Markov processes are used in molecular evolution to describe the change between nucleotides, aminoacids or codons over evolutionary time. Usually, time is measured as the number of substitutions because molecular sequence data does not allow the separate estimation of the rate and the time, but only of their product 53. In the context of forward simulation we are not interested in the transition after an arbitrary time t (branch length) but just in the transition from a nucleotide or codon to another, given that a mutation occurs. An advantage of this approach is that we need to compute the transition matrix just once at the beginning of the evolutionary process. Therefore, consider a given instantaneous substitution rate matrix Q, which allows for a complete definition of any Markovian substitution model 53, the matrix M = -qQ + I is the conditional transition matrix to go from i to j provided that a substitution occurs, where q = diagonal (1/q_i) and I is the identity matrix 54. Then, given an instantaneous substitution matrix Q, estimated for example using PAUP 55 or Hyphy 56 programs, we can obtain the corresponding transition matrix M that can be used to produce the necessary mutation process in a forward in time evolutionary model.

There are two basic biological models implemented in GenomePop, namely "viral" and "non-viral". The only difference that distinguishes them is just that in the viral model the initial sequences are different in each population, as the different viruses infect different individuals. Thus, the user can define a viral model indicating the percentage of sequence identity (0–100) between the sequences of the distinct populations. By default the sequence identity is zero i.e. the sequences at each population are randomly settled. In the non-viral model the initial sequence is the same for every population (identity of 100%).

There are different DNA models implemented in GenomePop (Table 1). In any of them, the user can decide to allow recurrent mutation, i.e. multiple site hits or not. Models can be haploid or diploid. Population size can be constant or variable. In the four-allele models, the sequences can be generated by the program or provided by the user. In the case of the 2-allele model (SNPs) just one or several chromosomes can be considered. In this same model, recombination can be constant or a hot spot recombination model can be defined. In the latter, the recombination rate r is per haploid region and generation. If no hot spots are defined, the expected number of recombination events between any two sites i and j will be 2rd_ij/(L-1) where d_ij is the implied region length and L is the chromosome length. The number of recombination events between the two chromosome extremes 0 and L -1 will be 2rd_ij/(L-1) = 2r. In GenomePop, the effect of natural selection can be modelled in two different ways: 1) by its effects on the dN/dS ratio i.e. by defining a codon model, and 2) via the fitness effect of mutation on specific loci. The user can run either of two models. The codon model option runs a MG94 codon model 57 with a given dN/dS combined with any defined nucleotide model. This model of codon evolution will be implemented by the instantaneous rate matrix to go from codon i to j. That is, Q_ij = θ_mnkπ_n where θ_mn accounts for biased nucleotide, m to n substitutions; k = 1 or ω for synonymous or nonsynonymous mutation rates respectively and π_n is the equilibrium frequency of the target nucleotide. This corresponds to the MG94 model 57 with the restriction of α = 1. Nucleotide equilibrium frequencies are used instead of codon frequencies. To simulate a given dN/dS we simply set ω = dN/dS. Alternatively, the user can set the codon model option to false (default option) and define specific sites under directional selection with a given selective coefficient which will apply when a mutation occurs at such site. The user can also force all sites to undergo selection. The selection coefficient, s, can be constant or sampled from a gamma distribution with user-defined shape parameter β and scale parameter β/s. The β parameter allows for modelling of the fitness effects distribution, e.g. a low value of β (0.1) will sample many mutations with low effect and few with high. A β parameter of 1 corresponds to the exponential distribution. If we set β to 0 then a constant effect model is applied. Moreover, GenomePop permits the combination of both kinds of models of selection, codon and fitness-based, though the biological meaning of such a mixture is not clear.

Two basic migration schemes, island model and one-dimensional stepping stone, are pre-defined in GenomePop. However, the user can define any migration model of interest (Figure 1). To do this, set the flow model to 'user' in the standard input file and then just introduce a scheme similar to that of Figure 1 in a file called MigrationModel.txt. In this file, the lines beginning with '#' are comments. To indicate how individuals will migrate from a given population just begin the line with the word "pop". The order of appearance of each population in the file will correspond with its index i.e. the first population that appear is the population number one, etc. The number below "pop" refers to the migration level, i.e. the number of different migration rates defined from this population. The next line should begin with a migration rate (between 0 and 1) followed, in the same line, by the target population(s). We should have as many of these kinds of lines as the migration level indicates, i.e. if the migration level is 2 we should have two lines beginning with a migration rate. More detailed explanation and specific examples are given in the program web page.

Clearly, the more complex the model defined, the slower the simulation. To avoid high computation times, GenomePop incorporates a scaling option based on the fact that, under neutral models, we can scale the population size N and the time t, provided the consequent correction to the mutation (μ), migration (m) and recombination (r) rates holds the corresponding compound products Nμ, Nr, Nm, etc., constant.

The input file should be called GenomePopInput.txt. In this file, lines beginning with '#' are comments and will be ignored. In Figure 2 we can see an example of an input file. Note that the input is flexible, i.e. the minimum input for GenomePop to work appropriately corresponds to the first line and the values below it. This line must begin with the identifier 'chromsize' and the line below with the corresponding desired values. Note that, in lines with identifiers, only the first word matters for the program.

Thus, the input in Figure 2 generates 100 datasets under a GTR model with substitution rates typical for HIV 58. Both recurrent and retromutation are allowed. The system will evolve 1 chromosome of 1 Kb under the given model over 20,000 generations. As can be seen in Figure 2, a scaling of 10 was used, which implies that both, population size and the number of generations, was divided by 10 and mutation was multiplied by the same factor. A more exhaustive explanation of the input facilities of GenomePop is provided at the program web page.

For each obtained dataset from the input in Figure 2, the best-fit model of nucleotide substitution under the Akaike information criteria (AIC) was estimated with Modeltest v3.6 59, using maximum likelihood (ML) estimates from PAUP* 55. The percentage of correct model estimation (GTR) was 97% although some datasets, about 29%, were also assigned invariable sites or rate heterogeneity among sites. The substitution pattern and equilibrium frequencies were correctly estimated.

As GenomePop has many different features and models it is difficult to validate every possibility or circumstance. However, strong effort has been made to validate the program as thoroughly as possible. For example, both unscaled and scaled simulations were performed under a Jukes-Cantor model with diversity θ = 4Nμ = 0.004 over 10⁴ generations and then θ was estimated using the finite-sites correction of Watterson θ 60. The accuracy was quite good, obtaining estimates of 0.0043 ± 0.00015 and 0.0037 ± 0.00016 for the unscaled and scaled cases respectively. Recombination was also tested by evolving datasets for 6N generations under a Jukes-Cantor 4-allele model with different values for the parameter ρ = 4NrL, where N is population size, r is recombination rate per site and L is the DNA sequence length (the corresponding parameter in GenomePop is 'Rec' = r × L). Namely, we ran cases with ρ equal to 0, 50 and 100. Recombination was then accurately estimated using the program Kpairwise 58. GenomePop allows also studying 2-allele SNPs at different frequencies in different populations. In Figure 3 we define a 2-allele model (JC2) with different initial composition at each population (viral model) and 10 independent SNPs (recombination 'Rec' = 10 × 0.5 = 5). The populations have different sizes (100 and 120) and migration occurs under the island model. Note that when defining different population sizes, the original population size provided in the 'chromsize' line under the 'popsizeKmax' identifier is overwritten.

We ran this example over 200 generations and then analyze the output with the GenePop 4.0 program 61. As expected the SNPs were detected as independent. We then changed the value of recombination to 0 ('Rec' = 0) and then GenePop 4.0 tell us that the 10 SNPs are linked, as expected. Note the many possibilities that the program provides in the context of studying SNPs under complex evolutionary situations. We can define any number of populations under any user-defined migration model. We can set any number of SNPs with the desired linkage relationships. The SNPs can be set at distinct initial frequencies in the different populations, for example, 'SNPfreqs' at 1.0 and 0.0 defines the first population with allele 1 fixed and the second with allele 2 fixed.

We performed a simple experiment to test the impact of recombination on dN/dS estimation. We ran 50 replicates, with and without population recombination per gene, 4Nr = 40 and 0, respectively. The runs were performed under a MG94 × JC model both with dN/dS = 1 and dN/dS = 2.5 evolving 333 codons for 10N generations with an effective population size of N = 10³ to get samples of 20 sequences. The dN/dS ratio was estimated with the FEL (Fixed effects Likelihood) model of Hyphy 62 which computes global and site by site dN/dS ratio. A p value of 0.1 was used to infer sites under positive selection. As can be seen in Table 2 a dN/dS of 2.5 provokes the detection of some sites under positive selection (1 or 2, not shown) in only 30% of the replicates (NSS = 0.3 in Table 2). Furthermore in the strictly neutral case (dN/dS = 1), one positive selected site was assigned in 10% of the replicates as expected given the p value used. If we correct by this 10% of false positive tests then positive selected sites were detected only in 20% of the replicates under a dN/dS value of 2.5 and no recombination. This is in agreement with the conservative nature of the FEL method 62. Also noteworthy is that recombination had no impact on global dN/dS estimation but had important effects on the number of sites detected under positive selection as is evident upon inspecting Table 2. It seems also that the effect of intracodon recombination is negligible. Interestingly, it appears that the effect of recombination is somewhat higher under non-neutral dN/dS than in the neutral case. The impact of recombination on positive selection detection has already been studied 4546. However, as far as we know, the comparison of the impact of recombination under neutral or positve dN/dS jointly with the effect of intracodon recombination has never been studied before. The significance of this effect should be studied with more replicates and cases, which is out of the scope of the present work.