Dwayne R Roach

Chung Yin Leung

Marine Henry

Eric Morello

Devika Singh

James P Di Santo

Joshua S Weitz

Laurent Debarbieux
email: laurent.debarbieux@pasteur.fr

Synergy between the host immune system and bacteriophage is essential for successful phage therapy against an acute respiratory pathogen Authors

The rise of multi-drug resistant (MDR) bacteria has spurred renewed interest in the use of bacteriophages as therapy. However, mechanisms underlying contributing to phage-medicated bacterial clearance in an animal host remain unclear. We investigated the effects of host immunity on the efficacy of phage therapy for acute pneumonia caused by MDR Pseudomonas aeruginosa in a mouse model. Comparing efficacies of phage curative and prophylactic treatments in healthy immunocompetent, MyD88 deficient, lymphocyte deficient, and neutrophil depleted murine hosts, revealed that neutrophil-phage synergy is essential for the resolution of pneumonia. Population modeling of in vivo results further showed that neutrophils are required to control both phage-sensitive and emergent phage-resistant variants to clear infection. This 'immunophage synergy' contrasts with the paradigm that phage therapy success is largely due to bacterial permissiveness to phage killing. Lastly, therapeutic phages were not cleared by pulmonary immune effector cells, and were immunologically well tolerated by lung tissues.

Introduction

The global spread of antibiotic resistant bacteria threatens all aspects of modern medicine, including advances in treatment of infectious disease, surgery, transplantation, and chemotherapy [START_REF] Brogan | A critical analysis of the review on antimicrobial resistance report and the infectious disease financing facility[END_REF]). An acutely worrying trend is the spread of resistance to carbapenemsthe 'antibiotics of last resort'predominantly among Gram-negative bacteria including Pseudomonas aeruginosa, Klebsiella pneumoniae, and Escherichia coli [START_REF] Brogan | A critical analysis of the review on antimicrobial resistance report and the infectious disease financing facility[END_REF]CDC, 2013;[START_REF] Murray | Intrinsic Antimicrobial Resistance Determinants in the Superbug Pseudomonas aeruginosa[END_REF]. Carbapenem resistance is almost always associated with resistance to several other classes of antibiotics leading to the rise of multi-drug resistant (MDR) bacteria [START_REF] Brogan | A critical analysis of the review on antimicrobial resistance report and the infectious disease financing facility[END_REF][START_REF] Murray | Intrinsic Antimicrobial Resistance Determinants in the Superbug Pseudomonas aeruginosa[END_REF]. Of the MDR bacteria, P. aeruginosa is a nosocomial opportunistic pathogen critical to the differential diagnosis of many gram-negative infections, including pneumonia, bacteraemia, and urinary tract infections (CDC, 2013;[START_REF] Murray | Intrinsic Antimicrobial Resistance Determinants in the Superbug Pseudomonas aeruginosa[END_REF]. Individuals with chronic pulmonary disorders, such as cystic fibrosis (CF), are also at considerable risk of P. aeruginosa respiratory infections [START_REF] Salsgiver | Changing Epidemiology of the Respiratory Bacteriology of Patients With Cystic Fibrosis[END_REF]. With few, if any, replacements for β-lactam antibiotics in development, especially the carbapenems, physicians and scientists have renewed efforts to identify alternative antibacterial therapies [START_REF] Czaplewski | Alternatives to antibiotics-a pipeline portfolio review[END_REF].

Phage therapy is an antibacterial approach that involves introducing bacterial viruses (phages) that infect and lyse bacteria to cure or prevent infectious disease [START_REF] Knoll | Antibacterial bioagents based on principles of bacteriophage biology: an overview[END_REF][START_REF] Roach | Phage therapy: awakening a sleeping giant[END_REF][START_REF] Salmond | A century of the phage: past, present and future[END_REF]. In a growing era of precision medicine, phage therapy has the distinct advantage over broad-spectrum antibiotics of being highly specific towards target bacterial pathogens without adversely affecting the host or host commensal microbiota (Sarker et al., 2016a). Pre-clinical animal studies have also shown success rates of phage therapy up to 100% for curing infections caused by MDR pathogens, including P. aeruginosa [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF][START_REF] Pabary | Antipseudomonal bacteriophage reduces infective burden and inflammatory response in murine lung[END_REF]. Furthermore, a limited number of clinical trials evaluating phage therapy in humans have shown promising results [START_REF] Abedon | Phage treatment of human infections[END_REF][START_REF] Vandenheuvel | Bacteriophage Therapy: Advances in Formulation Strategies and Human Clinical Trials[END_REF]. However, other clinical trials have produced conflicting results, e.g., demonstrating safety of phage but failing to provide evidence for improved clinical outcomes [START_REF] Rhoads | Bacteriophage therapy of venous leg ulcers in humans: results of a phase I safety trial[END_REF]Sarker et al., 2016b). These data suggest the need to investigate the basis for therapeutic effectiveness, e.g., when and how is phage therapy effective, who should receive phage therapy, and how will phages be immunologically tolerated?

A mechanistic understanding of phage therapy and the inherent tension between clinical aims and the biological processes underlying phage-bacteria dynamics has yet to be resolved. Phages are obligate intracellular parasites and the frequency of phage infection decreases with decreasing target host abundance [START_REF] Adams | Bacteriophages[END_REF]. Consequently, therapeutic phage populations may reach a dynamic equilibrium with a target bacterial host population rather than eradicating it [START_REF] Campbell | Conditions for the Existence of Bacteriophage[END_REF][START_REF] Levin | Resource-Limited Growth, Competition, and Predation: A Model and Experimental Studies with Bacteria and Bacteriophage[END_REF][START_REF] Weitz | Quantitative viral ecology: Dynamics of viruses and their microbial hosts[END_REF]. In addition, in vitro studies have shown that exposure to high densities of phage often selects for phage-resistant bacteria (Luria and Delbrück, 1943), which can lead to the elimination of administered therapeutic phages [START_REF] Dennehy | What Can Phages Tell Us about Host-Pathogen Coevolution?[END_REF][START_REF] Meyer | Repeatability and contingency in the evolution of a key innovation in phage lambda[END_REF]. The emergence of phage resistance represents a major conceptual and practical problem in moving from in vitro studies to the development of clinical phage therapy.

In a therapeutic setting, phage-bacteria dynamics unfold in tandem with pressures from the mammalian host immunity. Hence, host immune status will both influence susceptibility to pathogen infection, as well as the effectiveness of antimicrobial therapeutic agents, such as phages. Mathematical models have previously explored the potential consequences of interactions among bacterial infection, phage killing, and host immunity [START_REF] Levin | Phage Therapy Revisited: The Population Biology of a Bacterial Infection and Its Treatment with Bacteriophage and Antibiotics[END_REF][START_REF] Levin | Population and evolutionary dynamics of phage therapy[END_REF]. However, these mathematical models do not account for the selflimitation of healthy host immune responses that prevents excessive tissue damage. In addition, the models assume that host immune responses are unaffected by pathogen densities, whereas in vivo immune defenses have been shown to become saturated at high pathogen densities [START_REF] Drusano | Saturability of granulocyte kill of Pseudomonas aeruginosa in a murine model of pneumonia[END_REF]. Together, such assumptions drive model predictions that hosts can eliminate infections without therapeutic intervention.

Building on the modeling work of [START_REF] Levin | Population and evolutionary dynamics of phage therapy[END_REF], we recently proposed the incorporation of a maximum capacity and density-dependent saturation of host immune defenses [START_REF] Leung | Synergistic elimination of bacteria by phage and the immune system[END_REF]. With these additions, our proof-of-principle (PoP) in silico population model predicts that host immune defenses alone cannot eradicate infection at high bacterial burden. Furthermore, combining mathematical analysis and numerical simulation, our PoP model predicts that the added antibacterial activity of phages combined with the host's own immune defenses leads to asymptotic eradication of bacterial infections [START_REF] Leung | Synergistic elimination of bacteria by phage and the immune system[END_REF].

Although, our in silico model predicted that phage therapy effectiveness is not inevitable and dependent on active host immune defenses, it did not consider the proliferation of phage resistance.

P. aeruginosa is particularly well suited to investigate the interactions between phage, a bacterial pathogen, and the immune response in an animal model of phage therapy. This bacterium predominately causes pneumonia in patients with weakened or deficient immune defenses, which may have drastic effects to the efficacy of respiratory phage therapy. To investigate the effect of different host immunological states on the efficacy of phage therapy, we combine in vivo experiments and in silico modeling of phage-treated acute pneumonia caused by MDR P.

aeruginosa. Together, we demonstrate that acute respiratory infections can be cured and prevented with phage therapy, provided there is a synergy with host innate immunity, in particular with neutrophils. Phage therapy was also immunologically well tolerated by the hosts.

Results

In vivo phage therapy of acute respiratory infection

From previous work, the pseudomonas phage PAK_P1 has been characterized as being a double stranded DNA virus belonging to the Myoviridae family with a strictly virulent (lytic) life-cycle [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF][START_REF] Henry | The search for therapeutic bacteriophages uncovers one new subfamily and two new genera of Pseudomonas-infecting Myoviridae[END_REF]. Phage PAK_P1 has been shown in vitro to be well suited for lysing P. aeruginosa strains isolated from acute exacerbations rather than those strains found to chronically infect cystic fibrosis lungs. In addition, we have demonstrated using in vivo imaging that phage PAK_P1 can be an efficient cure for acute pneumonia caused by the MDR P. aeruginosa in mice [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF].

Here, we examine the efficacy of curative treatment of acute pneumonia in mice with innate and/or adaptive immunity deficiencies by quantifying the light emitted by the bioluminescent P. aeruginosa strain PAK in live mice (Fig. 1A). First, we demonstrated that a single inhaled phage PAK_P1 dose at a multiplicity of infection (MOI) of 10 (i.e. 10 8 plaque forming units [PFU]) delivered after a 2h delay rescued 100% of P. aeruginosa lung infected immunocompetent wild-type mice (Fig 1B). By quantifying light emitted from the chest regions (Fig. 1C and S1), we could monitor the proliferation and decline of bacterial densities as a function of bioluminescence in each lung of live animals in real-time (Fig. S1) [START_REF] Andreu | Noninvasive biophotonic imaging for studies of infectious disease[END_REF][START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF]. Imaging showed that bacterial bioluminescence began to decrease in as little as 2h post treatment in immunocompetent animals and was no longer detectable after 48h.

In contrast, bioluminescence increased unimpeded post infection to an bacterial density of ~2.4 × 10 9 CFU/g of lung tissue (in vivo radiance of ~4.8 × 10 4 p/s/cm 2 /sr) at 24h post infection in PBS mock-treated control animals (Fig. 1C) [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF]. Shortly thereafter mocktreated controls died (Fig. 1B). These experimental results provided as a baseline prognosis for comparison of in vivo and in silico modeling the effects of immunodeficiency on phage therapy efficacy.

In silico modeling of respiratory phage therapy

We adapt our PoP population model (Fig 2A) developed previously [START_REF] Leung | Synergistic elimination of bacteria by phage and the immune system[END_REF] to the phage treatment and infection challenge doses used in our in vivo acute pneumonia experiments (Table S1 and Eq. 1-3). Contrary to expectations, the initial in silico simulations of the PoP model predicted a highly-accelerated clearance of infection (<1h) (Fig. S2A), which is independent of host immunity and happens even in its absence (Fig. S2B). This rapid clearance timescale was inconsistent with our in vivo observations (Fig. 1C). We postulate that the inconsistency is due to overestimating phage lysis rates as a result of the assumption that phage and bacteria populations were 'well-mixed' in vivo. In a well-mixed environment, the phage lysis rate is expected to be proportional to phage population densities (Fig. S2C). Instead, we hypothesize that the rate of bacteria lysis in vivo increases sublinearly with increasing phage densities.

We identified two potential mechanisms in vivo that could be responsible for a sublinear phage lysis rate: spatial heterogeneity and phage saturation. Anatomical barriers in the mouse respiratory track could reduce encounters between phage and bacteria populations [START_REF] Florens | An anatomical and functional model of the human tracheobronchial tree[END_REF][START_REF] Weibel | Morphometry of the human lung[END_REF]. We account for such spatial heterogeneity by introducing power-law density dependence to the phage lysis rate (Fig. S2D and Eq. 4), following the convention of spatial epidemiological models [START_REF] Roy | On representing network heterogeneities in the incidence rate of simple epidemic models[END_REF]. We termed this the Heterogeneous Mixing (HM) model. Simulations of the HM model predict that the combined host immunity and phage lysis lead to bacterial population elimination over a longer course of 24h (Fig. 2B), which is consistent with our in vivo results (Fig. 1C). Without phage treatment, bacterial growth would rapidly saturate the host immune response and reach fatal levels within 48h (Fig. S3). This was also consistent with our mock-treated in vivo measurements (Fig. 1C). Another potential cause of sublinear phage lysis rates is phage saturation. This occurs when multiple phages adsorb to the same target bacterial cell when at a high phage population density [START_REF] Abedon | Kinetics of phage-mediated biocontrol of bacteria[END_REF][START_REF] Brown | Attenuation of virus production at high multiplicities of infection in Aureococcus anophagefferens[END_REF]. Therefore, we developed a second population model, termed the Phage Saturation (PS) model, to alternatively account for phage saturation (Fig S2H and Eq. 5). Simulations of the PS model (Fig. 2C) were also consistent with in vivo bioluminescence outcomes (Fig. 1C), as well as yielded qualitatively equivalent results as simulations with the HM model (Fig. 2B).

The synergistic elimination of bacterial pathogens over relevant time-scales was a robust feature of both in silico models. To probe robustness, we systematically varied the nonlinear phage adsorption rate and power-law exponent for the HM model (Figs. S2E andS2F), while for the PS model, we systematically varied the phage adsorption rate and phage density at half saturation (Figs. S2I and S2J). Both models yield a large region of synergistic clearance of bacteria, in which phage and host immunity can eradicate the bacterial population only when combined (Figs. S2G and S2K). Despite this qualitative agreement, there were some differences in the in silico predictions of these two models. For example, the PS model predicts a relatively lower maximum and faster elimination of bacterial burden in the lungs of immunocompetent hosts, when compared to the HM model. In addition, the HM model exhibited greater robustness of the synergistic clearance regime to changes in variation in lysis parameters over the range of parameter scenarios we investigated than did the PS model.

In silico modeling of phage resistance

Leveraging our findings of synergistic clearance in both the HM and PS models given a phagesusceptible population of pathogenic bacteria, we then investigated if the emergence of phageresistant bacteria can lead to monophage (use of a single phage type) therapy failure (Fig. 2D).

Both the HM and PS models were extended to include the outgrowth of phage-resistant bacterial mutants with a fixed probability per cellular division. We also applied a modest 10% trade-off between phage resistance and cellular growth rate of the mutants [START_REF] Bohannan | Trade-offs and coexistence in microbial microcosms[END_REF] (Table S1 and Fig. S2D). These extended models were termed the HM-R and PS-R, respectively (Eq. 6-9). Both the HM-R (Fig. 2E) and PS-R (Fig. 2F) models predicted similar curative outcomes in immunocompetent hosts despite the transient presence of phage-resistant bacteria. That is, phage lysis in both HM-R and PS-R models can initially decrease the phage-sensitive bacterial population giving way to phage-resistant mutant outgrowth. However, the phage-resistant bacteria are eliminated by host innate immunity. This suggests that asymptotic clearance of infection requires the synergy of both phage lysis and innate immunity. We utilized both models to further investigate the effects of immunodeficiency on phage therapy, given their compatibility with initial in vivo experiments.

Immune signaling deficiency allows for phage resistance outgrowth

Invading bacteria encounter the hosts' first line innate immune defenses, e.g., neutrophils and innate lymphocytes that sense microbes via pathogen recognition receptors (PRRs), such as the Toll-like receptors (TLRs). When activated by ligand binding, TLRs induce several effector functions to contain and eradicate infectious threats. To provide insight into which immune component was responsible for immunity-phage synergism, we first evaluated phage therapy efficacy in hosts with Myeloid Differentiation factor 88 (MyD88) deficiency. MyD88 is a central signaling transduction protein for most ligand-activated TLRs eliciting both innate and adaptive immune responses [START_REF] Warner | MyD88: a critical adaptor protein in innate immunity signal transduction[END_REF].

MyD88 deficiency has been shown to cause impaired innate immune responses toward P. aeruginosa [START_REF] Skerrett | Cutting edge: myeloid differentiation factor 88 is essential for pulmonary host defense against Pseudomonas aeruginosa but not Staphylococcus aureus[END_REF]. Here, we confirm that MyD88 -/-mice are highly susceptible to P. aeruginosa respiratory infection compared to wild-type mice. An infectious inoculum of 10 7 colony forming units (CFU) caused highly rapid mortality in MyD88 -/-mice. Therefore, we reduced the infectious inoculum 100-fold to decrease the time to mortality (Fig. 3A). Despite a more modest clinical course, phage treatment only improved survival by 15% over PBS mocktreated controls (p= 0.0287). Notably, bioluminescence in the lungs of Myd88 -/-mice showed that phage treatment significantly reduced bacterial burden to near undetectable levels by the 24h time point post infection (p.i.) compared to the mock-treated mice (p< 0.001) (Fig. 3B).

However, treatment failed to prevent fatality due to subsequent bacterial outgrowth after 24h.

We surmise that the resurgent bacterial outgrowth was not due to a low abundance of phages in the lungs, since a reduction in bioluminescence implies sufficient phage replication occurred.

Rather, we found that 100% of P. aeruginosa clones recovered from phage-treated Myd88 -/- lungs at 24h p.i. were resistant to phage lysis with in vitro plaque analysis.

In silico modeling was used to determine if emergence of phage-resistant population in hosts that lack Myd88 immune activation would support phage treatment failure seen in vivo. Myd88 deficiency has been shown to prevent immune cell activation and recruitment to the site of infection [START_REF] Kawai | Unresponsiveness of MyD88-deficient mice to endotoxin[END_REF][START_REF] Warner | MyD88: a critical adaptor protein in innate immunity signal transduction[END_REF]. In the models, we account for Myd88 deficiency by maintaining immune response intensity at a basal level (𝐼 = 𝐼 0 ) (Eq. 10). This still allows for marginal immune defenses during infection by the present immune effector cells in the lungs [START_REF] Barletta | Leukocyte compartments in the mouse lung: distinguishing between marginated, interstitial, and alveolar cells in response to injury[END_REF]. Without Myd88 immune signaling, both the HM-R (Fig. 3C) and PS-R (Fig. 3D) models predict that phage lysis will reduce phage-sensitive bacteria, with subsequent unimpeded outgrowth of phage-resistant mutants until mortality. Importantly, the predicted phage-resistant outgrowth rate was comparable to the regained bioluminescence seen after 24h with live imaging of Myd88 -/-mice (Fig. 3B). We surmise that without immune activation, phage-resistant mutants overwhelm the basal immune defenses leading to a resurgence of the bacterial population that ultimately causes mortality.

In the clinic, patients can suffer from various degrees of weakened and deficient immunity.

Building upon our initial results, we modeled in silico the effects of intermediate levels of immune activation on phage therapy efficacy by modifying the maximum immune response. The maximum immune response is scaled such that at 0% immune activation, the immune response stays at the basal level, while at 100% immune activation the immune response can reach the maximum level in healthy individuals (Eq. 10). These simulations predicted that phage therapy will be effective and prevent outgrowth of emergent phage-resistant mutants provided immune activation exceeds 50% of that of a healthy immune response (Figs. 3E andS4A). However, our in silico models also predict that even without phage resistance, successful phage therapy still requires at least 20% immune responses to completely eradicate the phage-sensitive bacterial population in the lungs (Figs. S4B andS4C).

Null effect of lymphocyte deficiency for phage therapy

As shown in figure 3, MyD88 signaling deficiency strongly reduced the efficacy of monophage treatment of P. aeruginosa respiratory infection. Next, we sought to explore which innate effector cell type(s) contribute to successful respiratory phage therapy in immunocompetent mice. MyD88 dependent TLR-activation triggers several immune pathways that together orchestrate innate immune responses while also initiating adaptive immunity [START_REF] Warner | MyD88: a critical adaptor protein in innate immunity signal transduction[END_REF]. Innate immune effector cells can be divided into numerous subsets, but are broadly categorized as being either from lymphoid or myeloid lineages. Innate lymphoid cells (ILCs)including natural killer cells (NK cells) and recently discovered ILC1, ILC2 and ILC3 are enriched at mucosal surfaces and exert crucial defensive functions [START_REF] Eberl | Innate lymphoid cells. Innate lymphoid cells: a new paradigm in immunology[END_REF].

Rag2 -/-Il2rg -/-mice carry a double germline knockout in recombination activating gene 2 and common gamma chain that prevent the production of ILCs, and antigen-specific T and B cells [START_REF] Colucci | Dissecting NK cell development using a novel alymphoid mouse model: investigating the role of the c-abl proto-oncogene in murine NK cell differentiation[END_REF]. Here, we demonstrate that the same phage treatment dose (10 8 PFU) used to cure acute pneumonia in immunocompetent mice, also rescued >90% of 10 7 CFU infected Rag2 -/-Il2rg -/-mice (Fig. 4A). Likewise, in vivo imaging revealed a qualitatively similar bioluminescence profile (Figs. 4B andS1) as phage-treated wild-type mice (Figs. 1C andS1). In Rag -/-Il2rg -/-mice, the lack of both innate and adaptive lymphoid cells does not seem to be a confounding factor for phage therapy of acute respiratory infection.

Neutrophil-phage synergy is required for effective phage therapy

Myeloid cells, including neutrophils, are major players for the innate inflammatory response to invading pathogens [START_REF] Kolaczkowska | Neutrophil recruitment and function in health and inflammation[END_REF]. We found that P. aeruginosa lung infected wild-type mice depleted of their neutrophils were completely unresponsive to inhaled phage treatment (Fig. 5A), even after reducing the bacterial challenge by 100-fold (10 5 CFU) of that given to immunocompetent wild-type animals (Fig 1). Compared to Myd88 immune signaling deficient animals (Fig. 3B), phage treatment in neutrophil depleted hosts caused no significant reduction in bioluminescence after phage treatment (Fig. 5B).

Next, we probed the effect of neutrophil depletion on phage therapy using in silico modeling by setting host innate immunity to zero (Eq. 11). The HM-R math model predicts similar dynamical outcomes between severe neutropenia (Fig 5C ) and that predicted for Myd88 deficient hosts (Fig. 3C). That is, the HM-R model predicts a significant reduction in bacterial density before phage-resistant outgrowth after 24h. However, this reduction was not seen in vivo (Fig. 5B). On the other hand, PS-R simulations of phage-treated neutrophil depleted hosts did not predict a significant reduction in the phage-sensitive bacteria, but rather a decline in their population growth rate before being replaced by phage-resistant outgrowth (Fig. 5D). This was more in line with in vivo observations, although the decline was not significantly different from mock-treated animals (Fig. 5B). The PS-R model also predicts that host neutrophils are needed to synergistically eliminate the bacteria during phage therapy. Overall, in vivo experiments and both models suggest that phage therapy of acute respiratory disease may be inadequate for patients with severe neutropenia.

Neutropenia encompasses a wide range of diagnoses from normal variants to life-threatening acquired (e.g. chemotherapy) and congenital disorders. It is generally defined as having blood neutrophil levels less than 10 9 /L, but the functional consequences is largely dependent on the severity of neutrophil depletion [START_REF] Newburger | Evaluation and management of patients with isolated neutropenia[END_REF]. For instance, hosts with an estimated 12-23% absolute neutrophil count (ANC) of healthy levels, may only have a slightly increased risk of infection. However, ANCs below 5% (referred to as "agranulocytosis") carries a high risk of severe infection [START_REF] Newburger | Evaluation and management of patients with isolated neutropenia[END_REF]. We in silico modeled the effect of different neutrophil counts in the lungs on the efficacy of phage therapy (Eq. 11). Both the PS-R (Fig. 5E) and HM-R (Fig. S4D) models predict that successful phage therapy could be achieved with emergent phage resistance, provided at least 50% of healthy lung neutrophil counts were present. Moreover, if phage resistance could be prevented entirely, effective phage therapy would require a lower lung neutrophil counts of at least 20% and 10% of healthy patients as predicted by the PS-R (Fig. S4E) and HM-R (Fig. S4F) models, respectively.

Phage prophylaxis of acute respiratory infection

Antimicrobial prophylaxis is commonly used for the prevention of numerous infectious diseases, including P. aeruginosa respiratory infection [START_REF] Stanojevic | Effectiveness of inhaled tobramycin in eradicating Pseudomonas aeruginosa in children with cystic fibrosis[END_REF]. We first determined that phage PAK_P1 was gradually eliminated from the lungs of mice with an exponential decay constant of approximately 0.5 log/d after a single intranasal dose of 10 9 PFU (Table S2).

Leveraging this result, we find that inhaled phages could provide prophylaxis by preventing pneumonia in 100% of challenged wild-type immunocompetent mice if given up to 4d prior to bacterial challenge (Fig. 6A). Prophylaxis could also be achieved in Rag2 -/-Il2rg -/-mice (Fig. S5).

We estimate that initial phage densities would be about 10 6 PFU at the time of infectious challenge given prophylactic dose and decay rates (Table S2). This prophylactic dose yields an effective treatment MOI of 0.1, which is 100-fold lower than our curative treatment dose (Fig. 1). Nonetheless, the relative changes within in vivo bioluminescence dissemination, proliferation, and resolution profiles were similar between phage prophylaxis and 'curative' phage therapy (compare Figs. 6B and 1C,respectfully). We utilized both HM-R (Fig. 6C) and PS-R (Fig. 6D) models to evaluate the prophylactic potential of phages, using the same parameters as with 'curative' phage therapy except adjusting the point of phage administration to -4d and using a prophylactic dose equivalent to 10 9 PFU (Table S1). Both mathematical models similarly predict effective phage prophylaxis in immunocompetent hosts. We demonstrate that the lower phage dose at the time of infectious challenge does not exert a strong effect on efficacy, likely due to the rapid replication of resident phages when bacteria are introduced. It remains unclear as to whether prophylactic application also provides greater time for penetration into the lung, potentially reducing the slowdown of phage lysis due to spatial heterogeneity.

Absence of evidence for direct activation of the immune system by phage PAK_P1

Previous studies have yet to adequately address if phages themselves promote recruitment of immune effector cells to the site of infection, or if phage prophylaxis primes immune cells to a state of enhanced defense [START_REF] El-Benna | Priming of the neutrophil respiratory burst: role in host defense and inflammation[END_REF]. As cytokines control the dynamics of immune cell behavior and initiate inflammation [START_REF] Eberl | Innate lymphoid cells. Innate lymphoid cells: a new paradigm in immunology[END_REF], we sought to examine whether phage PAK_P1 at the high dose of 10 9 PFU could stimulate the production of a panel of cytokines in immunocompetent mouse lung tissues. We found that a high phage dose did not significantly increase the production of cytokines compared to the PBS control over 4d (Figs. 6E andS6).

More importantly, phage-exposed lungs harbored significantly lower cytokine levels than lungs assaulted with lipopolysaccharide (LPS). Our data coincides with other studies [START_REF] Bocian | The Effects of T4 and A3/R Bacteriophages on Differentiation of Human Myeloid Dendritic Cells[END_REF]. This suggests that phage PAK_P1 was immunologically well tolerated, and it was unlikely that phage presence promoted further recruitment or primed immune cells during treatment. Accordingly, we did not include direct activation of host immunity by phages in our mathematical population models. Importantly, phage immunotolerance may be phage type and setting dependent.

A further concern regarding phage therapy is that the host immunity will neutralize the administered phages [START_REF] Lusiak-Szelachowska | Antiphage activity of sera during phage therapy in relation to its outcome[END_REF]. It has been suggested that innate immunity may be sufficient for rapid clearance of phages from circulation [START_REF] Merril | Long-circulating bacteriophage as antibacterial agents[END_REF].

However, we find no evidence of this, as phage PAK_P1 demonstrated a high degree of persistence in the airways of all tested uninfected mice (Table S2). However, immunodeficiency did prolong persistence of phage in the mouse lungs by day 4 (p<0.001). Some caution is warranted given that mice received only a single phage treatment dose. Repeated phage doses might promote a stronger antigen-specific neutralizing response, but previous studies are equivocal as to whether this would occur [START_REF] Bruttin | Human volunteers receiving Escherichia coli phage T4 orally: a safety test of phage therapy[END_REF][START_REF] Lusiak-Szelachowska | Phage neutralization by sera of patients receiving phage therapy[END_REF][START_REF] Majewska | Oral application of T4 phage induces weak antibody production in the gut and in the blood[END_REF]. For example, weeks of oral application of phages did result in anti-phage neutralizing antibodies in mice [START_REF] Majewska | Oral application of T4 phage induces weak antibody production in the gut and in the blood[END_REF], but long-term oral application in humans did not [START_REF] Bruttin | Human volunteers receiving Escherichia coli phage T4 orally: a safety test of phage therapy[END_REF]. Nonetheless, if and how the host immune system senses and responds to phage warrants further investigation.

Discussion

In this study, we show that successful phage curative and prophylactic therapies of acute respiratory infections depends on an essential synergy between phage lysis and the host's own immune defenses, a concept we term "immunophage synergy". In comparing phage treatment efficacies in murine hosts with distinct types of immunodeficiency, namely MyD88 deficient, lymphocyte deficient, and severe neutropenia, we have identified that phage-neutrophil synergy leads to resolution of acute infection. Utilizing in silico population modeling of phage therapy given systematic modulation of host innate immune responses, we provided further support for a mechanism by which host neutrophils are required to control both phage-sensitive and emergent phage-resistant variants to ensure infection eradication. Our in vivo results also demonstrate the efficacy of phage prophylaxis. In silico population modeling implies that both the slow elimination of phage from the lungs and the self-amplification of phage contribute to the prophylactic success. Further, we find that phages are immunologically well tolerated and were not significantly neutralized in the lungs by immune defenses. Our data supports a distinct interpretation of phage therapy than the prevailing paradigms. From a translational perspective, the current findings also suggest the need to revisit appropriate patient immunological inclusion criterion before phage therapies are employed.

Resistance development is a major hurdle with all antimicrobial therapies. Here, we modeled monophage therapy with a single virus type, namely phage PAK_P1. Our results provide a mechanistic basis to explain experimental observations that phage resistance can emerge but does not result in therapeutic failure in healthy immunocompetent hosts [START_REF] Bull | Dynamics of success and failure in phage and antibiotic therapy in experimental infections[END_REF][START_REF] Smith | Successful treatment of experimental Escherichia coli infections in mice using phage: its general superiority over antibiotics[END_REF]. Rather, treatment failure is caused when host innate immune defenses fail to eliminate phage-resistant subpopulations. We therefore speculate that resistance would not render phage therapy ineffective as long as a substantial ratio of the targeted bacterial population is sensitive to phage infection. While our study is limited in scope to monophage therapy, polyphage therapy (i.e. a mixture of different phage types) is often used to reduce the risk of emergent phage resistance [START_REF] Pabary | Antipseudomonal bacteriophage reduces infective burden and inflammatory response in murine lung[END_REF]. However, our in silico analysis also predicts that even if the bacterial population was entirely phage-sensitive (i.e. no resistant mutants), phages alone would still be unlikely to eradicate the bacterial burden in the lungs [START_REF] Levin | Resource-Limited Growth, Competition, and Predation: A Model and Experimental Studies with Bacteria and Bacteriophage[END_REF][START_REF] Weitz | Quantitative viral ecology: Dynamics of viruses and their microbial hosts[END_REF]. This suggests polyphage therapy will also need immunophage synergy to resolve acute infections. Our study also does not evaluate the feasibility and effect of combining phage with other antibacterials, such as small molecule antibiotics [START_REF] Chan | Phage selection restores antibiotic sensitivity in MDR Pseudomonas aeruginosa[END_REF]. The effects of host immunity on antibiotic efficacy will also need to be considered alongside immunophage synergy. In addition, combination therapies may be able to augment host immune responses in settings of weakened immunity to combat both phage-sensitive and phage-resistant populations.

Immunophage synergy is of relevance to ongoing efforts to understand the basic mechanisms and clinical potential of phage therapy. Our results suggest that ILCs are not key players in the immunophage synergy to resolve acute infection. At the mucosa of the lungs mouse ILCs are thought to respond to pathogens indirectly by sensing myeloid-cell-or epithelial-cell-derived cytokines and to further promoting immune defenses by further secreting immune effector mediators [START_REF] Eberl | Innate lymphoid cells. Innate lymphoid cells: a new paradigm in immunology[END_REF]. ILC absence might also cause a reduction in overall innate immune activation and recruitment similar to Myd88 deficiency, but at a less drastic level.

Modest immune response reductions due to ILC loss need not lead to therapeutic failure, given that our population models predict that 20-50% of a healthy innate immune response would be sufficient for successful phage therapy (Fig. 3E andS4). In generalizing the current framework for humans, a limitation of our study is that mouse ILCs lack PRRs [START_REF] Eberl | Innate lymphoid cells. Innate lymphoid cells: a new paradigm in immunology[END_REF]. In contrast, human ILC3s can express PRRs, like TLRs, which could receive additional signaling directly through bacterial stimulation. Together, we cannot rule out that ILCs have an additive contribution to immunophage synergy. Furthermore, adaptive humoral responses (B cells) and cellular responses (T cells) are expected to play a greater role in immunophage synergy in resolving chronic infections.

We have not determined whether the observed failure of phage therapy due to innate immunity deficiencies was due to a loss of neutrophil effector functions (e.g., phagocytosis and NETosis) or the absence of neutrophil-dependent activation of other immune effector components that also promote clearance of P. aeruginosa. As both likely play a role, future experiments studying mild neutropenia and neutrophil dysfunction could elucidate the innate immune mechanisms involved in efficient phage therapy. This may in turn lead to alteration of phage therapy formulations, or even novel phage-antibiotic combinations, to enhance treatments in hosts with weakened immunity. Besides likely extending beyond acute respiratory infections to other infectious diseases, the status of the innate immune response, including concentrations and activity of neutrophils and other effector cells, depend on disease status, environment, and genetic factors [START_REF] Heron | Bronchoalveolar lavage cell pattern from healthy human lung[END_REF].

There are multiple ways that the current immunophage synergy framework could be extended.

First, the current experimental and modeling framework could be used to test the efficacy of polyphage therapies, given trade-offs between coverage of types and relative densities in the initial therapeutic phage doses. The adaptation of the in silico population model to the in vivo context included the hypothesis that killing rate of bacteria by phage increases sublinearly with phage densities. Our prior analysis of the PoP phage therapeutic model showed that explicit modeling of the infected cell state did not change qualitative outcomes of therapy [START_REF] Leung | Synergistic elimination of bacteria by phage and the immune system[END_REF]. Future analysis of the current therapeutic model may also consider how predictions are affected by fixed delays between infection and lysis [START_REF] Levin | Resource-Limited Growth, Competition, and Predation: A Model and Experimental Studies with Bacteria and Bacteriophage[END_REF] and/or changes in cell state at high densities that suppress phage replication [START_REF] Levin | Resource-Limited Growth, Competition, and Predation: A Model and Experimental Studies with Bacteria and Bacteriophage[END_REF][START_REF] Weitz | Alternative stable states in host-phage dynamics[END_REF]. Moving forward, it will be worthwhile to develop population models and experimental methods to test the "microscale" mode of action, e.g., by resolving the joint, spatial dynamics of bacteria, phage, and neutrophils during treatment. In doing so, we anticipate that incorporation of stochastic, individual-based models may shed light on how microscale heterogeneity influences the quantitative time-course of dynamics of therapeutic intervention at population scales. Efforts to resolve spatially-explicit phage-bacteria interactions have already shed light on how viruses disrupt micro-colonies or biofilms in vitro and could be productively adapted for in vivo studies [START_REF] Azeredo | The use of phages for the removal of infectious biofilms[END_REF].

To conclude, association of in vivo and in silico modeling of human infectious disease provided a framework in studying outcomes arising from the tripartite interactions between phage, bacteria and host immunity. Although, this framework was built here to support the development of phage therapies, it will also be of value in a more general context to comprehensively decipher phage, bacteria and host immune cell population dynamics in human microbiomes (Debarbieux, The dotted compartment denotes the total bacterial population, which is the sum of the sensitive (S) and resistant (R) subpopulations. Predicted population densities with (E) HM-R or (F) PS-R models. All simulations predict phage-sensitive and phage-resistant bacterial elimination during phage therapy in hosts with a fully functioning innate immune response. See Table S1 for in silico simulation parameters.

Figures legends

METHOD DETAILS

In vivo acute pneumonia mouse model

We used the luminescent P. aeruginosa PAKlumi strain and in vivo imaging with the IVIS platform (PerkinElmer), as a noninvasive assessment of acute infection in mouse lungs [START_REF] Andreu | Noninvasive biophotonic imaging for studies of infectious disease[END_REF][START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF][START_REF] Munder | In vivo imaging of bioluminescent Pseudomonas aeruginosa in an acute murine airway infection model[END_REF]. Subject groups consisted of: (i) immunocompetent wild-type; (ii) lymphocyte deficient Rag2 -/-Il2rg -/-; (iii) pathogen associated pattern recognition receptor deficient MyD88 -/-; and (iv) neutrophil depleted wild-type mouse strains. Neutrophils were depleted 24h prior to infection by intraperitoneal injection of anti-Gr1 antibody (200 µg; BioXcell) as previously described [START_REF] Albanesi | Neutrophils mediate antibody-induced antitumor effects in mice[END_REF].

Mice were infected with intranasally instilled mid-log growing PAKlumi that was washed 3 times in sterile dH2O and suspended PBS at an infectious dose of 1.0 x 10 7 colony forming units (CFU) in 30 µL. We have previously shown that this infectious challenge and route of administration causes wild-type mouse fatality within approx. 48h [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF]. Due to the increased susceptibility to infection, Myd88 -/-and neutrophil depleted mice were challenged a lower infectious dose of 1.0 x 10 5 CFU PAKlumi. Twenty-four hours after bacterial challenge, lungs from wild-type mice were aseptically removed, homogenized in PBS and parameters corresponding to in vivo phage treatment in immunocompetent host (Fig. S2A). The rapid bacteria elimination is apparent even in the absence of innate immunity (Fig. S2B), suggesting that the linear model overestimates the rate of phage killing. We modify the functional form of 𝐹(𝑃) to account for spatial heterogeneity in the lungs (which we term the heterogeneous mixing (HM) model) and the effects of saturation of phage infection at high phage dose (which we term the phage saturation (PS) model); see text for more details.

In the HM model, spatial heterogeneity is modeled by a power-law dependence of the phage infection rate [START_REF] Bansal | When individual behaviour matters: homogeneous and network models in epidemiology[END_REF][START_REF] Liu | Dynamical behavior of epidemiological models with nonlinear incidence rates[END_REF][START_REF] Roy | On representing network heterogeneities in the incidence rate of simple epidemic models[END_REF] where 𝜙 ̃ is the nonlinear phage adsorption rate and 𝛾 is the power-law exponent. In the PS We also extended the model to include emergence of phage resistance in the bacterial population, which was experimentally observed in the Myd88 deficient mice. In this phage resistance model, the bacterial population consists of a phage-sensitive subpopulation 𝑆 and phage-resistant subpopulation 𝑅 (Fig. 2D). The phage-sensitive bacteria are infected and lysed by the phage, while the phage-resistant bacteria do not interact with the phage (Eq. 6-8). Similar to the model without resistance, infection of sensitive bacteria by phage is either subjected to heterogeneous mixing (Eq. 4) or saturation (Eq. 5), which we term as the heterogeneous mixingresistance (HM-R) model and phage saturation-resistance (PS-R) model respectively. Phage resistance is acquired by phage-sensitive bacteria through mutation with a fixed probability 𝜇 per cellular division (see Eq. 6-7) [START_REF] Luria | Mutations of bacteria from virus sensitivity to virus resistance[END_REF]. Both phage-sensitive and phageresistant bacteria are killed by the immune response (Eq. 6-7) and stimulate immune activation (Eq. 9). In general, the phage-resistant strain can have a different maximum growth rate 𝑟 ′ (Eq. 7) from the phage-sensitive strain. For example, phage resistance can come with evolutionary trade-offs as reflected by a reduction in growth rate [START_REF] Bohannan | Trade-offs and coexistence in microbial microcosms[END_REF][START_REF] Weitz | Quantitative viral ecology: Dynamics of viruses and their microbial hosts[END_REF].

Parameter estimation

See Table S1 for in silico model parameters. The life history traits of the bacteria and phage are obtained from empirical values from a P. aeruginosa murine pneumonia model (Drusano et al.,

Fig. 1 .Fig. 2 .

 12 Fig. 1. Immunophage therapy efficacy in the immunocompetent host. (A) Post treatment

Fig. 3 .Fig. 4 .

 34 Fig. 3. Phage therapy is inefficient in the innate immunity activation deficient host. (A)Myeloid differentiation primary response gene 88 deficient mice (MyD88 -/-) had a 15% survival probability of acute respiratory infection by Pseudomonas aeruginosa (10 5 CFU) when inhaled

Fig. 6 .

 6 Fig. 6. Efficient non-immune priming phage prophylaxis in the immunocompetent host. (A)Wild-type mice received a single inhaled monophage dose (10 9 PFU) which gave prophylaxis for 4d against P. aeruginosa (10 7 CFU; n=6 per group). (B) Luminescence plotted as mean radiance (p/s 2 /cm 2 /sr) from bacteria over time demonstrates phages maintain antibacterial activity after

  Figure 1

  model, saturation effects are modeled by the saturating Hill function where 𝑃 𝐶 is the phage concentration at which the phage infection rate is half-saturated and the expression reduces to the linear form when 𝑃 ≪ 𝑃 𝐶 . The phage infection rates as functions of the phage density for the well-mixed linear model (𝐹(𝑃) = 𝜙𝑃), HM model, and PS model are shown in Figs. S2C, S2D and S2H, respectively. The well-mixed linear model likely overestimates the phage infection rate, reaching about 40 h -1 at the initial phage dose for curative treatment (7.4 × 10 8 PFU). In contrast, both the HM model and PS model result in lower estimates for the phage infection rate more likely to be biologically relevant.

Acknowledgements:

We thank N. Dufour, C. Archambeau, and E. Atse (Microbiology, Institut Pasteur), H. Strick-Marchand (Immunology, Institut Pasteur), M.A Nicola (Imagopole, Institut Pasteur) and Center for Human Immunology at Institut Pasteur for their technical support. We thank S. Chollet-Martin (U Paris-Sud) for her comments about the manuscript. We thank J. Bull (UT-Austin) and another reviewer for their feedback. This study was supported by the Fondation EDF, the Association Vaincre la Mucoviscidose [IC1011], the Association CA. ZO. LA. Luttons contre la mucoviscidose, and the U.S. Army Research Office [W911NF-14-1-0402]. DRR is the recipient of a European Respiratory Society Fellowship [RESPIRE2-2015-8416].

DATA AND SOFTWARE AVAILABILITY

The source code for the mathematical models is included in the supplemental files as Data S1 and will also be deposited in Github.

Author contributions

DRR, JDS, JSW and LD conceived the experiments. DRR, EM and MH performed experiments.

CYL and DS performed mathematical modeling. CYL, DRR, JDS, JSW and LD analyzed the data. CYL, DRR, JSW and LD wrote the manuscript.

STAR Methods Text

CONTACT FOR REAGENT AND RESOURCE SHARING

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Laurent Debarbieux (laurent.debarbieux@pasteur.fr)

EXPERIMENTAL MODEL AND SUBJECT DETAILS

Microorganisms

The P. aeruginosa PAKlumi strain expressing the Photorhabdus luminescens luxCDABE operon [START_REF] Moir | A highthroughput, homogeneous, bioluminescent assay for Pseudomonas aeruginosa gyrase inhibitors and other DNA-damaging agents[END_REF] was cultured in lysogeny broth (LB) at 37 °C. Pseudomonal phage PAK_P1 [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF][START_REF] Henry | Predicting in vivo efficacy of therapeutic bacteriophages used to treat pulmonary infections[END_REF] was amplified in mid-log growing PAKlumi overnight after inoculation at the multiplicity of infection (MOI) 0.001. Cell lysate supernatants containing amplified phages were 0.22 µm filter sterilized and concentrated by ultrafiltration (Vivaflow 200,Sartorius) in phosphate buffer saline (PBS). Phage PAK_P1 was purified by two cesium chloride density gradient centrifugations [START_REF] Debarbieux | Bacteriophages can treat and prevent Pseudomonas aeruginosa lung infections[END_REF], dialyzed in Tris buffer (10 mM Tris, 150 mM NaCl, pH 7.5) and endotoxin removed by 3 passages through an EndoTrap HD column (Hyglos, Germany). Phage titer were determined from stocks or lung tissue homogenates by spotting serial dilutions on seeded lawns of mid-log growing PAKlumi at OD600 0.2. A single phage PAK_P1 stock stored at 4°C was used for all experiments.

Animals and Ethics

Wild-type BALB/c (C), wild-type C57Bl6/J (B6), recombination-activating gene 2 and interleukin 2-receptor γchain double-deficient (Rag2 -/-Il2rg -/-) [START_REF] Mazurier | A novel immunodeficient mouse model--RAG2 x common cytokine receptor gamma chain double mutants--requiring exogenous cytokine administration for human hematopoietic stem cell engraftment[END_REF], and myeloid differentiation primary response gene 88 (Myd88 -/-) [START_REF] Kawai | Unresponsiveness of MyD88-deficient mice to endotoxin[END_REF] male and further diluted to obtain calibration data on CFU/photons emitted. Phage resistance was determined by spotting serial dilutions of fresh phage PAK_P1 on seeded lawns of 10 isolated CFUs from treated lung tissue homogenates per MyD88 -/-mouse.

Curative and prophylactic phage therapies

In wild-type mice, treatment with phage PAK_P1 was administered intranasally at a curative dose of 1.0 x 10 8 plaque forming units (PFU; MOI 10) in 30 µL PBS at 2h p.i., or at a prophylactic dose of 1.0 x 10 9 PFU in 30 µL PBS 4d prior bacterial challenge. In immunodeficient mice, treatment with phage PAK_P1 was administered intranasally at a curative dose of 1.0 x 10 7 (PFU; MOI 100) in 30 µL PBS at 2h p.i. PBS was used as the mocktreatment control.

Lung tissue cytokines

Soluble cytokines and chemokines were measured in lung tissue of wild-type mice that received only intranasally instilled phages at the 10 9 PFU in 30 µL PBS prophylactic dose. Cytokine differential expressions were compared to PBS or 10 µg LPS from P. aeruginosa (Sigma) controls also administered intranasally. Mouse lungs were harvested, weighed and homogenized in PBS supplemented with protease inhibitor (cOmplete Ultra Tablets, Roche) with the gentleMACS Octo Dissociator (Miltenyi Biotec) and centrifuged at 500 rpm to remove large debris. Cytokines were detected and quantified in lung tissue homogenates using a multiplex cytokine and chemokine magnetic microsphere-based immunoassay (LMC0006, Life Technologies) on the Luminex platform and xPONENT software as per the manufacture's recommendations.

Mathematical model formulation

The schematic of the model (Fig. 2A) represents a system of nonlinear, coupled differential equations (Eq. 1-3) that governs the time evolution of the bacterial density 𝐵, phage density 𝑃 and immune response intensity .

The parameters 𝑟 and 𝐾 𝐶 are the maximum growth rate and carrying capacity of the bacteria respectively. The function 𝐹(𝑃) is the per-capita rate of infection. If the environment is wellmixed, then the functional form 𝐹(𝑃) = 𝜙𝑃 is linear where 𝜙 is the adsorption rate [START_REF] Leung | Synergistic elimination of bacteria by phage and the immune system[END_REF]. The maximum per-capita killing rate of the immune response is given by 𝜀. When the bacteria is at high concentration, the immune response is less effective due to the immune evasion, with 𝐾 𝐷 being the bacterial density at which the immune response is half as effective.

The phage replicate with an effective burst size of 𝛽 and decay at a rate of 𝜔. The immune response is stimulated by the presence of bacteria with a maximum activation rate 𝛼 until it reaches the maximum capacity 𝐾 𝐼 . 𝐾 𝑁 is the bacterial density at which the immune intensity growth rate is half its maximum.

Preliminary in silico analysis of this model found that if the environment is well-mixed, then inclusion of the linear form 𝐹(𝑃) = 𝜙𝑃 leads to extremely rapid elimination of bacteria using 2011) and P. aeruginosa strain PAKlumi and phage PAK_P1 used in this study. In wild-type mice, we measured the bacterial density 24h post infection (2.4 × 10 9 CFU/g, see results section for details) and used it to estimate the carrying capacity 𝐾 𝐶 in hosts with healthy immune responses. The decay rate 𝜔 of phage is estimated from the fitted phage half-life. The immune response intensity is taken to be the neutrophil density in the lungs, given the present identification of neutrophils as critical for therapeutic efficacy. Therefore the killing rate parameter of the immune response 𝜀 is estimated from the maximum killing rate of neutrophils [START_REF] Li | A critical concentration of neutrophils is required for effective bacterial killing in suspension[END_REF].

For the immunocompetent mice, the maximum activation rate of immune response 𝛼, initial immune response 𝐼 0 and maximum immune response 𝐾 𝐼 are estimated from fitting of neutrophil recruitment data in the lungs [START_REF] Reutershan | Sequential recruitment of neutrophils into lung and bronchoalveolar lavage fluid in LPS-induced acute lung injury[END_REF]. For mice that are Myd88 deficient, their innate immune activation is disrupted and we model that by reducing the maximum (activated) immune response 𝐾 𝐼 to the level of the initial immune response 𝐼 0 . We further investigate the effects of restoring intermediate levels of immune activation by setting where 𝑠 is a scaling parameter from 0% (no immune activation) to 100% (normal immune activation).

For mice with neutrophils depleted by anti-Gr-1 antibodies, we assume negligible primary innate immunity and set 𝐼 = 0. We further model intermediate levels of neutrophil depletion by setting the immune intensity as a fixed percentage of the normal immune intensity, i.e.

where 𝑠 ′ is a parameter that goes from 0% (complete lack of innate immunity) to 100% (normal innate immunity). The bacteria concentration 𝐾 𝐷 at which immune response is half as effective is assumed to be close to the bacterial inoculum that leads to a lethal infection (5.5 × 10 6 CFU/lung). The bacterial density 𝐾 𝑁 at which immune growth rate is half its maximum is obtained from in vitro data of the response of a pathogen recognition receptor (i.e. Toll-like receptor 5) to the P. aeruginosa strain PAK [START_REF] Zhang | Human airway epithelial cells sense Pseudomonas aeruginosa infection via recognition of flagellin by Toll-like receptor 5[END_REF]. Initial bacterial load and phage concentrations are set by the corresponding doses used in the experiments.

We select the parameters of the nonlinear phage infection rates in the HM and PS model by first performing a sensitivity analysis. As a reference point, the linear adsorption rate for the phage PAK_P1 is about 5.4 × 10 -8 g/h PFU measured at an MOI of 0.001. For the HM model, we expect the nonlinear adsorption rate 𝜙 ̃ would be close to this value when 𝛾 → 1. We vary both 𝜙 ̃ and the power law exponent 𝛾 and measure the minimum bacterial density reached in a simulation within 72h post infection in the absence and presence of innate immune response (Figs. S2E and 2F,respectively). When 𝜙 ̃ and 𝛾 are increased, the minimum bacterial density reduces until it reaches the point of extinction, i.e., when bacterial density is below 1 CFU/g. The parameters 𝜙 ̃ and 𝛾 are chosen to be within the phage-immune synergistic regime, where the bacterial population only becomes extinct in the presence of immune response (Fig. S2G).

Outside the phage-immune synergistic regime, the phage either eliminates the bacteria on its own (phage sufficient regime), or fail to do so even when combined with immune response (phageimmune insufficient regime). A similar analysis is conducted for the PS model by varying the adsorption rate 𝜙 and half-saturation phage concentration 𝑃 𝐶 in the absence and presence of innate immune response (Figs. S2I and S2J,respectively). The values of 𝜙 and 𝑃 𝐶 are chosen accordingly to be within the phage-immune synergistic regime (Fig S2K ).

For the phage resistance model, the probability 𝜇 of producing a phage-resistant mutant per cellular division is estimated from in vitro experimental measurements [START_REF] Luria | Mutations of bacteria from virus sensitivity to virus resistance[END_REF]. We assume a moderate trade-off between phage resistance and growth rate [START_REF] Bohannan | Trade-offs and coexistence in microbial microcosms[END_REF][START_REF] Weitz | Quantitative viral ecology: Dynamics of viruses and their microbial hosts[END_REF] with the maximum growth rate of the resistant strain 10% lower than its phage-sensitive counterpart.

Simulation of mathematical models

The in silico model equations in Eq. 1-3 and Eq. 6-9 are numerically integrated using ODE45 in MATLAB to obtain the bacteria, phage and immune response time series. Bacteria and phage are assumed to become extinct in the mice lung when their population densities drop below a threshold of 1 g -1 .

QUANTIFICATION AND STATISTICAL ANALYSIS

Statistical analysis

Statistical analyses were conducted with Prism 5 software (GraphPad). Photon emissions were log10 transformed prior to analysis by the Mann Whitney or the two-way ANOVA, followed by Bonferroni post-tests. Log-rank (Mantel-Cox) test compared survival curves and Welches t-test compared differential expression of cytokines in lung tissue homogenates. p< 0.05 was considered statistically significant.