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{\center \large \bf The Two Phases of the Coalescent and Fixation Processes\\}

{\center\bf Introduction \\}

% There is a lineage of the unique individuals each generation for whose
%descendants the population is destined to become fixed.

The coalescent process which traces back the current population to a common 
ancestor and the fixation process which follows an individual until the 
population is fixed for its descendants are heuristically inverse processes, 
yet the time reversal of one is seldom the other.  This is because several 
generations will share the same most recent common ancestor, and several 
generations will first achieve fixation for one of their genes in the same 
generation.  If 
the original individual is the most recent common ancestor of the present 
generation, and the present generation is the population 
in which the original individual becomes fixed, then the processes are 
inverses of each other.  In general, however, if a gene is followed to 
fixation, the most recent common ancestor of the generation in which it 
becomes fixed will be more recent than the original gene.  Similarly, if the 
present generation is traced back to its most recent common ancestor, that 
gene will have become fixed prior to the present generation.  But a 
fixation/coalescence inverse process from most recent common ancestor to first 
generation of fixation (or its reverse) will be a subset of any fixation or 
coalescent process. 

The present work considers this aspect of the structure of the coalescent and 
fixation processes.  The inverse process shall be referred to as the 
transition phase, since it manifests the actual increase from a single copy 
to the entire population (or contraction from the entire population to a 
single individual).  The several generations of a coalescent process which 
share the same most recent common ancestor, and the several generations of a 
fixation process which attain fixation in the same generation shall be called 
the stasis phase.  Because the expected fixation and coalescent times are 
equal, and those processes share the transition phase, the expected lengths of 
the stasis phase are the same for the coalescent and fixation processes. 

\newpage  
{\center\bf Notation \\}

The previous concepts can be elucidated by introducing appropriate notation.  
Start at some generation $t$, and let $T_i$ be the first generation that the 
population is fixed for some gene in generation $t$, then the expected 
fixation time is the expected value of $T_i - t$. Next let $t_i$ be the 
generation of the most recent common ancestor of the population in generation 
$T_i$, then the expected length of the transition phase will be the expected 
value of $T_i - t_i$.  $T_{i+1}$ ($T_{i-1}$) can be defined as the next 
(previous) generation when the population first became fixed for a different 
most recent ancestor, and $t_{i+1}$ ($t_{i-1}$) the generations of the 
respective most recent common ancestors.  Then all the generations from $T_{i-
1}$ to $T_i -1$ share the same most recent common ancestor (in generation 
$t_{i-1}$), and all the generations from $t_i +1$ to $t_{i+1}$ first attain 
fixation for one of their genes in the same generation ($T_{i+1}$).  The same 
notation could have been defined starting at an arbitrary generation, and 
going back to the generation of its most recent common ancestor rather than 
forward to its fixation.

The intervals $T_{i-1}$ to $T_i -1$ and $t_i +1$ to $t_{i+1}$, which I shall 
denote as $\Delta T$ and $\Delta t$, contain the stasis phases for coalescence 
and fixation, respectively.  Hence I shall call them stasis intervals.  Note 
the usage of ``phase'' and ``interval'':  the stasis phase is the realized 
stasis period, it is a stasis interval truncated at the initial (or present) 
generation.  Because the initial generation can be anywhere in the stasis 
interval, the average fixation (coalescent) time should be half the expected 
value of $\Delta t$ ($\Delta T$) (weighted by the lengths of the intervals) 
added to the expected value of the transition phase ($E[T_i - t_i]$).  Hence 
the expected fixation time (which is equal to the expected coalescent time) is 
$\frac{1}{2} E[ (\Delta t)^2 ] / E[ \Delta t] + E[T_i - t_i ]$. 

The adjacent figure illustrates these definitions for a simulation of a 
haploid population of six individuals.  The gene first becomes fixed in 
generation $T_1$, for which generation the most recent common ancestor is in 
generation $t_1$.  Generations $t_1$ to $T_1$ and $t_2$ to $T_2$ are 
transition phases; generations $t_1 +1$ to $t_2$ are a stasis interval for 
fixation; and generations $T_1$ to $T_2 -1$ are a stasis interval for 
coalescence.  The original generation $t$ would have occurred somewhere in a 
stasis interval for fixation.

\newpage

{\center\bf Characterization of the Stasis Phases \\}

The transition phase is the actual increase of a gene from a single copy to 
the entire population for fixation, and the reverse for coalescence. The 
length of the transition phase is the difference between the generation in 
which the ancestral gene becomes fixed, and the generation of the most recent 
common ancestor of that population.

The stasis phase of fixation heuristically has the ancestral gene as a single 
copy before it branches to spread to the population; actually the gene may 
branch and have several copies during that phase, and the branches may
persist during part of
the transition phase, but those branched lineages will die out before fixation 
occurs.  From the coalescent perspective of going back in time, the fixation 
stasis phase precedes the most recent common ancestor.  The length of the 
stasis phase is the difference between the initial generation and the most 
recent common ancestor of the population in which the original gene became 
fixed. 

The stasis phase of coalescence is generation(s) when the entire 
population shares the same most recent common ancestor; the transition phase 
(which precedes the stasis phase in real time) begins (going backward in 
time) when the population contains an 
individual not descended from that ancestor (i.e., there is a more ancient 
branch in the pedigree).  From the fixation perspective of going forward in 
time, the coalescence stasis phase is the generations after fixation for the 
most recent common ancestor of the population until the present generation.  
The length of the coalescence stasis phase is the difference between the 
present and the first generation in which the population has the specified 
most recent common ancestor. 

Note that the stasis phase for a given coalescent (or fixation) process will 
be a subset of a stasis interval, which includes all generations sharing the 
same most recent common ancestor (or generation of first fixation), including 
generations after the present generation (or before the initial generation). 

\newpage
{\center\bf Extreme Examples \\}
                                
If every member of the population replaces itself for $n-1$ generations, and 
then one individual parents the entire next generation,  the length of the 
transition phase will be 1, and the length of the stasis interval ($\Delta T$ 
or $\Delta t$)  will be $n-1$.  
The average fixation/coalescence time will be $(n+1)/2$ generations.
If the length of the stasis phase were a random variable $X$, the weighted 
expected value would need to be calculated as noted above. 

A stasis phase of length 0 is obtained if the member of the ancestral lineage 
(individuals whose descendants will not go extinct) produces two progeny every 
generation, every other individual produces one progeny, except that the 
individual whose ancestors left the ancestral lineage furthest in the past 
does not reproduce.  This follows since every generation will contain a most 
recent common ancestor for some future generation, and every generation will 
be the first generation of fixation for some previous generation.  If the 
population has $N$ individuals, then an ancestral gene will become fixed in 
$N-1$ generations; that will be the transition time, fixation time, and 
coalescent time.  

\newpage 

{\center\bf Poisson Progeny Distribution \\}

The binomial or Poisson progeny distribution is employed with the assumption 
that the future depends only the present, and not previous generations.  In 
particular, at the time of a fixation event ($T_i$), the time until the next 
fixation event ($T_{i+1} - T_i$) will be less than or equal to the expected 
fixation time, because at time $T_i$ there may be multiple copies of the next 
gene destined for fixation.  Therefore, the average length of the stasis 
interval will be less than the expected fixation time; however, this refers to 
the unweighted average of the length of the stasis interval.

Numerical simulations were performed for 1000 fixations each in haploid 
populations of 100 and 200 individuals.  The average times until fixation were 
199 and 390 generations, respectively, which are approximately equal to $2N$.  
At fixation, the average times since the most recent common ancestor were 97 
and 195 generations, respectively.  Hence the average length of the transition 
phase was half of the fixation time, and the weighted average of the stasis 
intervals was equal to the average fixation time.

\newpage

{\center\bf Discussion \\}

This study was motivated by the need to clarify the relation between 
coalescent and fixation events. Indeed, the expected coalescent and fixation 
times are equal, but the expected time since a 
common ancestor at the generation when fixation occurs is not the same as the 
expected coalescent time in general, nor is the expected time until fixation 
of a gene which is a most recent common ancestor equal to the expected 
fixation time in general. When studying the fixation of a gene or the 
coalescence of a population, the actual transition from a single copy to the 
entire population or from an entire population to the single copy will be less 
than the fixation or coalescent time. 

One implication of these results is that hitchhiking occurs in half of the 
fixation time (for random mating with Poisson progeny distribution) because 
it is only during the transition phase that crossing over could affect 
monomorphism at a linked locus.  Of course, this does not address the role of 
mutation in polymorphism.

In fact, these results are really not important  to the general questions of 
polymorphism and evolution.  Dead end lineages contribute to the variation of 
a population.  The breadth of the coalescent process as well as the coalescent 
time impacts how much mutation (which provides variation) occurs during 
fixation.  The minimal genetic history of a population is the lineage of 
single genes which eventually become fixed, for such a lineage there is no 
concept of variation or coalescence.   

The concise genetic history of a population is the lineage of the single 
genes in each generation which are destined for fixation.  The genetic 
diversity which we study is the embellishment of that lineage.  The present 
work provides another perspective on the nature of this embellishment.

\newpage

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