The Next Step in Understanding Population Dynamics: Comprehensive Numerical Simulation

mathematical modeling. Such simulation should enable us to obtain a more biologically integrated picture of how real populations change. Seven years ago one author (JCS) had the opportunity to oversee the development of a comprehensive numerical simulator for the aforementioned purposes. Since that time, a group of biologists and computer scientists have been collaborating to develop a numerical simulator that can simultaneously model all the major known factors that affect genetic change, as well as their relevant interactions, to better approximate what occurs in the real world. The resulting program, Mendel’s Accountant (Mendel), appears to be the first program that has seriously endeavored to do this. Mendel has been described in previous publications (Sanford et al., 2007a, 2007b), and is now beginning to be used for both research and teaching. This tool should not be viewed as a replacement for previous tools already developed within this field, but it is clear that it represents a major step forward. 2. Mendel’s Accountant Mendel’s Accountant simulates genetic change within a population as it moves forward through time. Mendel does this by establishing a virtual population of individuals, and then precisely simulates mutation, selection, and gene transmission through many generations, always in the most biologically realistic manner possible. Mendel is unique in that it attempts to treat all aspects of population dynamics simultaneously and comprehensively, thereby ushering in for the first time the prospect of simulating reasonable approximations of biological reality. Mendel’s Accountant is an apt name for this program because it is largely a “genetic accounting“ program. Every generation, huge numbers of specific mutations are introduced into a population, spread over the genomes of many individuals. Through the ensuing generations, some of these mutations are lost, while others increase in frequency. Each mutation must be tracked through many individuals and through many generations, along with all data that apply to that mutation (each has an allelic ID, mutational fitness effect, degree of dominance, and chromosomal location). During a large run, Mendel can track Studies in Population Genetics 120 hundreds of millions of different mutations. Not only does Mendel do the genetic accounting associated with tracking individual mutations, it simultaneously does the genetic accounting associated with tracking: 1) linkage blocks as they recombine; 2) net fitnesses of each individual; 3) the distribution of the fitness effects of all the accumulating mutations; and 4) the resulting distribution of allele frequencies. Genetic accounting via numerical simulation is possible because the underlying processes (Mendelian inheritance, random mutations, differential reproduction) are all relatively simple and mechanistic in nature and are therefore subject to straightforward accounting procedures. Furthermore, all the relevant biological variables are easily specified as parameters for use in simulation (e.g., population size, mutation rate, distribution of mutational fitness effects, heritability, and amount of selective elimination each generation). Like many high-performance numerical simulations, the core of the Mendel program is written in Fortran 90, allowing the execution of tasks that are extremely demanding computationally, making it possible to process huge amounts of genetic data. To explain how Mendel works in the simplest way possible, it is useful to consider the series of decisions that an experimenter must make. Firstly, the experimenter must define the species and its reproductive structure. Is it haploid or diploid? How big is the genome, and what fraction is funcional? Is its reproduction sexual or clonal? Does the species ever selffertilize? All these biological factors can be modeled by Mendel, and must be specified by the user, because they have a substantial impact on population dynamics. These parameters determine how reproduction and gene transmission will occur within the virtual species. Secondly, the experimenter must define the characteristics of a particular population within the species. How big is the population before and after selection? Are there subpopulations? How many generations do we wish to observe? These parameters define the actual scope and architecture of a particular experiment. Thirdly, the user must specify reproductive details. The reproductive rate must always be high enough to create a population surplus each generation, such that this surplus can then be selectively removed each generation. For example, the default reproduction rate is 3. In this case the number of offspring generated each generation is always 3 times larger than the specified population size. This creates a surplus population large enough for selection to remove two of every three offspring in the next generation. If the population under study reproduces sexually, recombination will occur at this stage. The experimenter must specify the number of chromosomes (assuming two cross-overs per chromosome) and the number of linkage blocks (this affects segregation of linkage blocks during gamete production). Fourthly, the experimenter needs to specify the mutations that will be added to the population. After creating a new virtual population of offspring, Mendel then begins to add new mutations to those individual offspring. Mendel assigns mutations to individuals randomly, following a Poisson distribution. The experimenter specifies a mutation rate appropriate for the species under study (or one that is of theoretical interest). Likewise, the experimenter must specify a distribution of mutational fitness effects. Typically this distribution will include deleterious, neutral, and beneficial mutations. The mutations that are added to the population are drawn randomly from a user-specified pool of potential mutations (usually having a Weibull distribution of fitness effects). Drawing from such a distribution, some mutations will have large effects, but most will have small (nearlyThe Next Step in Understanding Population Dynamics: Comprehensive Numerical Simulation 121 neutral) effects (Kimura, 1983), as occurs in nature (Eyre-Walker & Keightley, 2007). Each new mutation has an identifier for tracking purposes, a fitness effect, a specified degree of dominance, and a chromosomal location (i.e., a designated linkage block). Lastly, the experimenter needs to specify the nature of the selection process. Once Mendel has created a newly mutated population of offspring, it must implement selective removal. To do this Mendel first calculates the combined effect of all mutations in each individual (initial individuals containing zero mutations having a fitness of one, with beneficial mutations increasing fitness and deleterious mutations reducing fitness). Mutations can be combined either additively or multiplicatively (or in alternative ways, i.e., epistatically). Once the fitness of each individual has been calculated, a certain fraction of the population is selectively eliminated based upon genetic fitness, usually eliminating the exact population surplus, so that the original population size is restored. Selective removal can be either by truncation selection, probability selection, or partial truncation. To add biological realism, the user can specify a heritability of less than one, such that fitness variations caused by environmental noise will be added to the genetic fitness to establish the fitness phenotype, which is then the basis for selection. The individuals that survive selection will then be ready to repeat the cycle of mutation, reproduction, and selection. During a single experiment, Mendel can routinely simulate hundreds of millions of newly arising mutations. Each mutation is tracked through all generations, until it is either lost or goes to fixation, or until the experiment is complete. Throughout the whole run Mendel is continuously monitoring, recording, and plotting the average number of mutations per individual, individual and average fitness, population size history, the fitness distributions of accumulating mutations, selection threshold histories, linkage block net fitness values, and mutant allele frequencies. 3. Forward-time population genetic numerical simulations There are numerous forward-time simulation tools currently in use within the field of population genetics. Detailed reviews on the subject are available elsewhere (e.g., see Carvajal-Rodgriguez, 2008; Kim & Wiehe, 2008; Liu et al., 2008; Carvajal-Rodriguez, 2010). It is useful to provide a general overview of these programs to properly appreciate the types of problems that can be addressed with such simulations. Every forward-time simulation is designed with a particular application in mind, and each is best suited to study a certain