Logistic growth-based regression models

To test whether the observed differences in growth curves were the result of growth rate differences, differences in sensitivity to IFN, or both, the in vitro propagation assays above were repeated over a targeted range of IFN concentrations. A maximal dose of 0.5 pg/μL was chosen, as in the initial IFN spreading assays this dose enabled a clear difference between the TF/CC pair with minimal IFN-associated toxicity (~80% live cells). The remainder of doses were spread at 0.1 pg/μL intervals to capture incremental differences in growth rate.

Data from these assays were modelled as a logistic growth process:

Where x_t,v,j,d is the number cells infected at time t by virus v, in replicate j of a given treatment with IFN dose d, and x_1,v,j,d is the initial number of infected cells in this replicate (as measured at the first timepoint, 24 hours post inoculation). To account for IFN-toxicity to cells at higher doses, the maximum number of cells available to be infected (i.e., the carrying capacity, k_j) was modelled as a function of IFN dose (d):

where β_k,0 is the mean carrying capacity when no IFN is present, β_k,1 is the effect of 1 pg/μl IFN, and u_j,d is a random effect allowing variation in the number of cells available between different replicates of a given treatment.

In the most complex model fitted (here termed the differential sensitivity model), the achieved growth rate of each virus, r′_v,d, was modelled as a function of IFN dose, a virus-specific adjustment allowing growth rates to vary between viruses, and an additional virus-specific adjustment for interferon-sensitivity:

Here, v = 0 for the TF virus and 1 for the CC virus. As a result, β_r,0 is the growth rate of the TF virus in the absence of IFN (here termed the baseline growth rate), while β_r,1 is the adjustment needed to achieve the baseline growth rate of the CC virus. Finally, β_r,2 measures the baseline effect of 1 pg/μl IFN on the growth rates of both viruses, while β_r,3 allows the CC virus to be more or less sensitive to a given IFN dose than the TF virus. The fit of this model was compared to one without the additional virus-specific adjustment for interferon-sensitivity (i.e., without the β_r,3dv term), here named the constant sensitivity model.

Models were fit by maximum likelihood using version 3.1–149 of the nlme library in R version 4.0.2 [80,81]. Confidence intervals for all parameter estimates were generated by re-fitting models to 1000 hierarchical bootstrap samples of the data. For each IFN dose, the available data were truncated as soon as growth curves declined by more than 30% relative to the previous timepoint, with models fitted to the remaining data only. This was needed to accommodate the long timescale of these experiments, where both the accumulation of dead cells due to virus infection and release, and the toxicity effects of long-term culture in the presence of IFN, results in a reduction in viable cells that can be infected (Fig 5B). The sensitivity of models to this exclusion was assessed by evaluating a range of cut-off points (including no data removal). Truncation affected primarily the estimated carrying capacity and associated effect sizes (β_k,0 and β_k,1), with carrying capacity under-estimated when the declining parts of growth curves were included. All other parameter estimates remained broadly similar with overlapping confidence intervals, regardless of the cut-off used, and the differential sensitivity model remained unsupported.