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<!DOCTYPE html>
<html>
<head>
<meta http-equiv=Content-Type content="text/html" charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
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<title>User Manual for Mendels Accountant</title>
<style>
#toc:before, #basic_param:before, #adv_mutn:before, #adv_sel:before, #adv_pop:before, #adv_parallel:before, #adv_comp:before, #special:before, #output:before, #glossary:before {
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<body data-spy="scroll" data-target=".navbar" data-offset="50">
<div class="container">
<!-- The navbar - The <a> elements are used to jump to a section in the scrollable area -->
<nav class="navbar navbar-inverse navbar-fixed-top hidden-xs">
<ul class="nav navbar-nav">
<li> <a href="#toc">Contents</a>
<li> <a href="#basic_param">Basic</a>
<li> <a href="#adv_mutn">Mutation</a>
<li> <a href="#adv_sel">Selection</a>
<li> <a href="#adv_pop">Population</a>
<li> <a href="#adv_parallel">Substructure</a>
<li> <a href="#adv_comp">Computation</a>
<li> <a href="#special">Special Apps</a>
<li> <a href="#output">Plots</a>
</ul>
</nav>
<h1 align=center><i>User Manual for Mendel’s Accountant</i></h1>
<div id="toc" class="shift">
<h1>Contents</h1>
<p>
<ul>
<li> <a href="#welcome">Welcome</a>
<li> <a href="#basic_principles">MENDEL's Basic Principles of Operation</a>
<li> <a href="#requirements">Computer Requirements</a>
<li> <a href="#downloading">Downloading Instructions</a>
<li> <a href="#opening">Opening and operating MENDEL</a>
<li> <a href="#input_param">MENDEL Input Parameters</a>
<li> <a href="#basic_param">Basic Parameters</a>
<li> <a href="#advanced">Advanced Parameters:</a>
<ul>
<li> <a href="#adv_mutn">Mutation</a>
<li> <a href="#adv_sel">Selection</a>
<li> <a href="#adv_pop">Population Biology</a>
<li> <a href="#adv_parallel">Population Substructure</a>
<li> <a href="#adv_comp">Computation</a>
</ul>
<li> <a href="#special">Special Applications</a>
<li> <a href="#output">MENDEL Output</a>
<li> <a href="#apps">Applications of MENDEL</a>
<li> <a href="#glossary">MENDEL Glossary</a>
</ul>
</p>
</div>
<hr>
<div id="welcome">
<h1>Welcome to <i>Mendel’s Accountant</i>.</h1>
<p>This user’s manual provides an introduction to the program <i>Mendel's
Accountant</i>. You may download the program at
<a href="http://sourceforge.net/projects/mendelsaccount" target="_blank">http://sourceforge.net/projects/mendelsaccount</a>.</p>
<p><i>Mendel”s Accountant</i> (MENDEL) is an advanced numerical
simulation program for modeling genetic change over time and was
developed collaboratively by Sanford, Baumgardner, Brewer, Gibson and
ReMine.</p>
<p>MENDEL is a genetic accounting program that allows realistic
numerical simulation of the mutation/selection process over
time. MENDEL is applicable to either haploid or diploid organisms,
having either sexual or clonal reproduction. Each mutation that
enters the simulated population is tracked from generation to
generation to the end of the experiment - or until that mutation is
lost either as a result of selection or random drift. Using a standard
personal computer, the MENDEL program can be used to generate and
track millions of mutations within a single population.</p>
<p>MENDEL's input variables include such things as mutation rate,
distribution specifications for mutation effects, extent of dominance,
mating characteristics, selection method, average fertility,
heritability, non-scaling noise, linkage block properties, chromosome
number, genome size, population size, population sub-structure, and
number of generations.</p>
<p>The MENDEL program outputs, both in tabular and graphic form,
provide several types of data including: the history of deleterious and
beneficial mutation counts per individual, the history of mean individual
fitness, distribution of accumulating mutation
effects, selection threshold history, distribution of linked
mutation effects, fitness distributions before and after selection,
and allele frequencies. </p>
<p>MENDEL provides biologists with a new tool for research and teaching, and
allows for the modeling of complex biological scenarios that would have
previously been impossible.</p>
<hr size=2 width="100%" align=center>
</div>
<div id="basic_principles">
<h2>MENDEL’s Basic Principles of Operation.</h2>
<p>The actual design and code features of MENDEL are described in detail
elsewhere <i>[Sanford et al., 2007. SCPE 8(2): 147-165]</i>. Following is a simple
outline of how the program operates.</p>
<ol>
<li><p>Based on user input, Mendel creates a virtual population with
the specified number of individuals, mean reproduction rate, mating
characteristics, and possible population sub-structure. Also specified
are the genome size, number of chromosomes, and linkage dynamics. </p>
<li><p>Based on user input, Mendel creates a pool of potential
mutations with precisely specified characteristics including: range
and frequency distribution of mutation effects, ratio of recessives to
dominants, and fraction of beneficial mutations. </p>
<li><p>Mutations are selected randomly from the pool of potential
mutations and are assigned randomly to the new offspring in each
generation, based upon the specified average mutation rate (Poisson
distribution). </p>
<li><p>Individual genetic fitness values are calculated based upon
each individual's total mutation inventory. Individual genetic
fitness is defined as 1.0, adjusted by the positive and negative
effects of all its mutations. To obtain phenotypic fitness the genetic
fitness is modified using the specified heritability to account for
non-heritable factors such as variations in the environment.</p>
<li><p>Based on phenotypic fitness values, Mendel applies selection of
a specified type (probability, truncation, etc.) to eliminate a
specified fraction of the individuals from the mating pool. The
fraction removed is determined by the fertility of the population
(average number of offspring per surviving individual).
Offspring in excess of the base population size
represent the population surplus that is eliminated via selection based on
individual phenotypic fitness scores. Unless otherwise specified,
population size is kept constant by selecting away the entire surplus
population each generation.</p>
<li><p>Gametes are extracted from those individuals that survive the
selection process and will reproduce. These gametes are generated based
upon a sampling of the linkage blocks within the chromosome pairs of each
reproducing individual. Reproducing individuals are paired off
randomly, and their gametes are fused to create the next generation of
individuals. For asexual populations, each individual remaining after
selection produces one or more clonal offspring. </p>
</ol>
<p>Steps 1-6 are repeated, for the specified number of generations. Output
reports and plots are updated at regular intervals.</p>
<hr size=2 width="100%" align=center>
</div>
<h2 id="requirements">Computer Requirements.</h2>
<p>MENDEL was designed to run on most computer platforms
and is currently supported on Microsoft Windows, most Linux systems,
and Mac OS X. A recommended hardware requirement is a minimum of
512GB of RAM. Although MENDEL can run with less RAM, the size of a run
(largely determined by population size and mutation rate), is limited
by the available memory. The Mac and Linux versions are currently
superior versions to Windows because the Windows version has more limited
capabilities. However, the Windows version should be very useful in
teaching environments. For both versions, performance is
significantly enhanced when a dual processor (or dual core) machine is
used, because the graphical user interface does not need to compete
for resources with the Mendel program itself. A stand-alone multi-core
processor machine is therefore strongly recommended.</p>
<h2 id="downloading">Downloading Instructions.</h2>
<p>MENDEL can be downloaded by clicking through a series of green
download buttons at <a href="http://sourceforge.net/projects/mendelsaccount" target="_blank"> http://sourceforge.net/projects/mendelsaccount</a>.
You must specify whether you want the Linux, Mac OS X, or Windows version.
If “Downloading”
appears on the screen but the download does not appear to be
proceeding — click <a>“this direct
link”</a> which is shown in blue. Once downloaded, carefully
follow the instructions for installation.</p>
<h2 id="opening">Opening and operating MENDEL.</h2>
<p>MENDEL is controlled through its own web-based graphical user interface.
Therefore, to access MENDEL, one must open a web browser. (<i>Google Chrome</i>
and <i>Apple Safari</i> are recommended. <i>Internet Explorer</i> is not
supported.)</p>
<ul>
<li><p>To open the program one must click on the icon (Windows), or
one can enter the following address in the browser —
http://127.0.0.1:8580/mendel, or select the website
for Linux systems (e.g. <a
href="http://myservername.com/mendel-X.Y.Z/index.html">http://myservername.com/mendel-X.Y.Z/index.html</a>), where XYZ represents the major, minor, and patch level numbers (e.g. 2.0.2). or select the local website for Linux systems (e.g. <a
href="http://myservername.com/mendel"
title="http://myservername.com/mendel">http://myservername.com/mendel</a>).
</p>
<li><p>To start a fresh run, simply press “Start”, and begin to
fill in the parameter options as desired (initially there are default
input parameters in all boxes). </p>
<li><p>To start a run using most or all of the parameters from a
previous run, enter the previous run’s case ID in the box (panel of
options on left of screen), and then press “Start”. The input
options will now have the parameters from that earlier run, which can
then be modified as desired prior to execution. </p>
<li><p>Once the parameters have all been entered, press
“submit”. Review the selected options, and if necessary use the
browser “back” button to go back and modify the parameters.
Check to see if the required memory is greater than the available
memory before pressing the “execute” button. If the
“required” memory is greater than the available memory, in many
cases the run will still be completed successfully, but in some cases
it will fail to execute or will “crash” before the run is
complete. </p>
<li><p>If the run has been successfully activated, a “Run
Status” button can be activated, specifying the run’s number and it
time of initiation, and its progress. </p>
<li><p>During the run, one can periodically refresh and examine the output
file (click “Output” button). </p>
<li><p>During the run, one can periodically refresh and examine the
output figures (click “Plot” button). </p>
<li><p>During the run one can review the “Input Parameters”, or
check the “Run Status”. (Run Status not available in downloadable
version.)</p>
<li><p>If desired, one can open “Run Status”, and terminate a
run by selecting the current run and then selecting “Stop”. (This
option requires using a PBS queueing system and thus is not available
in downloadable version). </p>
<li><p>During or after a run, one can click “Label” and add
comments to a run, for future reference. </p>
</ol>
<h2 align=center style='text-align:center'>
<hr size=2 width="100%" align=center>
</h2>
<h2 id="input_param">MENDEL Input Parameters.</h2>
<p>MENDEL‘s default input values are parameters that might apply to
a small human population. MENDEL‘s input parameters have been separated
into two categories: ‘basic’ and ‘advanced’. </p>
<div id="basic_param">
<h1>Basic Parameters</h1>
<p><em>The basic input parameters include the variables that most
obviously affect mutation accumulation, and are most frequently
changed. This parameter set is useful for initial familiarization with
Mendel, and for training beginning students</em>. </p>
<ul>
<li><p name="caseid" id="caseid">
<b>Case ID</b>. Since MENDEL allows each run to be saved, each experiment
needs to be identified for the purpose of data retrieval. The ID must
be alphanumeric and must have exactly 6 characters. We recommend users
begin their case ID with their own initials.
</p>
<li><p name="nmpo" id="nmpo" class="shift">
<b>Mutation rate</b>. This is the average number of new
mutations per individual. In humans,
this number is believed to be approximately 100. The mutation rate can be
adjusted to be proportional to the size of the <em>functional</em> genome. Thus
if only 10% of the human genome actually functions (assuming the rest to be
biologically inert), then the biologically relevant mutation rate would be just
10. Rates of less than 1 new mutation per individual are allowed—including
zero. The human default value is 10 new mutations per individual per
generation. </p>
<li><p name="fomb" id="fomb">
<b>Ratio of beneficial versus deleterious mutations</b>. While some sources
suggest this number might be as high as 1:1000, most sources suggest
it is more realistically about 1:1,000,000. The default setting is
1:10,000. For studying the accumulation of only deleterious
or only beneficial mutations, the number of beneficials can
be set to zero or one. </p>
<li><p name="opf" name="opf" id="opf">
<b>Reproduction rate</b>.
This is the number of offspring per reproducing individual. Since population
size in Mendel is usually constant, this variable defines the maximum
amount of selection. There must be an average of at least one
offspring per individual (after the selection process) for the
population to maintain its size and avoid rapid extinction. Except
where random death is considered (see advanced parameters),
the entire surplus population is removed based upon phenotypic
selection. The default value for humans is two offspring per
selected individual (or four offspring per reproducing female). </p>
<li><p name="popsize" id="popsize">
<b>Population size</b>.
This is the number of reproducing adults, after selection.
For parallel runs, this is the population size of each sub-population.
This number is normally kept constant, except where
fertility is insufficient to allow replacement, or where certain
advanced parameters are used. For smaller computer systems such as
PCs, population size must remain small (100-1000) or the program will
quickly run out of memory. The default value is 1,000, since
population sizes smaller than this can be strongly affected by
inbreeding and drift. We find increasing population size beyond 1000
results in rapidly diminishing selective benefit.</p>
<li><p name="ngen" id="ngen">
<b>Number of generations</b>.
The number of generations the program should run. The default is 500
generations. If there are too many generations specified, smaller
computers will run out of memory because of the accumulation of large
numbers of mutations, and the experiment will terminate prematurely.
This problem can be mitigated by tracking only the larger-effect
mutations (see advanced computation parameters). The program
also terminates prematurely if fitness reaches a specified
extinction threshold (default = 0.0) or if the
population size shrinks to just one individual. </p>
</ul>
<hr size=2 width="100%" align=center>
</div>
<h2 id="advanced"><em><u>Advanced Parameters</u>: </em></h2>
<p name="adv" id="adv">
<em>Advanced Parameters: The
advanced parameter settings are for knowledgeable researchers and more
advanced students, and should be set to apply to a specific species or
circumstance. The advanced parameters are distributed among the
general categories of: Mutation, Selection, Population Biology,
Population Sub-structure, and Computation. </em></p>
<div id="adv_mutn">
<h2>Advanced Mutation Parameters</h2>
<ul>
<li><p name="psddme" id="psddme">
<b>Parameters shaping the distribution of deleterious
mutations</b>. Deleterious mutations in the natural world
typically range from a few rare lethals to a large number of
nearly-neutral and neutral mutations. It is widely agreed
that the distribution of mutational effects is characterized by an
exponential-like function, where there are few high-impact mutations
and many mutations which are nearly-neutral. Mendel uses a
generalized exponential function, called the Weibull function, to
generate its distribution of mutation effects ranging from 1 (lethal)
down to nearly 0 (near-neutral). See Sanford et al., SCPE 8(2)
p.147-165 available at http://scpe.org), for the mathematical formula
that describes Mendel's mutation effect distribution, and also for
an explanation of the difference between “fitness effect” and
“selection coefficient”. The exact shape of the mutation
effect distribution produced by MENDEL can be controlled precisely,
using the variables shown below. A special-case mutation distribution is
also provided, which is never seen in nature but is of theoretical
interest - wherein each deleterious mutation has exactly the same
effect on fitness (as specified by the user). Likewise each beneficial
mutation has exactly the same effect on fitness (as specified by the
user). We are also developing a special bi-modal distribution of
mutation effects.</p>
<ul>
<li><p name="hgs" id="hgs">
<b>Functional genome size</b> - If we know the genome size of a
species, and we subtract that fraction of the genome which is
perfectly inert, we can know the size of the "functional
genome". The relative abundance of near-neutrals depends, in
part, upon the functional genome size. A simple viroid with a genome
size of just 100 nucleotides may have very few nucleotide positions
which when mutated will have only miniscule biological effects. In
such a viroid, we might assume that a low-impact deleterious mutation
would generally reduce genomic information by (very roughly) about one
part in a hundred. Alternatively, a human genome of three billion will
have very many near-neutral positions, and a typical low-impact
mutation might (very roughly) reduce genomic information by only one
part in three billion. Therefore, it is obvious that the distribution
of deleterious mutational effects must in some way be adjusted to
account for genome size. An approximate yet reasonable means for doing
this is to define the minimal mutational effect as being 1 divided by
the functional haploid genome size. The result of this adjustment is
that smaller genomes have “flatter” distributions of deleterious
mutations, while larger genomes have “steeper” distribution
curves. Because we consider all entirely neutral mutations separately,
we only consider the size of the functional genome, so we choose the
default genome size to be 300 million (10% of the actual human genome
size). </p>
<li><p name="himf" id="himf">
<b>Fraction of mutations having “major effect”</b> - Most
mutations have an effect on fitness that is too small to measure
directly. However, mutations will have measurable effects in the far
“tail” of the mutation distribution curve. By utilizing the
frequency and distribution of “measurable” mutation effects, one
can constrain the most significant portion of the distribution curve
as it relates to the selection process. For most species, there may
not yet be enough data, even for the major mutations, to accurately
model the exact distribution of mutations. When such data is not yet
available, we are forced to simply estimate, to the best of our
ability and based on data from other organisms, the fraction of “major
mutations”. The human default is 0.001. </p>
<li><p name="himt" id="himt">
<b>Cut-off point for defining “major effect”</b> - A somewhat
arbitrary level must be selected for defining what constitutes a
“measurable”, or “major”, mutation effect. MENDEL uses a
default value for this cut-off of 0.10. This is because under
realistic clinical conditions, it is questionable that we can reliably
measure a single mutation’s fitness effect when it changes fitness
by less than 10%. </p>
<li> <p name="rdbm" id="rdbm">
<b>Maximal beneficial mutation effects</b>—Parameters
shaping distribution of beneficial mutations.
The distribution of beneficial mutations should generally be a mirror
image of the distribution of the deleterious mutations, except that
the area under the distribution curve should be adjusted to reflect
the proportionately lower number of beneficial mutations compared to
deleterious mutations. Since the distribution of beneficials should be
affected by genome size (as with deleterious mutations), it is useful
to likewise define the minimal beneficial mutation effect as 1 divided
by the functional haploid genome size. In addition, beneficials should
have a reduced upper range. </p>
<p name="mffg" id="mffg">
A realistic upper limit must
be placed upon beneficial mutations. This is because a single
nucleotide change can expand total biological functionality of an
organism only to a limited degree. The larger the genome and the
greater the total genomic information, the less a single nucleotide is
likely to increase the total. Researchers must make a judgment for
themselves of what is a reasonable maximal value for a single base
change. The MENDEL default value for this limit is 0.01. This limit
implies that a single point mutation can increase total biological
functionality by as much as 1%. In a genome such as man's, assuming
only 10% of the genome is functional, such a maximal impact point
mutation might be viewed as equivalent to adding three million new
information-bearing base pairs each of which had the genome-wide
average fitness contribution. Researchers need to honestly define the
upper limit they feel is realistic for their species. However it
should be obvious that, in all cases, the upper limit for beneficial
mutation effects ought to correspond to a very small fraction of the
total genomic information (i.e. a small number relative to one).
Mutations such as antibiotic resistance need to be handled separately,
as uploaded mutations.</p>
</ul>
<!--
<li><p name="nifm" id="nifm">
<u>Number of initial beneficial mutations</u> This parameter lets
the researcher begin a run with a specified number of pre-existing
beneficial mutations already in the population. These pre-existing
mutations will follow the specified beneficial mutation
distribution. These alleles will all begin as single-copy mutations
within the population. <i>(Caution - at present, using pre-specified
mutations can cause errors in some of the output plots)</i>.</p>
-->
<li> <p name="cr" id="cr">
<b>Parameters involving recessive and dominant mutations.</b> It is
widely agreed that in diploid species, most mutations are recessive,
while a small fraction are dominant. However, because modeling
recessives and dominants can become computationally more intense, and
because the output figures can become hard to read when plotting both
dominant and recessive mutations, the default setting is
co-dominance. This means that all mutations behave additively (a
heterozygote will always have half the effect of a
homozygote). However, for greatest realism, the majority of mutations
should be made recessive, with a minority being dominant by default,
as described below: </p>
<ul>
<li><p name="fr" id="fr">
<b>Fraction of mutations recessive</b> — This parameter
simply specifies the percentage of mutations that are recessive. The default
is 0.0%. If set to 80%, then 80% of mutations are recessive, so the
remaining 20% will automatically be made dominant.</p>
<li><p name="rhe" id="rhe">
<b>Recessive expression in heterozygotes</b> — It is widely
believed that recessive mutations are not completely silent in the
heterozygous condition, but are still expressed at some low level.
Although the co-dominance default is 0.5 expression, a reasonable
setting would be 0.05. </p>
<li><p name="dhe" id="dhe">
<b>Dominant expression in heterozygotes</b> — It is widely believed
that dominant mutations are not completely dominant in the
heterozygous condition, but are only expressed only at some very high
level. Although the co-dominance default is 0.5, a reasonable setting
would be 0.95.</p>
</ul>
</li>
<li> <p name="cmenam" id="cmenam">
<b>Combining mutational effects non-additively</b> — When
there are two or more mutations within an individual, the effects of
these multiple mutations must be combined. The most straightforward
way to do this is additively, by just adding up the effects of all the
deleterious and beneficial mutations within an individual, and
adjusting original fitness (initially 1.0) by that net
amount. Alternatively, one can adjust fitness by multiplying the
fitness (initially 1.0) by the net effect of each mutation (the net
effect of a single mutation would be one minus the fitness effect of
that mutation). This multiplicative method is quite commonly used in
population genetics, although in our experience it seems inadequate
when modeling reality - since in this model no amount of deleterious mutation can
drive fitness completely to zero. The reason for this is that in the
multiplicative model, each additional deleterious mutation has less
and less effect on absolute fitness. For this input parameter, the
researcher can select an all additive model (0.0 multiplicative =
default), or an all multiplicative model (1.0, no additive component),
or a mixed model having any intermediate value between 0 and
1.0. MENDEL’s default setting is the simple additive method. A
third way to combine mutational effects is to use a synergistic
epistasis model (see #6 below).</p>
<li> <p name="se" id="se">
<b>Special case: synergistic epistasis
(SE)</b> <p name="fslp" id="fslp"> In modeling synergistic
epistasis (SE) in Mendel, we distinguish between SE contributions from
deleterious mutation pairs which are linked together within a linkage
block on a chromosome from those which are not. Linked mutations are
inherited together, and therefore the SE effects of all their mutual
interactions are as well. By contrast, genetic recombination
progressively tends to scramble mutations that are not linked
together. Hence, the total SE contribution from non-linked mutations
has a transient component. The SE effects arising from the non-linked
interactions which change from one generation to the next act like a
type of noise that interferes with the selection process. Therefore,
realistic modeling of SE requires that linked and non-linked SE
effects be treated separately. We therefore partition the SE effects
on fitness into two parts, one involving interactions between
mutations which occur in the same linkage block (linked interactions)
and the other part involving interactions of mutations on different
linkage blocks (non-linked interactions). SE effects from linked
interactions are inherited, while part of those from non-linked
interactions are transient and act, in effect, as a type of noise as
far as the selection process is concerned. </p>
<p>Another difference between linked and non-linked interactions is
the relative magnitude of their SE effects. Logically, the strongest
SE interactions should be within the same linkage block, even as two
misspellings in an encyclopedia are likely to interact more strongly
if they occur within the same paragraph or sentence or word. Two
mutations are most likely to interact if they occur within the same
protein-coding sequence or at least the same genic region. Therefore,
the treatment in Mendel includes separate scaling factors for each of
these two categories of SE effects, se_linked_scaling and
se_nonlinked_scaling. Normally, the scaling factor for linked
interactions, se_linked_scaling, should be much larger (for example,
by a factor of 1000) than the one for non-linked interactions,
se_nonlinked_scaling.</p>
<p>Let us now consider how Mendel actually treats the linked SE
interactions. We assume the amplitude of the linked SE effect of each
pair-wise interaction to be proportional to the product of
non-epistatic fitness effects of the two mutations in the pair. This
means that if a mutation's effect on the non-mutant genome is small,
then the SE contribution from its interactions with other mutations
likewise is small. Whenever a new mutation is added to a given
linkage block, Mendel computes and accumulates the SE contribution to
linkage block fitness for that new mutation. This contribution is
proportional to the non-epistatic fitness effect of the new mutation
times the sum of the non-epistatic effects of each of the individual
mutations already present on the block. As these SE contributions are
accumulated, each of the m(m-1)/2 unique pair-wise interactions is
accounted for, where m is the number of mutations in the linkage
block. These contributions, as mentioned above, are scaled by a
user-specified parameter se_linked_scaling. Also for these SE
interactions, we assume co-dominance, which implies each haploid
occurrence of a mutation gives 50% expression of the mutation's total
non-epistatic value. This reduces the SE effect by a factor of 0.25.
We note that, because mutations within a given linkage block are
passed intact from one generation to the next, the SE effects arising
from linked mutations are also passed intact from parent to offspring.
Note that because linked SE interactions are inherited perfectly, they
must always make the degeneration problem worse. This is because the
negative SE contributions add to the negative non-epistatic fitness
effects of the mutations on each linkage block, and in effect make the
non-epistatic effects even more negative.</p>
<p>Mendel treats the non-linked SE interactions in a similar manner.
Let M be the total number of mutations in the genome of a given member
of the population and n be the number of equal-sized linkage blocks.
The total number of unique pair-wise interactions between mutations is
M(M-1)/2, the mean number of mutations per linkage block is M/n, and
the approximate number of linked interactions is n(M/n)[(M/n)-1]/2 =
M(M-n)/2n. With this approximation, the number of non-linked
interactions becomes (1 - 1/n)M<sup>2</sup>/2 and the ratio of the
number of non-linked interactions to linked ones is n-1/(1-n/M). With
n typically 1000 or greater, as M becomes much greater than n, this
ratio approaches n. In other words, as the total number of mutations
becomes large relative to n, the number of non-linked mutations
approaches n times the number of linked mutations.</p>
<p>Let us denote by F the overall genotypic fitness, apart from any SE
effects, of a given individual member of the population. We assume
the amplitude of the non-linked SE effect of each pair-wise
interaction to be proportional to the product of non-epistatic fitness
effects of the two mutations in each pair. The total non-linked SE
fitness contribution is then proportional to the sum of the
non-epistatic fitness effects of all the individual mutations, (1-F),
but scaled to account for the portion of the mutations which are
linked using the factor (1 - 1/n), times the mean non-epistatic
fitness effect of these mutations, (1-F)/M, times the number of unique
pair-wise interactions, (1 - 1/n)M/2, that each non-linked mutation
has with the others. We again assume co-dominance, which implies each
haploid occurrence of a mutation gives 50% expression of the
mutation's non-epistatic value. This reduces the overall contribution
by a factor of 0.25. We scale this non-linked SE contribution with a
user-specified input parameter se_nonlinked_scaling. As already
mentioned, one expects that interaction between mutations within the
same linkage block will on average have much greater SE effects than
mutations which are more distant within the genome. Hence, a value
for se_nonlinked_scaling should typically be much less (by a factor of
0.001 or less) than se_linked_scaling. The resulting expression for
the non-linked SE contribution to individual fitness, to be subtracted
from F, is therefore se_nonlinked_scaling times
0.125(1-F)<sup>2</sup>(1 - 1/n)<sup>2</sup>.</p>
<p>We note that the negative SE contribution to fitness from all the
non-linked interactions is proportional to (1-F)<sup>2</sup>. Since
the number of linkage blocks is typically 1000 or greater, the factor
(1 - 1/n)<sup>2</sup> can usually be approximated as unity. The SE
contribution from non-linked interactions is larger for individuals in
the population with lower fitness and smaller for individuals with
higher fitness. It therefore tends to accentuate the spread in
fitness across the population and thus to enhance selection
efficiency. Since the mean mutation fitness effect is directly
proportional to (1-F), the overall impact of this SE contribution from
non-linked interactions is to increase the mean negative mutational
fitness effect, just as is the case for the SE contribution from the
linked interactions. Therefore, the net effect of SE for both linked
and non-linked interactions should be a higher rate of fitness decline
with time.</p>
<p>What sorts of absolute values for the parameters, se_linked_scaling
and se_nonlinked_scaling are biologically realistic? If we assume no
linkage at all, the total SE fitness contribution (all from non-linked
mutations) is given by the expression se_nonlinked_scaling times
0.125(1-F)<sup>2</sup>, where F is the genotypic fitness. A realistic
limit on the magnitude of se_nonlinked_scaling might be the value that
drives F to zero when, without SE, the fitness F is 0.5. In this
case, se_nonlinked_scaling = 0.5/(0.125x0.5<sup>2</sup>) = 16. This
means that, if the accumulated mutations in a given individual reduce
its fitness to 0.5 without SE, then with SE and se_nonlinked_scaling =
16, the fitness of this individual drops to zero. In our view, a
biologically realistic value for se_nonlinked_scaling should therefore
be on the order of 1.0 or less and for se_linked_scaling 1000 or less.
In our numerical experiments we see discernable SE effects only when
we use SE scaling parameters a factor of 10 or more larger when
selection intensity is weak and only with parameters a factor of 100
or more when selection intensity is moderate. In other words, SE
appears to play a negligible role in most biologically realistic
situations.</p>
</p>
<li> <p name="upload" id="upload">
<b>Uploading Specific sets of mutations</b> — A specific set
of mutations can be uploaded into the population before a run begins.
When this option is selected a template is shown which can be used to
identify mutations for uploading, or a set of mutations can be pasted
into a template.</p>
<li> <p name="abm" id="abm">
<b>Allow back-mutations</b> — In a large genome, the rate of back
mutations (mutations that arise at nucleoside sides that have already
mutated), is vanishingly small and of no consequence, but in small
genomes (i.e., viruses), a significant fraction of the genome can
become mutated, such that this parameter becomes useful.</p>
</ul>
<div id="adv_sel">
<h2>Advanced Selection Parameters</h2>
<ul>
<li><p name="frd" id="frd">
<b>Rate of “Random Death”.</b> A certain fraction of any
population fails to reproduce, independent of phenotype. This can be
expressed as the percentage of the population subject to random
death. This is a useful parameter conceptually, but the same effect
can be obtained by proportionately decreasing the number of
offspring/female, so the default is zero.</p>
<li><p name="h" id="h">
<b>Heritability.</b> Because a large part of phenotypic
performance is affected by an individual’s circumstances
(the “environment”), selection in nature is less effective than
would be predicted simply from genotypic fitness values. Non-heritable
environmental effects on phenotypic performance must be modeled
realistically. MENDEL’s default value for the heritability is
0.2. This implies that on average, only 20% of an individual’s
phenotypic performance is passed on to the next generation, with the
rest being due to non-heritable factors. For a very general character
such as reproductive fitness, 0.2 is an extremely generous
heritability value. In most field contexts, it is in fact usually
lower than this, typically being below the limit of detection.</p>
<li><p name="nsn" id="nsn">
<b>Non-scaling noise.</b> If a population’s fitness is
increasing or declining, heritability (as calculated in the normal
way), tends to scale with fitness, and so the implied
“environmental noise” diminishes or increases as fitness
diminishes or increases. This seems counter-intuitive. Also, with
truncation selection, phenotypic variance becomes un-naturally small.
For these reasons, it is desirable to model a component of environmental noise that
does not scale with fitness variation. The units for this non-scaling
noise parameter are based upon standard deviations from the initial
fitness of 1.0. For simplicity, the default value is 0.05, but reasonable
values probably exceed 0.01 and might exceed 0.1. This means
that there will always be at least a population phenotypic fitness
standard deviation of 0.05 even with a heritability of 1.0.
The specified heritability and non-scaling noise both represent non-heritable
variation, so non-scaling noise makes the effective heritability lower
than the specified heritability. Mendel outputs the standard
deviation for genotypic and phenotypic fitness before and after
selection (in the .hst file), so effective (“realized”)
heritability can be calculated if the user so desires.</p>
<li><p name="fdf" id="fdf">
<b>Fertility declines as fitness declines.</b> It is widely
recognized that when fitness declines, fertility also declines. This
in turn affects population surplus, which affects selection
efficiency, and can eventually result in “mutational
meltdown”. To model this, we have included an option wherein
fertility declines proportional to the square of the fitness
decline. The resulting fertility decline is initially very subtle, but
becomes increasingly severe as fitness approaches zero. The default
value is “Yes”, which means that fertility declines with
fitness, especially as fitness approaches zero.</p>
<li><p name="ss" id="ss">
<b>Type of selection.</b> MENDEL’s default mode for type of
selection is probability selection, wherein the probability of
reproduction is proportional to an individual’s fitness ranking
within the population. Two forms of probability selection are
provided‐ classic and unrestricted. In classic (textbook)
probability selection, rather counter-intuitively, strict
proportionality (relative to the most-fit individual) can combine with
high average fitness and mild selection (low reproductive rates) to
cause reductions in fitness and relatively rapid
extinction. In unrestricted probability selection, with
certain combinations of average fitness and offspring/female, a range
of the highest fitness values are “guaranteed” survival in order
to maintain population size. To give the researcher maximal
flexibility, we also provide an option where strict truncation
selection (i.e. artificial selection) is employed. An intermediate
option, involving a form of “broken-line” selection which we
have designated <span name="ptv" id="ptv">“Partial truncation”</span>
has also been added. With certain combinations of fitness
distribution and offspring/female, this selection mode involves either
truncation of the least fit individuals, guaranteed survival of the
most fit individuals, or both. Selection then acts on the balance of
the population. The partial truncation value, k, equals the fraction
of the population which is truncated. If k=1, this is the same as
full truncation selection. However, if k=0, this equals full
probability selection. k=0.5 is the immediate blending of truncation
and probability selection.</p>
</ul>
</div>
<div id="adv_pop">
<h2>Advanced Population Parameters</h2>
<ul>
<li><p name="clonal" id="clonal">
<b>Reproductive mode.</b> Normal sexual reproduction is the
default setting, but clonal reproduction can be specified. If clonal
reproduction is selected, there is no recombination (overriding #4
below), and the genome is treated as one large non-recombining
chromosome. There is no mating, and the same genome is transmitted
from female to offspring, with each offspring then being assigned its
own set of new mutations. </p>
<li><p name="ch" id="ch">
<b>Ploidy Level.</b> Diploidy is the default setting, but haploidy can
be specified. In this special case, selection occurs during the
haploid phase of the reproductive cycle, which means all mutations are fully expressed (no recessive mutations).</p>
<li><p name="fsf" id="fsf">
<b>Fraction self-fertilization.</b> Certain plants and lower animals
can self-fertilize. The percentage of self-fertilization (as opposed
to out-crossing) can be set to range from the default value 0%) up to
100%. As this value increases, there is a strong increase in
inbreeding and in the rate of mutation fixation. Consequently,
recessive loci have a much stronger effect on overall fitness than
normal.</p>
<li> <p name="dl" id="dl">
<b>Dynamic linkage.</b>
Linkage has a major effect on selection, and must be modeled
as accurately as possible. MENDEL’s default mode
involves dynamic linkage.
This requires specification of the haploid chromosome number and
assumes two random crossovers within each chromosome, at random locations between
linkage blocks - every generation. Because tracking every linkage block can
become computationally expensive, the number of linkage blocks must be
limited (default = 989, min=1, max=100,000). Furthermore, the number of
linkage blocks should be an integer multiple of the number of chromosome
(e.g. the default value of 989 is 43 times the default 23 chromosomes).
MENDEL will automatically adjust to the nearest integer multiple (e.g. if you input
1000 and 23 chromosomes, MENDEL will use a value of 989).
The number of linkage blocks is evenly
distributed over a user-specified haploid number of chromosomes
(default=23). We also offer the researcher the option (turn off "dynamic linkage")
of a simpler model
involving the specification of a fixed number of linkage blocks and fully
randomized recombination between all linkage blocks each generation (no
chromosome number is required).</p>
<li> <p name="pgm" id="pgm">
<b>Dynamic population growth</b> <em>(Linux/Mac OS X Only)</em>. <i>This
function is experimental and relatively untested.</i> By default Mendel
uses a static population size. However, two options are
provided to simulate dynamic population growth: (1) exponential growth
model (e.g. Figure 1), and (2) carrying-capacity model. For the
exponential growth model, two additional inputs need to be
entered: <i>population growth rate</i> and <i>maximum population
size</i>. <span name="pgr" id="pgr">The <b>population growth rate</b>
parameter determines the percent growth rate per generation. A
value of 1.0 represents static population size (no growth). To
grow the population 2% per generation, enter the parameter 1.02 (note:
one may need to manually convert published annual population growth
rates to population growth per generation by using a formula such as
1.02<sup>20</sup> = 1.48/generation — assuming a 20 year
generation time). If the exponential growth model is selected, the
“number of generations” input parameter (under <i>basic
parameters</i>) is alternatively employed to input the maximum population
size (the number of required generations being unknown). Number of
generations is no longer necessary, because the simulation will
automatically shutdown when the computer runs out of memory or the
maximum population size is reached. The population growth
parameter may vary between 1.0 ‐ 1.5. The user will notice
for low population growth rates (e.g. 1.001) starting with small
population sizes, the population will grow linearly until a sizeable
enough population is formed, and then true exponential growth will
occur. This initial linear phase is because population size in an
integer number.</p>
<p>Mendel’s second population growth model is called “the
carrying-capacity model”. Wikipedia.org <i>(accessed
</i><i>October 7, 2008</i><i>)</i> gives the following definition for carrying
capacity: “The supportable population of an organism, given the food,
habitat, water and other necessities available
within an environment is known as the environment's carrying capacity
for that organism.” The equation describing relating population growth
to the environment's carrying capacity can be given as:</p>
<p><center> dN/dt = rN(K-N)/K</center></p>
<p><i>[Reference: Halliburton, Richard. <u>Introduction to Population
Genetics</u>, Benjamin Cummings, 2003].</i> where <i>N</i> is the
population size, <i>r</i> is the maximum reproductive rate of an
individual, and <i>K</i> is the carrying capacity. This
equation is solved numerically as:</p>
<p align=center>
<i>N<sub>i+1</sub> = N<sub>i</sub> (1 + r</i><i>Δt [1 - N<sub>i</sub> / K])</i></p>
<p>
Where <i>i</i>
represents the generation number, <i>r</i><i>Δt</i>
is a constant which represents the maximum reproductive rate of an individual
times over a period of time (here about 20 years). Presently, this constant
may only range between <i>0.0</i> to <i>1.0</i>.</p>
<li><p name="by" id="by">
<b>Population bottlenecks.</b> Population bottlenecks can
dramatically affect mutation accumulation and mutation fixation.
MENDEL allows the modeling of population bottlenecks. The
researcher can cause a bottleneck to automatically begin after a
specified number of generations, resulting in a specified reduction in
population size, and ending after a specified number of bottleneck
generations. The reduction of population size occurs immediately at
the beginning of the bottleneck, by selecting a random sub-sample of
the population. When the bottleneck ends, the original offspring
number/female does not change but half of the population excess
(i.e. all offspring exceeding 2 per female) is used to increase
population size, and half of the excess continues to be eliminated by
selection. When the original population size is reached, normal
selection is restored. To simulate recurrent bottlenecking, the
negative sign can be placed before the first parameter (time to
bottleneck).</p>
</ul>
</div>
<hr>
<div id="adv_parallel">
<h2>Advanced Population Substructure Parameters</h2>
<p name="ip" id="ip">
<b>Population sub-structure.</b> <em>(Linux Only)</em>.
Perfectly random mating within a population probably
never happens, especially in larger dispersed populations. MENDEL allows
creation of multiple sub-populations (tribes), to account for this reality.
<ul>
<li> <p name="ht" id="ht">
<b>Homogeneous sub-populations</b> — If this option is selected, all Mendel
parameters will be applied to each single sub-population equally, so
that all sub-populations will start out the same. If this is
de-selected, each tribe can have its own parameters defined
separately.</p>
<li> <p name="nt" id="nt"><b>Number of tribes</b>
— The number of sub-populations can be specified.</p>
<li> <p name="mm" id="mm"><b>Migration model</b> —
One of three possible methods of migration can be selected: ring pass,
stepping stone, and island models. The ring pass is the simplest mode
of communications, and is typically used in testing parallel computing systems.
In ring pass, the number of tribes are arranged as a circle, and each tribe sends
the user-specified number of individuals to the neighbor to its right.
In the stepping-stone model, the tribes are also arranged as a circle
each tribe exchanges individuals with its neighber.
In the island model, every tribe exchanges individuals with every other tribe.</p>
<li> <p name="mr" id="mr">
<b>Migration rate</b> —
The rate of migration can be specified. The rate of migration can be
less than one (one migration event every X generations), and also can be
zero.</p>
<li> <p name="tc" id="tc">
<b>Tribal competiton</b>
— Tribal competition can be specified (differential
growth/shrinkage of tribes). Tribal competition works by first computing
the global weighted average genetic fitness of all the tribes. Then, the
tribal_fitness_factor is computed which is
each tribes fitness relative to the global genetic fitness is computed.
For example, you may have two tribes, one tribe may have a
tribal_fitness_factor of 0.8 (representing 80% of global genetic fitness),
and 1.2 (representing 120% of the global genetic fitness). This value
is then subracted by 1 to get the reproductive_advantage_factor (e.g.
the two tribes would become -0.2 and 0.2). The
reproductive_advantage_factor is then multiplied by the user-specified
tc_scaling_factor. Then the next generations population size is
computed by multiplying the fertility_factor times the reproductive_rate.
One caveat: with a reproductive_rate of 1, and a low tc_scaling_factor
(e.g. 0.1), the the tribes will actually start shrinking over time.
If this is undesireable, one may solve this problem by increasing the
reproductive_rate.
<ul>
<li id="gssf"> A scaling factor specifies the strength of tribal competition.
<li id="gh"> Group heritability species the amount of environmental effect on
differential tribal growth/shrinkage.
</ul>
<!--
<li> <p name="alt" id="alt">
<b>Upload altruistic
mutations?</b> — This option is currently under development,
and is not ready for production use at this time. This option
allows the user to upload a set of