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Cell["\<\
Path Evolution by Natural Selection 
C. Fernando, V. Vasas, E. Szathmary, P. Husbands  
27th Dec 2010 \
\>", "Section",
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Cell["\<\
The following analysis compares path evolution with standard natural \
selection of non-overlapping genotypes. \
\>", "Subsubtitle",
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Cell[TextData[StyleBox["First we model a simple single bypass mutant as in \
the network at the top of the above diagram. Imagine that the offspring green \
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weight to the bypass (offspring) node, and (1-x) is the outflow weight to the \
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Cell[TextData[StyleBox["This is also the probability that the winning path \
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due to the weight change rules which multiply the weight of the winning path \
(higher reward obtaining path) by a factor (1+L) and weaken the weight of the \
loosing path by a factor (1-L) followed by normalization. For the winning \
path this gives the difference equation \
\>", "Subsubtitle",
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Cell["\<\
We now plot the vector field for the winning and the loosing paths for L = \
0.1, the standard lambda values used in the simulations. \
\>", "Subsubtitle",
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Cell["\<\
It shows how the new winning bypass mutant reaches fixation and the old lower \
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\
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Now let us consider selection at two loci, each with two alleles. An example \
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Cell["\<\
This shows two serial mutants. There are four possible paths through the \
system. Let us call these, A, B,C,D from top to bottom. Let us assume that A \
> B = D > C in terms of fitness. For example, we may want to maximize the \
number of unfilled nodes in a path. Let the first bifurcation weight and its \
mutant be x and (1-x) and the second bifurcation weight and its mutant y and \
(1-y). Then there exist the following probabilities of traversing each path. 
\
\>", "Subsubtitle",
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Recall that only non-overlapping parts of a winning and loosing paths undergo \
weight change. Later we may model a diversity maintenance method which \
actually punishes overlapping parts between the looser and the winner. 

A winning edge will be strengthened and a loosing edge will be weakened to \
the same values as previously described, i.e. \
\>", "Subsubtitle",
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Cell["\<\
We wish to write a difference equation that gives the  value of x(t+1) in \
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generation. The value of x only changes if the following path pairs are \
traversed... (A & C), (A & D) and (B & C), because only in these cases will \
the fitness of the two paths be different. If the fitness of the two paths is \
the same, nothing happens in that generation.  \
\>", "Subsubtitle",
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When the other pairs are travered, either fitness is identical and there is \
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edge, e.g when pair (D&C) or (A&B) are taken. Looking at each case in tern, A \
beats C , A beats D, and B beats C, and so x will always be strengthened or \
not changed at all in each generation according to the following eqaution. \
\>", "Subsubtitle",
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Similarly for weight y... the new value of that weight IF it is changed is \
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\
\>", "Subsubtitle",
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Cell[CellGroupData[{

Cell["\<\
In this two allele model, both edges go to fixation. The above graph shows \
the fixation of the two fitter alleles at each locus over time. The allele \
with the lower initial probability takes longer to reach fixation. 

We are also able to plot the frequency of phenotypes over time because we \
know the probability of the traversal of a path as a function of the x and y \
values... 
\
\>", "Subsubtitle",
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Cell["\<\
We observe that whilst the fittest path (purple) goes to fixation, and the \
other paths become extinct, the dynamics of the intermediate fitness paths (B \
and D) are not monotonic. In fact, B and D actually increase for a brief \
period. This is because the intermediate fitness paths are partially \
benefited by the fitness obtained by the fittest path A. \
\>", "Subsubtitle",
 CellChangeTimes->{{3.5026229333300333`*^9, 3.502623105119337*^9}}],

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The two wining and equal paths 01 and 10 have the highest frequency. Which \
one has a higher frequency depends on the initial condition x[0]. 
\
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Now consider the possibility that the phenotype 10 is fitter than the \
phenotype 01
The only difference in the equations will be in the change of x. 

X looses with probabilty PAC, and now wins with probability PBC + PBD. The \
others are unchanged. \
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