Table 1.
List of parameters, their dimensions, and the values or ranges used in figures. The epidemiological parameters correspond to a basic reproduction number of 3 (in the plausible range for S. pneumoniae [44]). However, the specific values are not important: the key results relating the k parameters to the effect on strain structure (Equations 5 and 8) do not depend on the epidemiological parameters. The bounds on the k parameters (0,1) arise from the assumption that co-colonisation is not as efficient as primary colonisation, in line with empirical findings (e.g., [33]).
Fig 1.
NFDS models of metabolic niche or competition-colonisation trade-offs.
We represent the two epidemiological models for the case of one bi-allelic locus. In the metabolic niche model (left), co-colonisation rates are reduced when the resident and incoming strains share the same allele, modeled by parameters . In the competition-colonisation model (right), allele a is “colonising”, leading to an increased rate of primary colonisation m, while the competitive allele A leads to higher rates of co-colonisation on average. Co-colonisation rates are dependent on the difference in number of competitive alleles between the incoming and the resident strain
through the parameters
.
and
represent respectively the partial forces of infection of allele a and A, taking into account that strains in co-colonisation have reduced transmission rates through parameter q. Solid orange lines represent colonisation and dashed blue lines represent clearance.
Fig 2.
Different competition regimes lead to maintenance or abolishment of strain structure.
In (a), we show how different values of the competition efficiencies k0, k1 and k2 influence epistasis and therefore strain structure according to Eq (5). Equilibrium D’ (t = 5000) is shown as a function of the competition parameters k1 and k2 in the absence (b) and the presence (c) of recombination. Müller plots in panel (b) show the dynamics of the cumulative frequencies of all genotypes through time, depending on the competition parameters, starting for an initial LD of D’ = 0.4. Some specific values of D’ are noted on panel (c). Parameter values used: , (b)
and (c)
. Initial allele frequencies were initialised to
, and this was applied to every figure of this work unless stated otherwise.
Fig 3.
The geometry of competitive interactions shapes epistasis and allelic associations in a competition-colonisation trade-off model.
(a) A linear relationship between all the competition parameters leads to additivity in how fitness effects combine across loci and thus no epistasis, see Eq (8). However a concave or a convex geometry will lead to epistasis of constant sign, as shown with solid lines. More complex geometries which are not purely concave or convex can produce epistasis which sign is frequency-dependent. The red dashed line scenario leads to PFDS wherein the most frequent pair of non-overlapping strains (those associated with either positive or negative linkage) is favoured. In contrast, the blue dashed line geometry leads to NFDS at the genotype level, favouring the least frequent pair of strains and thus abolishing allelic associations. D’ trajectories are shown in (b,c) using the parameter sets shown in panel (a), starting from both positive and negative initial D’. Other parameter values:
. See Fig K in S1 File Supplementary Information and L in S1 File Supplementary Information for equilibrium LD from simulations for a wider range of parameters.
Fig 4.
Observed patterns of linkage disequilibrium in several populations.
(a) shows, according to the KEGG annotation of a focal gene, the probability to be in strong linkage disequilibrium (|D’| > 0.6) with any given gene. An asterisk on the diagonal signifies that there is significantly more linkage for that combination (two genes with a shared KEGG category) compared to the rest of the row/column (one gene of that category with genes of different categories). See Fig R in S1 File Supplementary Information for the extension of this figure to all genes, without the 100kb distance filtering. In (b,c), we study the distributions of absolute difference of LD across populations. We show for gene pairs highly linked in the Massachusetts dataset (|D’| > 0.6) the distribution of change in linkage compared to (b) the Southampton dataset or (c) the MaeLa dataset. The observed changes distributions are shown in green and the purple distributions show the changes in the simulated population. Note how there are fewer pairs of genes being shown for the simulated populations, consistent with the observation that there is overall less LD in the simulated than the real dataset (Fig P in S1 File Supplementary Information). See Fig S in S1 File Supplementary Information for all the population pairwise comparisons.
Table 2.
Significance of LD Conservation across populations and time (Two-Sided Empirical Test).
Table 3.
Significance of LD Reversal across populations and time (Two-Sided Empirical Test).