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Host-to-host variation of ecological interactions in polymicrobial infections

Host-to-host variability with respect to interactions between microorganisms and multicellular hosts are commonly observed in infection and in homeostasis. However, the majority of mechanistic models used in analyzing host-microorganism relationships, as well as most of the ecological theories proposed to explain co-evolution of host and microbes, are based on averages across a host population. By assuming that observed variations are random and independent, these models overlook the role of inter-host differences. Here we analyze mechanisms underlying host-to-host variations, using the well-characterized experimental infection model of polymicrobial otitis media (OM) in chinchillas, in combination with population dynamic models and a Maximum Entropy (MaxEnt) based inference scheme. We find that the nature of the interactions among bacterial species critically regulates host-to-host variations of these interactions. Surprisingly, seemingly unrelated phenomena, such as the efficiency of individual bacterial species in utilizing nutrients for growth and the microbe-specific host immune response, can become interdependent in a host population. The latter finding suggests a potential mechanism that could lead to selection of specific strains of bacterial species during the coevolution of the host immune response and the bacterial species.


Introduction:
Consequences of a pathogen exposure or diversity of resident microbiota often vary from individual to individual in a population. This becomes evident when only one of two colleagues sharing the same office falls sick to a flu outbreak, or in experiments studying infection by specific pathogens in animals kept in controlled facilities, where bacterial or viral titers as well as abundances of biomarkers associated with the host immune response display wide ranges of variation between animals [1][2][3] . Similar variations between individuals are also observed in the structure of the community of microbes residing in homeostasis with the immune system 4 .
However, despite the ubiquity of such host-to-host variations of the host-microorganism relationship, our mechanistic understanding of such relationships or ecology of hostmicroorganisms 5,6 are based primarily on average values obtained from experiments done on a host population. The variations around the averages are usually assumed to arise due to independent inter-host variations of phenomena that affect the host-microorganism relationship, such as the host immune response or availability of nutrients for the microorganisms, and, variations between hosts are often represented merely as error bars in data summaries 7,8 . But this overlooks the fact that the differences between hosts themselves may provide valuable clues regarding perturbations of the underlying mechanistic framework in a natural setting, and may relate directly to evolutionary selection of a particular host-pathogen or host-microbiota relationship based on sustaining the observed diversity in a population 9 .
Here we seek mechanistic insights into host-to-host variations of the host-microorganism relationship by using the well characterized model of polymicrobial otitis media (OM) in adult chinchillas (Chinchilla lanigera). OM is a common childhood polymicrobial infection of the middle ear involving one or more of three predominant bacterial species that are normally carried within the microbiota in the upper respiratory tract (URT) 10,11 : Nontypeable Haemophilus influenzae (NTHI), Streptococcus pneumoniae (Sp), and Moraxella catarrhalis (Mcat). OM provides an excellent model system to dissect host-microbiota relationships because of the relatively small number of species in the relevant microbial community, and also because it offers practical advantages such as culturability of the three main bacterial species 11 . While chinchillas are not a natural host for the bacteria or viruses that cause human OM, they can be infected and/or colonized with all three of the predominant bacterial OM pathogens 11 .
Using an in silico approach based on Maximum Entropy (MaxEnt) and population dynamics, combined with samples recovered from the chinchilla middle ear, we quantified ecological interactions that regulate kinetics of bacterial infection and the host immune response in individual hosts. We show here that the nature of interspecies interactions (e.g., competition, co-operation or neutral) between the bacterial species NTHI and Sp, which is not directly related to the immune response, critically regulates the host-to-host variations of the ecological interactions. More importantly, seemingly independent ecological interactions, such as the ability of the bacterial species to utilize resources and the rate at which the host immune response eliminates specific bacterial species, become inter-dependent in hosts. This suggests evolutionary selection of interspecies interactions in microbial communities through host-bacteria interactions.

Variations of kinetics of polymicrobial infection
Animal-to-animal variations of kinetics of bacterial species are clearly observed in experiments studying OM in rodents such as rats 1 or chinchillas 2 . For instance, in the experiments reported by Weimer et al. 2 , the population of Sp showed an almost bimodal behavior at three days post inoculation with mixed NTHI and Sp strains; the Sp population fell below the detectable limit in a few animals, but varied between 10 4 to 10 6 CFUs in other animals (Fig. 1). The population kinetics of NTHI, although less dramatic, showed animal-to-animal variations up to three orders of magnitude in the experiments with single and mixed species inoculations (Fig.1). The bacterial species, NTHI and Sp, have been observed to interact with each other and with the host and these interactions affect the growth of the bacterial species. For example, in in vitro cultures certain strains of Sp eliminate NTHI by secreting the toxin hydrogen peroxide generated during aerobic metabolism 12 , or, NTHI can trigger mobilization of neutrophils in the epithelial layer that eliminate Sp but not NTHI via complement-mediated opsonization 13,14 . In addition, the secretion of quorum sensing molecules by these bacterial species has been found to affect the growth of multiple bacterial species participating in the infection 15 . The bacterial species also depend on the host for extracting essential nutrients such as metals for their growth. E.g., the Gram-negative NTHI and Gram-positive Sp require iron extracted from the serum generated by the host during inflammation 16,17 . Therefore, it is plausible that variations of these factors across hosts would lead to differences in infection kinetics between hosts. Here we quantify ecological interactions in the system and model the mechanisms that lead to the infection kinetics observed in the experiments reported by Weimer et al. 2 .

MaxEnt based method to quantify variations of ecological niches
A. Population dynamic model: We constructed ordinary differential equation (ODE) based kinetic models to describe the time evolution of populations of NTHI and Sp bacterial cells (Fig. S1). The equations are based on Lotka-Volterra (LV) type models 7 , which describe the growth of two or more bacterial species interacting with each other to access available resources. These models have been successfully applied to characterize kinetics of bacterial populations in chemostat experiments 7,18 . We modified the LV models to include the host immune responses during the acute infection phase, which is primarily regulated by innate immunity 11 . In our models, the bacterial species consume nutrients from the local environment and replicate. NTHI and Sp can compete for a common nutrient (e.g., iron) for their growth, and additionally each species can indirectly help in the growth of the other species by generating more inflammation. In addition, NTHI and Sp can affect each other's growth by secreting small molecules, e.g., toxins or quorum sensing molecules. Therefore, NTHI and Sp can potentially oppose, help, or remain uninvolved in each other's growth depending on the nature of inflammation or the concentration of secreted molecules in the microenvironment. We considered all 9 possibilities (see Table I) for inter-species interactions affecting the growth rates of NTHI and Sp.
In addition, both species induce innate immune responses (antimicrobial proteins 19 or influx of neutrophils in epithelial layer 14 ) in the middle ear. In our models, we do not distinguish between antimicrobial proteins or neutrophils, and immune response is represented by a single variable, I, that eliminates NTHI and Sp with different rates. The dynamics of the abundances of NTHI and Sp in the presence of the host immune response in the middle ear of a particular animal (indexed by a) can be described by a pair of coupled ODEs: where, N 1 (a) and N 2 (a) denote the population sizes of NTHI and Sp, respectively. f 1 (a) (N 1 (a) ,N 2 (a) ) and f 2 (a) (N 1 (a) ,N 2 (a) ) describe the growth rate of NTHI and Sp, respectively, regulated by available resources and inter/intra species interactions. Both NTHI and Sp interact with the immune response elicited by the host that eliminates the bacteria, and g 1 (a) (N 1 (a) ,N 2 (a) ) and g 2 (a) (N 1 (a) ,N 2 (a) ) describe the rate of elimination of NTHI and Sp, respectively, by the immune response. To keep the notation simple, we will drop the superscript in the rest of the equations where all the variables and the parameters describe the kinetics in an individual animal or a trial in culture experiments. Following the LV model for interspecies interaction we use 7 , f 1 (N 1 , N 2 )=r 1 N 1 (K 1 -α 11 N 1 -α 12 N 2 ) and f 2 (N 1 , N 2 )=r 2 N 2 (K 2 -α 22 N 2 -α 21 N 1 ). competing, co-operating, and neutral nature of inter-species interactions respectively. The inter-species interactions are generally not reciprocal, i.e., α ij ≠ α ji . We considered nine different models, each denoting a specific type of interspecies interaction (see Table I for the list), e.g., M +-describes the model where the inter-species interactions are given by α 12 > 0 and α 21 < 0. The immune responses are described by monotonically increasing functions with increasing values of N 1 and N 2 representing concentrations of antimicrobial proteins or neutrophils attracted to the infection site, i.e., g 1 (N 1 ,N 2 )=k d1 N 1 I, and, g 2 (N 1 ,N 2 )=k d2 N 2 I, where the immune response, I, is generated due to the immune response induced by N 1 and N 2 , and, is assumed to be additive, i.e., I=I 1 +I 2 , where, I 1 =k 1 N 1 /(K M1 +N 1 ) and I 2 =k 2 N 2 /(K M2 +N 2 ). We write g 1 (N 1 ,N 2 ) and g 2 (N 1 ,N 2 ) as, g 1 (N 1 ,N 2 )=k d11 (N 1 ) 2 /(K M1 +N 1 ) + k d12 N 1 N 2 /(K M2 +N 2 ), and, Depending on the values of the parameters, the kinetics described by the ODEs in Eq. (1) produce multiple fixed, e.g., N 1 is present but N 2 is absent, N 1 is absent but N 2 is present,  } and   Table II and Table S1) as 'ecological interactions'. We hypothesize that these ecological

Interspecies interactions regulate animal-to-animal variations of microbial kinetics
We quantified the extent of variation of the ecological interactions between animals by calculating the minimum value of the relative entropy, MinRE defined in Eq. (3) in Materials and Methods, for all nine models ( Fig. 2A). All the models were constrained to reproduce the average values and variances of NTHI and Sp populations measured at 7 days post inoculation when the animals were infected with NTHI and Sp simultaneously.
The model in which Sp helps NTHI to access nutrients but NTHI competes with Sp (model M -+ ) produces the smallest MinRE, i.e., this model is consistent with the broadest parameter variations. The next best (MinRE) model was M 0+ , in which Sp stays neutral to NTHI growth and NTHI competes with Sp for resources. In contrast, the models M +-and M 0-, in which NTHI helps Sp to access resources, are consistent with only a very small amount of variation in the parameters. The results can be understood in the following way. The experiments show that at 7 days after co-inoculation (NTHI~1000 CFU, Sp~150 CFU), the average population of NTHI (~10 7 CFU) is substantially higher than that of Sp (~10 CFU). In contrast, when the animals are infected with either NTHI or Sp alone, both species reach high population (~10 7 CFU). Therefore, the models that will produce high growth for NTHI and low growth for Sp at later times (~ 7 days) across a wider range of parameter variations will turn out to be the models with smaller MinRE.
In the model M -+ , the interspecies interaction supports a higher NTHI and a lower Sp growth since Sp co-operates with NTHI in its growth, but the presence of NTHI counteracts Sp growth. The immune response, regardless of the type of interspecies interaction, can also support a larger NTHI population than Sp population by killing Sp at a higher rate compared to NTHI. However, in the model M -+ when the elicited immune response kills NTHI at a higher rate compared to Sp, which can lead to kinetics opposite to those observed in experiments, higher values of the interspecies interactions (e.g., 12 and 12 ) can counteract effects induced by the immune response and produce a pattern similar to that observed in experiments. So that the MaxEnt probability distribution is heavily concentrated on the subset of vectors of the ecological interaction parameters for which the immune response is able to counteract this effect and produce higher growth in NTHI compared to Sp. These patterns also indicate how seemingly unrelated ecological interactions, such as the interspecies interactions and the immune response, can become correlated. This is discussed in greater detail in the next section. Therefore, the models that contain neutral interactions between the species allow for more variation in underlying parameters compared to the other models.
Testing Predictions: We used the estimated MaxEnt distributions to generate predictions for measurements that were not used as constraints in fitting the MaxEnt models.
Specifically, we predicted the average values of populations of NTHI and Sp at day 3 when the animals were co-inoculated with these species. In addition, we also predicted the correlation between NTHI and Sp at day 7. The predictions from model (M -+ ), which was the best (MinRE) model under the original set of constraints, were in reasonable agreement with the data (Table S2). The models with larger MinRE values produced less agreement with the additional measurements compared to model M +- (Table S2). In general, predictions were better for NTHI than Sp. The disagreement between the model predictions and the data for Sp could point to the importance of spatial structures such as biofilms in regulating the bacterial kinetics. This point is further deliberated in the discussion section.

Specific ecological interactions become inter-dependent
In order to further characterize the structure of the MinRE model, we first checked whether the inferred distribution P ({e i }) of the model parameters could be well approximated by a multivariate normal distribution (Eq. 4), which would imply that average values and pair-correlations between the parameters capture most of the variations in the system. Since the distribution appeared to be well approximated by a multivariate normal distribution ( analyzing the ODEs. E.g., the increase in the NTHI bacterial load required to induce the maximum immune response that favors an increase in the NTHI population is compensated for by the corresponding decrease in the available resources or the carrying capacity ( Fig. 3A). This implies that in order to be consistent with the experimental data, a particular strain in NTHI that is less efficient in stimulating the immune response will also undergo changes that reduce its capability to utilize the nutrients. Further explanations regarding the other interdependencies are provided in the supplementary material (Table S3).
These in vitro analyses show (Fig. 3C,D) that parameters describing inter-species interactions, in contrast to intra-species interactions, become more dependent during infections, e.g., a decrease in the carrying capacity for Sp, which would support a higher population of NTHI due to lower competitive interspecies interaction, is compensated by a decrease in the strength in interspecies competition between NTHI and Sp.

Discussion
We The structure of the inferred variations of ecological interactions reveal that seemingly unrelated ecological variables, such as the carrying capacities and the host immune response, become interdependent. For example, in the model M -+, which shows the largest variation in the ecological interactions, a mutually co-operative relationship between carrying capacity and the rate of bacteria elimination would imply selection of an NTHI strain that can use the same resources more efficiently, whose growth is also better suppressed by the host immune response (Fig. 3). We primarily studied models that approximated and simplified interspecies interactions in terms of a relatively small number of parameters. Therefore, these models need to be modified in order to investigate interspecies interactions such as quorum sensing, which increases fitness of the same strain, or the formation of spatial structures such as biofilms, which help bacterial species to evade the host immune response. The importance of these effects, in particular biofilm formation, becomes apparent as the predictions from the two-species models differ from the measurements of abundances of bacterial populations at higher inoculation doses. The two-species model could be extended to include additional strains associated with biofilms found in the chinchilla middle ear 2 .
Investigation of the role of these additional strains in host-to-host variations of infection kinetics would be an interesting future direction.
Our analysis showed that host-to-  Table II and Table S1) that represent respectively the parameters solely regulating bacterial kinetics for experiments with single species and the additional parameters required to describe the kinetics for the mixed co-infection/culture experiments. We described the kinetics in terms of dimensionless parameters { e i } constructed (Table S1 and  were cultured, alone or together in equivalent ratios, in brain-heart infusion (Difco) supplemented with hemin and NAD, and containing 10% horse serum (HemoStat Laboratories), essentially as described previously 30 . Bacterial counts were derived by plate-count.

Quantification of the relationship between the model parameters:
We approximate the distribution P ( M ) ({e i }) by a multivariate normal distribution (Fig. S3), i.e.,

Acknowledgements:
We are grateful to Lauren Bakaletz for a critical reading of the manuscript. The work is supported by a grant from the NIH to WES, WCR, VJV, CJ, and, JD. JD is also partially supported by the Research Institute at the Nationwide Children's Hospital and a grant from the Ohio Supercomputer Center (OSC). Tables   Table I:   In vitro culture K 1 , K 2 , K lag1 , K lag2 , α 11 , α 22 , α 12 , α 21

Fig. 3 Characteristics of the inferred distribution of the ecological interactions. (A)
Inter-dependencies between ecological interactions (shown in terms of the dimensionless parameters shown in Table S1)

Supplementary Material for "Host-to-host variation of ecological interactions in polymicrobial infections"
Section 1: Fixed points and the stability of the ODE models

1.1: In vitro model and the conditions for stability
The ODEs for the growth of NTHI (N 1 ) and Sp (N 2 ) populations in in vitro culture are given by, where γ' s and the α's represent the doubling rate of the bacteria and the inter and intra species competition for resource respectively. Using the same scheme for notation as in the main text, the parameters are defined as, γ 1 = r 1 K 1 , γ 2 = r 2 K 2 , α 11 = r 1 α 11 (main text), α 12 = r 1 α 12 (main text), α 21 = r 2 α 21 (main text) and α 22 = r 2 α 22 (main text) respectively. K lag1 and K lag2 are the parameters that determine the respective lags in the bacterial growth observed in in vitro experiments. For K lag1 , K lag2 >> N 1 0 , N 2 0 the initial growth of the bacteria is stunted. Only after N 1 and N 2 overcome the thresholds set by K lag1 and K lag2 , the bacteria can transition to an exponentially growing phase. The system described by Eqn (S1) allows for four fixed points, namely, 1.   The stability of the fixed points can be studied by analyzing the stability matrix S given by Assuming γ 1 , γ 2 > 0 , we can see that the first fixed point (N 1 s =N 2 s =0) is unstable if we take the third derivative of the RHS of Eqn S1. The second fixed point is stable in the N 1 direction while the third fixed point is only stable in the N 2 direction as can be seen by substituting the steady state values of N 1 and N 2 in the stability matrix S. When both NTHI and Sp co-exist the characteristic equation (λ being the eigenvalue) can be written as For N 1 s and N 2 s greater than K lag1 and K lag2 , respectively, (See Fig S2) we have a=b∼1.
Let us for the time being let us assume that γ 1  γ 2 (the doubling rate for NTHI is roughly one hour whereas the doubling rate of Sp is half an hour). Then the eigenvalues are given by The conditions for the stability of the co-existence fixed point are summarized below.

No interaction (model M 00 )
1. In the absence of interspecies interaction, we can see that the fixed point is always stable. The two species evolve irrespective of one another.
For a co-operation/competition model, the co-existence is guaranteed provided the interspecies competition is small compared to the intra species competition. The analysis for α 21 < 0 is similar.
Like the competition model, as long as the inter species coupling is small compared to the intra species competition we have co-existence. The model M +0 can be analyzed in a similar fashion.
In a nutshell, as long as the inter species competition for resource is not fierce we can have a stable co-existence.

1.2: Infection model and the conditions for stability
The kinetics of two species co-infection is modeled using the ODEs given by where the γ' s and the α's are defined in the same way as the previous section. We will focus our attention to the steady state relevant for us (see Weimer et al. Fig 1), namely N 1 The steady state for N 1 is The stability matrix S is given by The eigenvalues of S are given by, We have looked at two possible scenarios.
Case I: The decay rates k d11 , k d 21 ≈ ε < γ 1 . The immune response plays a marginal role in this case. α21 drives the extinction of N 2 . The steady state in Eqn (S3) can be recast as Substituting Eqn (S5) in the expression of λ1 in Eqn (S4) we have The condition of stability demands that both λ1 and λ 2 < 0. Thus to O(ε) λ1< 0 implies

S7
The extinction of N 2 is mainly driven by the interspecies competition α 21 . As long as γ 2 < α 21 γ 1 α 11 and Eqn (S6) are simultaneously satisfied we have a stable fixed point. Case II: γ 1 , k d 21 >> k d11 ,α 21 ≈ ε . In the absence of a strong interspecies interaction the immune response drives the decay of N 2 . The condition for stability (Eqn (S4)) dictates that

1.3: The condition for the existence of a maximum in Sp kinetics as observed by Weimer et al.
The results in Weimer et al. (1) showed that the average population of NTHI grows and saturates while the average population of Sp increases initially and then falls back to an undetectable range by 7 days. In the following section we delineate the condition for a transient growth of Sp followed by its decay. We can rewrite Eqn (S2) as The fact that N 2 reaches a maximum value (N 2 max ) at a finite time (t=t m ) and then starts decaying demands that dN 2 /dt | t=tm = 0 and d 2 N 2 /dt 2 | t=tm < 0 and N 2 =N 2 max , which yields The second term in the expression above is always negative. So as long as N 1 competes with N 2 for resource (α 21 > 0) the derivative above is negative. Thus dN 1 dt has to be greater than zero at t m for N 2 to have a maximum. For a competition model, the growth in Sp is arrested and reversed owing to the growth in NTHI, which elicits an immune response killing Sp at a rate of k d21 N 1 /(K M1 +N 1 ). In order to get the transient kinetics in N 2 in a co-operative model (α 21 < 0), either k d21 or N 1 (t m ) or both have to be much larger such that,

Section 2: Derivation of the MaxEnt distribution
We provide details regarding the MaxEnt scheme that we use to estimate the distribution . The parameters, {e i }, determine the kinetics of N 1 and N 2 in the ODEs (Eq. 1) used to explain the observed NTHI and Sp mono infection and co-infection. The parameters, {e i }, can be decomposed into two subsets (see Table S1), {e i (S) } and {e i (M) }, that represent the parameters required to describe experiments with single bacteria species and the additional parameters required to describe the kinetics in the mixed co-infection/culture experiments, respectively. The constraints on the distribution are imposed by the 3 days and 7 days average population of NTHI and Sp for single species inoculation and by the 7 days average values and variances of populations of NTHI and Sp for co-inoculation. We show the derivation for a smaller set of constraints. The calculations can be easily generalized.
Let us assume that we know the average values of NTHI and Sp measured at 7 days post infection in single bacteria and two bacteria experiments. Therefore, the constraints for the single bacteria experiments are given by, where We maximize the entropy, , subject to the above constraints and the normalization constraint, . Therefore, the estimated distribution can be obtained from the equation given below. (S11) The solution of the above equation is, and (S14) Since the interspecies interactions are described by , all the models considered will have the same dependence on {e i (S) } via . Therefore, the differences in the variations of the ecological niches in the models will be given by and we quantify variations for each model by the relative entropy, . (S15) where q U ({e i }) is the uniform distribution over all the parameters.

Section 3: Numerical scheme to evaluate the MaxEnt distributions A: Infection models
The rate equations Eqn (S2) for the mixed infection can be recast in a dimensionless form given below are obtained by dividing the respective k d 's in Eqn(S2) by γ 1 + γ 2 .

Evaluation of the MaxEnt distribution for the single species kinetics data
For single species inoculation (either by 10 3 CFU of NTHI or by 150 CFU of Sp) the inter species interaction parameters α  12 , α  21, k  d12 , k  d 21 are set to zero. represents an animal in our simulation. We have drawn these eight parameter tuples for 100,000 times in order to simulate a cohort of 100,000 animals.
We sought a P (S) ({e (S) },{0}) (for convenience we will drop the {0} in the argument and denote the probability distribution as P (S) ({e (S) })) that will maximize the Shannon Entropy (S17) (Note the time units are in days and the RHS is in CFU) Maximizing Shannon entropy with the constraints in Eqn (S17) yields

Evaluation of the MaxEnt distribution for the co-infection kinetics data
We draw the animals (tuple of eight parameters) that are most likely to yield the single species inoculation data from the MaxEnt distribution P  e S ( ) { } ( ) . Upon drawing we inoculate that animal with both the bacteria simultaneously by assigning non zero values to the interaction parameters α  12 , α  21, k  d12 , k  d 21 . The values are drawn from a uniform distribution U(0,10). We then solve ODEs in Eqn (S9). The values of mixed species N 1 and N 2 referred to as N 1 M and N 2 M respectively, are read out at 3 and 7 days. We constrain the mean and the second moments of NTHI and Sp abundances at 7 days.
The constraints used are given below.
is the MaxEnt distribution over all the parameters and the RHS is in CFU. B. In vitro culture models Eqn (S1) can be recast in the dimensionless form given below ( ) (S20) where K  lag1 and K  lag2 are ! α 2 11 K lag1 and ! α 2 22 K lag 2 respectively. The rest of the tilde variables are same as the one defined before in Eqn (S16).

Evaluation of the MaxEnt distribution for the single species kinetics data
For single species culture the inter species interaction parameters α  12 , α  21 are set to zero. The initial populations are set to 1.0x10 6 for NTHI and 4.5x10 5 for Sp. These numbers are obtained by averaging the three experimental trials shown in Fig (S2). Like the infection model, γ 1 , γ 2 are chosen from a uniform distribution U(0,10) and then the dimensionless variables, γ  1 and γ  2 are calculated. K  lag1 , K  lag2 are drawn from U(3.6x10 7 ,3.6x10 5 ) and U(1.0x10 8 , 5.0x10 6 ) respectively, while α 11 and α 22 are drawn from U(4.7x10 11 , 4.7x10 9 ) and U(3.1x10 -11 , 3.1x10 9 ) respectively. Upon drawing all the six parameters we solve the ODEs in Eqn (S20) and read out the values of and at 4 and 8 hours respectively. Each tuple of six parameters, referred to as {e (S) }, represents an experiment trial in our simulation. We have simulated 100,000 such trials.
As in the case of the infection models, we constructed a maximum entropy distribution P e S

Evaluation of the MaxEnt distribution for kinetics in co-culture experiments
We use the MaxEnt distribution in Eqn (S21) to draw the most likely experiments. Then we introduce co-culture interaction by drawing numbers for α  12 , α  21 from a uniform distribution U(0,10). Then we rerun the ODEs given by Eqn (S20) and read out the values of and at 4, 6 and 8 hours. Like the infection model, we constrain the mean and variances of the abundance of NTHI and Sp at 8 hours. The constraints are given by,  The relation between the dimensionless parameters and the original parameters are shown in Section 1. Predictions for mean values (µ 1 = N 1 , µ 2 = N 2 ) variances (σ 1 2 = N 1 2 − µ 1 2 , σ 2 2 = N 2 2 − µ 2 2 ), covariances (Cov(ij)= N i N j − N i N j , i≠j), and correlations (ρ ij =Cov(ij)/σ i σ j , i≠j) between abundances of NTHI (N 1 ) and Sp (N 2 ).