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Abstract
Motivation: The investigation of common disorders of
polygenic inheritance using both population-based
association designs and non-parametric linkage analysis
within families is gaining increasing importance. We have
created a program that allows for the flexible simulation of
populations as a tool to investigate the properties of
population-based mapping approaches.
Results: We have created a population simulation program,
POPSIM, that (i) creates a virtual representation of every
individual, (ii) makes no prior assumptions but the Mendelian
rules and (iii) allows populations of several million
individuals in size to be generated and to be followed over
hundreds of generations. The parameters of the disease
model, population structure and population expansion rate
can be specified. Flexible sampling options exist that allow
samples of families and individuals to be drawn at any given
point during the population history. The program may be a
useful tool in the study of the influence of genetic drift,
recombination and admixture on the generation and
maintenance of linkage disequilibrium in populations, as well
as the evaluation of stochastic sampling characteristics of
families and individuals conditional on a complex genetic
phenotype from homogeneous and heterogeneous
populations.
Availability: The source code as well as Sun and Windows
NT4.0 console executables of the program are available under
http://www.ukrv.de/ch/medgen/html/benutzer/j.hampe/popsi
m.html.
Contact: J.Hampe@mucosa.de
Introduction
Simulation techniques play an important role in applied
genetics and in the development of new statistical methods
for linkage and association analysis. Currently available
4To whom correspondence should be addressed at: 1st Department of
Medicine, Christian-Albrechts-University Kiel, D-24105 Kiel, Germany
458
programs include SIMULATE (Terwilliger and Ott, 1994),
SIMLINK (Boehnke, 1986) and SLINK (Weeks et al.,
1990), and the more general GASP (Wilson et al., 1996)
programs which simulate the genotypes of families under a
defined structure and preset linkage parameters. A
simulation of a given pedigree using SLINK is now a
common prerequisite before embarking on a genome-wide
linkage analysis in disorders where reasonable parametric
estimates on the disease model exist. However, for complex
genetic disorders like diabetes, arteriosclerosis, bipolar
disorder or inflammatory bowel disease, the genotype to
phenotype relationship is much more complex (Risch, 1990;
Cox et al., 1992; Orholm et al., 1993; Risch and Botstein,
1996). Possible approaches for the ultimate mapping of
susceptibility genes responsible for these conditions include
non-parametric linkage analysis (using recombination
events within the families under investigation) and linkage
disequilibrium mapping (association studies—using
recombination events over the past generations in the
population). In these approaches, population structure and
sampling effects may have profound effects on the results
(Spielman et al., 1993, 1994; Suarez et al., 1994; Thompson
and Neel, 1997). Several analytical models for defined
scenarios of population history and structure have been
developed for the prediction of allele frequencies and
disequilibrium parameters (Kaplan and Weir, 1995; Kaplan
et al., 1995), which have been applied successfully in
isolated populations (Hästbacka et al., 1992, 1994).
Simulation-based statistics have been used to assess the
sensitivity and specificity of non-parametric linkage analysis
(Almasy et al., 1995; Devlin and Risch, 1995; Davis et al.,
1996) which are each directed at a specific family structure.
The use of predefined statistical distributions for the
prediction of certain population parameters in the mentioned
parametric simulation approaches may not be generally
applicable.
Another approach uses direct simulation, whereby a
representation of every individual is generated. The first
implementation of this approach by MacCluer (1967;
POPGEN) has evolved into a set of Fortran programs that
Oxford University Press
Population simulation with POPSIM
have been used by the ‘Genetic Analysis Workshops’
(MacCluer et al., 1995) and allow—by using different
components from these programs—a variety of scenarios to
be implemented. LINKERS, SNAPPERS and GRASPERS
(Yang et al., 1990; Prasad, 1992; Ackerman et al., 1993; Goay
et al., 1995) for the VMS environment allow very detailed
simulation scenarios, but are limited to two major loci and two
alleles per locus. The underlying simulation concept has been
ported to a C++ library ‘SIMEX’ that enables the user to build
custom models in C++ which may also be applied to
infectious disorders like spread of influenza (Peterson et al.,
1993). SMALLPOP (Fournet et al., 1995) is limited to small
populations and is directed at animal breeding using external
selection criteria.
The population-based programs are very resource intensive
and often require additional programming for efficient use.
We therefore implemented a program under the following
aims: (i) to enable the direct simulation of populations of a
realistic size (i.e. several million individuals); (ii) making no
prior assumptions but the Mendelian rules; (iii) to enable the
specification of relevant population parameters; (iv) to deliver
the output in a format that is easily processed by analysis
programs for both parametric and non-parametric linkage
analysis; (v) to include all experimental information into a
set-up file that is suitable for batch processing.
System and methods
POPSIM was written in ANSI C and uses C standard
libraries. System-specific versions for Sun Solaris and a
Windows NT4.0 console application are also available.
Efforts to minimize the memory requirements of the
program have included the use of half-bytes for the storage
of genotype information and the immediate replacement of
the parents by their children in the population space. The
central ‘breeding’ routines use exclusively integer and bit
operations, all float point operations have been moved into
initialization routines.
Specification of the experiment
In order for the program to be as flexible as possible and to
be easily used in batch processing, all information on the
simulation experiment is specified in a text file (extension
*.pop_init). Here, a variety of settings pertaining to the
structure of the population, susceptibility locus and marker
information, disease model, population propagation and
sampling characteristics can be specified (Table 1). These
data are read into an ‘experiment’ structure during the
experimental set-up routines.
Table 1. Variables specified in the experiment set-up file
Population generation
Number of population replicates
Initial size of the population
Number of subpopulations and their ratio in the final population
Random seed for population generation
Population propagation
Number of generations
Average number of kids per family and the proportion of the population attempting to reproduce—a method of specifying population growth or shrinkage
Random seed for propagation
Locus structure and disease model
Number of susceptibility loci (all unlinked to each another)
Initial allele frequencies of the risk allele for each locus and subpopulation
Recombination fraction for the marker loci linked to each susceptibility locus (no interference)
Initial association of a founder haplotype with the risk allele of the susceptibility locus
Disease model (additive polygenic, classical inheritance, 2 locus penetrance matrix)
Weighting factors for each locus, weight of the environmental factor for the additive disease model
Cut-off value for affection
Sampling and I/O streams
For each Checkpoint: (i) number of generations after which sampling is performed; (ii) type of sample to be drawn
Sample types: Sib-pairs and nuclear families with the number of affected individuals per family to be specified
Output either on standard output or in a dedicated file for each type of sample
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J.Hampe et al.
Fig. 1. Implementation of a virtual individual: The internal storage
format is symbolized in boxes, giving the type of information and the
format of storage (0/1: bit; 0..15: half-byte; 0..255: byte) on the top
and bottom, respectively. Risk locus information contains the
genotype of the susceptibility gene itself and the genotypes of the
linked marker loci (up to four in the standard compilation). The
thetas between marker and risk locus are specified in the experimental set-up file.
Population implementation
The program uses non-overlapping populations. The population is stored in three principal data structures: person, family
and population. The central data structure of the program is
a person, for which the sex, a random environmental factor
and genetic information on risk genes and linked markers are
stored. In order to minimize memory requirements, the allele
numbers of the genotypes are packed in half-bytes (Figure
1). The family is a temporary dataset of pointers, for which
a male and a female are drawn from the population, and
children are generated as determined by a random draw from
a binomial distribution, the mean of which is given in the
experimental set-up (see below). Since the program uses
non-overlapping generations, memory requirements can be
minimized by immediately replacing the parents with their
children. Thus, the next generation grows in the same population space at the expense of the old one (Figure 2).
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Fig. 2. Implementation of the population. The old and the successive
population share the same population space in the main memory. As
the simulation of a generation progresses, the parent population is
gradually replaced. The bounds are managed by pointer variables (in
open arrows). The four steps in the creation and destruction of a
family are symbolized, the pseudo code for these operations is given
in the text.
Population replicates
Two streams of pseudo random numbers are used in the program. One stream provides all random numbers for mating,
meiosis, etc., and the second stream is used for population
set-up. The latter is reset every time a population replicate is
required, giving the opportunity to subject the same population to different mating scenarios repeatedly. After the experimental set-up, the complete experiment structure is put
on the stack and reconstituted for each population replicate.
Sampling during simulation
Sample information is either directed to the standard output
or to files with characteristic extensions (Figure 3). The
sampling timetable, i.e. after which generation what type of
sample is to be drawn, is given in the experimental set-up file
and read into the ‘checkpoints’ structure. No analysis functions are incorporated into the POPSIM program itself.
Rather, independent analysis programs like the LINKAGE
(Lathrop et al., 1984) package or the companion program
CHISQUARE (which calculates χ2 statistics from case–control data) and the TDT (transmission disequilibrium test)
Population simulation with POPSIM
Fig. 3. Flow of information in a simulation experiment using
POPSIM. As specified in the experimental set-up file, the flow of
sample information is directed into files with specific extensions.
(Spielman et al., 1993, 1994) statistic from triplet data are
used for data analysis.
Family creation and meiosis
A stream of integer random numbers is used to make all decisions in the mating and meiosis routines, and is designated
as ‘random’ in the pseudo code below. In order to avoid floating point operations, the cut-off values were normalized during the experimental set-up routines on the intervals of the
equally distributed pseudo random numbers. In the case of
the number of offspring to be created for a given family, the
mean of a binomial distribution is specified in the experimental set-up file. The density curve of this distribution is
translated into values of an integer vector and a random
number used as the index for an individual family.
We now give the pseudo code of the central mating and
meiosis routines.
breed
(struct
*experiment,
struct
*population, struct *checkpoints) {
for (all generations) {
reset_population_pointers;
total_sampling(population, check
points);
// total population sampling
happens here
while (still families needed) {
mother = get_male(population,
random);
father = get_female(population,
random);
if (get_number_of_kids(binomial
distribution, random))!=0) {
make_space_for_kids(population
experiment, kidnumber);
create_kids(mother, father,
population, experiment);
ascertain(mother, father,
population, checkpoints);
// family sampling is
performed here
destroy_parents(mother,
father, population); }
} } } }
create_kids(struct *population, struct
*experiment, kidnumber) {
while (still kids to create) {
assign_sex(random);
assign_environmental_factor
(random);
assign_paternal_haplotype(random)
assign_maternal_haplotype(random)
} } }
assign_maternal_haplotype(random){
for (all markers) {
assign_principal_chromo
some(first_bit_of_random);
// grandfather or grandmother
if (last_15_bits_of_random >=
normalized_theta)
recombination;
else no_recombination;
} }
Results and discussion
Efficiency of the implementation
The direct simulation approach leads to a heavy load on both
main memory and processor capacities. The bulk of processor resources are used during the simulation of meiosis.
Here, the packaging of genotype information into half-bytes
results in a slowdown of ∼50% (data not shown). However,
in contrast to the use of integer data types, this reduces (de-
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J.Hampe et al.
Fig. 4. Decline of initial linkage disequilibrium in an at random mating population of 200 000 individuals. It is evident that the recombination
fraction θ determines the rate of this process. The analytical prediction is plotted in parallel to the simulation results. A population with an initial
haplotype frequency (1D) of 0.87 and a population allele frequency of marker 1 of 1/16 was chosen. Fifty replicates of this population were
generated and subjected to successive generations of random mating. The mean of the observed haplotype frequency (1D) was plotted as a
function of time. Note the relative discrepancy of simulation and theoretical prediction in very small thetas (see the text).
pending on the compiler settings) the memory requirements
of a marker genotype from 4–8 bytes down to 1 byte. This
way, for instance, a three-susceptibility locus simulation with
four markers at each locus needs 18 MBytes for 1 million
individuals. A further speed-up was achieved by moving all
floating point operations into the experimental set-up functions and performing all operations on normalized integer
values. The final program version needs ∼50 min for the simulation of 100 generations in a population of size 1 million
individuals on a Sun SparcStation 10/30 and ∼10 min on a
Sun Ultra 170.
Current limitations and future development
In all population simulation programs, a balance between abstraction from the individual biographies and the amount of
genetic data and population size has to be struck. Programs
such as GRASPERS (Prasad, 1992) and SIMEX allow for a
very detailed specification of the individual biographies and
are therefore also suitable for the simulation of largely environmental disorders like infectious diseases, but are limited
for population size and the number of loci to be handled.
POPSIM does not consider the individual biographies in detail, but rather concentrates on problems like the evolution of
haplotypes over long periods of time, in large populations,
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Fig. 5. Distribution of the haplotype frequency 1D in an at random
mating population (θ = 1 cM). The same data as in the experiment
in Figure 4. Instead of the mean of f1D , the distribution of
observations from the 50 experiments (N on the Z-axis) is shown.
Population simulation with POPSIM
and allows for a theoretically unlimited number of susceptibility loci. Specialized simulations like changing population
expansion patterns, ongoing admixture, conditional mating,
differential environmental factors, interference and the use
of linked susceptibility loci are presently only possible by
modifying the respective routines in the source code. As a
future development, the respective procedures will be made
available as options in the experimental set-up files.
Error elimination
Errors are an inevitable part of all software from a certain
degree of complexity. However, in simulations of large datasets and complex problems, errors are very hard to trace
back and may corrupt the integrity of the whole computer
experiment. The present version of the program has been
used as a tool with gradual enhancement of functionality
since May 1996 in our laboratory. Pedigrees and recombinational events were traced by hand in several simulations. Furthermore, the program was used to simulate data under
known parametric models, under which the expected results
were obtained (comparison with SLINK under recessive and
dominant models; data not shown). It was our policy to leave
the central breeding routines untouched in the evolution of
the program, since they had been extensively debugged.
Application example: decay of linkage
disequilibrium in a population
We investigated the decay of linkage disequilibrium in a
large population, which can be well described by mathematical models (Robbins, 1918; Thompson and Neel, 1978) in
order to validate the simulation (Figure 4). Linkage disequilibrium ∆0 between a marker allele 1 of a biallelic marker M
(1,2) and a disease locus (D,d), that existed at a certain point
in population history, is expected to decline under random
mating during consecutive generations (n) at a rate that is
dependent on the recombination fraction (θ) between marker
M and the disease locus. In a population of indefinite size,
this relationship is given by ∆n = (1 – θ) n ∆0, where ∆n was
calculated as ∆ = f1D f2d – f1d f2D (f denoting the population
frequencies of the respective haplotypes with 1D being the
haplotype of interest in this setting) (Devlin and Risch,
1995). The figure shows the mean of 50 replicate simulation
experiments in good concordance with theoretical estimates.
As shown in Figure 4, the relative discordance of simulation
and prediction in very small recombination fractions (0.1
cM) shows that genetic drift rather than recombination determines the haplotype frequency. Here, the simulation offers
the possibility to look at the distribution of haplotype frequency 1D in replicate experiments (Figure 5), which are
difficult to derive analytically (Neel and Thompson, 1978;
Weir and Hill, 1980; Hill and Weir, 1988, 1994).
Possible applications
The program allows for a very flexible specification of population simulation experiments. Special options exist that
allow the disease model to be fitted empirically as close as
possible to the known epidemiological data. Once this model
is specified, the program can serve as a tool for the study of
genetic drift, recombination and admixture on the generation
and maintenance of linkage disequilibrium in populations, as
well as the evaluation of stochastic sampling characteristics
of families and individuals conditional on a complex genetic
phenotype from homogeneous and heterogeneous populations. The program may, therefore, serve both as an instrument in theoretical research and applied genetics in allowing
the generation of datasets for power estimations of mapping
projects, as well as the evaluation of genetic study designs in
complex genetic disorders.
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