Antibiotics, Vol. 11, Pages 1709: Modeling the Directed Evolution of Broad Host Range Phages

To provide a more quantitative sense of selection, we numerically analyze ordinary differential equations similar to those in Equation (1a,b) that encode phage growth, bacterial growth, and bacterial death from phage attack (see Methods for the structure of equations, see Data Availability for access to the files). Our approach is to compare the different host-presentation protocols, but it should be appreciated that, even with just two hosts, many variations are possible within one host-presentation protocol, affecting the dilution at the end of a cycle, the duration of a cycle, initial conditions, and bacterial growth. Our approach is to consider a limited set of conditions for one presentation protocol, then to vary some of those conditions to measure the impact on the types of phages retained. A systematic analysis that varies many conditions together would be unmanageable. The intent is to develop a sense of what set of conditions an empiricist might use to steer the outcome toward generalists or specialists, but any empirical application will likely benefit from numerical analyses tailored specifically to it.

3.2.1. Parallel PresentationRecall that, with Parallel presentation, phages are grown on each host separately and then pooled for the next cycle. Each panel in Figure 2 is a heat map illustrating the fate of one of the three phages under Parallel presentation with a fixed-count dilution (the phage pool was diluted to a density of 1000 every cycle). For a single pair of specialist burst sizes, the figure shows whether the generalist or specialists are maintained across a range of generalist burst sizes. Note that phage maintenance will generally depend both on evolution and demography, but for these trials, the outcomes are due entirely to evolution.The results are clear (Figure 2) and in agreement with the impressions from the heuristic results in Equation (2a,b). The fate of the generalist is governed by its burst sizes relative to the coordinates of the pair of specialist burst sizes (bA|A =15, bB|B = 15, given by the coordinates of the black-on-white ring): the generalist is lost if neither of its burst sizes is as good as that of the respective specialist. Likewise a specialist is lost if the generalist has a superior burst on that host. There are intermediate zones, and the generalist is not completely lost until its burst sizes fall somewhat below those of the specialists. Each specialist is lost as the respective generalist burst size exceeds 15, but there is a slight ‘shoulder’ effect in which the specialist is more prone to loss as the generalist performs well on the other host. This latter effect is consistent with the ratios in Equation (2a,b), that the generalist is somewhat better at suppressing one specialist as it does relatively better against the other specialist. Overall, Parallel presentation is a ‘fair arbiter’ of phage performance on each host: the best-growing phages are retained on each host.

For the parameter values considered, the generalist is retained in approximately ¾ of the space shown. The space shown does not necessarily reflect the spectrum of biological possibilities, of course. If the generalist was constrained so that it could not outperform specialists on each host, for example, the relevant space would be limited to the lower left quadrant, for which the generalist would be mostly absent.

Now consider an asymmetric case when the specialists have different burst sizes (bA|A = 20, bB|B = 10, Figure 3). This case could arise if one host is better for phage growth than the other, or just if one specialist happens to be poor at growth. The pattern for the generalist from Figure 2 is shifted to the new specialist coordinates but is otherwise largely the same as before: the generalist is maintained if its burst size on at least one host exceeds that of the specialist for that host. Again, the heuristic results in Equation (2a,b) agree with the numerical outcomes.We next consider how these outcomes depend on details within Parallel presentation (a summary Table of the different cases is offered in Section 3.3). The trials in Figure 2 and Figure 3 assumed a bacterial growth rate of r = 0.3 and a fixed-count dilution to density 1000 every cycle. As phage density often reached 109 or more, this dilution allowed phage growth by many logs every cycle. Those analyses were modified in either or both of two ways: (i) bacterial growth rate was reduced to 0.1, and (ii) the pool was diluted by a fixed volume (5% of the phage density). The time between phage pools remained the same as before, at 20 steps. Only the asymmetric case was considered (bA|A = 20, bB|B = 10).Merely lowering the bacterial growth rate (with no change in the dilution protocol) has a large effect on the outcome: only the A-specialist is retained in much of the space, and its effect at purging the generalist is symmetric despite the asymmetry in specialist burst sizes (Figure 4). That is, despite the presence of two phages that can grow on B, both are lost in much of the space, and the B-specialist is lost everywhere. This radical change is necessarily due to the reduction in bacterial growth rate because that is the only difference from the previous analysis.On first impression, this fundamental change in outcome is wholly unintuitive. Merely reducing bacterial growth rate is expected to allow faster clearance by the phage because there are fewer bacteria in the culture. The observed effect is the opposite: none of the phages grows fast enough to exhaust its host by cycle’s end. The two panels of Figure 5 contrast phage growth dynamics between the r = 0.1 and r = 0.3 cases (A and B, respectively), where it is easily seen that the lower bacterial growth prevents phages from exhausting their hosts by cycle’s end. With this change in phage growth dynamics combined with the use of a fixed-count dilution, the fastest-growing phage in the pool (whether that phage grows on just host A or B or both) sets the dilution limit, and all other phages are diluted more than they grow. Hence, all other phages progressively disappear. Thus, once the burst size of the generalist exceeds 20—on either host—it now displaces the A-specialist for the same reason the A-specialist was displacing the other phages when it had the superior burst. Thus the effect is symmetric despite the asymmetry in specialist burst sizes.This radical change in outcome is due to demographic effects of the protocol rather than to evolution. Ironically, reducing bacterial growth rate limits phage growth and thereby also prevents slow-growing phages from exhausting their hosts in 20 time steps. Phage growth depends on the product P·C, so phage growth increases with bacterial density and more quickly exhausts the cells at high cell density. With lower bacterial growth, phage numbers never get high enough to exhaust cells by cycle’s end. By this logic, the problem should be reversible in various ways. Indeed, the effect can be reversed without changing bacterial growth rate. For example, if the cycle length s increased, eventually the fastest-growing phage runs out of hosts, allowing slower-growing phages (on the other host) to exhaust their host and catch up to the dilution limit. Thus, increasing the cycle to 40 steps while maintaining r = 0.1 restores the pattern approximately to that of Figure 3.

This case reveals that non-independence of phage growth on different hosts can arise through the dilution protocol (also dependent on culture conditions). This outcome is not due to a change in the relative advantage of one phage over another (i.e., not due to selection and thus not reflected in the ratios of Equation (2a,b)) but rather stems from the demographic consequences of the dilution protocol. Phages can be lost because they cannot grow fast enough to maintain themselves, even though they may be ‘evolutionarily’ superior to their competitors who are also lost. The host-presentation protocol thus has ramifications for evolution and, separately, for demography.

The second modification of Parallel presentation considered here is to change the dilution mode while retaining the low bacterial growth rate of 0.1. Previously, with fixed-count dilution and low bacterial growth rate (Figure 4 and Figure 5A), the phage density reached just over 106 by cycle’s end, so the dilution to 1000 phage was a 10−3-fold reduction in phage density. Phage density could not get high because the starting phage density was always the same number, and bacterial densities did not support phage growth rapid enough to overwhelm the culture. With the fixed-volume dilution, each new cycle is started with 5% of the phage density in the phage pool. This means that any gains in total phage density in one cycle directly increase the density of phages transferred into the next cycle, which in turn results in even more phage at the end of the next cycle, and so on. This has the potential for phage concentration to become so high that bacterial density becomes limiting before the cycle’s end. Indeed, whereas phage density under fixed-count dilution ultimately reached and was maintained at just over 106 with transfers of 1000 phage, phage density reached almost 109 when transferring 5% of the pool, at which point hosts became limiting. When bacterial density becomes limiting, phages from host A no longer drive the demographic extinction of phages on host B. The patterns with fixed-volume dilution and low bacterial growth are now closer to those of fixed-count dilution with high bacterial growth (Figure 6), although there are now also broad intermediate zones.Although the use of a fixed-volume dilution protocol (combined with low bacterial growth) largely avoids the demographic purging of phages and restores the pattern of phage retention seen in Figure 3, the patterns in Figure 3 and Figure 6 have a striking difference: the generalist drives specialist extinction over broader parameter ranges in Figure 6. In Figure 6, the generalist is extinguishing specialists that are decidedly superior on the respective host. The reason for this change in outcome is not immediately clear. We conjecture that the difference lies in the lesser phage growth per cycle in Figure 6 due to a combination of low bacterial growth rate and fixed-volume dilution. Thus, the dilution in Figure 3 allowed approximately 6 orders of magnitude phage growth per cycle, whereas the dilution in Figure 6 allows approximately 20-fold phage growth per cycle. This reduced growth per cycle has the effect of reducing the impact of burst size differences on the Ni|j in the heuristic Equation (2a,b). With smaller Ni|j, the arithmetic advantage of the generalist looms ever larger.

This change in outcome (from fixed-count dilution to fixed-volume dilution, both with low r) is readily seen to stem from a restoration of favorable demography—phages are no longer being diluted to extinction. Avoiding demographic extinction is merely a matter of growing the phages long enough relative to dilution, an outcome that can be achieved either by longer growth or lesser dilution. As noted above, there are multiple ways to achieve this outcome, and an important one is presented next.

These few variations on the Parallel presentation protocol point to manifold complexity stemming from protocol details such as cycle duration, bacterial growth rate and dilution. They likewise motivate ways to avoid ‘unfair’ recovery of phages. One modification that could ensure maintenance of phages on each host despite differences in bacterial growth rate is to set the dilution separately for each host and then pool the samples. This protocol would also mitigate a tyranny from the generalist and prevent phage loss due to poor growth on one host even with high dilution rates. Numerical analyses support this intuition (Figure 7 for comparison to Figure 4).

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