Quantification of Lysogeny Caused by Phage Coinfections in Microbial Communities from Biophysical Principles

The association of temperate phages and bacterial hosts during lysogeny manipulates microbial dynamics from the oceans to the human gut. Lysogeny is well studied in laboratory models, but its environmental drivers remain unclear. Here, we quantified the probability of lysogenization caused by phage coinfections, a well-known trigger of lysogeny, in marine and gut microbial environments. Coinfections were quantified by developing a biophysical model that incorporated the traits of viral and bacterial communities. Lysogenization via coinfection was more frequent in highly productive environments like the gut, due to higher microbial densities and higher phage adsorption rates. At low cell densities, lysogenization occurred in bacteria with long duplication times. These results bridge the molecular understanding of lysogeny with the ecology of complex microbial communities.

these two ecosystems allowed us to test the hypothesis across a wide range of microbial densities.

RESULTS
Relationship between COI and phage-to-bacterium ratios. The model introduced in equation 1 was reexpressed to estimate the average number of phage (co)infections (COI) in terms of the phage-to-bacterium ratios (P i /B i ) for a single phage-host pair (Fig. 1a). The phage-to-bacterium ratio (P i /B i ) was used as a proxy for the operational multiplicity of infection (MOI ϭ P 0 /B 0 ) widely used in the phage field. The bacterial densities, adsorption rates, and commitment times were plotted as a function of COI and P i /B i (Fig. 1b to d). To illustrate the relationship of COI with different physical parameters, only one value was varied at a time covering typical environmental ranges (see meta-analysis section for environmental ranges). COI was higher for higher bacterial concentrations (Fig. 1b), phage adsorption rate constants (Fig. 1c), and lysogenic commitment times (Fig. 1d). For the typical adsorption rate and commitment time of lambda, the model showed that an average of two or more phage infections (COI 2) was unlikely to occur at bacterial densities below 10 6 cells/ml, even for phage-to-bacterium ratios above 10 (bottom right values in Fig. 1b). Due to the high adsorption rate of lambda, the average number of coinfections generated per phage-to-bacterium ratio was near the upper range of environmental values (Fig. 1c). Due to the short lysogenic commitment time of lambda, the average coinfections per phage-to-bacterium ratio unit were below the environmental values (Fig. 1d). These results show that as one departs from ideal experimental conditions, the proxy for MOI did not capture the average number of coinfections.
The probability of lysogeny as a function of average coinfections (COI) was compared with lambda-Escherichia coli MOI experiments. The percentage of lysogenized cells increased as a function of MOI and was best described by a sigmoidal Hill-Langmuir equation of order n ϭ 2, compared to orders n ϭ 1 and n ϭ 3 (Fig. 2a). This equation implied that two phages cooperated in producing lysogeny, in agreement with single-molecule experiments (25). This empirical model was functionally similar to the predicted probability of lysogeny from the average coinfection Poisson model, equation 2, which assumed that at least two infections were necessary to produce lysogeny (Fig. 2b). The MOI and COI values were similar. The discrepancy between the maximum percentages of lysogeny for the MOI and COI models was due to the fact that the COI model was a function of the initial phage-to-bacterium ratio and did not account for the rapid removal of phage particles due to cell adsorption in the course of the experiments. The MOI-COI equivalence in lambda-E. coli experiments was due to the fact that the original MOI experiments were set up to capture the number of coinfections, which was only possible for those specific growth conditions (Fig. 1).
Meta-analysis of COI physical parameters from marine and animal ecosystems. To apply the biophysical COI model to microbial communities and estimate lysogeny generated by phage coinfections, the ranges of phage adsorption rate constants, lysogenic commitment times, and phage-bacterium pair abundances were determined for marine and animal ecosystems. The range of adsorption rates was 7.2 · 10 Ϫ10 to 3.7 · 10 Ϫ7 ml/h for marine phages infecting Prochlorococcus sp., Roseobacter sp., Pseudoalteromonas sp., Synechococcus sp., and Vibrio sp. (Fig. 3a) (28,29). The lines correspond to fitted Hill-Langmuir cooperation models of order n ϭ1 (dashed), n ϭ 2 (solid), and n ϭ 3 (dotted). (b) Percentage of lysogeny estimated from the coinfection probability model as a function of COI, equation 2 (solid black line). Hill-Langmuir model of order n ϭ 2 from panel a as a function of MOI (solid gray line). rates was 5.9 · 10 Ϫ8 to 1.2 · 10 Ϫ6 ml/h for gut phages infecting E. coli. The median adsorption rate for gut phages was 1 order of magnitude higher (4.2 · 10 Ϫ7 ml/h) than the median for marine phages (3.4 · 10 Ϫ8 ml/h [ Fig. 3a], and t test P ϭ 7.23 · 10 Ϫ10 ). Lysogenic commitment times were longer in marine communities, 11 to 808 h, than in the mammalian gut, 2.74 to 7.27 h. This was a consequence of the long duplication times of marine communities in their natural environment. The phage and bacterium pair abundances were determined by combining the total and relative abundances of phage and bacteria in each ecosystem. Phage abundances were 1.4 · 10 5 to 3.7 · 10 7 phages/ml (marine) and 5.1 · 10 6 to 1.1 · 10 10 phages/ml (animal) (Fig. 3b). Bacterial abundances ranged from 3.8 · 10 4 to 6.8 · 10 6 cells/ml (marine) and from 3.5 · 10 5 to 7.7 · 10 9 cells/ml (animal) (Fig. 3c). The total abundances were at least 2 orders of magnitude higher in animal-associated mucosa than in the free-living communities of surface marine environments (t test P ϭ 7.02 · 10 Ϫ15 for phage [ Fig. 3b], and P value ϭ 4.17 · 10 Ϫ7 for bacteria [ Fig. 3c]). The most abundant phage genotype (P 1 ) in marine environments comprised only 0.8% of the total phage community, while in the gut, the dominant phage comprised just over 1% of the community (Fig. 3d). In the bacterial community, this pattern was inverted, with the dominant bacterial species (B 1 ) reaching 19% in marine environments, but only 15% in the gut (Fig. 3e). Lysogeny by coinfection in microbial communities. The community model assumed a direct phage-host network, where each phage rank infected the same rank in the bacterial community, that is, P i infected B i (Fig. 4a). The biophysical COI model quantified the percentage of lysogeny generated by phage coinfections for each pair by stochastically sampling the parameter ranges from the meta-analysis of marine and gut ecosystems. The percentage of lysogeny caused by coinfections increased with total bacterial density ( Fig. 4b and Fig. S1). Lysogeny was more frequent in the gut, where 25% of the simulated communities displayed at least 25% of bacteria becoming lysogens by coinfection (Fig. 4c, Table 1, and Table S1). The median percentage of lysogeny in these communities was 47.8%. Given the median bacterial abundances  Cells/ml 5.6 · 10 6 6.8 · 10 8 1.8 · 10 9 2.4 · 10 9 3.9 · 10 9 7.6 · 10 9 Phage concentration Phages/ml 2.0 · 10 8 8.0 · 10 8 1.3 · 10 9 1.9 · 10 9 2.2 · 10 9 1.1 · 10 10 Phage adsorption rate ml/h 5.9 · 10 Ϫ9 1.3 · 10 Ϫ7 2.6 · 10 Ϫ7 3.7 · 10 -7 5.6 · 10 -7 1. (1.7 · 10 9 cells/ml), duplication times (4.75 h), and volume of the human colon (400 ml [39]), we estimated that a median of 1.8 · 10 12 lysogens is potentially formed in the human gut every day via coinfection. Among marine communities, 90% displayed 10% or fewer bacteria becoming lysogens by coinfection (Fig. 4c, Table 2, and Table S1).
For communities with lysogeny above 1%, the most abundant phage-host pairs contributed an average of 67% Ϯ 12% (standard deviation [SD]) for marine and 51% Ϯ 16% for gut to the total lysogeny ( Fig. 4d and Fig. S2). This was significantly higher than the contribution from the second most abundant phage-host pair, which yielded 13% Ϯ 1% for marine and 15% Ϯ 2% for gut. For communities with lysogeny above 1%, the most abundant phage-host rank displayed median COI of 1.00 (one infection on average) for marine (Fig. 4e, Fig. S2, and Table 3) and COI of 2.35 for gut (Fig. 4f, Fig. S2, and Table 4).
Physical parameters contributing to the formation of lysogens in communities. Communities with at least 1% lysogeny caused by coinfection were analyzed to extract the distribution of physical parameters yielding lysogeny. The distribution of bacterial abundances favoring lysogeny in marine communities was skewed toward high densities with a median of 1.5 · 10 6 cells/ml ( Fig. 5a and Table 3). In gut communities, low bacterial abundances did not contribute to lysogeny, with the first quartile of the probability distribution at 9.2 · 10 7 cells/ml ( Fig. 5a and Table 4). Phage concentrations yielding lysogeny in marine communities were also skewed toward higher densities (median of 9.1 · 10 6 phages/ml) but more centered than the bacterial density distribution ( Fig. 5b and Table 3). In the gut, low phage concentrations did not contribute to lysogeny, displaying a first quartile of the probability distribution at 2.8 · 10 8 phages/ml ( Fig. 5b and Table 4).
Phage adsorption rates in marine communities producing lysogeny were skewed toward high values with a median of 1.1 · 10 Ϫ7 ml/h ( Fig. 5c and Table 3). In the gut, instead, the full range of adsorption rates contributed to communities with lysogeny ( Fig. 5c and Table 4). The lysogenic commitment time in marine communities producing lysogeny was again skewed toward long time windows, with a median of 262 h ( Fig. 5d and Table 3). For the gut, the full range of lysogenic commitment times contributed to producing lysogens, but larger values had a higher likelihood of contribution, displaying a median lysogenic commitment time of 0.92 h (Fig. 5d and Table 4).

DISCUSSION
The stochastic biophysical COI model introduced here estimated an increase in coinfections in the highly productive mammalian gut microbial communities (Fig. 4).  4 7.9 · 10 5 2.0 · 10 6 2.5 · 10 6 3.9 · 10 6 6.7 · 10 6 Phage concentration Phages/ml 6.2 · 10 5 6.3 · 10 6 1.3 · 10 7 1.6 · 10 7 2.5 · 10 7 3.9 · 10 7 Phage adsorption rate ml/h 6.2 · 10 -9 7.8 · 10 -8 1.6 · 10 -7 1.7 · 10 -7 2.5 · 10 -7 3.  This higher frequency of coinfections predicts considerable levels of lysogeny in animal mucosa microbiomes, as observed empirically (9,11,12,14,16,17), and supports the Piggyback-the-Winner framework (15,16). In the murine gut, empirical genomic data show that 65.8% of the bacterial genomes are lysogens and that 83.2% of the prophages observed in these lysogens are active (40). Together, these lysogens reach relative abundances of 53.6% to 78.6% of the bacterial community. This frequency of lysogeny is consistent with the output of the coinfection model, which predicts lysogeny levels between 40% and 70% for abundances above 10 9 cells/ml (Fig. 4b, solid line). The model also predicted that 41.89% of the gut communities displaying lysogeny above 1% had virus-to-microbe ratios (VMR) lower than 1 (see Fig. S3 in the supplemental material), indicating that the formation of lysogens through coinfection was compatible with the observation of low virus-to-microbe ratios in high-density animal mucosa (12,41). This was possible because coinfections were not required to occur  simultaneously. Instead, they occurred within the lysogenic commitment time, which was assumed to be proportional to the duplication time of bacteria in vivo (42)(43)(44). The lysogenic commitment time played a paramount role in the findings of the model because the duplication time of gut bacteria in vivo is significantly slower than the duplication time of gut bacterial isolates in pure cultures (42)(43)(44). For laboratory E. coli duplication times, the model predicted an average number of coinfections almost an order of magnitude lower than that in vivo (Fig. 1d). This result aligned with the low number of coinfections observed even at high MOIs in a recent stochastic model that used parameters similar to lambda and E. coli (38). The influence of the lysogenic commitment time was even more pronounced in marine communities (Fig. 5d). Lysogeny levels above 10% were favored for communities that displayed long lysogenic commitment times, which compensated for the lower adsorption rates and phage and bacterial abundances (Fig. 3 and 5).
The relationship between the lysogenic commitment time () and duplication time in the biophysical COI model was based on lambda-E. coli single-cell experiments, which identified the relationship ϳ20% of the duplication time (25,45). This pattern arises from the voting phenomenon, where each coinfecting phage genome independently votes for lytic or lysogenic commitment (46). In cells that undergo lysogeny, phages cooperate for host cell resources (25). Late phage genome arrivals contribute less to the cell-level decision, and the time window for the contribution of the second phage is proportional to the transcription level of phage repressors, which vary with the host growth rate (45). Future studies addressing the relationship between the lysogenic commitment time and host duplication time in other phages would refine the model and further test its validity in different ecosystems.
Only 10% of the simulated marine communities displayed levels of lysogeny above 10%. These communities were characterized by high phage and bacterial concentrations, displaying medians of 1.3 · 10 7 phages/ml and 2.0 · 10 6 cells/ml ( Fig. 5a and b; Table 2). A direct comparison between empirical data and our model outputs was not possible due to the lack of estimates of percent lysogenic bacterial cells in marine environments using genomic data, which remains a bioinformatics challenge. Previous studies assessed the percentage of lysogeny in marine samples using mitomycin C induction, but this method has been proved inaccurate, and the density relationships derived from it are unclear (47). Alternatively, we pursued an indirect approach to compare the results of the model with empirical data by analyzing indicators of lysogeny that were available for both marine and gut ecosystems. In the free viral particle metagenomes, only 5% to 20% of the identified viral contigs are predicted to be temperate in marine samples (7), 3 to 15 times less than that observed in the gut, which ranges from 53% to 72% (40). When comparing genomes from isolates, marine bacteria encode 5 Ϯ 2 (mean Ϯ SD) prophages per genome, three times less than human gut bacteria, which encode 14 Ϯ 5 prophages per genome (17). In the model, marine communities displayed, on average, 5 to 10 times less lysogeny than gut communities (solid lines in Fig. 4b). This ratio is on the same order of magnitude as the change in the two empirical indicators for lysogeny (temperate phage particles and prophage abundances) in marine and gut communities.
The stochastic community simulations assumed that the most dominant phages infected the most dominant bacteria (10,11,(48)(49)(50)(51). This empirically-based assumption allowed us to bridge the environmental data on viral and bacterial abundances with the species distributions from genomic data (Fig. 3). In the model, the most abundant phage-host pairs dominated the formation of lysogens (Fig. 4d and Fig. S2). If temperate phages do not occupy the first rank, lysogeny via coinfection will decrease by about 50% in marine and 70% in gut communities, unless there is a cross-infection network. Metagenomic data support that temperate phages and their hosts are likely to occupy high ranks in the community. For example, pelagiphages infecting SAR11, the most abundant marine bacterium, account for 35% of the phage particles in the virioplankton, representing the most abundant phages in the oceans, and 11 out of 16 isolated pelagiphages are temperate (50,52,53). If these closely related phages cooperate during the lysogenic decision, their summed abundances would be much higher than the abundance of the highest rank in our model, 0.8% for marine communities. One challenge in building accurate phage-bacterial infection networks to test these comparisons is that the hosts of the majority of phages identified from metagenomic analyses are unknown (8,54,55). The reconstruction of accurate infection networks will, in the future, improve the model's predictive power on the contribution of coinfections to lysogeny (56,57).
The average percentage of lysogens formed by coinfection decreased by almost 2 orders of magnitude as total bacterial concentrations dropped from 10 6 to 10 5 cells/ml (Fig. S1b). This contrasts with viral metagenomic studies from deep oceans with microbial abundances ranging from 10 4 to 10 5 cells/ml showing an increase in lysogeny compared to more productive surface waters (7,20). In these environments, lysogeny has been proposed to serve as a low-density refugium for temperate phages in conditions of poor host growth and scarce resources for viral particle production (58)(59)(60)(61). In the biophysical COI results, lysogeny in these low-cell-density communities occurred at very long commitment times (Fig. S4), which is likely the case in the natural environment. The model did not incorporate assumptions relating bacterial densities and commitment times. Adding this relationship would increase the percentage of lysogeny predicted in deep oligotrophic waters. Additional mechanisms could also contribute to the increase of lysogeny at these low concentrations, such as the favored phage integration in starved cells as observed in lambda due to the reduced degradation of the lytic repressor (62).
The COI model assumed that two phage infections occurring within the commitment time were necessary for lysogeny. This assumption was based on the observation that most temperate phages seem to encode a repressor system that is functionally similar to lambda's cro/cI (33,34). This includes phages in marine environments, such as temperate phages infecting SAR11, suggesting that lessons learned from lambda can be extended to the marine environment (7,52,53). The model introduced here, however, does not capture lysogeny from a smaller fraction of temperate phages, such as P1 and P4-like, that do not respond to coinfections (63,64). Further work will be necessary to assess alternative mechanisms for the control of lysogeny and refine the model predictions.
The lysogenic commitment time provides plasticity for phage adaptation to different ecosystems, which may be the reason why the response to coinfection has been selected in disparate environments. Phage densities are higher in the gut where bacterial replication times (and commitment times) are shorter than in marine environments. Above environment-specific density thresholds, communities would be driven to extinction by lysis unless immunity mechanisms emerge (65). Because of its superinfection immunity, lysogeny could be selected through the plastic coinfection response that depends on growth rates. Other density-dependent mechanisms, such as quorum sensing, may also contribute to maintaining population stability when phage densities are relatively high (66,67). Coinfections might also act together with other molecular defenses, such as bacterial restriction-modification systems, which delay infection onset until bacteria reach densities that favor lysogeny via coinfection (68). Our model did not incorporate mechanisms that would lead to stability over long-term evolutionary dynamics (69). Further work will be necessary to assess these stability mechanisms.
Conclusion. The stochastic biophysical COI model proposed here identified the ranges of physical parameters that drive phage coinfections in complex microbial communities. The model predicted a high frequency of lysogeny caused by phage coinfections in the mammalian gut. This finding was a consequence of high phage and bacterial densities and high phage adsorption rates in comparison with marine communities. Longer lysogenic commitment times in vivo, compared to laboratory isolates, also contributed to high lysogeny in the gut. The simulated marine communities showed a lower frequency of lysogeny by coinfections. Those communities that displayed a high fraction of lysogeny were characterized by long lysogenic commitment times. Our findings bridge the main molecular mechanism causing lysogeny in laboratory systems with metagenomic observations of lysogeny in complex microbial communities.

MATERIALS AND METHODS
Phage coinfection model. The average number of phage (co)infections (COI) was derived from physical properties of phage and bacteria (Fig. 1a). The rate of phage infections on a single bacterium can be estimated by solving the Smoluchowski coagulation equation (37). In a well-mixed community, this rate is the product of the phage concentration (P i ) and the phage adsorption rate (␣), which depends on the mobilities and sizes of both the phage particle and the bacterium. The adsorption rate constant (␣) expresses how fast a single phage adsorbs to a single bacterium given a volume (37), and its units are expressed here in ml/h. The subindex i specifies a single phage-bacterium pair in the community.
The number of (co)infections (COI) was defined as the number of phages infecting a cell within a given time window. This number was the product of the infection rate and the time window. In the case of lysogeny, this time corresponds to the lysogenic commitment time (), when the second phage can still interfere with the decision (lysis or lysogeny) of the first infecting phage (25). This led to the average phage coinfection equation (Fig. 1a) Therefore, COI ϭ 1 means one infection per cell on average within the window time, and COI ϭ 2 means two phage infections. The average probability of coinfections was calculated assuming that each infection was independent and that, in a given environmental community, the changes in phage concentration (P i ) and bacterial concentrations (B i ) were small (within 20%) during the lysogenic commitment times (). This assumption is consistent with the typical changes of abundances in the environment (58,70), but it does not apply during rapid changes in abundances observed under laboratory conditions (38). In the community model described below, the variance in COI due to the variance in (P i ) and (B i ) in a given community is negligible compared to the variance in COI resulting from the stochastic sampling across the ranges of microbial traits (see meta-analysis and stochastic sampling sections below). The average number of coinfections was also expressed as a function of the phage-to-bacterium ratio, COI ϭ ␣ · · B i · (P i /B i ), as a proxy for comparison with lambda-to-E. coli ratio in MOI experiments. Numerically, the phage-to-bacterium ratio (P i /B i ) was explored for the range 0.01 to 100. The median values extracted for the lambda adsorption rate (␣ 0 ϭ 5.6 · 10 Ϫ7 ml/h), lysogenic commitment time ( 0 ϭ 0.1 h), and bacterial concentration (B 0 ϭ 5 · 10 8 cells/ml) were used as reference values (see section on meta-analysis for lambda parameters). Two parameters were fixed at these reference values, and the third was explored over a range of values based on the meta-analysis of microbial communities (see details below). These ranges were 10 5 to 10 10 cells/ml for bacterial concentrations, 10 Ϫ11 to 10 Ϫ6 ml/h for the phage adsorption rates, and 10 Ϫ3 to 10 2 h for the lysogenic commitment times.
Percentage of lysogeny for the coinfection model. A lysogen was formed when a cell was infected within the lysogenic commitment time by two or more phages from the same phage-host pair. This was based on the effect of cooperative infection by phages on the production of lysogens (24,25,46). Thus, the probability of lysogenization, p lys , was determined by p lys ϭ 1 Ϫ p(0) Ϫ p(1), where p(k) was the probability of having k infections within the commitment time. The probability of k infections with average (co)infection COI, equation 1, was given by a Poisson distribution p(k) ϭ COI k e ϪCOI /k!. The probability of forming a lysogen via coinfection was p lys ϭ 1 Ϫ e ϪCOI Ϫ COI e ϪCOI (2) The model assumed that a higher probability of lysogenization resulted in a higher prevalence of lysogeny. This assumption was supported by experimental data (68,69). The probability of lysogenization was compared with the percentage of lysogeny obtained from lambda and E. coli MOI experiments (28,29). The empirical data for the percentage of lysogeny and MOI were fitted using the nonlinear least-squares method for Hill-Langmuir cooperation models, f(x) ϭ ax/(b ϩ x n ), with cooperation orders n ϭ 1, n ϭ 2, and n ϭ 3.
(ii) Lysogenic commitment times. The lysogenic commitment time () was assumed to be 20% of the bacterial duplication time (25,45). The ranges of bacterial duplication times were obtained from in situ data sets for marine ecosystems (83) and from in vivo data sets for mammalian gut ecosystems (42-44) (Data Set S2).
(iv) Phage and bacterial diversity. The rank-abundance curves of phage genotypes were constructed from the median slope and intercept of power-law functions fitted to 192 marine viromes and 1,158 human-associated viromes (89) (Data Set S4). Phage genotypes were defined as unique viral contigs at 98% sequence identity. The rank-abundance curves of bacterial species in marine communities were obtained from operational taxonomic unit (OTU) tables constructed by clustering universal, protein-coding, single-copy phylogenetic marker genes into metagenomic OTUs (which can be inter-preted as species-level clusters) from the Tara Oceans data set (Data Set S5) (90,91). For animalassociated bacterial microbiomes, rank-abundance curves were constructed using OTU tables obtained by mapping metagenomic reads from 11,850 human gut metagenomes to 92,143 metagenomeassembled genomes (92). Consensus rank-abundance curves were obtained by averaging the frequency of bacteria in the same rank across the metagenomes within each ecosystem.
Quantification of lysogeny through phage coinfection in communities. The biophysical COI model, equations 1 and 2, was applied to predict the probability of lysogenization in marine and gut ecosystems as a result of coinfection. The model generated stochastic communities that sampled empirical phage and bacterial concentrations, relative abundance of the top 100 members of the community, phage adsorption rates, and lysogenic commitment times obtained from the meta-analysis of marine and gut ecosystems described above.
(i) Stochastic sampling. The model generated 100,000 stochastic communities for both marine and gut ecosystems using Latin hypercube sampling (LHS). For each ecosystem, the ranges of bacterial concentrations, adsorption rates, and lysogenic commitment times were each divided into equal intervals in logarithmic scale (base 10), generating 100,000 points per coordinate. These coordinates defined the hypercube. One hundred thousand random values were sampled from the hypercube without repeating any coordinate value, that is, all coordinate values were sampled, following the standard LHS implementation (17,93,94).
(ii) Parameter ranges. The ranges of bacterial concentrations used were 3.78 · 10 4 to 6.75 · 10 6 bacteria/ml for marine communities and 3.45 · 10 5 to 7.60 · 10 9 bacteria/ml for gut. The ranges of phage concentrations used were 1.45 · 10 5 to 3.80 · 10 7 phages/ml for marine and 5.09 · 10 6 to 1.05 · 10 10 phages/ml for gut. The ranges of phage adsorption rate constants used were 7.2 · 10 Ϫ10 to 3.7 · 10 Ϫ7 ml/h for marine and 5.9 · 10 Ϫ8 to 1.2 · 10 Ϫ6 ml/h for gut. The ranges of lysogenic commitment times used were 11 h to 808 h for marine and 2.74 h to 7.27 h for gut. All parameter ranges were obtained in the meta-analysis described above.
(iii) Assumed relationships. Based on environmental data of microbial communities, the total phage concentration (P) was modeled following a power function relationship with the total bacterial concentration (B) (15,17,39,95): P(B) ϭ a (B/B u ) b . The bacterial concentration was given in units of B u ϭ bacteria/ml. The prefactor a and exponent b were obtained by fitting the power function to the viral and microbial counts obtained in the marine and gut meta-analyses. A linear regression fit was applied using the least-squares method to the log-log data in base 10. The parameters obtained were a ϭ 10 2.50 phage/ml and b ϭ 0.712 for marine and a ϭ 10 5.35 phage/ml and b ϭ 0.388 for gut. To reproduce the noise observed in empirical communities, the value log 10 P(B) was weighted by a normal distribution, N(mean, SD), with mean 1 and standard deviation 0.05 in logarithmic space (base 10), that is, log 10 P ϭ N(1,0.05) · log 10 P(B). The final value of the phage concentration was constrained within the empirical phage abundance range, that is, P min Յ P Յ P max . The community model also assumed that the most dominant phages infected the most dominant bacteria (10,11,(48)(49)(50)(51). This led to a phage-host network where the phage of rank i infected the bacteria with the same rank i. Data availability. The codes for the model are available in the GitHub repository at https://github .com/luque82/Luque_and_Silveira_2020.git.

SUPPLEMENTAL MATERIAL
Supplemental material is available online only.