Understanding and predicting evolutionary dynamics is an ongoing challenge across all fields of biology. Microbial populations offer relatively simple model systems for investigating adaptation on multiple length scales, from the molecular to the population level, on timescales ranging from a few generations to decades. Yet even these simplest of systems exhibit rich and often counterintuitive evolutionary dynamics. Despite enormous progress, both theoretical and experimental, predicting evolution remains exceedingly difficult, in part because it is challenging to identify the phenotypes, selective gradients, and environmental factors shaping adaptation. In turn, controlling those dynamics – for example, by judicious manipulation of environmental conditions – is often impossible. These challenges represent open theoretical questions but also underlie practical public health threats – exemplified by the rapid rise of antibiotic resistance (Davies and Davies, 2010; Levy and Marshall, 2004) – where evolutionary dynamics are fundamental to the challenge, and perhaps, to the solution.

In recent years, evolution-based strategies for impeding drug resistance have gained significant attention. These approaches have identified a number of different factors that could modulate resistance evolution, including spatial heterogeneity (Zhang et al., 2011; Baym et al., 2016a; Greulich et al., 2012; Hermsen et al., 2012; Moreno-Gamez et al., 2015; Gokhale et al., 2018; De Jong and Wood, 2018; Santos-Lopez et al., 2019); competitive (Read et al., 2011; Hansen et al., 2017; Hansen et al., 2020), cooperative (Meredith et al., 2015b; Artemova et al., 2015; Sorg et al., 2016; Tan et al., 2012; Karslake et al., 2016; Yurtsev et al., 2016; Hallinen et al., 2020), or metabolic (Adamowicz et al., 2018; Adamowicz et al., 2020) interactions between bacterial cells; synergy with the immune system, especially in the context of adaptive treatment (Gjini and Brito, 2016); epistasis between resistance mutations (Trindade et al., 2009; Borrell et al., 2013; Lukačišinová et al., 2020); plasmid dynamics (Lopatkin et al., 2016; Lopatkin et al., 2017; Cooper et al., 2017); precise tuning of drug doses (Lipsitch and Levin, 1997; Yoshida et al., 2017; Meredith et al., 2015a; Nichol et al., 2015; Fuentes-Hernandez et al., 2015; Coates et al., 2018; Iram et al., 2021); cycling or mixing drugs at the hospital level (Bergstrom et al., 2004; Beardmore et al., 2017); and statistical correlations between resistance profiles for different drugs (Imamovic and Sommer, 2013; Kim et al., 2014; Pál et al., 2015; Barbosa et al., 2017; Rodriguez de Evgrafov et al., 2015; Nichol et al., 2019; Podnecky et al., 2018; Imamovic et al., 2018; Barbosa et al., 2019; Rosenkilde et al., 2019; Apjok et al., 2019; Maltas and Wood, 2019; Maltas et al., 2020; Hernando-Amado et al., 2020; Roemhild et al., 2020; Ardell and Kryazhimskiy, 2020).

Drug combinations are a particularly promising approach for slowing resistance (Baym et al., 2016b), but the evolutionary impacts of combination therapy remain difficult to predict, especially in a clinical setting (Podolsky, 2015; Woods and Read, 2015). Antibiotics are said to interact when the combined effect of the drugs is greater than (synergy) or less than (antagonism) expected based on the effects of the drugs alone (Greco et al., 1995). These interactions may be leveraged to improve treatments – for example, by offering enhanced antimicrobial effects at reduced concentrations. But these interactions can also accelerate, reduce, or even reverse the evolution of resistance (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008; Pena-Miller et al., 2013; Dean et al., 2020), leading to tradeoffs between short-term inhibitory effects and long-term evolutionary potential (Torella et al., 2010). In addition, resistance to one drug may be associated with modulated resistance to other drugs. This cross-resistance (or collateral sensitivity) between drugs in a combination has also been shown to significantly modulate resistance evolution (Barbosa et al., 2018; Rodriguez de Evgrafov et al., 2015; Munck et al., 2014).

Collateral effects (Pál et al., 2015; Roemhild et al., 2020) and drug interactions (Bollenbach et al., 2009; Chevereau et al., 2015; Lukačišin and Bollenbach, 2019; Chevereau et al., 2015), even in isolation, reflect interactions – between genetic loci, between competing evolutionary trajectories, between chemical stressors – that are often poorly understood at a mechanistic or molecular level. Yet adaptation to a drug combination may often reflect both phenomena, with the pleiotropic effects that couple resistance to individual drugs constraining, or constrained by, the interactions that occur when those drugs are used simultaneously. In addition, the underlying genotype space is high-dimensional and potentially rugged, rendering the genotypic trajectories prohibitively complex (de Visser and Krug, 2014).

In this work, we attempt to navigate these obstacles by modeling evolutionary dynamics on lower-dimensional phenotype spaces that leverage scaling relations between the drug-response surfaces of ancestral and mutant populations. Our approach is inspired by the fact that multiobjective evolutionary optimization may occur on surprisingly low-dimensional phenotypic spaces (Shoval et al., 2012; Hart et al., 2015). To develop a similarly coarse-grained picture of multidrug resistance, we associate selectable resistance traits with changes in effective drug concentrations, formalizing the geometric rescaling assumptions originally pioneered in Chait et al., 2007; Hegreness et al., 2008; Michel et al., 2008 and connecting evolutionary dynamics with a simple measurable property of ancestral populations. We show that evolved resistance to the component drugs – and in turn, the adaptation of growth rate – is governed by a Price equation whose covariance terms encode geometric features of both (1) the two-drug-response surface in ancestral populations (the drug interaction) and (2) the correlations between resistance levels to those drugs (collateral effects). In addition, we show how evolutionary trajectories within this framework reduce to a type of weighted gradient dynamics on the two-drug landscape, with the drug interaction dictating the shape of the underlying landscape and the collateral effects constraining the motion on those landscapes, leading to deviations from a simple gradient descent. We also illustrate two straightforward extensions of the basic framework, allowing us to investigate the effects of both constrained mutational pathways and sequential multidrug treatments. Our results clarify the complex relationship between drug interactions and collateral effects in multidrug environments and illustrate how specific dosage combinations can shift the weighting of these two effects, leading to different selective outcomes even when the available genetic routes to resistance are unchanged.

Our goal is to understand evolutionary dynamics of a cellular population in the presence of two drugs, drug 1 and drug 2. These dynamics reflect a potentially complex interplay between drug interactions and collateral evolutionary tradeoffs, and our aim is to formalize these effects in a simple model. To do so, we assume that the per capita growth rate of the ancestral population is given by a function $G(x,y)$, where $x$ and $y$ are the concentrations of drugs 1 and 2, respectively. We limit our analysis to two-drug combinations, but it could be extended to higher-order drug combinations, though this would require empirical or theoretical estimates for higher-dimensional drug-response surfaces (see, e.g., Wood et al., 2012; Zimmer et al., 2016; Zimmer et al., 2017; Russ and Kishony, 2018; Tekin et al., 2016; Tekin et al., 2017; Tekin et al., 2018). At this stage, we do not specify the functional form of $G(x,y)$, though we assume that this function can be derived from pharmacodynamic or mechanistic considerations (Engelstädter, 2014; Bollenbach et al., 2009; Wood and Cluzel, 2012) or otherwise estimated from experimental data (Greco et al., 1995; Wood et al., 2014). In classical pharmacology models (Loewe, 1953; Greco et al., 1995), the shape of these surfaces – specifically, the convexity of the corresponding contours of constant growth (‘isoboles’) – determines the type of drug interaction, with linear isoboles corresponding to additive drug pairs. In this framework, deviations from linearity indicate synergy (concave up) or antagonism (concave down). While there are multiple conventions for assigning geometric features of the response surface to an interactions type – and there has been considerable debate about the appropriate null model for additive interactions (Greco et al., 1995) – the response surfaces contain complete information about the phenotypic response. The manner in which this response data is mapped to a qualitative interaction type – and therefore used to label the interaction as synergistic, for example – is somewhat subjective, though in what follows we adopt the isobole-based labeling scheme because it is more directly related to the geometry of the response surface than competing models (e.g., Bliss independence; Greco et al., 1995).

Resistance as a continuous trait and rescaling in a simple model

The primary assumption of the model is that the phenotypic response (e.g., growth rate) of drug-resistant mutants, which may be present in the initial population or arise through mutation, corresponds to a simple rescaling of the growth rate function $G(x,y)$ for the ancestral population. As we will see, this scheme provides an explicit link between a cell’s level of antibiotic resistance and its fitness in a given multidrug environment. Specifically, we assume that the growth rate (g_i) of mutant $i$ is given by

(1) ${\displaystyle g_i=G({\alpha }_{i}x,{\beta }_{i}y),}$

where ${\alpha }_i$ and ${\beta }_i$ are rescaling parameters that reflect the physiological effects of mutations on the growth rate. In some cases – for example, resistance due to efflux pumps (Wood and Cluzel, 2012) or drug degrading enzymes (Yurtsev et al., 2013) – this effective concentration change corresponds to a physical change in intracellular drug concentration. More generally, though, this hypothesis assumes that resistant cells growing in external drug concentration $x$ behave similarly to wild-type (drug-sensitive) cells experiencing a reduced effective concentration $\alpha x$. Similar rescaling arguments were originally proposed in Chait et al., 2007, where they were used to predict correlations between the rate of resistance evolution and the type of drug interaction. These arguments have also been used to describe fitness tradeoffs during adaptation (Das et al., 2020) and to account for more general changes in the dose-response curves, though in many cases the original two-parameter rescaling was sufficient to describe the growth surface in mutants (Wood et al., 2014).

When only a single drug is used, this rescaling leads to a simple relationship between the characteristic inhibitory concentrations – for example, the half-maximal inhibitory concentration (IC₅₀) or the minimum inhibitory concentration (MIC) – of the ancestral (sensitive) and mutant (resistant) populations. In what follows, we refer to these reference concentrations as $K_i^j$, where $i$ labels the cell type and, when there is more than one drug, $j$ labels the drug. Conceptually, this means that dose-response curves for both populations have the same basic shape, with resistance (or sensitivity) in the mutant corresponding only to a shape-preserving rescaling of the drug concentration ($D\to D/K_i$; Figure 1A). In the presence of two drugs, the dose-response curves become dose-response surfaces, and rescaling now corresponds to a shape-preserving rescaling of the contours of constant growth. There are now two scaling parameters, one for each drug, and in general they are not equal. For example, in Figure 1B, the mutant shows increased sensitivity to drug 1 ($\alpha \equiv K_{\text{WT}}^1/K_{\text{Mut}}^1>1$) and increased resistance to drug 2 ( $\beta \equiv K_{\text{WT}}^2/K_{\text{Mut}}^2<1$), where superscripts label the drug (1 or 2) and subscripts label the cell type (wild type, WT; mutant, Mut).

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The power of this rescaling approach is that it directly links growth of the mutant populations to measurable properties of the ancestral population (the two-drug-response surface) via traits of the mutants (the rescaling parameters). Each mutant is characterized by a pair of scaling parameters, $({\alpha }_i,{\beta }_i)$, which one might think of as a type of coarse-grained genotype (Figure 1C). When paired with the ancestral growth surface, these traits fully determine the per capita growth rate of the mutant at any external dosage combination $(x,y)$ via Equation 1. While the scaling parameters are intrinsic properties of each mutant, they contribute to the phenotype (growth) in a context-dependent manner, leading to selection dynamics that depend in predictable ways on the external environment (Figure 1).

We assume a finite set of $M$ subpopulations (mutants), (${\displaystyle i=1,\dots M}$), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, initially we assume each of these mutants is present in the original population at low frequency and neglect mutations that could give rise to new phenotypes, though later we show that it is straightforward to incorporate them into the same framework. We do not specify the mechanisms by which this standing variation is initially generated or maintained; instead, we simply assume that such standing variation exists, and our goal is to predict how selection will act on this variation for different choices of external (true) drug concentrations. As we will see, statistical properties of this variation combine with the local geometry of the response surface to determine the selection dynamics of these traits.

Population dynamics of scaling parameters

The mean resistance trait to drug 1, which in our case is the scaling parameter $\overline{\alpha }(t)\equiv {\sum }_{i=1}^M{\alpha }_i{f}_i(t)$, evolves according to

(2) ${\displaystyle \frac{d\overline{\alpha }}{dt}=\sum _{i=1}^M{\alpha }_i\frac{d{f}_i(t)}{dt},}$

where $f_i(t)$ is the population frequency of mutant $i$ at time $t$ in the population. Assuming that each subpopulation grows exponentially at the per capita growth rate ($d{n}_i/dt=g_i{n}_i$, with n_i the abundance of mutant $i$ and g_i the per capita growth rate given by Equation 1), the frequency $f_i(t)$ changes according to

(3) ${\displaystyle \frac{d{f}_i}{dt}=\frac{d}{dt}\left(\frac{n_i}{\sum _i{n}_i}\right)=f_i(g_i-\overline{g}),}$

where $\overline{g}={\sum }_{i=1}^M{f}_i{g}_i$ is the (time-dependent) mean value of g_i across all $M$ subpopulations (mutants). Combining Equations 2 and 3, we have

(4) ${\displaystyle \frac{d\overline{\alpha }}{dt}={\mathrm{C}\mathrm{o}\mathrm{v}}_{\mathit{x}}(\alpha ,g),}$

where $\mathrm{Cov}{(\alpha ,g)}_{\mathit{𝒙}}\equiv {\sum }_{i=1}^M{\alpha }_i{f}_i(g_i-\overline{g})$ is the covariance between the scaling parameters ${\alpha }_i$ and the corresponding mutant growth rates g_i. The subscript $\mathit{𝒙}$ is a reminder that the growth rates g_i and $\overline{g}$ that appear in the covariance sum depend on the external (true) drug concentration $\mathit{𝒙}\equiv (x,y)$. An identical derivation leads to an analogous equation for the scaling parameter with respect to drug 2, mean susceptibility to drug 2, $\overline{\beta }$, relative to the original population, and the full dynamics are therefore described by

(5) ${\displaystyle \begin{array}{l}{\displaystyle \frac{d\overline{\alpha }}{dt}={\mathrm{C}\mathrm{o}\mathrm{v}}_{\mathit{x}}(\alpha ,g),}\\ \frac{d\overline{\beta }}{dt}={\mathrm{C}\mathrm{o}\mathrm{v}}_{\mathit{x}}(\beta ,g).\end{array}}$

To complete the model described by Equation 5, one must specify the external (true) concentration of each drug ($\mathit{𝒙}$); a finite set of scaling parameter pairs ${\alpha }_i,{\beta }_i$ corresponding to all ‘available’ mutations; and an initial condition $(\overline{\alpha }(0),\overline{\beta }(0))$ for the mean scaling parameters. When combined with the external drug concentrations, the scaling parameters directly determine the effective drug concentrations ($D_1^{\text{eff}},D_2^{\text{eff}}$) experienced by each mutant according to

(6) ${\displaystyle \begin{array}{l}{\displaystyle D_1^{\text{eff}}={\alpha }_{i}x,}\\ D_2^{\text{eff}}={\beta }_{i}y.\end{array}}$

We note that drug concentrations can be above or below the MIC contour, with higher concentrations leading to population collapse ($G<0$) and lower concentrations to population growth ($G>0$) in the ancestral (drug sensitive) cells. However selection dynamics remain the same in both cases as selection depends only on differences in growth rates between different subpopulations.

Equation 5 is an example of the well-known Price equation from evolutionary biology (Price, 1970; Price, 1972; Frank, 1995; Lehtonen et al., 2020), which says that the change in the (population) mean value of a trait is governed by the covariance of traits and fitness. In general, fitness can be difficult to measure and, in some cases, even difficult to define. However, the rescaling assumption of our model replaces fitness with $g$, which can be directly inferred from measurable properties (the two-drug-response surface) in the ancestral population. In what follows, we will sometimes casually refer to Equation 5 as a ‘model,’ but it is important to note that the Price equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5– the Price equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between the model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.

Equation 5 encodes deceptively rich dynamics that depend on both the interaction between drugs and the collateral effects of resistance. First, it is important to note that αs and βs vary together in pairs, and the evolution of these two traits is not independent. As a result, constraints on the joint co-occurrence of ${\alpha }_i$ and ${\beta }_i$ among the mutant subpopulations can significantly impact the dynamics. These constraints correspond to correlations between resistance levels to different drugs – that is, to cross-resistance (when pairs of scaling parameters simultaneously increase resistance to both drugs) or to collateral sensitivity (when one scaling parameter leads to increased resistance and the other to increased sensitivity). In addition, $g$ contains information about the dose-response surface and, therefore, about the interaction between drugs. The evolution of the scaling parameters is not determined solely by the drug interaction or by the collateral effects, but instead by both – quantified by the covariance between these rescaled trait values and the ancestral dose-response surface.

As an example, we integrated the model numerically to determine the dynamics of the mean scaling parameters and the mean growth rate for a population exposed to a fixed concentration of two drugs whose growth surface has been fully specified (Figure 2A). The dynamics can be thought of as motion on the two-dimensional response surface; if the initial population is dominated by the ancestral cells, the mean scaling parameters are approximately 1, and the trajectory therefore starts near the point representing the true drug concentration (in this case, $(x_0,y_0)$), where growth is typically small (Figure 2A). Over time, the mean traits evolve, tracing out a trajectory in the space of scaling parameters (Figure 2B). When the external concentration of drug is specified, these dynamics also correspond to a trajectory through the space of effective drug concentrations, which, in turn, can be linked with an average growth rate through the drug-response surface (Figure 2B). The model therefore describes both the dynamics of the scaling parameters, which describe how resistance to each drug changes over time (Figure 2C), and the dynamics of growth rate adaptation in the population as a whole (Figure 2D).

Figure 2

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Selection dynamics depend on drug interaction and collateral effects

This rescaling model indicates that selection is determined by both the drug interaction and the collateral effects, consistent with previous experimental findings. As a simple example, consider the case of a fixed drug interaction but different qualitative types of collateral effects – that is, different statistical relationships between resistance to drug 1 (via α) and resistance to drug 2 (via β). In Figure 3A, we consider cases where resistance is primarily to drug 2 (black), primarily to drug 1 (cyan), strongly positively correlated (cross-resistance, pink), and strongly negatively correlated (collateral sensitivity, green). Using the same drug interaction surface as in Figure 2, we find that a mixture of both drugs leads to significantly different trajectories in the space of scaling parameters (Figure 3B) and, in turn, significantly different rates of growth adaptation. In this example, cross-resistance (pink) leads to rapid increases in resistance to both drugs (rapid decreases in scaling parameters, Figure 3C, D) and the fastest growth adaptation (Figure 3E). By contrast, if resistance is limited primarily to one drug (cyan or black), growth adaptation is slowed (Figure 3E) – intuitively, purely horizontal or purely vertical motion in the space of scaling parameters leads to only a modest change in growth because the underlying response surface is relatively flat in those directions, meaning that the rescaled concentration, in each case, lies near the original contour (Figure 3). As a result of the contour shapes (drug interaction), resistance to both drugs can develop at approximately the same rate (for example), even when collateral structure suggests resistance to one drug will dominate (e.g., cyan case in Figure 3). We note that the dynamics will depend on the (true) external drug concentrations $(x_0,y_0)$, even if the rescaling parameters remain identical, because a given rescaling transformation will lead to different effective drug concentrations, and therefore different growth rates, for different values of $(x_0,y_0)$.

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When both collateral effects and drug interactions vary, the dynamics can be considerably more complex, and the dominant driver of adaptation can be drug interaction, collateral effects, or a combination of both. Previous studies support this picture as adaptation has been observed to be driven primarily by drug interactions (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008), primarily by collateral effects (Munck et al., 2014; Barbosa et al., 2018), or by combinations of both (Baym et al., 2016b; Barbosa et al., 2018; Dean et al., 2020). Figure 4 shows schematic examples of growth rate adaptation for different types of collateral effects (rows, ranging from cross-resistance [top] to collateral sensitivity [bottom]) and drug interactions (columns, ranging from synergy [left] to antagonism [right]). The growth adaptation may also depend sensitively on the external environment – that is, on the true external drug concentration (blue, cyan, and red). In the absence of both drug interactions (linear isoboles) and collateral effects (uncorrelated scaling parameters), adaptation is slower when the drugs are used in combination than when they are used alone, consistent with the fact that adaptation to multiple stressors is expected to take longer than adaptation to each stressor alone (Figure 4; middle row, middle column). As in Figure 3, modulating collateral effects with drug interaction fixed can have dramatic impacts on adaptation (Figure 4, columns). On the other hand, modulating the drug interaction in the absence of collateral effects will also significantly impact adaptation, with synergistic interactions leading to accelerated adaptation relative to other types of drug interaction (Figure 4, middle row; compare green curves across row). Similar interaction-driven adaptation has been observed in multiple bacterial species (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008; Dean et al., 2020).

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Model predicts experimentally observed adaptation of growth and resistance

Our model makes testable predictions for adaptation dynamics of both the population growth rate and the population-averaged resistance levels to each drug (i.e., the mean scaling parameters) for a given drug-response surface, a given set of available mutants, and a specific combination of (external) drug dosages. To compare predictions of the model with experiment, we solved Equation 5 for the 11 different dosage combinations of tigecycline (TGC) and ciprofloxacin (CIP) used to select drug-resistance mutants in Dean et al., 2020. We assumed that the initial population was dominated by ancestral cells ($\alpha =\beta =1$) but also included a subpopulation of resistant mutants whose scaling parameters were uniformly sampled from those measured across multiple days of the evolution (see Materials and methods). The model predicts different trajectories for different external doses (selective regimes: red to blue, Figure 5, top panel), leading to dramatically different changes in resistance (IC₅₀) and growth rate adaptation (Figure 5, bottom panels), similar to those observed in experiment. Specifically, the model predicts (and experiments confirm) a dramatic decrease in CIP resistance as TGC concentration is increased along a contour of constant growth. As TGC eclipses acritical concentration of approximately 0.025 g/mL, selection for both CIP resistance and increased growth is eliminated. We note that comparisons between the model and experiment involve no adjustable parameters, and while the model captures the qualitative trends, it slightly but systematically underestimates the growth across TGC concentrations – perhaps suggesting additional evolutionary dynamics not captured by the model. Intuitively, the experimental results can be explained by the strongly antagonistic interaction between the two drugs, which reverses the selection for resistance to CIP at sufficiently high TGC concentrations (compare CIP resistance at TGC = 0 and TGC = 0.04 g/mL, which both involve CIP ≈ 0.2 g/mL; similar results have been seen in other species and for other drug combinations; Chait et al., 2007). We also compared model predictions with experimental adaptation to two additional drug pairs (Figure 5—figure supplements 1 and 2) and again found that the model captures the qualitative features of both resistance changes to the component drugs and growth adaptation.

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Effects of mutations

While we have focused on selection dynamics, the model can be extended to include mutation events linking different phenotypes (i.e., linking different pairs of scaling parameters). These mutational pathways may be necessary to capture certain evolutionary features – for example, historical contingencies between sequentially acquired mutations (Barbosa et al., 2017; Card et al., 2019; Das et al., 2020). To incorporate mutational structure and processes, we modify the equation for each subpopulation $n_i$ to

(12) ${\displaystyle \frac{d{n}_i}{dt}=g_i{n}_i-\mu n_i+\mu \sum _j^M{m}_{ji}{n}_j}$

where μ is the mutation rate and $m_{ji}$ is the probability that strain $j$ mutates to strain $i$, given that there is a mutation in strain $j$. We will refer to the matrix formed by the parameters $m_{ji}$ as the mutation matrix as it contains information about allowed mutational trajectories linking different phenotypes (Day and Gandon, 2006). The Price equation for the mean traits (the mean scaling parameters) now becomes

(13) ${\displaystyle \begin{array}{l}{\displaystyle \frac{d\overline{\alpha }}{dt}={\mathrm{C}\mathrm{o}\mathrm{v}}_{\mathit{x}}(\alpha ,g)-\mu (\overline{\alpha }-{\overline{\alpha }}_m),}\\ \frac{d\overline{\beta }}{dt}={\mathrm{C}\mathrm{o}\mathrm{v}}_{\mathit{x}}(\beta ,g)-\mu (\overline{\beta }-{\overline{\beta }}_m),\end{array}}$

where ${\overline{\alpha }}_m$ and ${\overline{\beta }}_m$ are the mean trait values in all mutants that arise, and are given by

(14) ${\displaystyle \begin{array}{l}{\overline{\alpha }}_m=\sum _{ij}{\alpha }_i{m}_{ji}{f}_j,\\ {\overline{\beta }}_m=\sum _{ij}{\beta }_i{m}_{ji}{f}_j.ll\end{array}}$

When the mutation matrix has a particularly simple structure, the effects of mutation will be similar to those of selection. For example, when mutations occur with uniform probability between the ancestral strain and each mutant, the evolutionary dynamics are qualitatively similar to those in the absence of mutation (Figure 6—figure supplement 1, compare to Figure 4). On the other hand, certain mutational structures can lead to new behavior. As an example, we consider a toy model consisting of four phenotypes defined by the level of resistance (sensitive, S, or resistant, R) to one of two drugs. The phenotypes are designated SS $(\alpha =1,\beta =1)$, RS $(\alpha =1/5,\beta =1)$, SR $(\alpha =1,\beta =1/5)$, or RR $(\alpha =1/5,\beta =1/5)$. Note that these phenotypes do not assume, a priori, any particular relationships between the underlying genotypes. For example, each phenotype could correspond to a single mutation – in which case the RR phenotype would be said to exhibit strong cross-resistance – or alternatively, the phenotypes could correspond to single drug mutants (SR and RS) and double mutants (RR), which implies a particular sequence in which the mutations can be acquired. These two situations would lead to significant differences in the expected structure of the mutational matrix and, as we will see, in the evolutionary dynamics.

For simplicity, we consider two different mutational structures: one corresponding to direct and uniform pathways between the ancestor (SS) and all mutant phenotypes (SR, RS, and RR), and the second corresponding to sequential pathways from ancestor to single mutants (SR, RS) and then from single mutants to double mutants (Figure 6). To illustrate the impact of mutational structure, we consider two drugs that interact antagonistically and choose the external drug concentrations such that $D_1>D_2$ (Figure 6C). While both mutational structures lead eventually to a population of double-resistant phenotypes (RR), the sequential pathway leads to slower adaptation of growth as the population evolves first toward the most fit single mutant (in this case, RS because the drug combination contains a higher concentration of drug 1) before eventually arriving at RR (Figure 6C–E). The specific trajectory will of course depend on both the drug interaction (the shape of the growth contours) and the specific dosage combination (see Figure 6—figure supplement 2).

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There are other types of mutational constraints that can be implemented in the formalism, including explicit functional dependencies between different antibiotic resistance phenotypes. As an example, we assumed a distance-based mutation matrix where the mutation probability between different strains depends on the Euclidean distance between their scaling parameter pairs (${\alpha }_i,{\beta }_i$) according to $m_{ji}\sim e^{-d_{ji}}$, where $d_{ji}=\sqrt{{({\alpha }_i-{\alpha }_j)}^2+{({\beta }_i-{\beta }_j)}^2}$. Intuitively, this structure means that mutations are more likely between strains with similar rescaling parameters (and therefore similar levels of resistance to the component drugs). As expected, this constrained mutational structure leads to slower growth rate adaptation than a model with a simple uniform mutation model (Figure 6—figure supplement 3). These effects are further compounded by statistical properties of the collateral structures (e.g., positive or negative correlations between the possible αs and βs; Figure 3), the specific (stochastic) realization of those scaling parameters (Figure 6—figure supplement 4), and the precise shape of two-drug growth surface. Note, however, that these dynamics depend fundamentally on the global mutation rate μ, which modulates the relative balance between selection and mutation in a given environment.

Evolutionary dynamics under temporal sequences of drug combinations

Evolutionary dynamics in the presence of multiple drugs can be extremely complex. While past work has focused primarily on the use of either temporal sequences or (simultaneous) combinations of antibiotics, more complex scenarios are possible, in principle, but difficult to analyze even in theoretical models. The simplicity of the Price equation framework allows us to investigate evolutionary dynamics in response to a more complex scenario: temporal sequences of antibiotic combinations. As proof of principle, we numerically studied time-dependent therapies consisting of two sequential regimes (treatment A and treatment B; Figure 7) characterized by different dosage combinations of tigecycline (drug 1) and ciprofloxacin (drug 2) (the growth surface was measured in Dean et al., 2020). The two dosage combinations were chosen to lie along a contour of constant growth (Figure 7A), meaning that the net inhibitory effects of A and B are the same when applied to ancestral cells. Using experimental estimates for $\alpha ,\beta$ (Figure 3—figure supplement 1C), we find that both the resistance levels to the two drugs and the growth rate increase during treatment, as one might expect. However, the dynamics of these changes depend on both the relative duration of each treatment and total treatment length (Figure 7—figure supplements 1 and 2). For example, consider a treatment of total length $T$ consisting of an initial period of treatment A followed by a final period of treatment B. If we vary the relative length of the two epochs while keeping the total treatment length fixed, we find that the resistance to each drug and the growth rate increase monotonically as the fraction of time in B increases (Figure 7H–J). This is a consequence of the interplay between the distribution of available mutants and the shape of the growth surface under these particular two drugs; the mutants tend to feature higher levels of resistance to drug 2 than drug 1, yet the benefits of this resistance – that is, the increase in growth rate due to rescaling the concentration of drug 2 – are favored much more so under condition B than condition A (where rescaling tends to produce effective drug concentrations that lie near the original growth contour). It is notable, however, that effects (both resistance levels and final growth) change nonlinearly as the fraction of time in B is increased – that is, even in this simple model, the effects of a two-epoch (A then B) treatment cannot be inferred as a simple linear interpolation between the effects of A-only and B-only treatments.

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Antibiotic resistance is a growing threat to modern medicine and public health. Multidrug therapies are a promising approach to both increase efficacy of treatment and prevent evolution of resistance, but the two effects can be coupled in counterintuitive ways through the drug interactions and collateral effects linking the component drugs. Our results provide a unified framework for incorporating both drug interactions and collateral effects to predict phenotypic adaptation on coarse-grained fitness landscapes that are measurable features of the ancestral population (Table 1). Special cases of the model reproduce previous experimental results that appear, on the surface, to be contradictory; indeed, adaptation can be driven primarily by drug interactions, primarily by collateral effects, or by a combination of both, and the balance of these effects can be shifted using tunable properties of the system (i.e., the ratio of drug dosages). Our model was inspired by rescaling arguments that were originally introduced in Chait et al., 2007 and have since been shown to capture phenotypic features of dose-response surfaces or adaptation in multiple bacterial species (Michel et al., 2008; Hegreness et al., 2008; Torella et al., 2010; Wood and Cluzel, 2012; Wood et al., 2014; Dean et al., 2020; Das et al., 2020). Our results complement these studies by showing how similar rescaling assumptions, when formalized in a population dynamics model, lead to testable predictions for the dynamics of both growth adaptation and phenotype (resistance) evolution. Importantly, the model also has a simple intuitive explanation, with evolutionary trajectories driven by weighted gradient dynamics on two-dimensional landscapes, where the local geometry of the landscape reflects the drug interaction and collateral effects constrain the direction of motion. We have also illustrated how de novo mutation can be integrated in the same framework, provided the mutational structure among a pool of possible phenotypes is known or if specific assumptions are made regarding functional links and constraints for shifts between different resistance levels. Mutation at constant rate is indeed a classic version of the complete Price equation. However, more complex mutational effects – including stress-dependent modulation of mutation rate (Kohanski et al., 2010; Vasse et al., 2020) – could be included as a more flexible tunable term in Equation 5.

It is important to keep in mind several limitations of our approach. The primary rescaling assumption of the model is that growth of a drug-resistant mutant is well approximated by growth of the ancestral strain at a new ‘effective’ drug concentration – one that differs from the true external concentration. This approximation has considerable empirical support (Chait et al., 2007; Wood and Cluzel, 2012; Wood et al., 2014; Das et al., 2020) but is not expected to always hold; indeed, there are examples where mutations lead to more complex nonlinear transformations of the drug-response surface (Wood et al., 2014; Munck et al., 2014). In addition, it is possible that selection can act on some other feature of the dose-response curve characterizing single-drug effects – modulating, for example, its steepness (rather than merely its scale). While these effects could in principle be incorporated into our model – for example, by assuming transformations of the ancestral surface, perhaps occurring on a slower timescale, that go beyond simple rescaling – we have not focused on those cases. For simplicity, we have also neglected a number of features that may impact microbial evolution. For example, we have assumed that different subpopulations grow exponentially, neglecting potential interactions including clonal interference (Gerrish and Lenski, 1998), intercellular (Koch et al., 2014; Hansen et al., 2017; Hansen et al., 2020) or intra-lineage (Ogbunugafor and Eppstein, 2017) competition, and cooperation (Yurtsev et al., 2013; Sorg et al., 2016; Estrela and Brown, 2018; Frost et al., 2018; Hallinen et al., 2020), as well as potential effects of demographic noise and population extinction (Coates et al., 2018). These complexities could also be incorporated in our model, perhaps at the expense of some intuitive interpretations (e.g., weighted gradient dynamics) that currently apply. In addition, we have not explicitly included a fitness cost of resistance (Andersson and Hughes, 2010) – that is, we assume that growth rates of mutants and ancestral cells are identical in the absence of drug. This assumption could be relaxed by including a prefactor to the growth function, $g_i\to (1-{\gamma }_i({\alpha }_i,{\beta }_i))g_i$, where ${\gamma }_i({\alpha }_i,{\beta }_i)$ is the cost of resistance, which in general depends on the scaling parameters (if not, it can be easily incorporated as a constant). While such fitness costs have traditionally been seen as essential for reversing resistance (with, e.g., drug-free ‘holidays’; Dunai et al., 2019), our results underscore the idea that reversing adaptation relies on differential fitness along a multidimensional continuum of environments, not merely binary (drug/no drug) conditions. Our results indicate resistance and bacterial growth can be significantly constrained by optimal tuning of multidrug environments, even in the absence of fitness cost. Finally, our model deals only with heritable resistance and therefore may not capture phenotypic affects associated with, for example, transient resistance (El Meouche and Dunlop, 2018) or cellular hysteresis (Roemhild et al., 2018).

Our goal was to strike a balance between analytical tractability and generality vs. biological realism and system specificity. But we stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose-response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g., 'synergistic combinations always accelerate resistance') that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.

Our approach also has pedagogical value as it connects evolutionary effects of drug combinations and collateral effects with well-established concepts in evolutionary biology and population genetics (Price, 1970; Price, 1972; Day and Gandon, 2006). Approximations similar to Equation 9 have been derived previously in quantitative genetics models (Abrams et al., 1993; Taylor, 1996) and other applications of the Price equation (Lehtonen, 2018). While the gradient approximation does not require that the population be monomorphic with rare mutants or a particular form for the phenotype distribution, it does require that the majority of trait values (here scaling parameters) are contained in a regime where $G(x,y)$ is approximately linear; in our case, that linearity arises by Taylor expansion and neglecting higher-order deviations from the population mean. More generally, the direction of evolutionary change in our model is determined by the gradient of the fitness function $\nabla G$ evaluated at the mean trait $(\overline{\alpha },\overline{\beta })$; when the gradient vanishes, this point corresponds to a singular point (Waxman and Gavrilets, 2005) or an evolutionarily singular strategy (Geritz et al., 1998). Whether such point may be reached (convergence stability) and how much variance in trait values can be maintained around such point (whether evolutionarily stable [ESS]) depend on other features, such as higher-order derivatives (Otto and Day, 2011; Eshel et al., 1997; Lehtonen, 2018; Smith, 1982; Parker and Smith, 1990). In the case of drug interactions, the fitness landscape in $(\alpha ,\beta )$ space will typically have a single maximum at (0, 0) corresponding to effective drug concentrations of zero. However, whether that point is reachable in general or within a given time frame will depend on the available mutants preexisting at very low frequencies in the population, or on the speed and biases in the mutational process itself, if such mutants are to be generated de novo during treatment. In principle, the model also allows for long-term coexistence between different strains; in that case, the rescaled effective drug concentrations experienced by both strains would fall along a single contour of constant growth. Hence, while variance in the population growth will necessarily decrease over time, variance in the traits (scaling parameters) themselves can change non-monotonically.

The framework is sufficiently flexible to integrate different strands of empirical data, and our results underscore the need for quantitative phenotype data tracking resistance to multiple drugs simultaneously, especially when drug combinations are potentially driving selection dynamics. At an epidemiological level, the dominant approach in describing resistance has been to use fixed breakpoints in MIC and track the percentage of isolates with MIC above (drug-resistant) or below (drug-sensitive) such breakpoint (e.g., ECDC, 2019; Chang et al., 2015). By missing or decoupling patterns of co-occurrence between MICs to different drugs across isolates, this approach remains incomplete for mechanistic predictions. Our framework suggests that going beyond such binary description towards a more continuous and multidimensional phenotype characterization of drug resistance is possible, with applications not just in microbiology but also in the evolutionary epidemiology of drug resistance (Day and Gandon, 2012; Day et al., 2020). In the long run, these advances may yield better and more precise predictions of resistance evolution at multiple scales, and, in turn, optimized treatments that balance short-term inhibitory effects of a drug cocktail with its inseparable, longer-term evolutionary consequences.

Perhaps most importantly, our approach provides a low-dimensional approximation to the high-dimensional dynamics governing the evolution of resistance. In contrast to the classical genotype-centric approaches to resistance, our model uses rescaling arguments to connect measurable traits of resistant cells (scaling parameters) to environment-dependent phenotypes (growth). This rescaling dramatically reduces the complexity of the problem as the two-drug-response surfaces – and, effectively, the fitness landscape – can be estimated from only single-drug dose-response curves. Such coarse-grained models can help extract simplifying principles from otherwise intractable complexity (Shoval et al., 2012; Hart et al., 2015). In many ways, the classical Price equation performs a similar function, revealing links between trait-fitness covariance and selection that, at a mathematical level, are already embedded in simple models of population growth. In the case of multidrug resistance, this formalism reveals that drug interactions and collateral effects are not independent features of resistance evolution, and neither, alone, can provide a complete picture. Instead, they are coupled through local geometry of the two-drug-response surface, and we show how specific dosage combinations can shift the weighting of these two effects, providing a framework for systematic optimization of time-dependent multidrug therapies.

Data used in this paper was taken from a public repository: Dean, Ziah; Maltas, Jeff; Wood, Kevin (2020), Antibiotic interactions shape short-term evolution of resistance in Enterococcus faecalis, Dryad, Dataset, https://doi.org/10.5061/dryad.j3tx95x92 There are no restrictions on any new results.

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Acceptance summary:

Your manuscript offered a provocative integration of the Price equation with fundamental theory in population genetics, in the context of a relevant biomedical problem. We found the approach creative and rigorous, and is the type of cross-disciplinary theoretical work that evolutionary medicine could benefit from.

Decision letter after peer review:

Thank you for submitting your article "Price equation captures the role of drug interactions and collateral effects in the evolution of multidrug resistance" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, one fo whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by George Perry as the Senior Editor. The reviewers have opted to remain anonymous.

Summary:

I thank the authors for submitting a compelling and well-executed study. The reviewers and reviewing editor agree that the manuscript offers an interesting take, is technically sound and provides a provocative synergy between modern questions in the evolution of antibiotic resistance, and classical theory in the form of the Price equation.

Some of the discussion between the reviewers involved the question of whether the paper's main argument, as outlined in the draft, was too strongly bound to the usage of the Price equation. That is, that the main result can be couched in terms of the Price equation may or may not mean that the finding is a true contribution to modern examinations of the evolution of antibiotic resistance, a field that is increasingly sophisticated in its use of both theory and technology.

I especially applaud the author's transparency: the issue is not that they oversold their result. To the contrary-the limitations were responsibly articulated. A lingering question is whether the result, as reported, is anchored to a slice of reality that is too thin to be broadly useful. This is an admittedly subjective area of discussion, but was a theme from the reviews and discussions.

In order to assuage some of the concerns, I have provided a list of recommended changes, collated from the formal; reviews and further exchanges with the reviewers, that could make the manuscript stronger.

I thank the authors for their hard work. I want to especially acknowledge that we are all working in times that are challenging for many reasons. I congratulate the authors on producing a strong piece of science under the circumstances.

Essential Revisions:

1) Please propose some treatment of the setting in the result that includes mutation and other stochastic processes. Several reviewers made mention of the "model" that was used. Some had specific issues with the notion that the rescaling in the form of the Price was truly a model at all. Others had the related concern that the paper, as it stands, applies to the very strict circumstance: that of a small population of equal numbers of every potential mutant in the model. For example, the authors state:

"We assume a finite set of M subpopulations (mutants), (i = 1…M), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, we assume each of these mutants is initially present in the population at low frequency and neglect mutations that could give rise to new phenotypes, though it is straightforward to incorporate them into the same framework"

If it is indeed "straightforward," then it would be great to see this explored, with new mutations and other stochastic processes. We recognize that the strength of the study is in its elegance re-scaling, however, there are concerns that the use case is small enough that it limits its applicability.

2) Somewhat relatedly, the reviewers had some issues with use of the term "model" to describe the Price equation. While the mathematics and framing themselves are quite responsible, there were some particular issues raised about the framing. I want to especially point you towards the comments from reviewer #2, who explained issues with usage of the term "model" and "prediction" throughout.

3) There are several issues with figures, figure labels, captions and descriptions that require the author's attention. As this paper does hinge on an understanding of the particulars, it is especially important that figures clearly communicate the results. Please see the individual reviewer's comments for specifics.Reviewer #1:

This study utilizes the price equation in order to describe evolution of drug resistance, recognizing the cutting-edge problem of predicting the evolution of resistance in light of collateral effects and drug-drug interactions.

The manuscript offers a formalism that cleverly integrates theory across paradigms, from quantitative genetics to the evolution of antibiotic resistance. As an intellectual exercise, this is courageous, the type of integrative thinking necessary in the modern iteration of evolutionary medicine.

I am not, however, of the mindset that the Price equation is so important a framing that its invocation/application to a different paradigm (antibiotic resistance) is especially interesting by itself. Said differently, I am not personally intrigued by the a study that is successful at connecting antibiotic resistance to the Price equation. Alternatively, I am much much more intrigued by the possibility that the Price equation might add a richer and more nuanced overall perspective to cutting-edge problems in the evolution of antibiotic resistance.

And I can say that, on the balance, the manuscript was successful in offering a new take on relevant problem.

Also, this study carries out its methods carefully, and its formalisms are well-written. The essential idea is that the vagaries of drug interactions and collateral effects can be collapsed into a formalism that considers both when predicting the trajectory of drug resistance, is a fascinating and important one.

The study is especially transparent (refreshingly so) about the limitations of its scope. And while this transparency is a legitimate positive of this story, I did find myself concerned about the small number of cases that theory proposed in this study applies to. For example, the Price equation as applied in the study applies to selection on standing variation, and not for the step-wise acquisition of resistance that fitness landscapes are often constructed to study. The authors argue that this type of nuance can be engineered into the formalism, which is not obvious to me (I'm not suggesting that it isn't true, only that it isn't obvious).

But that this study has a relatively thin application space does not make the study irrelevant. For the outlined problem space, the arguments appear sound.

I must say that given the current state of the problem of antimicrobial resistance in 2021, sometimes featuring complex, highly dynamic, epistastic fitness seascapes, for example, I was hoping for framework that could be used for fitness landscapes as they are currently constructed and studied.

On the balance, however, I found the study to be well-constructed, written, and the formalisms to be correct. The main concerns regarding the scope might be a matter of preference. And even I must recognize that responsible theoretical biology often requires a narrow scope in order to generate formalisms. The idea here is that from this contribution, the community can then extend the study to other such systems and data sets.

I would like the authors to consider, and possibly cite, if they are relevant, more recent papers by Pamela Yeh on this topic:

Tekin, Elif, Pamela J. Yeh, and Van M. Savage. "General form for interaction measures and framework for deriving higher-order emergent effects." Frontiers in Ecology and Evolution 6 (2018): 166.

Tekin, Elif, Van M. Savage, and Pamela J. Yeh. "Measuring higher-order drug interactions: A review of recent approaches." Current Opinion in Systems Biology 4 (2017): 16-23.

Tekin, Elif, et al., "Enhanced identification of synergistic and antagonistic emergent interactions among three or more drugs." Journal of The Royal Society Interface 13.119 (2016): 20160332.

There are some small issues with the figures that need to be resolved:

I found Figure 6 to be complicated. I would suggest the authors re-generate it with the goal of explaining it to someone not well-versed in the system.

I had some issues with the legends in Figures 4 and 5. Ensure that they describe the information in the figures properly.Reviewer #2:

This paper examines the interaction between pharmacological and evolutionary factors that mediate the emergence of resistance to combination antimicrobial therapy. The authors introduce and demonstrate the utility of a mathematical equation that describes how both (a) the pharmacological interaction between two drugs (e.g. synergy vs antagonism) and (b) the interaction between the fitness effects of mutations that confer resistance to one drug or the other (or both), jointly determine the direction of selection in the presence of two-drug therapy. They show that the equation they describe follows the same form as a two-trait version of the classical "Price equation" in population genetics. They also examine a "weak selection" limit (when evolution proceeds by selection of a sequence of strains each with only a slight change in resistance level from the previous). In this limit their equation has an even more intuitive physical meaning as movement along an weighted version of the gradient of the drug's dose-response curve, where the weighting reflects the nature of the cross-resistance profile. Many examples of the implications of this equation for different drug interactions and different fitness landscapes are presented. The results of this paper unite findings from quite a few previous studies in a common framework. For example, the authors show how to understand (i) when resistance to one drug in a combination will be more strongly selected than resistance to the other and how this direction of selection can change based on the relative levels of two drugs in a combination – even if total inhibition is the same, (ii) when drug antagonism (vs synergy) is morel likely to slow down the rate of adaptation, and (iii) how the relative ordering and frequency of cycling between two drugs in a combination impacts the rate of resistance emergence.

It is important for readers to understand that the Price equation – and thus by extension the result in this paper – is not really a mathematical "model" that includes assumptions about the underlying biological processes and makes predictions about potential experimental results. Instead, the Price equation is a mathematical truth that simply involves decomposing certain statistical relationships between variables and expressing them in an intuitive way. Roughly, in this case the Price equation describes how the average level of resistance in a population of microbes will change over time depending on the frequencies of different strains in the population and their resistance levels, including the correlation between resistance to each drug. Thus, this paper is primarily purely theoretical/mathematical work.

The authors do apply their equation to some real-life drug profiles and fitness landscapes using dose-response curves for two-drug antibiotic combinations to calculate the direction of selection for different dose combinations. Later in the paper, they do also apply their equation as a "model" to predict entire evolutionary trajectories, by additionally assuming that they can define a priori all the available resistance mutations and ignore any additionally generated mutations. Then, they show that the level of resistance selected after a specific time as predicted by their equations agrees relatively well with the results of in vitro experiments containing these same mutations. This is actually quite a surprising and nice result.

Overall this paper provides a useful new framework for understanding drug resistance evolution and provides multiple examples of insights offered by this framework. They provide a nice connection between the mainly mathematical nature of their results and scenarios involving antibiotic treatment and resistance evolution in the lab. The introduction of this paper is very comprehensive and summarizes a huge body of work on the evolution of drug resistance. The discussion is equally thorough and mentions most of the caveats I describe here.

Although the authors of the paper focus on bacterial infections and antibiotic resistance, I think the results are much more general, and likely also apply to multi-drug therapy to viral infections (e.g. HIV, HCV), parasitic infections (e.g. malaria), and even anti-cancer therapy.

While not a weakness that is specific to this paper, it is important for readers to understand that a Price equation only describes the act of selection acting on existing genetic variation in a population, and does not describe the act of generating that variation (i.e. mutations). Thus, while this approach can be helpful for understanding the direction and speed of selection in one vs two-drug combinations of different types, it cannot be used to understand the overall risk of resistance emergence, since the rate at which mutations are generated may also differ. For example, it often takes two separate mutations to generate resistance to two drugs with different mechanisms of action, so resistance to two-drug therapy is often slowed by the rate of generating resistant mutations. In addition, the rate at which mutations are generated during therapy depends on the population size over time, which might differ along different selective paths described by the Price equation for different drug combinations. Therefore, there are some important results from previous work related to factors that accelerate or inhibit multi-drug resistance which cannot be understood in the framework presented in this paper.

One assumption that the authors use in order to get a simplified Price-equation-like result is that the effect of a drug resistance mutation on the dose-dependent effect of a drug on microbial growth can be described by a single parameter, which just shifts the drug concentration to a lower value. This is equivalent to saying that a mutation simply alters the IC50 of the drug. However, many studies have shown that in general mutations have at least one and often two other effects on dose-response curves: firstly, they have a fitness cost, meaning that the maximum fitness in the absence of drug is lower, and secondly, they can also alter the slope of the dose response curve. It is unclear how these effects change the equation the authors have derived.

The impact of the authors approach on studying resistance evolution in any particular system depends on the regime that system is in. If resistance tends to evolve by selection slowly acting on an extremely diverse population containing strains with many different resistance levels, then this approach will have the most utility. Alternatively, if resistance evolution tends to be limited by rare mutational events that produce strains with large differences in resistance levels from the parental strain, then this approach will have less use.

To summarize, the most important caveat for readers to understand is that the mathematical expressions that come out of this paper should not be said to be able to "predict" evolution, since they say nothing about the availability of possible resistance mutations or the rate at which they could be generated.

Concerns and suggested changes/clarifications:

– It is a bit misleading to call the Price Equation a mathematical model. It's not a model in the way that word is generally used in biology – it doesn't descibe a mechanism or make any assumptions, and it's not subject to experimental verification. It's just a mathematical statement about a statistical relationship among variables, based on the definition of those variables and basic laws of calculus. The authors should be very clear about this throughout the paper.

– The parts of the paper that are more like a "model" are the parts where you use the Price Equation to describe an evolutionary trajectory, assuming a particular distribution of mutants from standing genetic variation and ignoring additional mutations, or, when you assume a weak selection regime.

– Overall the authors should be more careful about the use of the word "model" and claims about "predictions" throughout.

– I'm not sure about Equation 2 – 5 requires the assumption that the effect of a mutant can be described by a rescaling of drug concentrations. Instead, I think you could replace the α parameters with another parameter r for example that simply means "resistance level" but does not specify how the "resistance level" changes the shape of the dose-response curve.

– Equation 5 actually is only half of the regular Price equation. It is the "selection" component but is missing the "transmission" component. The authors should be clearer about what processes relevant to the evolution of drug resistance they are ignoring that allow them to set this second term to zero.

Comments on Figures and results:

Figure 2 – What are the units of the axes on all these graphs?

a) I don't understand this statement in the caption: "We then simulated an external condition where the combination drug concentration is held constant at 0.6 of the maximal concentration of drug 1 and 0.1 of the maximal concentration of drug 2.". There is no such thing as a "maximal" drug concentration … you can always keep adding more and more drug. It would be clearer just to state the drug level used in absolute terms. Or maybe you mean the "highest tested drug level"?

Figure 3 – Some of the results shown here are a little confusing and hard to understand. In the black scenario, there are few available resistant strains with lower α, so we expect selection to proceed in the direction of Drug 2/β.. that agrees with the results in B. But for the blue case where, despite lots of available α but little available β, the trajectory in B seems relatively symmetrical in both drugs … why? Also I was surprised that the case of collateral sensitivity (green) wasn't the worst case … but I realized it's a bit misleading because the four scenarios don't only differ in the nature of the correlations in resistance levels, but also in the magnitude of the variance in α or β and hence the best possible available mutation. I think that it would be a fairer comparison if for the green and magenta scenarios the variance was narrower around the lines of perfect correlation/anticorrelation.

Figure 4 – This is nice figure that really summarizes the combined effects of drug interactions and cross-resistance profile. However this figure also summarizes the major limitations of this approach. The early slopes of each growth rate vs time curve do show how these factors interact to determine selection strength. However, the fact that the maximum growth rate varies by curve is really just an artifact of the arbitrary and limited available mutants the authors have created for selection to act on. In reality, mutations are continually and sequentially generated over time, and this rate of generation of mutants also impacts the increase in growth rate over time and might remove any "maximum" resistance level that is achievable.

Figure 5 – Are these predicted trajectories using the weak selection limit or the full equations without this additional assumption?

– I'm assuming there was some fixed time limit that selection was allowed to proceed for in these experiments and that it was the same in all cases?

– B: is the y axis absolute IC50 or change in IC50? The axes titles just say IC50 but the figure caption says "relative change in IC50". Also it would be helpful if the baseline IC50 (ie for the wild type strain) where shown on these graph as well. Finally, I don't understand how the results for the TGC IC50 agree with the results in A. From B, the IC50 looks totally flat with TGC concentration. But from A, it looks like there is a large increase in TGC resistance (increase in TGC IC50) for high TGC concentrations, but little TGC resistance for low TGC concentrations.

– C: I don't understand what the y axis is here. If this is the increase in growth rate, then why is it highest for low TGC concentrations, yet both panels in B show no change in IC50 to either drug at low TGC concentrations? And any idea what could be causing the systematic difference between the predicted and observed changes in growth rate?

Figure 6 – This figure was quite confusing, it needs more annotation on the figure itself (not just written in the caption), and some panels are unnecessarily small (with unnecessary white space between them). For panels H and K, are there better axis limits that could be used so that 95% of the figure is not all one color? Also I found it is hard to see in this figure any of the differences between the outcomes of the two different orderings that are described in the text.

Reviewer #3:

In this work, Gjini and Wood create a model to simulate evolutionary dynamics of antibiotic resistance arising as a response to drug combinations. They demonstrate that evolution to these drug combinations is a variation of the Price equation and show that some evolutionary paths experimentally correspond to a weighted gradient descent on the growth landscape.

The work appears mathematically sound, though we have concerns about the scope and generality of the results. In particular this work appears to rely fundamentally on the assumption that drug resistance acts as a linear rescaling of the two-drug dose response curve. We believe this work would be strengthened by making it more clear what, beyond the mathematical description of the rescaling hypothesis they cite extensively, this work adds to the literature. Since the model appears to also require an a priori description of the resistance characteristics of every possible accessible mutant, which typically is not known, it isn't clear the utility of the model.

– This paper models evolution in the presence of physiological interactions between antibiotics but not evolutionary interactions. The fundamental assumption that evolution of resistance scales the surface but does not change its shape, which is central to the findings, is true in some cases, though violated for many mutations (the authors cite numerous papers on cross-resistance and collateral sensitivity that are examples of this).

– While the authors describe it as "straightforward" to include discrete mutational events instead of a small population of equal numbers of every potential mutant in the model, it is not clear that this is the case. For example, a population exposed to an above-MIC concentration of a bacteriostatic drug will no longer generate additional mutants, and so the standing diversity before drug application matters critically. As it stands the model is really about differential growth and not evolution per se, since neither mutation nor complete extinction can occur.

– It isn't clear the experiments are representative of nontrivial behavior predicted by the model. For example in figure 5 no adaptation to TGC was seen, so it seems the only real evolution was rescaling CIP, which wouldn't yield the suppression-induced selection against resistance they then discuss. These results could be entire an artifact of the selection being entirely on CIP and not at all on TGC. The evolution experiments in Figures S3 and S4 seem much more compelling, though.

– If it is indeed straightforward to add mutational events, this would make the model appreciably more realistic.

Essential Revisions:

1) Please propose some treatment of the setting in the result that includes mutation and other stochastic processes. Several reviewers made mention of the "model" that was used. Some had specific issues with the notion that the rescaling in the form of the Price was truly a model at all. Others had the related concern that the paper, as it stands, applies to the very strict circumstance: that of a small population of equal numbers of every potential mutant in the model. For example, the authors state:

"We assume a finite set of M subpopulations (mutants), (i = 1…M), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, we assume each of these mutants is initially present in the population at low frequency and neglect mutations that could give rise to new phenotypes, though it is straightforward to incorporate them into the same framework"

If it is indeed "straightforward," then it would be great to see this explored, with new mutations and other stochastic processes. We recognize that the strength of the study is in its elegance re-scaling, however, there are concerns that the use case is small enough that it limits its applicability.

In the revised manuscript, we include a new section (including 1 additional figure, Figure 6, and several associated supplemental figures) that incorporates mutation. Under these conditions, the Price Equation includes an additional term that depends on a mutation matrix that describes the probability of mutation between any pairs of strains (Equation 13). Under some conditions (e.g. when mutations occur with uniform probability between the ancestor cells and each mutant), the effects of mutation are qualitatively similar to those of selection (Figure 4 and Figure 6 figure supplement 1). On the other hand, mutational matrices can be chosen to reflect known features of the mutation process or fitness landscapes. As examples, we consider several illustrative cases, including (1) a case where mutations can only be acquired sequentially (Figure 6, Figure 6 figure supplement 2) and (2) a case where mutations occur in a distance-dependent manner in the space of rescaling parameters, in essence making mutations between strains with similar scaling parameters more likely than those with vastly different scaling parameters (Figure 6 figure supplements 3-4).

Similar to the case with selection only, it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. The relationship between these features and the evolutionary dynamics is complex, but as these examples illustrate, the framework can be extended to incorporate these features and produce testable predictions. In the discussion, we discuss this point as follows:

“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”

2) Somewhat relatedly, the reviewers had some issues with use of the term "model" to describe the Price equation. While the mathematics and framing themselves are quite responsible, there were some particular issues raised about the framing. I want to especially point you towards the comments from reviewer #2, who explained issues with usage of the term "model" and "prediction" throughout.

We thank the reviewers for their comments, and we have taken additional care to discuss these important, but subtle, distinctions in the revised manuscript. We agree that the manuscript is not merely about the Price Equation, which (as the reviewers point out) is really just an algebraic restatement of generic statistical features of this (and similar) models. Our underlying model formalizes the concept of concentration rescaling as a general description of resistance, a continuous trait that evolves in a manner that depends on the multi-component environment, and that model could indeed be analyzed in the absence of the Price Equation connection. At the same time, we believe the connection to the Price Equation has conceptual value, not only because it links our work with a classical result in evolutionary biology, but also because it provides a convenient form for disentangling the effects of collateral resistance and drug interactions (at least in the stated limits). We now include the following statement in the Results, just after the brief derivation of the Price Equation for our model:

“In what follows, we will sometimes casually refer to Equation 5 as a "model", but it is important to note that the Price Equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5 – the Price Equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between those model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.”

In addition, we now consistently refer to the “model” rather than the “Price Equation” when describing our simulations and results (e.g. in the figure captions).

3) There are several issues with figures, figure labels, captions and descriptions that require the author's attention. As this paper does hinge on an understanding of the particulars, it is especially important that figures clearly communicate the results. Please see the individual reviewer's comments for specifics.

Thank you for these suggestions. The particulars indeed make this a challenging story to tell, and we agree that some parts of the initial presentation added unnecessary confusion. We have revised all the figures (and captions) in question in an effort to clarify, and in some cases simplify, the presentation. Most notably, we have completely remade Figure 7 (previously Figure 6) while correcting minor errors and moving additional details to the SI.

Reviewer #1:

This study utilizes the price equation in order to describe evolution of drug resistance, recognizing the cutting-edge problem of predicting the evolution of resistance in light of collateral effects and drug-drug interactions.

The manuscript offers a formalism that cleverly integrates theory across paradigms, from quantitative genetics to the evolution of antibiotic resistance. As an intellectual exercise, this is courageous, the type of integrative thinking necessary in the modern iteration of evolutionary medicine.

We thank the reviewer for the positive evaluation.

I am not, however, of the mindset that the Price equation is so important a framing that its invocation/application to a different paradigm (antibiotic resistance) is especially interesting by itself. Said differently, I am not personally intrigued by the a study that is successful at connecting antibiotic resistance to the Price equation. Alternatively, I am much much more intrigued by the possibility that the Price equation might add a richer and more nuanced overall perspective to cutting-edge problems in the evolution of antibiotic resistance.

The reviewer makes a good point. We agree that the manuscript is not merely about the Price Equation, but instead about formalizing the concept of concentration rescaling as a general description of resistance as a continuous trait that manifests itself in environmentally-dependent ways. Note that this is quite distinct from previous applications of the Price Equation, which typically cast resistance as a binary trait (0 or 1) and where the primary goal is predicting mean trajectories for the frequency of resistance in a population. In addition, we believe the connection to the Price Equation within our rescaling model has conceptual value, not only because it links our work with a classical result in evolutionary biology, but also because it provides a convenient form for disentangling the effects of collateral resistance and drug interactions (at least in the stated limits).

And I can say that, on the balance, the manuscript was successful in offering a new take on relevant problem.

Also, this study carries out its methods carefully, and its formalisms are well-written. The essential idea is that the vagaries of drug interactions and collateral effects can be collapsed into a formalism that considers both when predicting the trajectory of drug resistance, is a fascinating and important one.

The study is especially transparent (refreshingly so) about the limitations of its scope. And while this transparency is a legitimate positive of this story, I did find myself concerned about the small number of cases that theory proposed in this study applies to. For example, the Price equation as applied in the study applies to selection on standing variation, and not for the step-wise acquisition of resistance that fitness landscapes are often constructed to study. The authors argue that this type of nuance can be engineered into the formalism, which is not obvious to me (I'm not suggesting that it isn't true, only that it isn't obvious).

But that this study has a relatively thin application space does not make the study irrelevant. For the outlined problem space, the arguments appear sound.

I must say that given the current state of the problem of antimicrobial resistance in 2021, sometimes featuring complex, highly dynamic, epistastic fitness seascapes, for example, I was hoping for framework that could be used for fitness landscapes as they are currently constructed and studied.

On the balance, however, I found the study to be well-constructed, written, and the formalisms to be correct. The main concerns regarding the scope might be a matter of preference. And even I must recognize that responsible theoretical biology often requires a narrow scope in order to generate formalisms. The idea here is that from this contribution, the community can then extend the study to other such systems and data sets.

We thank the reviewer for recognizing our efforts to be transparent, for highlighting the value of the formalism – and also recognizing the difficulty of developing theoretical biology with broad scope – and for the generally positive appraisal. The revised version extends the formalism to include mutations, which allows us to incorporate empirical constraints on the mutational pathways linking different phenotypes (similar, though not identical, to the information contained in fitness landscapes). The trade-off with this added complexity is, of course, the need for additional assumptions, and while we have focused on simple illustrative examples (e.g. sequential mutations, Figure 6 and Figure 6 figure supplement 2, and distance-dependent mutations in parameter space, Figure 6 figure supplements 3-4) we believe they illustrate that the approach is flexible and amenable to new empirical constraints as they become available. We discuss these points in the revised manuscript, and (most notably) we also added the following to the discussion:

“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”

I would like the authors to consider, and possibly cite, if they are relevant, more recent papers by Pamela Yeh on this topic:

Tekin, Elif, Pamela J. Yeh, and Van M. Savage. "General form for interaction measures and framework for deriving higher-order emergent effects." Frontiers in Ecology and Evolution 6 (2018): 166.

Tekin, Elif, Van M. Savage, and Pamela J. Yeh. "Measuring higher-order drug interactions: A review of recent approaches." Current Opinion in Systems Biology 4 (2017): 16-23.

Tekin, Elif, et al., "Enhanced identification of synergistic and antagonistic emergent interactions among three or more drugs." Journal of The Royal Society Interface 13.119 (2016): 20160332.

Thank you for these suggestions. We have cited all 3 papers, which focus on higher-order (more than 2) drug interactions and therefore may be useful for extending our framework to higher dimensions.

There are some small issues with the figures that need to be resolved:

I found Figure 6 to be complicated. I would suggest the authors re-generate it with the goal of explaining it to someone not well-versed in the system.

We have simplified and remade Figure 6 (now Figure 7), which we agree was overly complicated in the initial draft.

I had some issues with the legends in Figures 4 and 5. Ensure that they describe the information in the figures properly.

We have clarified the legends of Figures 4 and 5, which were indeed confusing in the initial draft.

Reviewer #2:

This paper examines the interaction between pharmacological and evolutionary factors that mediate the emergence of resistance to combination antimicrobial therapy. The authors introduce and demonstrate the utility of a mathematical equation that describes how both (a) the pharmacological interaction between two drugs (e.g. synergy vs antagonism) and (b) the interaction between the fitness effects of mutations that confer resistance to one drug or the other (or both), jointly determine the direction of selection in the presence of two-drug therapy. They show that the equation they describe follows the same form as a two-trait version of the classical "Price equation" in population genetics. They also examine a "weak selection" limit (when evolution proceeds by selection of a sequence of strains each with only a slight change in resistance level from the previous). In this limit their equation has an even more intuitive physical meaning as movement along an weighted version of the gradient of the drug's dose-response curve, where the weighting reflects the nature of the cross-resistance profile. Many examples of the implications of this equation for different drug interactions and different fitness landscapes are presented. The results of this paper unite findings from quite a few previous studies in a common framework. For example, the authors show how to understand (i) when resistance to one drug in a combination will be more strongly selected than resistance to the other and how this direction of selection can change based on the relative levels of two drugs in a combination – even if total inhibition is the same, (ii) when drug antagonism (vs synergy) is morel likely to slow down the rate of adaptation, and (iii) how the relative ordering and frequency of cycling between two drugs in a combination impacts the rate of resistance emergence.

We thank the reviewer for the concise summary and for recognizing the intuitive value of the framework and its potential to unify findings from several previous studies.

It is important for readers to understand that the Price equation – and thus by extension the result in this paper – is not really a mathematical "model" that includes assumptions about the underlying biological processes and makes predictions about potential experimental results. Instead, the Price equation is a mathematical truth that simply involves decomposing certain statistical relationships between variables and expressing them in an intuitive way. Roughly, in this case the Price equation describes how the average level of resistance in a population of microbes will change over time depending on the frequencies of different strains in the population and their resistance levels, including the correlation between resistance to each drug. Thus, this paper is primarily purely theoretical/mathematical work.

We agree with this point, and we have now been careful to clarify this distinction. Most notably, we added the following to the Results section, just after the introduction of the Price equation (copied from response to Evaluation Summary, above):

“In what follows, we will sometimes casually refer to Equation 5 as a "model", but it is important to note that the Price Equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5 – the Price Equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between those model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.”

In addition, we now consistently refer to the “model” rather than the “Price Equation” when describing our simulations and results (e.g. in the figure captions).

The authors do apply their equation to some real-life drug profiles and fitness landscapes using dose-response curves for two-drug antibiotic combinations to calculate the direction of selection for different dose combinations. Later in the paper, they do also apply their equation as a "model" to predict entire evolutionary trajectories, by additionally assuming that they can define a priori all the available resistance mutations and ignore any additionally generated mutations. Then, they show that the level of resistance selected after a specific time as predicted by their equations agrees relatively well with the results of in vitro experiments containing these same mutations. This is actually quite a surprising and nice result.

Overall this paper provides a useful new framework for understanding drug resistance evolution and provides multiple examples of insights offered by this framework. They provide a nice connection between the mainly mathematical nature of their results and scenarios involving antibiotic treatment and resistance evolution in the lab. The introduction of this paper is very comprehensive and summarizes a huge body of work on the evolution of drug resistance. The discussion is equally thorough and mentions most of the caveats I describe here.

Although the authors of the paper focus on bacterial infections and antibiotic resistance, I think the results are much more general, and likely also apply to multi-drug therapy to viral infections (e.g. HIV, HCV), parasitic infections (e.g. malaria), and even anti-cancer therapy.

While not a weakness that is specific to this paper, it is important for readers to understand that a Price equation only describes the act of selection acting on existing genetic variation in a population, and does not describe the act of generating that variation (i.e. mutations). Thus, while this approach can be helpful for understanding the direction and speed of selection in one vs two-drug combinations of different types, it cannot be used to understand the overall risk of resistance emergence, since the rate at which mutations are generated may also differ. For example, it often takes two separate mutations to generate resistance to two drugs with different mechanisms of action, so resistance to two-drug therapy is often slowed by the rate of generating resistant mutations. In addition, the rate at which mutations are generated during therapy depends on the population size over time, which might differ along different selective paths described by the Price equation for different drug combinations. Therefore, there are some important results from previous work related to factors that accelerate or inhibit multi-drug resistance which cannot be understood in the framework presented in this paper.

Thank you for these points. We have now included additional discussion of the limitations of the model, potential applications and, more importantly, we now illustrate several examples where mutational dynamics are included. We copy below our response to Point 1 in the Evaluation Summary (above):

In the revised manuscript, we include a new section (including 1 additional figure, Figure 6, and several SI figures, S8-S10) that incorporates mutation. Under these conditions, the Price Equation includes an additional term that depends on a mutation matrix that describes the probability of mutation between any pairs of strains (Equation 13). Under some conditions (e.g. when mutations occur with uniform probability between the ancestor cells and each mutant), the effects of mutation are qualitatively similar to those of selection (Figure 4 and Figure 6 figure supplement 1). On the other hand, mutational matrices can be chosen to reflect known features of the mutation process or fitness landscapes. As examples, we consider several illustrative cases, including (1) a case where mutations can only be acquired sequentially (Figure 6, Figure 6 figure supplement 2) and (2) a case where mutations occur in a distance-dependent manner in the space of rescaling parameters, in essence making mutations between strains with similar scaling parameters more likely than those with vastly different scaling parameters (Figure 6 figure supplements 3-4).

Similar to the case with selection only, it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. The relationship between these features and the evolutionary dynamics is complex, but as these examples illustrate, the framework can be extended to incorporate these features and produce testable predictions. In the discussion, we discuss this point as follows:

“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”

One assumption that the authors use in order to get a simplified Price-equation-like result is that the effect of a drug resistance mutation on the dose-dependent effect of a drug on microbial growth can be described by a single parameter, which just shifts the drug concentration to a lower value. This is equivalent to saying that a mutation simply alters the IC50 of the drug. However, many studies have shown that in general mutations have at least one and often two other effects on dose-response curves: firstly, they have a fitness cost, meaning that the maximum fitness in the absence of drug is lower, and secondly, they can also alter the slope of the dose response curve. It is unclear how these effects change the equation the authors have derived.

Thank you for this suggestion. We now include additional discussion of this point in the discussion.

The impact of the authors approach on studying resistance evolution in any particular system depends on the regime that system is in. If resistance tends to evolve by selection slowly acting on an extremely diverse population containing strains with many different resistance levels, then this approach will have the most utility. Alternatively, if resistance evolution tends to be limited by rare mutational events that produce strains with large differences in resistance levels from the parental strain, then this approach will have less use.

To summarize, the most important caveat for readers to understand is that the mathematical expressions that come out of this paper should not be said to be able to "predict" evolution, since they say nothing about the availability of possible resistance mutations or the rate at which they could be generated.

We agree with these points. We have added brief discussions of these points in the new section on mutations as well as in the discussion (as we have noted in the previous responses).

Concerns and suggested changes/clarifications:

– It is a bit misleading to call the Price Equation a mathematical model. It's not a model in the way that word is generally used in biology – it doesn't descibe a mechanism or make any assumptions, and it's not subject to experimental verification. It's just a mathematical statement about a statistical relationship among variables, based on the definition of those variables and basic laws of calculus. The authors should be very clear about this throughout the paper.

– The parts of the paper that are more like a "model" are the parts where you use the Price Equation to describe an evolutionary trajectory, assuming a particular distribution of mutants from standing genetic variation and ignoring additional mutations, or, when you assume a weak selection regime

– Overall the authors should be more careful about the use of the word "model" and claims about "predictions" throughout

Thank you for these important suggestions. In the revision, we have been careful to distinguish between the assumptions of our model (e.g. concentration rescaling, statistical features of collateral effects, or particular mutational structures) and the Price Equation, which the reviewer rightly points out is simply a restatement of statistical relationships emerging from the underlying model. We have also been more careful in our use of “prediction”, which can indeed be misleading in this context. Please see also our responses to point 2 in the Evaluation Summary above.

– I'm not sure about Equation 2 – 5 requires the assumption that the effect of a mutant can be described by a rescaling of drug concentrations. Instead, I think you could replace the α parameters with another parameter r for example that simply means "resistance level" but does not specify how the "resistance level" changes the shape of the dose-response curve.

The reviewer is correct that selection need not occur only on the drug concentration, but can instead alter any feature of the dose response curve, and we now discuss this briefly in the discussion.

– Equation 5 actually is only half of the regular Price equation. It is the "selection" component but is missing the "transmission" component. The authors should be clearer about what processes relevant to the evolution of drug resistance they are ignoring that allow them to set this second term to zero.

As described above, the revised manuscript now includes a new analysis including the “transmission” (in this case, mutation) terms. Please see our response to point 1 in the Evaluation Summary above.

Comments on Figures and results:

Figure 2 – What are the units of the axes on all these graphs?

a) I don't understand this statement in the caption: "We then simulated an external condition where the combination drug concentration is held constant at 0.6 of the maximal concentration of drug 1 and 0.1 of the maximal concentration of drug 2.". There is no such thing as a "maximal" drug concentration … you can always keep adding more and more drug. It would be clearer just to state the drug level used in absolute terms. Or maybe you mean the "highest tested drug level"?

Thank you for the suggestion. We have now rewritten the caption for clarity.

Figure 3 – Some of the results shown here are a little confusing and hard to understand. In the black scenario, there are few available resistant strains with lower α, so we expect selection to proceed in the direction of Drug 2/β.. that agrees with the results in B. But for the blue case where, despite lots of available α but little available β, the trajectory in B seems relatively symmetrical in both drugs … why? Also I was surprised that the case of collateral sensitivity (green) wasn't the worst case … but I realized it's a bit misleading because the four scenarios don't only differ in the nature of the correlations in resistance levels, but also in the magnitude of the variance in α or β and hence the best possible available mutation. I think that it would be a fairer comparison if for the green and magenta scenarios the variance was narrower around the lines of perfect correlation/anticorrelation.

Indeed these results can be counterintuitive. In short, they arise from a combination of the collateral effects and the drug interaction. While the reviewer is correct that the four different cases are not a precise “apples to apples” comparison, the seemingly strange results do make sense in light of the rescaling hypothesis (and are actually not driven by the somewhat smaller differences in correlation parameters generating the collateral profiles). For example, the black trajectory in B is not dominated by resistance to drug 2, which is what one would expect from the collateral effects alone. The reason is that additional resistance to drug 2 does not offer significant growth advantages at this dosage combination due to the drug interactions. Intuitively, one can see this by noting that a rescaling of drug 2 concentration would correspond to moving downward on the contour plot. But moving downward at this location would not change growth dramatically because the rescaled concentration would fall near the same contour as the original drug concentration (i.e. the growth contour itself is nearly vertical at this location). We have added a brief discussion of this point in the main text and also included the specific covariance parameters in the caption.

Figure 4 – This is nice figure that really summarizes the combined effects of drug interactions and cross-resistance profile. However this figure also summarizes the major limitations of this approach. The early slopes of each growth rate vs time curve do show how these factors interact to determine selection strength. However, the fact that the maximum growth rate varies by curve is really just an artifact of the arbitrary and limited available mutants the authors have created for selection to act on. In reality, mutations are continually and sequentially generated over time, and this rate of generation of mutants also impacts the increase in growth rate over time and might remove any "maximum" resistance level that is achievable.

The reviewer makes a good point: the steady state behavior in the no-mutation limit is likely to be determined by growth of a single mutant (or strictly, a series of mutants that fall along a single growth contour), and it is therefore highly dependent on the extremes in the set of scaling parameters. In practice, new dynamics are likely to arise on these long time scales, and the model is expected to fail as mutational processes (and potentially a breakdown of the scaling assumption, in general) become increasingly relevant. The upshot is that the no-mutation model is useful over a limited time horizon, though fortunately, that timescale is similar to that observed in many lab evolution experiments aimed at understanding drug interactions and collateral effects. Perhaps more importantly, we have also added a mutational process to the model in the revised manuscript. While the set of available mutants must still be assigned up front, different mutational structures can be chosen to mimic longer-time processes (e.g. by making mutation rate dependent on the distance between mutants in scaling parameter space, the evolutionary trajectory may proceed through increasingly fit phenotypes). We have also included new supplemental figures that illustrate the connection between the time of observation and potential saturation of the dynamics, though in a slightly different context (Figure 5 figure supplement 4; Figure 7 figure supplements 1-2).

Figure 5 – Are these predicted trajectories using the weak selection limit or the full equations without this additional assumption?

– I'm assuming there was some fixed time limit that selection was allowed to proceed for in these experiments and that it was the same in all cases?

– B: is the y axis absolute IC50 or change in IC50? The axes titles just say IC50 but the figure caption says "relative change in IC50". Also it would be helpful if the baseline IC50 (ie for the wild type strain) where shown on these graph as well.

We have clarified the caption to note that (1) these simulations are from the full equations and (2) these experiments were performed for a fixed time period of 72 hours, and (3) IC50 for each drug is measured in units of the ancestral strain IC50 (so they are indeed normalized IC50s). The wild type IC50 is therefore 1 for each drug (this was not clear in the original version because we didn’t properly indicate that the IC50 is normalized).

Finally, I don't understand how the results for the TGC IC50 agree with the results in A. From B, the IC50 looks totally flat with TGC concentration. But from A, it looks like there is a large increase in TGC resistance (increase in TGC IC50) for high TGC concentrations, but little TGC resistance for low TGC concentrations.

We have now revised the caption to hopefully clarify these points. In panel A, we show a collection of 100 trajectories at each condition, with each trajectory based on a uniformly sampled collection of approximately 10 parameter pairs. The black x marks indicate the mean value (across all trajectories for a given dosage combination) at the end of the simulation. In B, we plot these final-time average values (normalized IC50 corresponds to the reciprocal of the corresponding scaling constant) as a function of TGC concentration in the selecting environment (which here is just a proxy for the different external dosage combinations, the different conditions, as you move left to right on the contour). The discrepancy you note at high TGC concentrations arises from the fact that the trajectories with large increases in TGC are relatively rare and therefore contribute little to the trajectory average.

– C: I don't understand what the y axis is here. If this is the increase in growth rate, then why is it highest for low TGC concentrations, yet both panels in B show no change in IC50 to either drug at low TGC concentrations?

The axis shows increase in growth rate, which we have clarified in the caption. At low TGC concentrations, panel B shows that the change in TGC IC50 is minimal (TCG scaling parameter near 1) while the change in CIP IC50 is larger (it increases by a factor of roughly 3-4). In terms of rescaling, this trend makes intuitive sense: a vertical rescaling (increased resistance to CIP) leads to an increase in growth for conditions at low TGC, where the rescaling moves the effective concentration into a region of higher growth. But for higher TGC concentrations, such a vertical concentration rescaling would not be expected to have much effect on growth because a vertical shift in effective drug concentration does not lead to large deviations from the original growth contour (and in fact experiments in this regime did not select for large increases in CIP resistance). In words: TGC has a strongly suppressive effect on CIP (strongly antagonistic drug interaction). As a result, low TGC concentrations select for CIP resistance while higher TGC concentrations do not, even when the CIP concentrations are similar.

And any idea what could be causing the systematic difference between the predicted and observed changes in growth rate?

This is an interesting question, and we don’t know the answer. One potential explanation is that the time chosen for the simulation (72 hours) does not match the true time over which selection was taking place in the experiment. The experiment was performed for 72 hours, but various environmental variables (fluctuations in temperature, density-dependent changes in growth rate, etc) could mean that the evolution occurred, effectively, for more or less than 72 hours. The upshot is that the total number of generations may not match perfectly between model and experiment. We did not attempt to tune this parameter to fit the data, but indeed we can decrease the systematic error by choosing a different value of T (but we prefer to show the model without fitted parameters). It is also possible that the systematic error arises from other features, such as uncertainty in the estimates of the wild-type growth surface. We have now explicitly noted and briefly discussed the systematic discrepancy in the main text.

Figure 6 – This figure was quite confusing, it needs more annotation on the figure itself (not just written in the caption), and some panels are unnecessarily small (with unnecessary white space between them). For panels H and K, are there better axis limits that could be used so that 95% of the figure is not all one color? Also I found it is hard to see in this figure any of the differences between the outcomes of the two different orderings that are described in the text.

We have replaced this figure (now Figure 7) with a simplified version. Our hope is to highlight one way that the framework can be extended to make predictions in an otherwise intractably complex scenario. However, the previous figure was unnecessarily confusing. In the new figure, we choose a simpler example than in the previous version, and we focus on a smaller set of the dynamics.

Reviewer #3:

In this work, Gjini and Wood create a model to simulate evolutionary dynamics of antibiotic resistance arising as a response to drug combinations. They demonstrate that evolution to these drug combinations is a variation of the Price equation and show that some evolutionary paths experimentally correspond to a weighted gradient descent on the growth landscape.

The work appears mathematically sound, though we have concerns about the scope and generality of the results. In particular this work appears to rely fundamentally on the assumption that drug resistance acts as a linear rescaling of the two-drug dose response curve. We believe this work would be strengthened by making it more clear what, beyond the mathematical description of the rescaling hypothesis they cite extensively, this work adds to the literature. Since the model appears to also require an a priori description of the resistance characteristics of every possible accessible mutant, which typically is not known, it isn't clear the utility of the model.

Thank you for these suggestions. In our view, there are two main advances. First, we incorporate multiple features of the evolutionary process (collateral effects, drug interaction, and, in the new version, mutational structure) into a model that makes testable predictions. As with all models, the outputs are only as good as the inputs, and we can not use the model to predict resistance evolution without those inputs (e.g. the drug interaction and statistical properties of the collateral effects). However, when those properties are known, or when we can make reasonable assumptions about them, the model allows us to make predictions that would not otherwise be possible and to systematically investigate the impacts of different phenomena (e.g. collateral effects vs drug interactions). Second, we believe the approach has considerable conceptual value. It provides a direct link to the Price Equation, which connects the work with classical approaches in population genetics. Perhaps more importantly, the Price Equation formalism allows us to mathematically disentangle effects of collateral resistance and drug interactions, at least in the stated limits, in a way that highlights how the geometry of the response surface interacts with statistical properties of the available mutants.

We also note that in the new version, we now incorporate mutation between different phenotypes, which significantly extends the scope of the model (see Figure 6 and Figure 6 figure supplements).

Please also see our responses to points 1 and 2 in the Evaluation Summary (above).

We also copy below a (somewhat related) response to a suggestion from Reviewer 1:

We thank the reviewer for recognizing our efforts to be transparent, for highlighting the value of the formalism, and also recognizing the difficulty of developing theoretical biology with broad scope, and for the generally positive appraisal. The revised version extends the formalism to include mutations, which allows us to incorporate empirical constraints on the mutational pathways linking different phenotypes (similar, though not identical, to the information contained in fitness landscapes). The trade-off with this added complexity is, of course, the need for additional assumptions, and while we have focused on simple illustrative examples (e.g. sequential mutations, Figure 6, and distance-dependent mutations in parameter space, Figure 6 figure supplements 3-4) we believe they illustrate that the approach is flexible and amenable to new empirical constraints as they become available. We discuss these points in the revised manuscript, and (most notably) we also added the following to the discussion:

“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.“

– This paper models evolution in the presence of physiological interactions between antibiotics but not evolutionary interactions. The fundamental assumption that evolution of resistance scales the surface but does not change its shape, which is central to the findings, is true in some cases, though violated for many mutations (the authors cite numerous papers on cross-resistance and collateral sensitivity that are examples of this).

We agree that the rescaling assumption will not always hold, and we discuss these limitations extensively in the discussion. We also discuss the fact that the model does not incorporate interactions between cells (e.g. cooperation and competition), it is an avenue for future work, we hope! We would like to point out, however, that cross-resistance and collateral sensitivity in and of themselves are not indications that the scaling assumption is violated (testing this would require measuring the full dose response surface in the mutants and determining whether a simple rescaling of the two axes is sufficient to link the two surfaces, that is, whether the shape of the surface changes, not merely the axes scales). There is considerable experimental support suggesting the scaling assumption provides a reasonable approximation, at least for conditions of a typical lab evolution experiment, but indeed the assumption will not always hold, and we do note that important limitation.

– While the authors describe it as "straightforward" to include discrete mutational events instead of a small population of equal numbers of every potential mutant in the model, it is not clear that this is the case. For example, a population exposed to an above-MIC concentration of a bacteriostatic drug will no longer generate additional mutants, and so the standing diversity before drug application matters critically. As it stands the model is really about differential growth and not evolution per se, since neither mutation nor complete extinction can occur.

We have incorporated mutation in the revised manuscript (see response to point 1 in the Evaluation Summary, above). But the reviewer makes an important point: the model accounts only for differences in growth rates between different strains. It does not account for changes in total population size, though these effects would be particularly important as the population nears extinction. The model also neglects demographic stochasticity, which could play a significant role in some regimes. We’ve added a short discussion of these points in the discussion.

– It isn't clear the experiments are representative of nontrivial behavior predicted by the model. For example in figure 5 no adaptation to TGC was seen, so it seems the only real evolution was rescaling CIP, which wouldn't yield the suppression-induced selection against resistance they then discuss. These results could be entire an artifact of the selection being entirely on CIP and not at all on TGC. The evolution experiments in Figures S3 and S4 seem much more compelling, though.

We found this example quite compelling, and at first we were puzzled by your comment. But then we realized that there was a bug in the code that plotted the experimental results, which obscured the most interesting features of the experimental data (they were still present in the model). Correcting this plotting error also led to an improved agreement between experiment and model. Thank you for your comment, which helped us avoid a silly mistake (the bug was apparently introduced during an internal revision of the figure prior to submission; it wasn’t there in our original internal drafts but showed up in the version we submitted, and we simply missed it).

With that bug now fixed, hopefully it is clear why we find the example compelling. Selection for CIP resistance occurs only for sufficiently low concentrations of TGC, even in cases where inhibitory concentrations of CIP are present. For example, evolution in the far left condition (TGC~0, CIP~0.2) leads to a 2-3 fold increase in CIP IC50 (panel B) and significant growth adaptation (panel C). However, evolution at the 3rd condition from the right (TGC~0.04, CIP~0.2) shows very little resistance to CIP, despite the fact that a similar concentration of CIP was used. The fourth condition from the right is even more striking, as there the concentration of CIP is even higher (about 0.35) yet little CIP resistance is observed.

– If it is indeed straightforward to add mutational events, this would make the model appreciably more realistic.

We have now incorporated mutation in a new section of the manuscript. Please see response to point 1 in the Evaluation Summary (above).

Author details

1. Erida Gjini

   Center for Computational and Stochastic Mathematics, Instituto Superior Tecnico, University of Lisbon, Portugal, Lisbon, Portugal

   Contribution

   Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   erida.gjini@tecnico.ulisboa.pt

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0493-3102

2. Kevin B Wood

   Departments of Biophysics and Physics, University of Michigan, Ann Arbor, United States

   Contribution

   Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   kbwood@umich.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0985-7401

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