Multi-strain phage induced clearance of bacterial infections

agrees with the numerical results of the general model in Eq. 3 and with the findings of [16] where therapy worked in an immunocompetent cohort but failed in immunocompromised mice. The green and grey dashed lines in Fig. 5 show the quantitative impact of bacteria concentration at the treatment start, as they correspond to Eq. (9) with respectively a 3-fold increase and decrease in B0. We note that the highest B0 scenario brings the success to failure transition very close to the parameters that saw successful treatment in immunocompetent mice in [16]. D. Robust benefits of therapeutic phage cocktails within an in vivo infection model, modulating immune responses and phage-bacteria interactions Finally, we address the sensitivity of phage cocktail efficacy given variation in innate immune efficiency and phage life history traits. When adding a second phage to an in vivo infection model with a responsive modulated immune system and structured nonlinear phage-bacteria interactions, as described in Section II B 5, we predict that treatment would succeed if (see SI Sec. IV) I > max  r − ϕPc κ  1 + B0 KD  , r − pϕPc/2 κ  . (11) The first term in the maximum, the same as in Eq. (9), ensures that susceptible bacteria are killed, whereas the second term is the condition under which resistant types can be kept in check. Fig. 6 shows the infection clearance pattern as a function of I and ϕ for simulations of Model (4) with p = 1. The dashed lines show Eq. (11) for p = 0, 0.5 and 1 (blue, yellow, green). The black line represents a scenario where bacteria do not develop phage resistance, .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted September 7, 2024. ; https://doi.org/10.1101/2024.09.07.611814doi: bioRxiv preprint 13 in which case the success condition is given just by Eq. (10). Our simulations agree with the analytical prediction in Eq. (11), as shown in p = 1 in Fig. 6 and in Fig. S1 for the other cases. The simulations findings in Fig. 6 show that the treatment with two phages improves the therapeutic outcome in immunocompromised hosts if the adsorption rate is high enough, confirming the qualitative picture presented in Section II B 3. Eq. (11) quantifies how the therapy success depends on phage-bacteria evolutionary interactions through variation in the cross-phage resistance parameter p. There is a critical ϕc = 2r pPc above which therapy succeeds when I = 0. We interpret this finding to mean that higher p, i.e. using therapeutic phages with more complementary phenotypes, improves the therapeutic outcome. In contrast, when p = 0 bacteria can evolve simultaneous resistance to both phages leading to the same result as in the single-phage treatment. Hence it is crucial to pre-select phages against which it is less likely that bacteria could evolve combined resistance, for instance making sure that they do not share a common receptor target, or that receptors targets are not pleiotropically linked [42], like in our LD experiment scoring a potential cocktail of PAK P1 and LUZ19v against P. aeruginosa. When p = 1 and the phage combination covers all bacteria resistant mutants, there is a region of parameters, between the black and the green lines in Fig. 6, where therapy fails due to phage resistance despite the phage cocktail treatment. Here the only way to improve the treatment efficacy would be to use phages with stronger lytic features. Interestingly, when p < 2/3 and P2 is a generalist phage used alone, the maximum phage killing term in Eq. (4) against the most resistant bacteria type would be larger than when using two phages for the same value of p (see SI Sec. IV for the derivation). Therefore, it would be better to deliver P2 alone than together with another phage. Fig. S2 shows that, in the special case p = 0.5, a single P2 produces the same numerical results as a phage cocktail with p = 1, and indeed gives better outcomes than two phages with p = 0.5, as expected from the theory. This result is conditioned on the specific assumption we made in this model that phages compete to infect bacteria. Nevertheless it showcases an extreme situation where using a phage cocktail could be worse than a single phage, suggesting that it is crucial to quantify experimentally the effect of phage combinations in the system of interest. Even in this scenario, sub-optimal treatment with two phages with intermediate p still yields much better therapy outcomes in immunocompromised hosts than a treatment composed of a single or multiple phages that do not target all bacteria mutants in the pathogen population, as evident when comparing the dashed blue and yellow lines in Fig. 6. These

can be generalized to cases when the life history traits of the two phage differ from one another (see SI Sec. V). Altogether, this analysis reinforces the importance of selecting a combination of efficient phage against target bacteria when designing phage cocktails for treating infections in immunocompromised hosts. IV. DISCUSSION In this work, analyzed simple tripartite population dynamic models of single-phage and phage combination treat- ments to clear infections by multiple strains of bacteria capable of evolving phage resistance in immunomodulated hosts. In doing so we extended previous work that considered the impacts of phage therapy when bacteria were exclusively susceptible to phage [26] by considering the potential combined use of phage with complementary modes of infection. This assumption is supported by new, fluctuation test experiments in which P. aeruginosa was unlikely to randomly acquire double resistance mutations to phage LUZ19v and PAK P1. Beginning with monophage treat- ment and extending this to phage cocktails, we find that immunophage synergy underlies the curative treatment of bacterial infections given sufficiently efficient phage. Notably, the use of phage cocktails can extend the range of immunocompromised conditions in which phage therapy can clear a pathogen. Finally, we extended core theoretical findings to a realistic in vivo modeling contexts, showing the robustness of immunophage synergy given variation in immune state, phage adsorption rates, and asymmetry in phage effectiveness within cocktails. Our analytical results quantify the importance of a prompt infection treatment and of selecting phages that are highly effective against the target pathogens including potential resistant mutants [65], especially when dealing with immunocompromised hosts. We extrapolate therapy outcome predictions around parameters inferred in in vivo lung infections by P. aeruginosa in immunomodulated mice [16], showing that moderate adsorption rate variations in any employed phage can have drastic effects on therapy outcomes, potentially making the difference between a successful and a failing treatment in experimental applications. Our theoretical exploration of immunophage synergy builds on a set of assumptions that come with caveats to be addressed in future work - broadly speaking we categorize this in terms of simplifications in our representation of the immune system, phage infection and administration dynamics, and the evolutionary relationship between phage and bacteria. First of all, we consider a relatively simple impact of the host immune system on pathogens, through quantitative features that have been proposed in past infection models [16, 53, 59]. The first part of this study, focusing on non-responsive immunity is consistent with a signaling deficient immune system as in the case of myeloid differentiation primary response gene 88-deficient mice (MyD88 −/−) [16]. The predictions in this limit may also be tested in ex vivo experiments mimicking infections in the lungs [66] by inoculating therapeutic phage and .CC-BY-NC-ND 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted September 7, 2024. ; https://doi.org/10.1101/2024.09.07.611814doi: bioRxiv preprint 14 a fixed amount of immune cells. The innate immune responses considered in the second part of our work focuses on neutrophils, the first immune barrier against invading pathogens [67], while neglecting the other components of the innate immune response and the adaptive immune response altogether. Notably, inhibition of phage via immune responses [46] and by the reduction of circulating infectious phage by macrophages [25]. In the future it will be important to increase our quantitative understanding of the impact of the complex immune dynamics arising during infections. The overall modeling structure used here adopts an implicit view of complex spatial processes taking place during phage treatment of respiratory lung infections [68]. Although it is possible to include effective, nonlinear interaction terms to mimic spatial complexity [16], moving forward it will be paramount to evaluate the explicit impact of spatial structure on quantitative phage-pathogens dynamics, for instance leveraging ex vivo technologies [69, 70], so that future models can incorporate spatial components [71]. In this study we also do not address the impact of treatment timing on therapy outcomes. A theoretical work applying control theory on a phage cocktail model suggested that additional treatment improvements may be possible by optimizing the timing and distribution of phage titers [72]. Such optimum would depend on the pathogen population composition and likelihood of resistance mutations, as well as on the quantitative features of phage-bacteria interactions during an infection, such as the functional shape of phage adsorption profiles. Simultaneous administration has been shown to outperform sequential treatments as controlling the pathogen population size right away reduces the chances of multi-resistance [40, 73], even though the generality of these results is unclear. In the future it will be important to integrate empirically constrained population dynamics models of local infections with pharmacokinetics parameters describing the likelihood and delay of delivering phage in the desired infected tissue [24]. Finally, analysis of phage cocktail impacts here assumes relatively simple evolutionary interactions between phage and bacteria such that bacteria cannot evolve resistance to both phages at the same time. Our choice is motivated by previous works that suggested combinations of phages that target different bacteria receptors in order to improve treatment efficacy [41, 42, 54]. This hypothesis is supported by fluctuation test findings in which P. aeruginosa did not randomly develop simultaneous resistance to LUZ19v and PAK P1 (see SI Section III). It is important to note that although a wild type population susceptible to both phages need not necessarily evolve a double resistant mutant immediately, this does not preclude the potential for a population already selected to persist given exposure to one of the two phages may evolve to become double resistant. The current study shows that phage cocktails can perform robustly even when modulating rules regarding phage complementarity. In the future it will be crucial to further explore more complex eco-evolutionary processes that can arise between pathogenic bacteria and therapeutic phage during the course of an infection, whether in an acute or chronic infection context. Despite these caveats, the combined use of experiments, simulations and theory provide guidance on the expected range of phage therapeutic efficacy whether using monophage or phage cocktail treatments. Building upon earlier findings [16], our framework provides testable predictions on the quantitative impact of different modes of tripartite phage-bacteria-immune interactions on therapeutic outcomes over a range of host immune conditions and phage life history traits, highlighting the success of single-phage therapy in synergy with a strong enough immune system and the benefit of phage combination therapy in immunocompromised hosts. Importantly, the parameters inferred in [16] fall right at the boundary between treatment failure and success in immunodeficient hosts, which could result in drastic differences in the infection outcomes given small variations in phage and immune features. Such sensitivity makes it crucial to inform model development with in vitro and in vivo data to improve therapeutic design. More broadly, this work represents a further step towards quantitatively addressing the impact of evolutionary considerations on phage therapy outcomes. Here we showcased a specific example of how we can harness the evo- lutionary potential of phages to develop phage cocktails that target specific strains of multi-drug resistant bacteria, facing the challenge posed by the evolution of phage resistance [42, 43]. Whether we seek to exploit in vitro phage training [30, 43] or evolutionary trade-offs [31–36], in the future it will be essential to integrate further experimen- tal evidence into quantitative models, tackling different aspects of phage-bacteria evolutionary interactions during therapy. 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