Time-Delayed Spread of Viruses in Growing Plaques

The spread of viruses in growing plaques predicted by classical models is greater than that measured experimentally. There is a widespread belief that this discrepancy is due to biological factors. Here we show that the observed speeds can be satisfactorily predicted by a purely physical model that takes into account the delay time due to virus reproduction inside infected cells. No free or adjustable parameters are used.

Figure 1

One-step growth of virus T7 on E. coli versus the time elapsed after adsorption ($t=0$). In this experiment, the concentration is uniform and adsorption is negligible ($[B{]}_t\approx 0$), attained by dilution. Full line: least-square fit of the experimental data points [14] to Eq. (3), which yields $k_2=1.39{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$ and $\tau =18.4\mathrm{m}\mathrm{i}\mathrm{n}$. Before the rise in $[V]$, the experimental dots refer in fact to infected bacteria [19], and the yield is the ratio of the final value $[V{]}_{\max }$ to this initial concentration [19], in this case $Y=34.5$. Inset: T7 adsorption on E. coli, based on the data in Ref. 20. The assay was carried out in the presence of KCN, which is known to inhibit the virus reproduction. Then Eqs. (4) hold; thus, $[B]=[V]+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. From the initial concentrations (taking care of the multiplicity) in Ref. 20, we find $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}=1.39\times {10}^8{\mathrm{m}\mathrm{l}}^{-1}$. The solution for $[V]$ satisfies $g([V])\equiv \ln \{([V]+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})/[V]\}-\ln \{([V{]}_{t=0}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t})/[V{]}_{t=0}\}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\times k_{1}t$. Full line: least-square fit of the experimental data [20] to this equation. The slope yields $k_1=(1.29\pm 0.59)\times {10}^{-9}\mathrm{m}\mathrm{l}/\min $. This value is rather uncertain, but it turns out that the predictions are essentially the same throughout this range of values for $k_1$ (Fig. 2).

Figure 2

Speed of the growth of T7 virus plaques on E. coli as a function of the bacterial relative concentration $f$. Upper curves: predictions of the classical model [17], i.e., Eqs. (7) with $\tau =0$. Lower curves: time-delayed model ($\tau =18.4\mathrm{m}\mathrm{i}\mathrm{n}$, from Fig. 1). Symbols: experimental data [17]. The open and closed data symbols refer to initial bacterial concentrations of ${10}^7$ and ${10}^8{\mathrm{m}\mathrm{l}}^{-1}$, respectively. The mean value and error for $f=0.2$ has been computed by taking into account that there are some additional experimental data available for $f=0.2$ in Refs. 14, 15. In (b), the two lower curves are not distinguishable from each other at this scale, showing that the uncertainty in the value of $k_1$ (Fig. 1, inset) has very little effect on the predicted speed.