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The authors have declared that no competing interests exist.

Phenomenological relations such as Ohm’s or Fourier’s law have a venerable history in physics but are still scarce in biology. This situation restrains predictive theory. Here, we build on bacterial “growth laws,” which capture physiological feedback between translation and cell growth, to construct a minimal biophysical model for the combined action of ribosome-targeting antibiotics. Our model predicts drug interactions like antagonism or synergy solely from responses to individual drugs. We provide analytical results for limiting cases, which agree well with numerical results. We systematically refine the model by including direct physical interactions of different antibiotics on the ribosome. In a limiting case, our model provides a mechanistic underpinning for recent predictions of higher-order interactions that were derived using entropy maximization. We further refine the model to include the effects of antibiotics that mimic starvation and the presence of resistance genes. We describe the impact of a starvation-mimicking antibiotic on drug interactions analytically and verify it experimentally. Our extended model suggests a change in the type of drug interaction that depends on the strength of resistance, which challenges established rescaling paradigms. We experimentally show that the presence of unregulated resistance genes can lead to altered drug interaction, which agrees with the prediction of the model. While minimal, the model is readily adaptable and opens the door to predicting interactions of second and higher-order in a broad range of biological systems.

Applying multiple antibiotics simultaneously can boost treatment effectiveness and aid against rampant antibiotic resistance. Because of the impractically large number of possible combinations of drugs, those that are effective are found by trial and error. Hence, a predictive theory to characterize drug cocktails would be of enormous value. Recently identified phenomenological laws ease the construction of predictive models of bacterial growth. Here, we build a model of the effects of antibiotic combinations on bacteria and show that it makes reliable predictions for experimental outcomes. Our model takes responses to individual drugs as inputs and predicts their combined effect. This output determines the type of drug interaction, which can range from antagonistic (the combined effect is weaker) to synergistic (the combined effect is stronger). We broaden the model by including the direct physical interaction on the target, drug resistance genes that alter the drug interaction, and drugs that mimic poor growth environments by choking the supply of growth-essential components, which we test experimentally. Our results prove how biophysical models that use empirical laws can predict responses to drug combinations. Importantly, such models can successfully predict mechanisms underlying interactions of drug combinations. This approach is extensible to combinations of more than two drugs and diverse biological systems.

Antibiotics are small molecules that interfere with essential processes in bacterial cells, thereby inhibiting growth or even killing bacteria [

The combined effect of antibiotics emerges from the complex interplay of individual drug effects and the physiological response of the cell to the drug combination. Drug interactions are determined by the combined effect of multiple drugs on cell growth and survival. These interactions are defined with respect to an additive reference. By definition, additive drugs act as substitutes for each other; synergy occurs if the combined effect of the drugs is stronger than in the additive reference case, and antagonism occurs if the combined effect is weaker. An extreme case of antagonism–termed suppression–occurs when one of the drugs loses its potency in the presence of the other drug,

By measuring the growth rate λ over a two-dimensional matrix of drug concentrations (_{A}, _{B}), the dose-response surface _{A}, _{B}) = λ(_{A}, _{B})/λ_{0} is obtained; here, λ_{0} is the growth rate in the absence of antibiotics. The dose-response surface can be characterized by the shape of its contours,

The dose-response surface is given by the relative growth rate _{A}, _{B}). Here, the concentrations are normalized such that _{i} = 1 corresponds to 50% growth inhibition in the presence of drug

Higher-order interactions, which occur when more than two drugs are combined, can be predicted to some extent using mechanism-independent models [

Apart from their clinical importance, antibiotics targeting the bacterial ribosome (translation inhibitors) are particularly well-suited for biophysical modeling since the physiological response to perturbations of translation can be described quantitatively using bacterial growth laws [

Here, we present a complete analysis of the biophysical model that predicts bacterial growth responses to combinations of translation inhibitors and its non-trivial theoretical predictions. Starting from responses to single antibiotics, we derive approximate analytical solutions of this model and investigate the effects of direct physical or allosteric interactions between antibiotics on the ribosome. We discuss several relevant extensions of the model, in particular (1) interactions with antibiotics that induce starvation, (2) the effects of resistance genes, (3) the correspondence to non-mechanistic models of interactions between more than two drugs, and (4) predictions for interactions of translation inhibitors with antibiotics that alter growth law parameters. We validate several non-trivial predictions made by the biophysical model in experiments.

First, we recapitulate the biophysical model for a single translation inhibitor [^{3} [_{t} = 0.06 ^{−1}h^{−1}, _{u} and _{min} = 19.3 ^{−1}. However, when the growth rate is lowered by addition of a translation inhibitor in a constant nutrient environment, the total ribosome concentration _{tot} and growth rate become negatively correlated [_{max} = 65.8 _{max} − _{min} = 46.5 _{b} is the concentration of antibiotic-bound ribosomes, and λ_{0} is the maximal growth rate in the absence of antibiotics [

(A) Schematic of processes captured by the model. Ribosomes (double ovals) are synthesized with the rate _{u}). Unbound ribosomes contribute to growth. Antibiotics enter the cell (_{ex} → _{on} and _{off}, respectively. Bound ribosomes (_{b}) do not contribute to growth [_{min}) and maximal ribosome concentration (_{max}), respectively, which are Δ

When antibiotics enter the cell, they can bind to ribosomes. The net rate of forward and reverse binding of antibiotics to the ribosome is given by _{u}, _{b}, _{on} _{u} − _{min}) + _{off} _{b}, where _{off} and _{on} are first and second order rate constants, and _{u} − _{min}) can be bound by the antibiotics [

The intracellular antibiotic concentration is affected by the kinetics of antibiotic entry into the cell, which is given by _{ex}, _{in}_{ex} − _{out} _{ex} is the extracellular antibiotic concentration. Typical influx and efflux rates, _{in} and _{out}, for different translation inhibitors range from 1 − 1000 h^{−1} and from 0.01 − 100 h^{−1}, respectively. Typical rates of forward and reverse binding, _{on} and _{off}, are around 1000 ^{−1}h^{−1} and between 0 − 10^{5} h^{−1}, respectively [_{off} = 0 corresponds to antibiotics with effectively irreversible binding such as streptomycin [_{tot} = λ{_{max} − λΔ_{0} − 1/(_{t}Δ_{D} = _{off}/_{on}, and _{ex} = _{50}, λ = _{0} and _{50} is the extracellular antibiotic concentration that leads to 50% growth inhibition, a common measure of drug sensitivity (

Since

Steep dose-response curves (_{crit}) occur for antibiotics with tight binding to the ribosome (_{D} → 0) or inefficient efflux (_{out} → 0). Alternatively, if these two quantities are growth-rate invariant, dose-response curves become steeper with increasing growth rate in the absence of drug, as _{0}. For typical values of the relevant parameters (discussed above),

When combinations of two different translation inhibitors are present, each ribosome can be bound by either of them alone or by both simultaneously. To generalize the model described in the previous section to this situation, we need to introduce additional populations of ribosomes. Extending the mathematical model [_{i}(_{u}, _{b,i}, _{i}) and _{i}(_{ex,i}, _{i}) describe the first binding step and membrane transport of antibiotic _{off,i} _{off,i} _{d} and _{d} and the binding of antibiotic _{σ,i} with _{σ,i} = 1. When both antibiotics compete for the same binding site on the ribosome, _{on,i} = 0. In general, the parameters _{σ,i} can vary continuously to capture any changes in ribosome binding of one antibiotic due to the binding of the second, as long as the binding is described by mass-action kinetics.

What is the main consequence of including the double-bound ribosomes? Below, we show that in the absence of double-bound ribosomes, drug interactions are generally expected to be additive. If we assume that no double-bound ribosomes can form, _{on,i} = 0, _{u} = λ/_{t} + _{min} is constant. This implies that the total concentration of ribosomes bound by either antibiotic (_{b} = _{b,A} + _{b,B}) remains constant for all different concentration pairs (_{A}, _{B}) along the isobole. In steady state, the concentration of ribosomes bound by antibiotic _{b,i} = _{i} × _{i}, where _{i} = _{on,i}λ/[(_{off,i} + λ)_{t}]. The bound ribosome concentration reads
_{i} as a function of _{ex,i} from

The proportionality constant ϒ_{i} in this expression depends only on λ and kinetic parameters; in particular, it is independent of the concentration of the other antibiotic. Since _{b} = _{b,A} + _{b,B} = _{A} _{A} + _{B} _{B} [_{i} _{i} are independent of the other antibiotic. This argument shows that additivity generally occurs when double-bound ribosomes cannot form. Additionally, this confirms that the model correctly predicts additivity when the antibiotic is combined with itself–double-bound ribosomes cannot form in this case.

In the limit where _{off}, _{out} ≫ λ, _{A} + _{B} = λ_{0}/λ − 1, where _{i} = _{ex,i}/IC_{50,i}. To derive this expression, we noted the definitions of

To study the effect of double-bound ribosomes (_{add} is the response surface of the additive expectation, which is calculated directly from the responses to the individual drugs (see _{A}, _{B} of the two antibiotics that are combined, we determined the complete phase diagram of drug interactions (

(A) Schematic: Ribosomes already bound by a single antibiotic (black and white circles) can be bound by another one. If the binding is independent of the presence of an already bound antibiotic, the second binding step follows the same kinetics as for a single antibiotic. (B) Examples of dose-response curves of different steepness and corresponding dose-response surfaces calculated from the model. Top: Dose-response curves with low or high _{A}, _{B}; white dashed line shows additive interactions (

The fact that combinations of antibiotics that bind the ribosome irreversibly or are poorly pumped out of the cell yield antagonism can be understood intuitively. Consider a situation in which the antibiotics A and B are added to a bacterial population such that the concentration of antibiotic A far exceeds the concentration of antibiotic B. If at some point the majority of ribosomes is irreversibly bound by antibiotic A (due to its high intracellular concentration and/or irreversible binding), then antibiotic B is likely to bind to already inactivated ribosomes as well. Irreversibly-bound ribosomes thus effectively act as a “sponge” that soaks up antibiotics which can then no longer contribute to growth inhibition–a situation that results in antagonism.

What causes the transition from antagonism to synergy as _{D} _{out} → ∞). If the growth rate is low due to inhibition, then _{tot} ≈ _{max} as ribosome synthesis is upregulated to its maximum. In this case, the typical rate of dilution by growth is much slower than that of antibiotic-ribosome binding and we obtain _{i} ≈ _{ex,i} _{in,i}/_{out,i}. In this regime, we can derive an approximate solution that yields a synergistic dose-response surface, supporting the conclusion that qualitatively changing the binding kinetics alters the drug interaction type.

The system becomes linear and analytically solvable (see _{0} = λ_{max} and _{A}, _{B} as arguments, _{A})(1 + _{B})], which is independent of λ_{0} and

(A) Comparisons of numerically calculated dose-response surfaces and approximate solution. Purple isoboles (dashed and solid lines correspond to 50% and 20% relative growth rate, respectively) show the approximate solution on top of the dose-response surface calculated from the biophysical model (gray scale). Examples are shown for two pairs of antibiotics with identical (left) or different

The results above can be experimentally verified. To predict the entire dose-response surface, we require only the response parameters

We next asked how more general binding schemes, in which two different antibiotics can directly interact on the ribosome to stabilize or destabilize their binding affect the resulting drug interactions. Two antibiotics do not need to come into direct, physical contact to affect each other’s binding: Allosteric effects (

(A) Schematic of antibiotics symmetrically affecting their binding on the ribosome. (B) Changes in the shape of the dose-response surfaces for pairs of antibiotics with (i) identical ^{−5} and (ii) identical ^{2}, when _{A} = _{B}) through the phase diagram for different

We focused on pairs of antibiotics in which both drugs either have low or high _{on,i} = _{off,i} = 1 (

Experimentally, these results suggest that the presence of direct interactions on the ribosome between bound antibiotics would manifest in stronger synergy than predicted for independent binding if the response parameters of the combined antibiotics are sufficiently high (

To corroborate these numerical results, we investigated the limit of reversibly binding antibiotics with rapid binding kinetics at low growth rates as for _{off,i} = 0), _{on,i} = 0, yields
_{on,i} = _{off,i} = 1 the expression in

More generally, direct interactions between the antibiotics on the ribosome could be asymmetric. For example, binding of only one of the antibiotics could trap the ribosome in a conformation that facilitates the binding of the other antibiotic but not _{off,i} = 1 and varied _{on,i} for antibiotics with different response parameters

(A) Dose-response surfaces for different instances of asymmetric direct interaction and response parameters; insets on top show schematics of the type of direct interaction and top-right symbols correspond to those in (B). Antibiotics with shallow (_{A} = 2^{2}) and steep (_{B} = 2^{−3}) dose-response curves are shown by black and white disks, respectively. Left: Antagonism occurs when an antibiotic with a steep dose-response asymmetrically hinders the binding of another one with a shallow dose-response, which in turn promotes the binding of the former. Middle: Symmetrizing the direct interaction almost completely abolishes antagonism. Right: Inverting the scenario from the left-most panel results in mild synergy. (B) Phase diagram of drug interactions for asymmetric direct interactions between antibiotics with different response parameters. Different response parameters (_{A} = 2^{2} and _{B} = 2^{−3}) profoundly affect the resulting drug interaction: A continuous transition from antagonism to synergy occurs (white dashed line denotes

When antibiotics with identical response parameters _{on} enhances the drug interaction. For combinations of antibiotics with different dose-response curve shapes, asymmetric direct interactions on the ribosome result in a different behavior (

The biophysical model described above can predict the pairwise drug interactions that are needed to apply recently proposed mechanism-independent models for higher-order drug interactions [_{i}) and their pairwise combinations (_{ij}):

In the limit of slow growth used previously, it is straightforward to analyze the effects of higher-order drug combinations. The approximate results below are based on the assumptions that: (i) the growth rate is directly proportional to the concentration of unblocked ribosomes, (ii) growth rate is nearly zero, (iii) the intracellular drug concentration depends only on transport kinetics (_{ex} _{in}/_{out}), and (iv) the growth rate is directly proportional to the concentration of unblocked ribosomes (_{i} and pairwise combinations _{ij}. We obtain responses

(A) Assumptions that simplify the system to allow obtaining a closed solution. (B) Binding kinetics diagram shows allowed transitions between ribosome subpopulations. Different symbols on the ribosomes denote different antibiotics.

Next, we tested if the mechanism-independent formula for three drugs [

The assumptions described above can be generalized to antibiotics with other modes of action, provided that the combined antibiotics bind to the same target. For example, if growth is limited by a specific enzyme due to antibiotic inhibition, we can consider the growth rate to be proportional to the abundance of this limiting enzyme. If the enzyme concentration does not change, the approximative mathematical framework from this section is applicable to antibiotics targeting this enzyme. These results provide a potential mechanistic explanation for the apparent validity of the mechanism-independent model, at least for combinations of antibiotics binding the same target.

Other phenomena than those treated so far can shape drug interactions. Below, we discuss two cases in which (i) antibiotics perturb translation in orthogonal ways and (ii) the expression of antibiotic resistance genes alters a drug interaction. While certainly not exhaustive, these two cases illustrate relevant extensions of the model.

Translation inhibitors target the protein synthesis machinery, which is carefully regulated in response to changes in the nutrient environment [

Bacterial growth strongly depends on the availability and quality of nutrients. Protein synthesis requires that amino acids are delivered to the translation machinery (ribosomes) by dedicated proteins [elongation factors (EF-Tu)] [

(A) Schematic: Simplified regulation of translation coordination. Nutrients are transported into the cell, where they serve as a source of amino acids. These amino acids are required for tRNA charging. Oversupply of amino acids leads to down-regulation of the nutrient transport and processing machinery, and depletion of the intracellular signaling molecule ppGpp (guanosine tetraphosphate). This in turn de-represses the expression of the translation machinery, which increases the overall translation capacity, leading to faster growth. In contrast, if amino acids are in short supply, the translation machinery is down-regulated. (B) Translation inhibitors (TI) inhibit progression of the ribosome, while a starvation-mimicking antibiotic (SMA) perturbs the amino acid supply. The ribosome progresses along the mRNA (black wavy line), if charged tRNAs (black fork with gray circle) deliver amino acids (gray circles) at a sufficient rate to support the rapid synthesis. A starvation-mimicking antibiotic inhibits tRNA charging and thus mimics amino acid depletion, a hallmark of starvation. (C) Dependence of relative change in IC_{50} on SMA inhibition (_{50}/IC_{50,F}). Example solutions of _{F} = 1.04, white circles show experimental data) and streptomycin (STR; bottom solid line with _{F} = 0.46, gray squares show experimental data). The gray areas correspond to the confidence intervals as obtained from standard errors in _{s}). Error bars correspond to standard errors in IC_{50} as obtained from fitting; where error bars are not visible, they are smaller than the symbol size. (D) Measured (top) and predicted (bottom) dose-response surfaces for CHL-MUP (left) and STR-MUP (right). Insets show scatter-plots of predicted and measured non-zero growth rates.

We can capture the effect of an SMA in our model and thus make predictions for the drug interactions between an SMA and translation inhibitors. To this end, we assume that the growth rate in the absence of drug λ_{0}, which characterizes the quality of the nutrient environment in _{s} of the SMA only. Under this assumption, the growth rate in the simultaneous presence of an SMA and a translation inhibitor can be derived directly from the previous results for a single antibiotic [_{s}). Since IC_{50} ∝ (^{2} + 1)/_{F}/_{s}), the relative change in IC_{50} at constant SMA concentration becomes
_{F} and IC_{50,F} are _{50} in the absence of the SMA, respectively. It follows that _{F} > 1; this condition is obtained by solving ∂_{g} _{s})≤1. If _{F} ≤ 1, then the minimal _{s}) = _{F}. We further note that two functional limits exist: in the limits _{s}) and _{s}), respectively.

The dose-response curve for a single antibiotic is given by _{50} [_{s}), we can evaluate the entire dose-response surface:

A specific example of an SMA is the antibiotic mupirocin (MUP), which reversibly binds to isoleucin tRNA synthetase and prevents tRNA charging [_{50} of _{s}) for two translation inhibitors: chloramphenicol (CHL) and streptomycin (STR); see

Our results show that the steepness of the dose-response curve and the coupling between growth laws and antibiotic response play a key role in determining drug interactions. Dose-response curve steepness can change if genes that convey antibiotic resistance are present [

Bacterial resistance genes often code for dedicated enzymes that degrade the antibiotic or pump it out of the cell. Resistance genes can be constitutively expressed, _{max} = Δ_{t}, _{0}/λ_{max}) (_{0} (_{rem} is a Michaelis-Menten dissociation constant and _{max} is the maximal antibiotic removal rate, which bundles the maximal enzyme abundance and catalytic rate per enzyme. In

(A) Expression of unregulated (constitutive) gene depends on nutrient quality and degree of translation inhibition. Top: When growth is varied by nutrient quality, the expression of the gene decreases with increasing growth. The highest expression achieved in the limit of low growth rates is determined by _{max} (different shades of gray). When λ_{0} = λ_{max}, expression ceases, invariantly of _{max}. Bottom: In a fixed nutrient environment (circles), expression decreases as the growth rate decreases upon translation inhibition. The expression in the absence of antibiotic is _{0} (white disk)]. (B) Schematic of positive feedback loop for unregulated antibiotic resistance gene. A drug-resistance enzyme degrades the antibiotic, thus reducing growth inhibition and boosting its own expression. However, if the antibiotic concentration exceeds the capacity of removal by the enzyme, growth rate starts to drop and so does the expression of the resistance enzyme, amplifying the growth rate drop. The lightly drawn part (right) illustrates how two antibiotics can get coupled via the growth-rate dependent loop. (C) Examples of dose-response curves in the presence or in the absence of a constitutively expressed resistance gene (CERG). Black line shows dose-response curve for ^{−1}, _{rem} = 0.1 ^{−1}. Concentration axes were rescaled with respect to the increased IC_{50}. Note the qualitative change in dose-response surface shape.

Due to the linear relation between growth rate and the expression of the resistance gene, the rate of antibiotic removal decreases with decreasing growth rate under translation inhibition. This constitutes a positive feedback loop that leads to growth bistability (_{rem} ≫

By extending this scenario to a pair of antibiotics, we can directly test how the presence of resistance genes affects drug interactions. In the most relevant case, there are two CERGs each of which specifically provides resistance to one of the antibiotics. For simplicity, we assume that there is no cross-resistance,

To test this prediction, we constructed an

(A) Schematic showing two common resistance mechanisms: Resistance can result from degradation of the drug [left: chloramphenicol acetyltransferase (CAT) degrades chloramphenicol (CHL)] or from drug efflux [right: an antibiotic efflux pump (TetA) removes tetracycline (TET) from the cell]. (B) Change in dose-response curve shape due to a constitutively expressed resistance gene. CHL dose-response curves of sensitive (white circles) and resistant strain (gray circles). (C) Measured CHL-TET dose-response surfaces for (i) sensitive and (ii) resistant strain. Concentrations were normalized to the IC_{50} of respective strains. The strain with CERGs is 50.5 and 91.5 times more resistant to TET and CHL, respectively, as measured by increase of IC_{50}. Drug interaction changes from additive to antagonistic as suggested by theory (

In future work, this framework could be expanded to include resistance mechanisms other than the efflux and degradation of the drug. Other resistance mechanisms include target modification, overproduction of a target mimic (decoy), and factor-associated protection [_{0}, the expression of CERG should increase and thus its effects should manifest. If λ_{0} → λ_{max}, then the effect of CERG should be less prominent and the drug interaction should resemble the WT one. While these extensions are of high basic and clinical importance, they are outside of the scope of this study. Here we included only the best-characterized examples that required minimal genetic intervention in the system.

We constructed a minimal biophysical model of antibiotic interactions that takes into account the laws of bacterial cell physiology. Most parameters in our model are constrained by established results or by the dose-response curves of the individual antibiotics that are combined (

Our work highlights the advantages of a physiologically relevant “null model,” which captures all effects that are generally relevant for ribosome-binding antibiotics without trying to describe any molecular details of specific antibiotics (

We showed that direct physical (or allosteric) interactions of antibiotics on their target do not necessarily lead to synergy (

The predictions of our model are directly testable in experiments (shown above and in Ref. [

Discrepancies between experimental results and model predictions can expose cases in which more complicated mechanisms cause the observed drug interaction [

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B.K. is thankful to C. Guet for additional guidance and generous support which rendered this work possible. We thank all members of Guet, Bollenbach, and Tkačik groups for many helpful discussions and sharing of laboratory resources. B.K. additionally acknowledges the tremendous support from A. Angermayr and K. Mitosch with experimental work. We thank M. Hennessey-Wesen and M. Zagórski for constructive comments on the manuscript.

Dear Dr. Bollenbach,

Thank you very much for submitting your manuscript "Minimal biophysical model of combined antibiotic action" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

Both the reviewers found the paper interesting, correct, and timely. They included a list of suggestions for minor modifications. In addition, as suggested by the second reviewer, it is essential that the authors clearly explain why they have not tested experimentally some of the predictions.

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Both the reviewers found the paper interesting, correct, and timely. They included a list of suggestions for minor modifications. In addition, as suggested by the second reviewer, it is essential that the authors clearly explain why they have not tested experimentally some of the predictions.

Reviewer's Responses to Questions

Reviewer #1: This is an interesting and significant manuscript which presents and analyses a minimal model for the combined action of pairs of ribosome targeting antibiotics on a bacterial cell. Understanding the combined action of antibiotics is crucially important since many clinical therapies involve combinations of antibiotics, and it is well known that interactions can be either synergistic or antagonistic, or even suppressive. However, there are few quantitative predictive models for drug interactions (or even for the action of individual antibiotics). Therefore this work makes an important contribution to the field. The work presented is careful, thorough, mostly clearly described, and supported by some original experimental data. I support its publication in PLoS Computational Biology, although I have some suggestions for the authors to consider.

Suggestions:

1. The caption of Fig 2 should cite Scott et al (2010) [9] and Greulich et al (2015) [10].

2. It would be useful to give the expression for the bound ribosome concentration rb before Eq (2), since that is needed in Eq (7).

3. On line 94, ref [10] is not appropriate (just ref. [9])

4. There are blank spaces in place of section numbers in several places, eg before Eq 6, line 305, line 480.

5. Lines 151-153 state that “in general the parameters delta can vary continuously to capture any change in ribosome binding of one antibiotic due to binding of the second” – I don’t think this is quite true. For example if the binding becomes non-linear this would not be captured?

6. In lines 159-163 the wording is confusing where it refers to “bound to one antibiotic” and “single-bound antibiotic” – here we are anyway only talking about the case of single binding?

7. Before Eq (7) the parameter xi appears with no explanation, and in Eq (7) the expression for rb = Delta r(1-lambda/lambda0) appears without explanation – this should have been explained explicitly before Eq (2). Also it should be made clear that the final relation applies only along the isobole.

8. In Eq (8) it should be made clear this in the steady state. Also after Eq (8) it was not clear to me what is meant by the proportionality constant.

9. Eq (8) is not actually an equation for the isobole (that would be an equation that links the two antibiotic concentrations for fixed growth rate). I think the statement “Eq (8) corresponds to a linear isobole” would be clearer if an actual equation for the isobole can be stated.

10. Lines 171-173 seemed a little out of place and perhaps distracting.

11. Lines 187-188 a sentence or two to describe Fig 3c in more detail might be useful for clarity.

12. Lines 189-196: the wording here implies a time-line (antibiotic 1 binds before antibiotic 2) that is not accurate: the model does not specify which antibiotic binds first. The explanation here could be rewritten to reflect this.

13. Lines 203-204 the wording is a little hard to follow, e.g. “growth is inhibited close to zero” – what is the assumption here, that the antibiotic concentration is low so there is little inhibition?

14. In the caption of Fig 4b, it states that the dashed line is for antibiotics with different alpha values. But the axis label seems to imply the alpha values are the same.

15. Lines 225-230 the statement “it is unclear if mutual stabilization of binding necessarily leads to synergy” is not backed up by the text that follows. Actually it seems to imply that mutual stabilization of binding does lead to synergy; in contrast, it is if the two antibiotics mutually destabilize that the interaction is no longer synergistic.

16. Lines 324-326 – this statement that the model can be generalized to other modes of antibiotic action is not clear to me. The model is based around the growth rate-dependent regulation of ribosome abundance. It is not obvious to me that it can easily be generalized to other modes of action. I feel that this statement is interesting but deserves a more in-depth justification, for example in the discussion section.

17. Lines 329-330 The statement that the growth rate is invariant of the target details as it can be recovered from first order kinetics is unclear to me. What is meant by first order kinetics here? In my view the core of the model is specific to ribosome-binding antibiotics.

18. Line 377 onwards: it seems important to clarify here that the experiments were done with E. coli (rather than S. aureus which is mentioned early in the paragraph).

19. Lines 396-7 “recent work” – it is perhaps not that recent (2013).

20. Line 428 The word “drastic” here is perhaps overkill since the previous sentence states that the effect is “slight”.

21. The result that the presence of resistance genes can qualitatively change the nature of the drug interaction is very nice and could perhaps be made more prominent in the abstract (it is there but is rather buried under other things).

22. I also wonder whether how this phenomenon (effect of resistance genes) depends on the growth rate?

Reviewer #2: The manuscript studies a mathematical model for the combined action of two or more growth-inhibiting antibiotics. In these combinations at least one of the studied antibiotics is a translation-inhibiting antibiotic therefore the authors base their model on a previous model (Greulich et al, 2015) for the mode of action of a translation-inhibiting antibiotic, which takes into account the coupling between antibiotic action and bacterial growth.

The authors predict the interaction of antibiotics (antagonistic, independent, or synergistic) for different combinations of antibiotic types. In particular, for antibiotics which simultaneously bind to the ribosome, they study how binding interactions between antibiotics on the same ribosome (e.g. of allosteric origin) affect the combined drug efficacy. In that scenario, the combined effect is explicitly modelled as an extension of the model (Greulich et al. 2015), and it is predicted that the combined drug effects -- antagonistic or synergistic -- are enhanced if drug molecule binding is synergistic, and weakened if drug molecule binding is antagonistic. This is surprising when considering the naive view that synergistic drug binding should generally lead to a synergistic drug effect. This is an interesting prediction worth to be tested experimentally.

It is further studied how translation-inhibiting antibiotics interact with starvation-mimicking antibiotics and how the presence of resistance genes affects the drug interaction. Those predictions are also tested experimentally in bacterial growth assays where antibiotics are applied and genetically modified bacterial strains are used to study the effect of resistance genes.

This study is a very interesting and solid piece of work, with a thorough, comprehensive study of a realistic model (partially confirmed in the past) and experimentally testable predictions. Some of these predictions are directly tested experimentally in this work. Predictions about the effect and interaction of antibiotics (e.g. dose-response curves) as done in this work are of high medical value.

The only concern I have is that the authors decided to test some of their predictions experimentally, but others not. While it is generally a plus of this study that model predictions are experimentally tested, there is the risk of a confirmation- or selective bias if only some predictions are experimentally tested. It is therefore essential that the authors give a clear rationale why other predictions have not been tested experimentally, and argue thoroughly that no selection bias occurs. Of course, if possible, doing the corresponding experiments for those predictions would be preferable and strengthen this work even further.

Further minor comments:

1. line 105: For completeness, it should be said that the form if s(lambda) stated here is only valid in the steady state (the steady state is only mentioned in the following sentence).

2. below Eq. 6, index are sometimes named "A" and "B" and sometimes "1" and "2". It would be good to be consistent in the notation here.

3. line 199: it is stated that large alpha means that binding is more reversible, however, it could also mean that antibiotic outflux of the cell is large.

4. The authors prefer using Loewe's definition of drug interactions (antagonistic/independent/synergistic). Please argue why this definition is preferred, since in the following it is shown that this crucial classification depends sensitively on this definition. In particular, for the limit studied in Eq. (10), the interaction is synergistic in Loewe's view, but independent in Bliss' view, so the terms "synergistic" and "independent" are just semantics at this point.

5. line 323: It is stated that Eq. 16 is consistent with Eq. 14. This does not become clear at this point. Could the authors show this at this point or refer to an Appendix?

6. In Fig. 7c there are substantial deviations between experimental data and model predictions for streptomycin. Please comment on this deviation.

7. At several points in the text, references to Appendix 5 are made for experimental details, but these are actually given in Appendix 6.

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Large-scale datasets should be made available via a public repository as described in the

Reviewer #1: Yes

Reviewer #2:

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Dear Dr. Bollenbach,

We are pleased to inform you that your manuscript 'Minimal biophysical model of combined antibiotic action' has been provisionally accepted for publication in PLOS Computational Biology.

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As suggested by reviewer 1, please make sure that the experimental data are available.

Reviewer's Responses to Questions

Reviewer #1: The authors have addressed all my comments and I am happy to recommend publication.

Reviewer #2: The authors have addressed all my remarks satisfactorily. They explain well their choice of experiments and why the seen deviations between experiments and data are to be expected.

The authors state that "All relevant data are within the manuscript and its Supporting Information files", however, I cannot find the numerical values of the data. I do understand that all data is shown in the figures, but for reproducibility it would be good to have the experimental data also in numerical form, e.g. in a data table.

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Large-scale datasets should be made available via a public repository as described in the

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Reviewer #1: No

Reviewer #2: No

PCOMPBIOL-D-20-01159R1

Minimal biophysical model of combined antibiotic action

Dear Dr Bollenbach,

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