An Individual-based Simulation System for Population GeneticsCarolyn Leach and Anthony Pietsch
Faculty of Science, The University of Adelaide
Since its inception, population genetics has been an area that is difficult to teach. Like many other disciplines involving complex systems, the basic concepts have generally been communicated in the form of algebraic models. Typically, such a model is a single mathematical expression, representing some characteristic behaviour or property of an idealised population. There are many examples of these. However, in most cases they stand alone and cannot easily be combined, which presents a severe limit to the number of factors that can be dealt with at the same time. They also give little or no information about interactions between factors.
The algebraic approach presents difficulties for many students, particularly in introductory courses and even for those advanced students of biology without experience in advanced mathematics. The main problem is that algebraic models do not look anything like real biological populations. Therefore, it can be hard to make the conceptual leap from one to the other and, hence, to gain an intuitive understanding of the phenomenon that the model is intended to represent.
In the teaching of many scientific disciplines, physical experimental models are used as a supplement (or as an alternative) to mathematical models. Unfortunately, the opportunities for population genetics are severely limited, for a number of reasons. Firstly, the required time-scales span many generations. Secondly, even with short-lived laboratory organisms, the logistical constraints associated with sufficiently large numbers of individuals are usually prohibitive. Thirdly, the collection of genetic information on such a scale is also costly, where possible at all.
One-By-One is a computer simulation package for teaching and research in population genetics. Designed to be used in the same way that experimental populations would be, it addresses all of the problems mentioned above. At its heart is an individual-based model, which is a type of model where every individual in the population is represented. Such models are becoming increasingly popular in the field of ecology, but, to date, have received surprisingly little attention from population geneticists.
The One-By-One Model
Every individual in the One-By-One population has a gender, a genotype and an age. A simulation begins by generating a starting population, with a stable distribution of ages calculated by the program. Genes are normally assigned randomly according to specified frequencies. After this, the model operates on an annual cycle with two main stages, as illustrated in Figure 1.
Figure 1. Basic model structure
In the first stage, involving the 'birth' of new individuals, each adult female is assigned a mate according to a selected algorithm (set of rules). Another algorithm determines the number of progeny (if any) from this mating, while yet another governs the inheritance of genes from parent to offspring. All of these algorithms, in different ways, make use of a random number generator to represent stochastic processes.
The second stage of the annual cycle determines which individuals 'die' and are removed from the population. The program considers every individual, one by one, casting a random number (between 0 and 1) against their probability of survival (PS). In each case, PS depends primarily on the age of the individual, but may also be modified in response to its genotype, to the current population density, or to random environmental fluctuations (if the applicable options are selected). Only those whose random number is less than PS survive to the following year.
The algorithms governing each particular process are essentially independent of each other, allowing a great degree of flexibility in the way that the model is configured. For example, there is no theoretical limit to the number of separate factors that may concurrently affect survival. Because most of the algorithms represent a relatively simple elemental process that is well understood, they are easy to interpret and relate back to the system being modeled.
Model Parameter Options
One-By-One provides facilities to alter many different ecological and genetic characteristics of the simulated population. At the first level, the options presented reflect the standard assumptions of the classical, algebraic models of population genetics. They include:
In each case, the latter alternative activates secondary options which govern the details of the process concerned, such as selection coefficients, mutation rates etc. Non-random mating (which is still stochastic) includes lifetime pairing, [with/without] infidelity and [with/without] replacement, as well as harem and dominance systems. Selection (and also the regulation of population size) can act independently on survival and/or fecundity.
From a genetic perspective, the program can track genes at one or two loci, which may be autosomal, mitochondrial or sex-linked. There are two modes of operation. The first, involving two alleles at each locus, is designed for the study of specific issues such as selection or linkage disequilibrium. The other is geared towards general issues of genetic diversity. Here, every allele in the founding population is unique, which allows the calculation of an inbreeding coefficient for every simulation. Further options (which are too numerous to list in full) include settings for various aspects of the initial population, the reproductive characteristics and, of course, the population size.
Such a wide array of parameter options may seem daunting. However, the program is designed so that the user is not forced to deal with any more variables than is desired. This is achieved firstly by providing sensible default values for every parameter at all levels. Secondly, by choosing basic options, the required number of decisions can be reduced. For example, selecting 'discrete generations' obviates the need to specify a life span, survival curve and breeding age. Alternatively, selecting 'sexes the same' means that only one set of such values is required, rather than two. Thirdly, a complete set of parameter values can simply be loaded or saved as a named 'scenario file' (which is particularly useful in the group laboratory situation).
Simulation User Options
The user interface includes a range of output displays. Gene frequencies, demographic data and other information about the simulated population (e.g. coefficient of linkage disequilibrium) are available as the simulation runs, in either numerical or graphical form. Summary statistics (mean generation time, mean number of breeding adults, etc.) appear at the end of a simulation or can be called up on demand.
A key feature of One-By-One (which is not possible with programs based on algebraic models) is its capacity to display every individual in the population. With this feature, abstract phenomena such as random genetic drift can be viewed directly, in a concrete form. An animation option shows matings, births and deaths as they occur, which helps to illustrate the way that the model works. The 'individuals' display can also be used to provide an occasional snapshot of the population, which may include the genotype, age and/or the probability of survival (to name but a few) for each individual.
For advanced users, One-By-One provides a range of options for saving data to file. There are facilities to program multiple simulations and then calculate the mean output. There are also a range of special-purpose displays that provide additional information about age distributions, litter sizes, mating success or the regulation of population size. These may be used to check and analyse the inner workings of the model, or as an aid to the process of parameterisation.
One-By-One can be used in a number of ways. At the first level, in the lecture theatre, it may be used to provide 'live' demonstrations of phenomena such as random genetic drift, selection and the loss of genetic diversity. For example, it is easy to illustrate a key difference between dominant and recessive selection regimes by directly showing that, when an allele is rare, it is carried almost exclusively by heterozygotes (expressing the dominant phenotype).
Secondly, in the teaching laboratory, students can gain an intuitive understanding of population dynamics by running their own experiments. Typically, a single parameter (e.g. population size or initial gene frequency) will be systematically varied over a series of simulations, to explore its effect on some dependent variable (e.g. time to fixation, heterozygosity or number of alleles remaining). This shows more than just the average or typical relationship between the two variables. It also provides insights into the stochastic variability that is characteristic of biological systems but which often surprises students.
At the third level, there is considerable potential for postgraduates (and others) to use the program for research and theory development. Individual-based modeling offers a new approach to population genetics and allows the exploration of previously untestable hypotheses, with the potential for novel and unexpected discoveries. The wide array of parameter options means that the model can be tailored to suit many different animal and (to a lesser extent) plant species.
One-By-One promises to add a new dimension to the teaching of population genetics, which has traditionally had a heavily mathematical slant. In an entertaining way, it brings to life concepts that are often seen as abstract and difficult to understand. It can provide students with the hands-on experience necessary to develop a rich and full understanding of population genetic processes.
The simulation package has an emphasis on clarity, operational transparency and ease of use. The functional independence of the model's component algorithms imbues a high degree of flexibility, which is enhanced by the design of the interface. Thus, the program can be used for either general or specific applications, in a number of different roles, with students at all levels.
UniServe Science News Volume 17 November 2000
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