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\title{Critical states of fitness landscapes}
\author{\bf{Yukihiko Toquenaga}\\
               \small{Institute of Biological Sciences\\
                    University of Tsukuba\\
		toque@biol.tsukuba.ac.jp}}
\date{}

\abstract{The traffic cone hypothesis on evolving fitness landscapes is
reconsidered here to incorporate aspects of self-organized criticality.
This modification enables the model to explain coexistence of various
kinds of genetic operations at multiple selection levels as well as
conventional views of the distribution of mutational effects in
evolutionary biology.}

\maketitle
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\section{Introduction}

A fitness landscape is a metaphor that casts evolutionary trajectories
of phenotypes, genotypes, and gene frequencies of evolving populations.
This has been extensively used in studies of evolutionary biology, yet
there are several criticisms against it.

The most popular criticism is that a fitness landscape cannot be defined
if interaction dominates selection regimes.  This point of view has its
root in the tautology argument against utility of the phrase, ``survival
of the fittest.''  This criticism, however, can be overcome by defining
fitness landscapes as resultant snapshots rather than causes of
evolution.  Nevertheless, such a landscape can play a role of boundary
condition for further evolutionary processes.

Another criticism is that there is no clear-cut definition of
coordinates for projecting fitness values~\cite{Provine}.  Fitness
landscapes are mountainous images on which individuals can occupy
locations, and means of a population travel as they evolve. People who
discuss optimal evolutionary strategies often assign phenotype to basal
axes for plotting fitness values.  Some geneticists use genotype, and
others use allele frequency for the same purpose.  Some evolutionary
biologists even use a hyper cube of boolean network~\cite{Eigen, 
Kauffman}.

Evolutionary trajectories on the landscape should be continuous.
Otherwise, the mountainous landscapes are meaningless.  In that sense,
the basal axes should be additive.  This prerequisite is often dismissed
in the above theoretical arguments except for the boolean network
approach.  There are no specific guidelines for selecting appropriate
basal axes for a fitness landscape.

In spite of this potential obscurity, the fitness landscape is a useful
metaphor for evolutionary biology, and Wade and I proposed a hypothesis
of evolution of fitness landscapes\cite{tQ2}.  In the next section, I
explain this hypothesis, called the ``traffic cone hypothesis''(THC).
In the following sections, I point out a weakness of the THC and propose
a modification that incorporates the idea of self organized criticality.
In the final section, I proposed the direction of future studies.

\section{The Traffic Cone Hypothesis}

This hypothesis was developed in 1996 while I was a visiting scholar
with Michael J. Wade at the University of Chicago.  We discussed the
discrepancy between conventional assumptions of evolutionary biology and
those being applied in artificial life studies.

\begin{figure}
\begin{center}
  \resizebox{7cm}{!}{\includegraphics{figs/me.eps}}
  \hspace{2mm}
  \caption{Mutational Effects.  X- and y-axes represent absolute
 intensity of mutational effects and corresponding frequencies,
 respectively. In evolutionary biology, most mutational effects are
 believed to be neutral, and hence the distribution has its mode at the
 zero mutational effect.  In genetic algorithms used in artificial life,
 a single mutation can have large as well as small effects.  If the
 fitness values are randomly assigned, there is more instances with
 small mutational effects than large effects for a single mutation.  The
 expected distribution of mutational effects decreases monotonically
 from small to large effects as shown in the dotted line in the figure.}
  \label{me}
\end{center}
\end{figure}

The most prominent discrepancy between the two disciplines regards the
frequency distribution of mutational effects.  In evolutionary biology,
most mutations are assumed to be neutral or slightly deleterious.  In
contrast, artificial life simulations often assume that mutations of
large effect are relatively common (Fig.\ref{me}).  Nevertheless the two
disciplines share a common goal: to study evolution of life
systems\cite{tQ1}.  Based on the comparison between the two, we
developed the following general scenario of biological evolution (the
TCH) in terms of the fitness landscape~\cite{tQ2};

\begin{figure}
\begin{center}
  \resizebox{7cm}{!}{\includegraphics{figs/tch.ps}}
  \hspace{2mm}
  \caption{Traffic cone hypothesis. \(\mu\), \it{r}, and \(N_e\)
  stand for mutation, recombination, and inter-demic selection,
  respectively.}
  \label{tch}
\end{center}
\end{figure}

At the very beginning of life, organisms faced very rugged fitness
landscapes where point mutations in asexual reproduction frequently
moved organisms or populations from one hill to another in a genotypic
space (the mutation era).  Lives begin adaptive walks, and seek
successful ways of reproducing themselves.  Most successful lives are
those who can reproduce, or produce copies of themselves.  Their
reproductive way should be robust against drastic mutations.  One way to
achieve this robustness is to acquire genetic redundancy that maps
several different genotypes to a single phenotype, or a fitness value.
At the same time, the reproductive mechanism should keep a way for
continuous changing not to be trapped in a certain domain for ever.

As time goes by the landscape becomes locally correlated by acquisition
of genetic redundancy, or nonlinear relationships in mapping genotypes
to phenotypes.  Each domain of attraction in the genotypic space
(i.e. hills) enlarges, and some smaller ones are engulfed by larger
ones.  The hight of merged peaks is shorten because of genetic trade-off
among pooled genotypes.  At the same time, the shape of peak becomes
dull.  This dynamical change of the fitness landscape causes the
distance between domain centers to increase, and shorten the hight of
the peaks on average.  At this stage, a more powerful form of mutation,
sexual recombination, is invoked to permit continued evolutionary
exploration of the fitness landscape (the recombination era).

Further evolution of fitness landscapes calls for more dynamic exchange
of genotype configurations by group (inter-demic) selection described in
the shifting balance theory\cite{Wright}.  At this stage each domain
represents a species where a variety of genotypes have similar and
suboptimal fitness values, and mutations often have little or no effect
on fitness (the inter-demic selection era).  Figure~ \ref{tch}
schematically illustrates the whole scenario.  

\section{Power Balance of Genetic Operation at Different Selection Levels}

The above scenario bridges the traditional model of evolutionary biology
(most mutations are neutral) and assumptions of genetic algorithms in
artificial life studies (any degree of effect can be achieved by a
single mutation with equal probability).  Resultant fitness landscapes
consist of huge domains (very fat, short, and dull traffic cones) in the
genotypic space at the late evolutionary phase (Fig.\ref{tch}). Low
fitness domains can be easily invaded by higher ones.  It remains
unclear how the fatness of domains of attraction in the landscape would
be maintained.  There should be some mechanisms to prevent reverse
evolution to the rugged fitness landscapes.

\begin{figure}
\begin{center}
  \resizebox{12cm}{!}{\includegraphics{figs/muteffct.eps}}
  \hspace{2mm}
  \caption{Mutational effects in uniform (upper most) and fractal
  (lower two) fitness landscapes.  The fractal landscapes are
  generated by the midpoint displacement method~\cite{PeitgenSaupe}.
  The parameter H (Hurst exponent) describes roughness of the
  landscape at small scales.}
  \label{muteffct}
\end{center}
\end{figure}

Another problem in evolutionary biology is that the highest level
process (i.e. inter-demic selection) does not overwhelm the two at lower
levels.  Inter-demic selection is often thought to play a subordinate
role in evolutionary biology~\cite{Coyne}.  Point mutation and
recombination can explain much wider ranges of evolutionary phenomena
without invoking inter-demic selection.  The long-lasting controversy
between the Fisherian and Wrightian schools of thought~\cite{WG} is
based on these distinctions.  The TCH cannot explain such a power
balance among the three kinds of genetic operation at the later
evolutionary phases.

\section{TCH at a Critical State}

How can one achieve both genetic redundancy and maintenance of genetic
operation at the lower selection levels?  One possible explanation is
that the fitness landscape might be at a self-organized critical state
(SOC; \cite{BTW, Bak, Jensen}) rather than a simple cluster of smoothed
and flattened cones.

Single-mutational effects can be defined as the absolute difference
between the fitness value of a focal point on the landscape and a nearby
point that can be achieved by single mutation from the focal point.  If
the fitness landscape is maximally rugged (i.e., a uniform random
distribution of fitness across the landscape \cite{Kauffman}), the
frequency distribution of mutational effects becomes a monotonic
decreasing function of the effect as shown in the left panel in
Fig.\ref{muteffct}.

If the fitness landscape is fractal, the frequency distribution of
mutational effects piles up at lower mutational effects (see the center
and the right panels in Fig.\ref{muteffct}).  This distribution
corresponds to the prevalence of neutral mutations, which is widely
accepted in evolutionary biology (see Fig.\ref{me}).

Moreover, the fractal surface of landscape ensures that three different
genetic operations (mutation, recombination, and inter-demic selection)
can have the same or similar effects against fitness values.  There is
no domination of a genetic operation over the others.  This is believed
to be the essence of the shifting balance theory of Sewall
Wright~\cite{Wright, Provine}.

\section{Future Direction}

It is very difficult to detect critical states in real biological data.
Log-log plots are not enough to represent criticality, let alone
self-organized criticality~\cite{Jensen}.  I would recommend two
approaches to test the new TCH featuring SOC.

\subsection{Strong Artificial Life}

Strong Artificial Life (ALife) seeks to create and study theoretical
representations of life forms apart from existing living things on the
earth. One can examine snap-shots of fitness landscapes of such
artificial creatures.  However, the Bak and Sneppen Model~\cite{BS} is
hardly acceptable to biologists as an evolutionary model.

Alternatives are Avida~\cite{Adami}, and other ALife systems.  Lenski et
al. ~\cite{Lenski} is a good example of collaboration between ALifers
and evolutionary biologists.  They address the evolution of the
mutational effects (see Figure 2 in \cite{Lenski} and compare it with
Fig.\ref{me} and Fig\ref{muteffct}).  Evolutionary systems featured by
the simple genetic algorithm might be the good starting points.

\subsection{Mining in Genome Database}

Data from the rapidly growing genomic databases are best suited for
testing the modified TCH.  Ubiquitous observations of fractal base
sequences in real organisms may relate to the critical
state~\cite{Jeffrey, Korolev}.  Prevailing duplication and palindrome
structures in base sequences are believed to have some functions (e.g.,
folding and efficient replication) for the current as well as past
usage.  Introns rather than exons might be important targets if one
searches for fossil records of the RNA world~\cite{Forsdyke}.

\section*{Acknowledgements}

I am most grateful to Guy Hoelzer for his valuable and helpful comments
on the manuscript.  

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