Host-Pathogen Coevolution and the Emergence of Broadly Neutralizing Antibodies in Chronic Infections
Viruses perform best against antibodies from the past and perform worst against antibodies from the future due to the adaptation of antibodies. (A) Stationary rescaled binding affinity between viruses from time t and antibodies from time t + τ, averaged over all t: ετ = 〈∑α,γ Eαγ yγ(t)xα(t + τ)〉t/E0, and (B) time-shifted mean fitness of viruses Nv FV;τ = −sv ετ, are shown for three regimes of coevolutionary dynamics: strong adaptation of both populations , with sa = sv = 2 (blue), stronger adaptation of viruses with sv = 2, sa = 0 (red), and stronger adaptation of antibodies with sv = 0, sa = 2 (green). Wright-Fisher simulations (solid lines) are compared to the analytical predictions from eqs. (S102, S103) in S1 Text for each regime (dashed lines). The “S”-shape curve in the blue regime is a signature of two antagonistically coevolving populations sv θv ∼ sa θa. For large time-shifts, binding relaxes to its neutral value, zero, as mutations randomize genotypes. In the absence of selection in one population, the time-shifted binding affinity reflects adaptation of one population against stochastic variation in the other due to mutation and genetic drift. The slope of time-shifted fitness at lag τ = 0 is the viral population’s fitness flux (slope towards the past) and the transfer flux from the opposing population (slope towards the future), which are equal to each other in the stationary state. The slope of the dotted lines indicate the predicted fitness flux and transfer flux (eqs. (8 and 9)). Time-shifted fitness shown here does not include binding to the conserved region since that value is constant for all time-shifts in stationarity (see S6 Fig for non-stationary state). Simulation parameters are given in the Materials and Methods. (C) Empirical time-shifted fitness measurements of HIV based on a neutralization titer (IC50) , averaged over all time points with equal time-shift τ. Circles show averaged fitness ± 1 standard error, and crosses show fitness at time-shifts with only a single data point. Solid lines show analytical fits of our model to the data (see Materials and Methods and Section F of S1 Text). In patient TN-1, viruses and antibodies experience a comparable adaptive pressure, with a similar time-shift pattern to the blue “S-curve” in panel (B). In patient TN-3, however, adaptation in viruses is much stronger than in antibodies, resulting in an imbalanced shape of the time-shifted fitness curve, similar to the red curve in panel (B).