated at zero mutation rates.

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  4, 4 	& 3, 4  \\ \cline{2-3}
Mutation 		&  4, 3	& 3, 3	 \\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 1 and {\em defense-punishment} = 0.
When lineages 1 and 2 perfectly match, a mutation in the {\em defense}
gene produces these scores.}
\label{tab:p1e0match}
\end{center}
\end{table}

When the lineages mismatch, as in Table~\ref{tab:p1e0mismatch}, there is
pressure on the lineages to mutate to
reduce mismatches and then to stop mutating when a match is achieved.
Under this analysis, it is not surprising that occasionally a pair of
species fixated at a zero mutation rate when they had a good but
not perfect match.  As above, the case where {\em offense-reward\/} = 0
and {\em defense-punishment} = 1 follows an exactly equivalent analysis.

\begin{table}
\begin{center}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{}		& \multicolumn{2}{c}{\em Lineage 2} \\ 
\multicolumn{1}{c}{\em Lineage 1}	& \multicolumn{1}{c}{No Mutation} & \multicolumn{1}{c}{Mutation} \\ \cline{2-3}
 No Mutation	&  -4, -4 	& -4, -3 	\\ \cline{2-3}
Mutation 		&  -3, -4	& -3, -3	\\ \cline{2-3}
\end{tabular}
\caption{{\em Offense-reward\/} = 1 and {\em defense-punishment} = 0.  When lineages 1 and 2 perfectly mismatch, a mutation in the {\em offense} gene produces these scores.}
\label{tab:p1e0mismatch}
\end{center}
\end{table}


\subsection{Implications for Biology}

It is now clear that the results of the model follow the evolutionary
pressures that are applied to the lineages of organisms in the model.
But what has this got to do with biology?  The dramatic distinction in
the model is between the case where {\em offense-reward\/} =  {\em
defense-punishment} and the cases where {\em offense-reward\/} $\neq$
{\em defense-punishment}.  This distinction roughly corresponds to
zero-sum games versus non-zero-sum, or mixed-motive games.  In more biological terms, the
first case corresponds to interactions where the consequences are
roughly the same for each participant.  Intraspecific competition might
be a common example.  Under this form of interaction, the model predicts
that the participants will evolve relatively high mutation rates.  The
second case corresponds to an interaction where the consequences are
significantly different for the participants.  Here the model predicts
that the participant lineages will settle down into a mutualistic
association with relatively low mutation rates.  The dinner principle
may be a familiar example.  The cost/benefit analysis is dramatically
different for predator and prey, as long as the predator is not on the
brink of starvation.  The preceding results suggest that
such an imbalance in consequences should put the breaks on predator-prey
genetic arms races.

Mutation should be understood under this theory as phenotypic mutation.
Any heritable, causal factor that affects the rate of change in a
phenotypic trait should be considered as a possible adaptation to the
coevolutionary pressures on that trait.  In a zero-sum game, the
evolution of a relatively high ``mutation rate'' might be realized
through the evolution of a greater sensitivity of the relevant trait to
environmental conditions during development.  This would produce a lower
correlation between the trait of the parent and the trait of the
offspring.  On the other hand, the evolution of a high phenotypic
mutation rate might be realized as a ``hot spot'' in the genes that
influence the development of the relevant traits.

These results suggest that different forms of interactions put different
evolutionary pressures on the mutation rates of the participants.  This
leads to the prediction that different expressed genes should have
evolved different intrinsic mutation rates as adaptations to their
environment of interactions.  We should expect to find structures in the
cell and genome for selectively allowing some genes to mutate faster
than others.  This may mean actively boosting the mutation rates.  A
possible example might involve the selective methylization of cytosine
into 5-methylcytosine in a gene.  Conversely, it may simply mean that
some genes are the subject of better error correction than others.         

\bibliographystyle{named}
\bibliography{lewontin}

\end{document}