When does a Lotka-Volterra model represent microbial interactions? Insights from in vitro nasal bacterial communities

ABSTRACT To alter microbial community composition for therapeutic purposes, an accurate and reliable modeling framework capable of predicting microbial community outcomes is required. Lotka-Volterra (LV) equations have been utilized to describe a breadth of microbial communities, yet, the conditions in which this modeling framework is successful remain unclear. Here, we propose that a set of simple in vitro experiments—growing each member in cell-free spent medium obtained from other members—can be used as a test to decide whether an LV model is appropriate for describing microbial interactions of interest. We show that for LV to be a good candidate, the ratio of growth rate to carrying capacity of each isolate when grown in the cell-free spent media of other isolates should remain constant. Using an in vitro community of human nasal bacteria as a tractable system, we find that LV can be a good approximation when the environment is low-nutrient (i.e., when growth is limited by the availability of nutrients) and complex (i.e., when multiple resources, rather than a few, determine growth). These findings can help clarify the range of applicability of LV models and reveal when a more complex model may be necessary for predictive modeling of microbial communities. IMPORTANCE Although mathematical modeling can be a powerful tool to draw useful insights in microbial ecology, it is crucial to know when a simplified model adequately represents the interactions of interest. Here, we take advantage of bacterial isolates from the human nasal passages as a tractable model system and conclude that the commonly used Lotka-Volterra model can represent interactions among microbes well when the environment is complex (with many interaction mediators) and low-nutrient. Our work highlights the importance of considering both realism and simplicity when choosing a model to represent microbial interactions.

I feel that I sincerely believe the authors' conclusions*, with a single true caveat: In the current presentation, nothing really proves to me that the OD600 shown on many figures is indeed a "carrying capacity" -i.e. I am never shown the saturation of growth outside of the coculture experiments. It could well be that I am simply looking at a transient value of abundance in the exponential growth phase, in which case a positive correlation of abundance with growth rate is rather obvious. I do not really believe this to be the case, but it would be good to lay such a doubt to rest.
To conclude, I have a few recommendations, most importantly a figure demonstrating that it is reasonable to call these OD600 "carrying capacities", at least for a reasonable fraction of them if growth curves over time are not available for all.
* As a small note concerning my own bias, I am generally sympathetic to the idea that more complex environments can lead to simpler effective models because the idiosyncrasies of any given strain-metabolite interaction are less important in aggregation. This clearly makes me more likely to believe the authors' conclusion regarding differences between simple and complex media. where r_ij is the only factor being influenced by the fresh or spent medium, will always give proportionality between r and K, as the equilibrium abundance is K_ij=r_ij/alpha There is no necessity for r_ij to strictly be linear in K_j. To really test the Lotka-Volterra model, you would need to also vary K_j, i.e. do spent media experiments with various spent media from the same species (e.g. having started with various pHs), and show that r_ij is indeed r_ij = r_i -c_ij K_j with constant c_ij. This is not essential here, but it's worth considering that maybe a non-LV model compatible with the original observation would still work a bit better.
2) Building on this, I want to propose a slightly different way of presenting things than Lipson's explanation proposed in discussion.
First, I would like to mention that I don't think the high-nutrient results (Supp Fig 4 and 5) should be described in terms of correlations. The more striking fact is that there is little variance on the x-axis, while there is quite a lot of variance on the y-axis.
If there was little variation in both, you could have weak correlations simply due to noise and errors, but that would not say much. Here, the variance on the y-axis (and furthermore the ordering of points) mean that nutrient concentration is really doing something interesting to K.
The relationship between r and K can be clarified in the r-alpha formulation as above, i.e. dS/dt = r S -alpha S^2 with K=r/alpha. it seems that --at low nutrient concentration, alpha is roughly constant while r varies, hence K ~ r --at high nutrients, alpha really varies with nutrient concentration, while r is practically fixed.
This would be compatible while various recent studies finding strong negative per-capita interactions overall at high nutrients, e.g.

Ratzke et al (2020) Strength of species interactions determines biodiversity and stability in microbial communities -Nature ecology & evolution
Li et al (2022) Resource availability drives bacteria community resistance to pathogen invasion via altering bacterial pairwise interactions -Environmental Microbiology Quoting from Ratzke: "At high nutrient concentrations, extensive microbial growth leads to strong chemical modifications of the environment, causing more negative interactions between species. " Here this seems to be true for within-species interactions as well. (the difference with what is currently in the discussion is that there is no need to invoke a trade-off between r and K; r does not seem to be changing much anymore, except in a few instances --instead alpha is what really is impacted) 3) the choice to re-adjust pH in the spent media is not really discussed although it means removing an important vector of species interactions, as obvious from Fig 4 and many other works (see e.g. Ratzke's paper above). I do not think it is a bad thing in itself --indeed it might favor the expression of other mechanisms of interaction --but it is worth noting that it is not a completely benign choice, especially as such adjustment was not done for cocultures and this may explain some of the diverging results.
More generally, thinking of interactions entirely in terms of what is left in spent media is anchored in a particular assumption about possible interaction mechanisms: it ignores any kind of interaction that would involve a more dynamical response of species' metabolisms to each other, or that is not mediated by metabolites (e.g. mechanical interactions). Thus, a quantitative disagreement with co-culture results could point to a whole range of possibilities. Figure 2 as a matrix of interaction coefficients, instead of giving them only for the cocultured strains in Table 2? That would be an interesting result in itself for various readers (i.e. it provides an idea of the magnitude of competition or facilitation in these systems) 5) In the main text you say "we observe a strong linear and positive correlation between carbon concentrations and carrying capacity (Supplementary Figure 6), but not growth rate (Supplementary Figure 7)" I would not say that relationships in Sup Fig 7 are really that weak --in most other contexts, the R squared and p-values you are showing would be taken as good results. I would simply claim the relationship to carrying capacity is stronger (and even there you had to remove an inconsistent strain), but this particular experimental manipulation is likely one where both r and K or alpha are changing a bit. This is not at all a main result so it is not a problem of course. 6) Regarding Sup Fig 10, I would note that Lotka-Volterra does allow a dependency on initial conditions in the case of mutual exclusion, and you would have a hard time properly fitting the coefficients and getting the right results in that case. But in any case, LV does not allow this at all when species coexist. I agree that a different model is clearly appropriate here. 7) Sup Fig 11 has honeslty too little variation between media to really convince that the combined media behave like the average of the component media (rather than, say, like just one of hte two component media, or like a randomly drawn spent medium)

MINOR COMMENTS
Page 2: "To converge these distinct phenomenological and mechanistic inquiries" -> I am likely less of a proper English speaker than some authors, but this transitive use of the verb converge seems unusual. Figure 2: Clarify in caption that colors are the media, and panels are the strain being grown (it can be gathered from the fact that 10% THY is included as a color, but still, it is better made explicit) Reviewer #3 (Comments for the Author): I have read the paper entitled "When does a Lotka-Volterra model represent microbial interactions? Insights from in-vitro nasal bacterial communities" with interest.
The authors use the Lotka-Volterra (LV) framework to estimate interactions from cell-free spent-media (CFSM) assays, and confront the estimated interaction with actual co-cultures experiments, using the nasal microbiota as an experimental system. The authors first derive a criteria to assess when the LV framework can be used. The growth rate and carrying capacity of monocultures (growth assays) must be linearly positively correlated in different environmental conditions. Then, they use CFSM data to estimate pairwise interactions, and compare these predictions to cocultures. They show that LV models fitted on CFSM accurately capture the interaction when the environment is complex and limited in nutrients.
This study and the question it raises are very interesting, as the use of the LV framework to model bacterial interactions has been recently debated, in particular, in terms of its predictive ability (as referenced in the introduction). The comparison between CFSM estimations and co-cultures allows the authors to quantify this predictive power.
I do not have a lot of comments on the experimental parts (choice of medium, etc) but more on the assumptions of the modeling approach.
General comments: 1) I think there are two aspects to the study: a first one, regarding the use of LV framework to estimate interactions, and a second one, regardless of the modeling framework, regarding the use of CFSM vs cocultures. The first question is nicely introduced, and I would appreciate a bit more discussion or reference to the literature regarding the use of CFSM to estimate interactions, since it is then used as a reference to characterize the quality of the LV estimations, if I understood the approach correctly. What is the advantage of the CFSM method compared to directly fitting to the co-cultures or monocultures/co-cultures comparisons?
2) I wonder about the link between the carrying capacity as defined here in the end of introduction ("to what extent bacteria grow"), and the final biomass/yield of a Monod growth model? This comment is linked to minor comment 3. Are these two concepts equivalent?
3) A related comment on the comparison with consumer-resource models: would it be possible to estimate the interactions from fitting a Monod consumer-resource model to the monocultures and predict the co-cultures pattern with this type of data? I would be interested to see maybe in a supplementary analysis what a consumer-resource model would do in terms of fitting on top of the different curves of figure 5. Also, what framework do the authors suggest to use when LV is less relevant (i.e. when the environment is rich in nutrients or not complex)? 4) I appreciate how the authors introduce this idea of bacteria modifying their common habitat, thus inducing intra-and interspecific interactions (beginning of page 3 & final paragraph of the discussion 2) Page 8: K i and r i the "and " is in italics 3) Page 8: The authors state that the formulation is equivalent to the LV model in which a ij = c ij r i / K i . I wonder if the requirement that growth rate and carrying capacity must be linearly correlated simply comes from the fact that in the logistic formulation the carrying capacity is the ratio between an intrinsic growth and a self-regulation term, so that the two parameters are not independent. For instance, considering the alternative formulation dS/dt = (r -a S) S, the equilibrium density (effective carrying capacity) is S* = r/a. If I use this formulation and follow the same steps as in the Supplementary Material, I find that "r 12 "/"K 12 " = a 11 which is equivalent to what the authors find and is the strength of intraspecific competition. 4) Beginning of the discussion: I think there are two "historical" LV set of equations, one for predator-prey interactions (the harelynx model with cycles), and one for competitive interactions with coexistence theory built upon, which was further extended to positive interactions/multispecies with generalized LV models. Maybe add "and" in "originally developed to describe competitive [and] predator-prey interaction"?
We thank all reviewers for their constructive suggestions. A point-by-point response to the concerns raised by the reviewers is listed in the following. In addition to the changes suggested by the reviewer, we have moved the derivation of the linear relationship between growth rate and carrying capacity values from Supplementary Information to the main text, because this was a consistent suggestion in the feedback we received from our peers.
[Authors' response is labeled as blue. Modifications to the text are shown as quoted text.]

GENERAL ASSESSMENT
The manuscript is clear, well-written, free of errors as far as I can tell. As I am not an experimentalist, I will focus on the methodological and theoretical aspects of the work.
I think the main experimental idea, i.e. comparing growth in fresh media and cell-free spent media, is interesting and sound. The comparison to coculture experiments is what makes this work even more worthwhile in my eyes, and I really appreciate the range of information shown in Fig 5. The way of representing and testing the adequacy of the model, i.e. looking at the relationship between initial growth and final abundance in various fresh or spent media, is nice in its simplicity, though I note below that it is not strictly a mark of the Lotka-Volterra model specifically. I feel that I sincerely believe the authors' conclusions*, with a single true caveat:

Supplementary
In the current presentation, nothing really proves to me that the OD600 shown on many figures is indeed a "carrying capacity" --i.e. I am never shown the saturation of growth outside of the coculture experiments. It could well be that I am simply looking at a transient value of abundance in the exponential growth phase, in which case a positive correlation of abundance with growth rate is rather obvious. I do not really believe this to be the case, but it would be good to lay such a doubt to rest.
To conclude, I have a few recommendations, most importantly a figure demonstrating that it is reasonable to call these OD600 "carrying capacities", at least for a reasonable fraction of them if growth curves over time are not available for all.
* As a small note concerning my own bias, I am generally sympathetic to the idea that more complex environments can lead to simpler effective models because the idiosyncrasies of any given strain-metabolite interaction are less important in aggregation. This clearly makes me more likely to believe the authors' conclusion regarding differences between simple and complex media.
Thank you for your constructive feedback. You have a perfectly valid point regarding the link between OD600 and carrying capacity. We have included the growth curves ( Supplementary  Fig 1), along with the relationship between the OD and the actual cell numbers ( Supplementary  Fig 3) in the revised manuscript to give our readers the full picture. Other comments mentioned here are addressed below.

MAIN COMMENTS
1) Rather than just Lotka-Volterra, any model of the form where r_ij is the only factor being influenced by the fresh or spent medium, will always give proportionality between r and K, as the equilibrium abundance is

K_ij=r_ij/alpha
There is no necessity for r_ij to strictly be linear in K_j. To really test the Lotka-Volterra model, you would need to also vary K_j, i.e. do spent media experiments with various spent media from the same species (e.g. having started with various pHs), and show that r_ij is indeed r_ij = r_i -c_ij K_j with constant c_ij. This is not essential here, but it's worth considering that maybe a non-LV model compatible with the original observation would still work a bit better.
Thank you for bringing this to our attention. You are correct that there can be other models that would produce results consistent with the linear r-K relationship. As of now, our study is not ruling out that possibility. Our only definite claim is that if the linear relationship is contradicted, LV will not be the right choice. To address your point, we have added the following lines to our Discussions.
"We should emphasize that observing a linear relationship between the growth rate and carrying capacity is not sufficient to establish that a Lotka-Volterra model is appropriate for representing the community of such interactions-other models may produce the same relationship or higher-order interactions may make the multispecies LV inaccurate.
Instead we argue that if the linear relationship is not satisfied, LV will not be the right choice for representing the system." 2) Building on this, I want to propose a slightly different way of presenting things than Lipson's explanation proposed in discussion.
First, I would like to mention that I don't think the high-nutrient results ( Supp Fig 4 and 5) should be described in terms of correlations. The more striking fact is that there is little variance on the x-axis, while there is quite a lot of variance on the y-axis.
If there was little variation in both, you could have weak correlations simply due to noise and errors, but that would not say much. Here, the variance on the y-axis (and furthermore the ordering of points) mean that nutrient concentration is really doing something interesting to K.
The relationship between r and K can be clarified in the r-alpha formulation as above, i.e.

dS/dt = r S -alpha S^2
with K=r/alpha. it seems that --at low nutrient concentration, alpha is roughly constant while r varies, hence K ~ r --at high nutrients, alpha really varies with nutrient concentration, while r is practically fixed.
This would be compatible while various recent studies finding strong negative per-capita interactions overall at high nutrients, e.g. (2020)  Here this seems to be true for within-species interactions as well. (the difference with what is currently in the discussion is that there is no need to invoke a trade-off between r and K; r does not seem to be changing much anymore, except in a few instances --instead alpha is what really is impacted) Thank you for this insightful suggestion. Our understanding is that you are proposing a relationship between growth rates and carrying capacities, based on alpha. Even though mathematically alpha makes perfect sense as the second order term in the equation, we could not come up with a relatable interpretation of alpha. One interpretation of alpha is the per-cell rate of resource depletion, but we are not sure if that resonates with our typical readers with a microbiology background. For that reason, we have not explicitly applied this formulation in the rest of the manuscript to avoid confusion between different parameterizations (formulations with versus without alpha).

Ratzke et al
We think overall, an interpretation based on alpha is still in-line with Lipson's interpretation, with the basic idea that at low nutrient availability, all the resources are channeled into reproduction (fixed alpha), whereas at high nutrient levels when the growth rate approaches its maximum, the cell might take alternative "more wasteful" strategies or increase its energy expenditure to deal with waste/inhibition (Ratzke 2020, Li 2022). As a result, alpha will increase.
We have also included and discussed the two papers you have mentioned for completeness and to highlight why at high nutrient availability growth rates might drop (page 20 of the 'Marked Up' manuscript, first paragraph).
"There is also additional evidence that in nutrient-rich environments populations exhibit stronger inhibitory effects (74, 75), which can lead to a drop in growth rates in cocultures compared to monocultures and thus more deviation from LV equations." 3) the choice to re-adjust pH in the spent media is not really discussed although it means removing an important vector of species interactions, as obvious from Fig 4 and many other works (see e.g. Ratzke's paper above). I do not think it is a bad thing in itself --indeed it might favor the expression of other mechanisms of interaction --but it is worth noting that it is not a completely benign choice, especially as such adjustment was not done for cocultures and this may explain some of the diverging results.
More generally, thinking of interactions entirely in terms of what is left in spent media is anchored in a particular assumption about possible interaction mechanisms: it ignores any kind of interaction that would involve a more dynamical response of species' metabolisms to each other, or that is not mediated by metabolites (e.g. mechanical interactions). Thus, a quantitative disagreement with co-culture results could point to a whole range of possibilities.
Regarding the pH, you are correct that a change in pH can have a significant impact on other species. In fact, in our own data we have observed that pH, if not buffered, was one of the major drivers of interactions between our strains (data not shown). However, since the environment in the nose is fairly stable (although with a distinct spatial gradient), we chose to buffer our media to remove pH as a factor. The medium in our coculture experiments is also buffered to eliminate large shifts in pH as a source of discrepancy between CFSM and coculture data. We have updated the Methods section to make this clear.
We absolutely agree with you that spent media assessment of interactions misses a range of interactions, including those that depend on physical contact, those that are strictly triggered by the partner, and mechanical interactions as you mentioned. We have added a paragraph to make this point clear and to include some of the relevant citations.
"We note that a major motivation for assessing microbial interactions using CFSM experiments is the feasibility of performing these experiments in different contexts, only requiring the ability to measure growth properties in monocultures. However, these experiments will not comprehensively capture all possible interaction mechanisms. Notable interactions such as those that rely on physical/mechanical contact between cells or those that are triggered only when a partner is present will not be captured in basic CFSM experiments. Nevertheless, because many important interactions are represented in CFSM experiments, this approach is routinely used to assess microbial interactions (17,19,(52)(53)(54)." 4) Why not show results of Figure 2 as a matrix of interaction coefficients, instead of giving them only for the cocultured strains in Table 2? That would be an interesting result in itself for various readers (i.e. it provides an idea of the magnitude of competition or facilitation in these systems) Thank you for the suggestion. We agree with you about the importance of such a matrix. However, we have already published such results at 10% THY elsewhere (Dedrick et al., Frontiers in Microbiology, 2021; https://doi.org/10.3389/fmicb.2021.613109), so we decided not to repeat the table in this manuscript. Instead, we included plots such as Figure 2 for a visual representation of data that is more in line with the main conclusions of this manuscript.

5) In the main text you say
"we observe a strong linear and positive correlation between carbon concentrations and carrying capacity (Supplementary Figure 6), but not growth rate (Supplementary Figure 7)" I would not say that relationships in Sup Fig 7 are really that weak --in most other contexts, the R squared and p-values you are showing would be taken as good results. I would simply claim the relationship to carrying capacity is stronger (and even there you had to remove an inconsistent strain), but this particular experimental manipulation is likely one where both r and K or alpha are changing a bit. This is not at all a main result so it is not a problem of course.
We realized this was perhaps caused by our inaccurate language. In these experiments we observed that K was linearly and proportionally related to the total carbon concentration. In contrast, r deviated from being proportional to the total carbon concentration. We have updated the captions. Fig 10, I would note that Lotka-Volterra does allow a dependency on initial conditions in the case of mutual exclusion, and you would have a hard time properly fitting the coefficients and getting the right results in that case. But in any case, LV does not allow this at all when species coexist. I agree that a different model is clearly appropriate here.

6) Regarding Sup
We agree with your statement. However, the main point of this figure was to show that we cannot find a consistent LV model to represent different initial conditions. We do not think the issue here is that the populations are lop-sided, which would be the experimental challenge in mutual exclusion cases. Here, the measured populations are large enough to offer a reasonable model in each case, but those models are inconsistent because LV is not a suitable model. 7) Sup Fig 11 has honestly too little variation between media to really convince that the combined media behave like the average of the component media (rather than, say, like just one of the two component media, or like a randomly drawn spent medium) We agree with you that the evidence in the Supplementary Fig 11 (Supplementary Figure 10 in the revised version) was not strong, because the differences between the two components were only significantly different in a few cases. We have de-emphasized our related conclusions (page 18 of the 'Marked Up' manuscript, the last paragraph before Discussions) in the revised version (while keeping the data).
"Although these results need further investigations, in the majority of cases tested, the response to 1:1 combined CFSMs was indistinguishable from the average of responses to each of the CFSMs (Supplementary Figure 10)."

MINOR COMMENTS
Page 2: "To converge these distinct phenomenological and mechanistic inquiries" -> I am likely less of a proper English speaker than some authors, but this transitive use of the verb converge seems unusual.
Thank you. We have reworded the sentence by replacing "converge" with "reconcile" based on your suggestion. Figure 2: Clarify in caption that colors are the media, and panels are the strain being grown (it can be gathered from the fact that 10% THY is included as a color, but still, it is better made explicit) Thank you for the suggestion. We have now explicitly mentioned the colors and panels in the legends of Figures 2, 3, and 4.

Reviewer #3
I have read the paper entitled "When does a Lotka-Volterra model represent microbial interactions? Insights from in-vitro nasal bacterial communities" with interest.
The authors use the Lotka-Volterra (LV) framework to estimate interactions from cell-free spentmedia (CFSM) assays, and confront the estimated interaction with actual co-cultures experiments, using the nasal microbiota as an experimental system. The authors first derive a criteria to assess when the LV framework can be used. The growth rate and carrying capacity of monocultures (growth assays) must be linearly positively correlated in different environmental conditions. Then, they use CFSM data to estimate pairwise interactions, and compare these predictions to cocultures. They show that LV models fitted on CFSM accurately capture the interaction when the environment is complex and limited in nutrients.
This study and the question it raises are very interesting, as the use of the LV framework to model bacterial interactions has been recently debated, in particular, in terms of its predictive ability (as referenced in the introduction). The comparison between CFSM estimations and cocultures allows the authors to quantify this predictive power.
I do not have a lot of comments on the experimental parts (choice of medium, etc) but more on the assumptions of the modeling approach.
General comments: 1) I think there are two aspects to the study: a first one, regarding the use of LV framework to estimate interactions, and a second one, regardless of the modeling framework, regarding the use of CFSM vs cocultures. The first question is nicely introduced, and I would appreciate a bit more discussion or reference to the literature regarding the use of CFSM to estimate interactions, since it is then used as a reference to characterize the quality of the LV estimations, if I understood the approach correctly. What is the advantage of the CFSM method compared to directly fitting to the co-cultures or monocultures/co-cultures comparisons?
Thank you for the suggestion. We have expanded the manuscript in the revised version to mention some of the previous examples of the use of CFSM to estimate microbial interactions. This is a common approach because it is easy to implement. The only technical requirements for implementing such an assay are the ability to: (1) monitor the growth of individual species in monocultures, and (2) filter sterilize the spent media. Our choice of assay was highly driven by its simplicity, because we think this makes it the most accessible assay for a wide range of studies. We have included our motivation in the revised version.
"We note that a major motivation for assessing microbial interactions using CFSM experiments is the feasibility of performing these experiments in different contexts, only requiring the ability to measure growth properties in monocultures. However, these experiments will not comprehensively capture all possible interaction mechanisms. Notable interactions such as those that rely on physical/mechanical contact between cells or those that are triggered only when a partner is present will not be captured in basic CFSM experiments. Nevertheless, because many important interactions are represented in CFSM experiments, this approach is routinely used to assess microbial interactions (17,19,(52)(53)(54)." 2) I wonder about the link between the carrying capacity as defined here in the end of introduction ("to what extent bacteria grow"), and the final biomass/yield of a Monod growth model? This comment is linked to minor comment 3. Are these two concepts equivalent?
For the purpose of the discussions in this manuscript, we use the maximum OD within 48 hours of growth, adjust it using a calibration curve to account for the nonlinear relationship between OD and cell density, and use the adjusted value as an estimate of the carrying capacity. The value obtained from this approach is tightly linked to the carrying capacity defined in the basic logistic growth. The Monod growth model offers a relation between the growth rate and the concentration of a limiting resource. From the standard Monod equation, one can derive the carrying capacity as r/a*R0, where r is the maximum growth rate, R0 is the initial amount of the limiting resource, and a is the amount of R0 used to produce a cell. To the best of our knowledge, there is no direct link between the Monod coefficient (resource concentration at which growth rate reaches half of its maximum value) and the carrying capacity.
3) A related comment on the comparison with consumer-resource models: would it be possible to estimate the interactions from fitting a Monod consumer-resource model to the monocultures and predict the co-cultures pattern with this type of data? I would be interested to see maybe in a supplementary analysis what a consumer-resource model would do in terms of fitting on top of the different curves of figure 5. Also, what framework do the authors suggest to use when LV is less relevant (i.e. when the environment is rich in nutrients or not complex)?
Unfortunately, without a priori knowledge about the mechanisms of interactions, it is very difficult to construct a consumer-resource model based on coculture dynamics. There are many different ways to construct the model (the number of resources and their production/consumption links are the primary structural unknowns), and the measurements of population dynamics is not enough to adequately constrain the model. Thank you for bringing the link to our attention. We have incorporated this in our Introduction in our revised manuscript.
Minor comments: 1) Page 3: "This medium condition was selected since it moderately reduces the growth rate and the carrying capacity of each isolate, [...] " compared to what condition?
Compared to 100% THY, where the growth rate of species is close to its maximum possible value, at 10% THY, the growth rate is large enough to be measurable, but still significantly less than the maximum possible value. As a result, in CFSM experiments both increases and decreases in the growth rate (influenced by the CFSM of other species) can be reliably detected. This is stated in the Methods. The level of nutrients in 10% THY is also comparable to the concentrations directly measured in the nasal passage (Krismer et al., PLOS Pathogens, 2014; https://doi.org/10.1371/journal.ppat.1003862). This makes this choice relevant for investigations of microbial properties in vitro.

2) Page 8: K i and r i the "and " is in italics
We have fixed this in the revised version.
3) Page 8: The authors state that the formulation is equivalent to the LV model in which a ij = c ij r i / K i . I wonder if the requirement that growth rate and carrying capacity must be linearly correlated simply comes from the fact that in the logistic formulation the carrying capacity is the ratio between an intrinsic growth and a self-regulation term, so that the two parameters are not independent. For instance, considering the alternative formulation dS/dt = (r -a S) S, the equilibrium density (effective carrying capacity) is S* = r/a. If I use this formulation and follow the same steps as in the Supplementary Material, I find that "r 12 "/"K 12 " = a 11 which is equivalent to what the authors find and is the strength of intraspecific competition.
You are correct that r/K is the magnitude of the self-regulating term in the logistic formula. The relevance to LV equations comes into picture when growth in the cell-free spent medium of other species is considered. The same ratio of r/K is obtained for growth in the cell-free spent medium obtained from other species. This allows the simplification of considering the impact as a change in "habitat quality," regardless of which species is responsible for that change.
You are correct that a linear r-K relationship is not unique to the LV model. This was also mentioned by Reviewer #2, and we have added a paragraph to the Discussion section to explicitly clarify this point. 4) Beginning of the discussion: I think there are two "historical" LV set of equations, one for predator-prey interactions (the hare-lynx model with cycles), and one for competitive interactions with coexistence theory built upon, which was further extended to positive interactions/multispecies with generalized LV models. Maybe add "and" in "originally developed to describe competitive [and] predator-prey interaction"?
Thank you for bringing this typo to our attention. It is fixed in the revised version. Thank you for submitting your manuscript to mSystems. We have completed our review and I am pleased to inform you that, in principle, we expect to accept it for publication in mSystems. However, acceptance will not be final until you have adequately addressed the reviewer comments.
The reviewer comments, which you will find below, refer to 2 substantial concerns that need to be addressed before publication as they point out to misleading statements presented in the manuscript.
I would really appreciate if you could address this. I am sure you agree these changes will substantially improve the accuracy of your manuscript.
Thank you for the privilege of reviewing your work. Below you will find instructions from the mSystems editorial office and comments generated during the review.

Preparing Revision Guidelines
To submit your modified manuscript, log onto the eJP submission site at https://msystems.msubmit.net/cgi-bin/main.plex. Go to Author Tasks and click the appropriate manuscript title to begin the revision process. The information that you entered when you first submitted the paper will be displayed. Please update the information as necessary. Here are a few examples of required updates that authors must address: • Point-by-point responses to the issues raised by the reviewers in a file named "Response to Reviewers," NOT IN YOUR COVER LETTER. • Upload a compare copy of the manuscript (without figures) as a "Marked-Up Manuscript" file. • Each figure must be uploaded as a separate file, and any multipanel figures must be assembled into one file. • Manuscript: A .DOC version of the revised manuscript • Figures: Editable, high-resolution, individual figure files are required at revision, TIFF or EPS files are preferred ASM policy requires that data be available to the public upon online posting of the article, so please verify all links to sequence records, if present, and make sure that each number retrieves the full record of the data. If a new accession number is not linked or a link is broken, provide production staff with the correct URL for the record. If the accession numbers for new data are not publicly accessible before the expected online posting of the article, publication of your article may be delayed; please contact the ASM production staff immediately with the expected release date. I would suggest the following additional two minor modifications, which are simply trying to rectify statements that do not have much bearing on your main results, but are unfortunately incorrect and could be slightly misleading to your readers (not regarding the validity of your work, but more broadly). ================================================= 1) Most importantly, I had overlooked this previously, but you state: "If a LV representation of interactions is valid, we would expect the growth rate-carrying capacity ratio of an isolate grown in the presence of another isolate to be equivalent to that of the isolate grown in a monoculture." In fact that is not true in general, but only here because you are using spent media, so the focal isolate is not impacting the other one.
My recommendation: state that this proportionality of growth rate and carrying capacity with a constant slope is true here because of your experimental design using spent media, not a property of Lotka-Volterra in general. If species 1 is alone, and we write LV in the following way: dN1/dt = r_1 N_1 -A_11 N_1^2 K1 = r1 / A11 so r1/K1 = 1/A11 If the two species are changing in abundance together: then the initial growth rate of species 1 in presence of fully-grown species 2 (N2=r2/A22) would be R1 = r1 -A12 r2/A22 but species 1's carrying capacity in mixture would be K1 = (A22 r1 -A12 r2) /(A11 A22 -A12 A21 ) so R1/K1 is now obviously not at all simply 1/A11 anymore, but instead 1/(A11 -A12 A21/A22) (where you recover the original expression if species 2 is not impacted by species 1, i.e. A21=0) ================================================ 2) This is more minor, but in your reply (and relevant parts of the main text), you state "We think overall, an interpretation based on alpha is still in-line with Lipson's interpretation, with the basic idea that at low nutrient availability, all the resources are channeled into reproduction (fixed alpha), whereas at high nutrient levels when the growth rate approaches its maximum, the cell might take alternative "more wasteful" strategies or increase its energy expenditure to deal with waste/inhibition (Ratzke 2020, Li 2022). As a result, alpha will increase." My point was that it's not clear that you actually get an r/K *tradeoff* at high nutrients (only a few of your results are compatible with such a tradeoff, and then only because of a single outlier).
Perhaps what you see is still explainable with a biological mechanism such as that suggested by Lipson, but I think it should not be framed as a tradeoff between r and K, when K is changing a lot and r not so much without a clear negative correlation between the two.

Response to Reviewers' Comments:
We thank the reviewer for pointing out the inaccuracies in the language of the manuscript. We have incorporated their suggestions accordingly, as described in the following.
[Authors' response is labeled as blue. Modifications to the text are shown as quoted text.]

Reviewer #2
Dear Authors, Thank you very much for your responses and updates to the manuscript! I would suggest the following additional two minor modifications, which are simply trying to rectify statements that do not have much bearing on your main results, but are unfortunately incorrect and could be slightly misleading to your readers (not regarding the validity of your work, but more broadly). ================================================= 1) Most importantly, I had overlooked this previously, but you state: "If a LV representation of interactions is valid, we would expect the growth rate-carrying capacity ratio of an isolate grown in the presence of another isolate to be equivalent to that of the isolate grown in a monoculture." In fact that is not true in general, but only here because you are using spent media, so the focal isolate is not impacting the other one.
My recommendation: state that this proportionality of growth rate and carrying capacity with a constant slope is true here because of your experimental design using spent media, not a property of Lotka-Volterra in general. If species 1 is alone, and we write LV in the following way: dN1/dt = r_1 N_1 -A_11 N_1^2 K1 = r1 / A11 so r1/K1 = 1/A11 If the two species are changing in abundance together: dN1/dt = r1 N1 -A11 N1^2 -A12 N2^2 dN2/dt = r2 N2 -A21 N1^2 -A22 N2^2 then the initial growth rate of species 1 in presence of fully-grown species 2 (N2=r2/A22) would be R1 = r1 -A12 r2/A22 but species 1's carrying capacity in mixture would be K1 = (A22 r1 -A12 r2) /(A11 A22 -A12 A21 ) so R1/K1 is now obviously not at all simply 1/A11 anymore, but instead 1/(A11 -A12 A21/A22) "If a LV representation of interactions is valid, we would expect the growth rate-carrying capacity ratio of a focal isolate grown in the supernatant of another partner isolate to be equivalent to that of the focal isolate grown in a monoculture." ================================================ 2) This is more minor, but in your reply (and relevant parts of the main text), you state "We think overall, an interpretation based on alpha is still in-line with Lipson's interpretation, with the basic idea that at low nutrient availability, all the resources are channeled into reproduction (fixed alpha), whereas at high nutrient levels when the growth rate approaches its maximum, the cell might take alternative "more wasteful" strategies or increase its energy expenditure to deal with waste/inhibition (Ratzke 2020, Li 2022). As a result, alpha will increase." My point was that it's not clear that you actually get an r/K *tradeoff* at high nutrients (only a few of your results are compatible with such a tradeoff, and then only because of a single outlier).
Perhaps what you see is still explainable with a biological mechanism such as that suggested by Lipson, but I think it should not be framed as a tradeoff between r and K, when K is changing a lot and r not so much without a clear negative correlation between the two. Thank you for the suggestion. We agree with you that perhaps the current statement is too strong, given the data presented in this manuscript. We have revised the relevant sentences to keep the interpretations within the scope of our observations. "Indeed, when nasal isolates were grown in dilutions of a growth medium, we observed a positive growth rate-carrying capacity relationship at lower nutrient concentrations but not at higher nutrient concentrations. Our interpretation, consistent with Lipson's explanation, is that in low-nutrient environments, cells dedicate all of their resources to producing biomass, leading to a strong positive correlation between growth rate and carrying capacity. In nutrient-rich environments, growth rate no longer increases linearly with the nutrient availability and as a result the growth-rate carrying capacity trend deviates from a simple linear relationship."