Viral quasispecies profiles as the result of the interplay of competition and cooperation

Abstract

Background: Viral quasispecies can be regarded as a swarm of genetically related mutants. A common approach employed to describe viral quasispecies is by means of the quasispecies equation (QE). However, a main criticism of QE is its lack of frequency-dependent selection. This can be overcome by an alternative formulation for the evolutionary dynamics: the replicator-mutator equation (RME). In turn, a problem with the RME is how to quantify the interaction coefficients between viral variants. Here, this is addressed by adopting an ecological perspective and resorting to the niche theory of competing communities, which assumes that the utilization of resources primarily determines ecological segregation between competing individuals (the different viral variants that constitute the quasispecies). This provides a theoretical framework to estimate quantitatively the fitness landscape.

Results: Using this novel combination of RME plus the ecological concept of niche overlapping for describing a quasispecies we explore the population distributions of viral variants that emerge, as well as the corresponding dynamics. We observe that the population distribution requires very long transients both to A) reach equilibrium and B) to show a clear dominating master sequence. Based on different independent and recent experimental evidence, we find that when some cooperation or facilitation between variants is included in appropriate doses we can solve both A) and B). We show that a useful quantity to calibrate the degree of cooperation is the Shannon entropy.

Conclusions: In order to get a typical quasispecies profile, at least within the considered mathematical approach, it seems that pure competition is not enough. Some dose of cooperation among viral variants is needed. This has several biological implications that might contribute to shed light on the mechanisms operating in quasispecies dynamics and to understand the quasispecies as a whole entity.

Figure 1

Population fractions for the case of pure competition. Relative abundance of each variant after 500 generations for, respectively, σ = 0.2, σ = 0.15 and σ = 0.1. Corresponding temporal evolution of the fractions of each variant is shown (right panels).

Figure 2

Population fractions for the case f_C = 0.01. Analogous to Figure 1 for the case of f_C = 0.01. Relative abundance of each variant after 500 generations for σ = 0.2, σ = 0.15 and σ = 0.1. Corresponding temporal evolution of the fractions of each variant is also shown.

Figure 3

Population fractions when half of the interactions are cooperative. Relative abundance of each variant after 500 generations (Left) and temporal evolution of the fractions of each variant (Right).

Figure 4

RNA molecules niche position (ξ) and overlapping. Scheme showing two normal probability distributions centered at μ₁ = 0.4 and μ₂ = 0.6 of two RNA molecules, both with σ = 0.1. The region in black corresponds to the overlap between the two normal distributions.

Figure 5

The independence from the initial state. Panel (A) shows the evolution of the population profile for σ = 0.15 starting from only one viral variant at an arbitrary position μ_j on the niche axis (that is, x_i = 0 for all i ≠ j and x_j = 1). Panel (B) and (C) corresponds to the population profile after 250 and 500 generations, respectively. Emergence of three different lumps becomes clearer after 500 generations.