Fitness landscapes emerging from pharmacodynamic functions in the evolution of multidrug resistance.

doi: 10.1111/jeb.12355

Fitness landscapes emerging from pharmacodynamic functions in the evolution of multidrug resistance € TER J . E N G E L S T AD School of Biological Sciences, The University of Queensland, Brisbane, Qld, Australia

Keywords:

Abstract

adaptation; adaptive landscape; antibiotic resistance; cross-resistance; drug interactions; epistasis; pharmacodynamics-pharmacokinetics.

Adaptive evolution often involves beneficial mutations at more than one locus. In this case, the trajectory and rate of adaptation is determined by the underlying fitness landscape, that is, the fitness values and mutational connectivity of all genotypes under consideration. Drug resistance, especially resistance to multiple drugs simultaneously, is also often conferred by mutations at several loci so that the concept of fitness landscapes becomes important. However, fitness landscapes underlying drug resistance are not static but dependent on drug concentrations, which means they are influenced by the pharmacodynamics of the drugs administered. Here, I present a mathematical framework for fitness landscapes of multidrug resistance based on Hill functions describing how drug concentrations affect fitness. I demonstrate that these ‘pharmacodynamic fitness landscapes’ are characterized by pervasive epistasis that arises through (i) fitness costs of resistance (even when these costs are additive), (ii) nonspecificity of resistance mutations to drugs, in particular cross-resistance, and (iii) drug interactions (both synergistic and antagonistic). In the latter case, reciprocal drug suppression may even lead to reciprocal sign epistasis, so that the doubly resistant genotype occupies a local fitness peak that may be difficult to access by evolution. Simulations exploring the evolutionary dynamics on some pharmacodynamic fitness landscapes with both constant and changing drug concentrations confirm the crucial role of epistasis in determining the rate of multidrug resistance evolution.

Introduction Resistance to several drugs simultaneously is an alarming phenomenon observed in many important bacterial pathogens (Nikaido, 2009). Multidrug resistance is often caused by several mutations, each of which may be specific to one drug or may confer (partial) resistance to several drugs. For example, in Mycobacterium tuberculosis, resistance to the main drugs used for treatment can be caused by mutations in the genes rpoB (resistance to rifampicin), katG, inhA and others (resistance to isoniazid), and pncA (resistance to pyrazinamide), with additional resistance genes present against other drugs in some strains (Da Silva & Palomino, 2011). Correspondence: Jan Engelst€adter, School of Biological Sciences, The University of Queensland, Brisbane, Qld 4072, Australia. Tel.: +61 7 336 57959; fax: +61 7 336 51655; e-mail: [email protected]

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Likewise, a multitude of genes have been identified that provide resistance to different drugs in Staphylococcus aureus (Chambers & Deleo, 2009), Pseudomonas aeruginosa (Livermore, 2002), enterobacteriaceae such as Klebsiella pneumoniae (Bush, 2010), and others. Given this multilocus genetic basis of multidrug resistance, a central concept for studying the evolutionary dynamics of resistance mutations is that of a genotypic fitness landscape (Wright, 1932). When considered mathematically and not merely metaphorically, fitness landscapes are mappings assigning fitness values to a set of genotypes forming a mutational network, for example all combinations of resistance and drug-sensitive alleles under consideration. Fitness landscapes play a central role in evolutionary biology, determining for instance the trajectories and speed of adaptation (Maynard Smith, 1970; Kauffman & Levin, 1987; Poelwijk et al., 2007), the evolutionary impact of recombination (Bergman & Feldman, 1992; Kondrashov & Kondrashov, 2001;

ª 2014 THE AUTHOR. J. EVOL. BIOL. 27 (2014) 840–853 JOURNAL OF EVOLUTIONARY BIOLOGY ª 2014 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY

Pharmacodynamic fitness landscapes

Misevic et al., 2009; de Visser et al., 2009; Moradigaravand & Engelst€adter, 2012) and the evolution of reproductive isolation and speciation (Gavrilets, 2004). An important property of fitness landscapes is epistasis, a measure for deviations from independence of fitness effects of alleles at different loci (Weinreich et al., 2005; de Visser et al., 2011). Negative epistasis implies that beneficial mutations at different loci increase fitness less in combination than what is expected from their individual effects. Conversely, with positive epistasis, the fitness of a genotype combining several beneficial mutations is greater than the sum of their individual effects. In addition to this so-called ‘magnitude epistasis’, there can also be sign or reciprocal sign epistasis, in which not only the magnitude but also the direction of selection on one or both mutations is affected by the allelic state at the respective other locus. Because epistasis generates linkage disequilibria of the same sign, it determines how fast beneficial mutations spread at several loci simultaneously: positive epistasis accelerates adaptation, negative epistasis slows down adaptation and reciprocal sign epistasis may even prevent adaptation in the presence of recombination (Felsenstein, 1965; Crow & Kimura, 1970; Eshel & Feldman, 1970; Weinreich et al., 2005). Only recently have researchers begun to investigate fitness landscapes and epistasis for mutations conferring antibiotic resistance. Some of these studies have focused on measuring epistatic effects of resistance mutations in the absence of drugs, that is, epistasis in the costs of resistance (Trindade et al., 2009; Hall & MacLean, 2011; Silva et al., 2011). Although these studies revealed a wide distribution of epistatic effects, a tentative consensus appears to be a predominance of positive epistasis in fitness costs. Interestingly, Silva et al. (2011) also reported many instances of sign epistasis between chromosomal resistance mutations and resistance mutations located on plasmids. Epistasis is also prevalent between different mutations conferring resistance to a single drug in the presence of that drug. For example, characterizing the fitness landscape for resistance to the b-lactam cefotaxime in Escherichia coli, Weinreich et al. (2006) documented extensive reciprocal sign epistasis that made many evolutionary pathways to full resistance inaccessible. By contrast, sign epistasis was found to be rare or absent in mutations conferring resistance to rifampicin in Pseudomonas aeruginosa, but negative (i.e. diminishing returns) magnitude epistasis was pervasive (MacLean et al., 2010). The above studies have investigated fitness landscapes of drug resistance in either the absence or presence of drug pressure. However, populations of bacterial pathogens will generally be exposed to a continuum of varying drug concentrations. In particular, drug concentrations will vary over time within patients following drug-specific pharmacokinetics that comprise adsorption, distribution within the body as well as

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metabolization and excretion. Moreover, drug concentrations may vary between different host tissues and between successively infected host individuals. It is important therefore to also take the pharmacodynamics of drugs into account, that is, the relationship between drug concentration and bacterial growth or death rates. One common approach for single drugs is to employ Hill functions describing this relationship and to estimate the parameters of this function (e.g. Regoes et al., 2004), and this approach can also be extended to multiple drugs (e.g. Ankomah & Levin, 2012; Ankomah et al., 2013). Here, I put forward a mathematical framework that integrates the concepts of fitness landscapes and pharmacodynamics. Specifically, I construct fitness landscapes that emerge from Hill functions describing the pharmacodynamics of drugs for different drug-sensitive and resistant genotypes. I focus on the simplest case of two mutations conferring resistance to two drugs, thus yielding a two-locus fitness landscape for each combination of drug concentrations. The impact of various factors such as costs of resistance, cross-resistance and drug interactions on properties of these pharmacodynamic fitness landscapes is then analysed. Finally, I briefly investigate the evolutionary dynamics taking place on the pharmacodynamic fitness landscapes, both with constant and with changing drug concentrations.

Construction of pharmacodynamic fitness landscapes In what follows, I will outline the construction of the drug-dependent fitness landscapes in three steps. Step 1: Single drug Consider first only a single, drug-susceptible genotype and a single drug. Following Regoes et al. (2004), reduction in bacterial fitness depending on the drug concentration A will be modelled as a Hill function: j A f

H ð A Þ ¼ u j A f

wu w

(1)

Here, u is the maximum reduction in fitness, w is the growth rate in the absence of antibiotics, f is the minimum inhibitory concentration (MIC) of the drug (the drug concentration when there is zero net growth), and j determines the steepness of the Hill function. The fitness (or net growth rate) of the bacteria is then given by wð AÞ ¼ w H ð AÞ:

(2)

Figure 1a illustrates this fitness function and its parameters.


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Fig. 1 Illustration of the construction of the pharmacodynamic fitness landscape. (a) This plot Shows the pharmacodynamics with a single drug and single genotype, based on a Hill function (see eqns 1 and 2). Parameters take the values w = 0.1, u = 0.2, f = 1, j = 1. (b) These plots show the fitness and MIC curve (red line) of the fully susceptible genotype SS as a function of the two antibiotic concentrations (see eqn 3) as well as the corresponding functions for the two single-resistant genotypes (characterized by increased MIC and slightly reduced maximum growth rate) and the double-resistant genotype (with increased MIC to both drugs and additively reduced maximum growth). SR SS RS RS RR SR RR i Parameters take the values wSS = 0.1, wSR = wRS = 0.09, wRR = 0.98, fSS 1 ¼ f1 ¼ f2 ¼ f2 ¼ 1; f1 ¼ f1 ¼ f2 ¼ f2 ¼ 100, uj ¼ 0:2 and jij ¼ 1 for all i, j.

by Ankomah & Levin (2012) in the absence of drug interactions.

Step 2: Two drugs Let us now consider two different drugs and assume that these two drugs reduce bacterial fitness independently (i.e. there are no drug interactions; this assumption will later be relaxed). Denoting the concentrations of the two drugs by A1 and A2, bacterial fitness is then given by w ðA1 ; A2 Þ ¼ w H1 ðA1 Þ H2 ðA2 Þ

(3)

Here, each of the two Hill functions is characterized by its own set of parameters (u1, f1, j1 vs. u2, f2, j2). Barring differences in notation, this is the function used

Step 3: Resistant genotypes For each of the two drugs, let us now consider a gene conferring resistance to that drug. This could be a point mutation modifying the targets of the drugs, a gene acquired through horizontal gene transfer or a cassette of several genes. Denoting by S the susceptible and by R the resistance allele at each of these two loci, there are four different genotypes: the fully susceptible genotype SS, two single-resistant genotypes (SR and RS)



and one genotype resistant to both drugs simultaneously (RR). Each of these four genotypes is characterized by its own fitness function (3), where the genotype will be denoted as a superscript. For example, for the SR genotype, the fitness function would take the form wSR ðA2 ; A2 Þ ¼ wSR H1SR ðA1 Þ H2SR ðA2 Þ 2 3 !jSR !jSR 1 1 SR SR A w u 1 SR SR A1 1 4 5 ¼ w u1 fSR fSR wSR 1 1 2 3 !jSR !jSR 2 2 SR SR A w u 2 SR A2 2 4 5 u2 fSR fSR wSR 2 2 (4) Figure 1b shows the pharmacodynamic fitness landscape for a particular set of parameters. Table 1 summarizes all notation used in this model. All analyses and simulations were conducted using the software package MATHEMATICA, version 9.0 (Wolfram Research Inc., Champaign, IL, USA).

Characterizing the fitness landscapes In this section, the fitness landscapes that emerge from incorporating pharmacodynamics functions for all resistance genotypes will be investigated and in particular scrutinized for the presence of epistasis. Epistasis is defined as deviation from independent effects of alleles at two or more loci. Specifically, epistasis in fitness in our model, E, will in general depend on antibiotic concentrations A1 and A2 and is given by

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EðA1 ; A2 Þ ¼ wSS ðA1 ; A2 Þ þ wRR ðA1 ; A2 Þ wSR ðA1 ; A2 Þ wRS ðA1 ; A2 Þ (5) Before proceeding with this characterization, the important concept of drug specificity of resistance mutations as applied to the present model needs to be defined. The two resistance mutations will be called drug-specific if x1SR ¼ x1SS ; x1RS ¼ x1RR ; x2RS ¼ x2SS ; and x2SR ¼ x2RR ;

(6)

where x stands for any of the three parameters u, f and j. Intuitively, this condition means that each of the two resistance mutations affects only those parameters that are associated with the drug that it confers resistance to. Note, however, that condition (6) does not imply that the corresponding Hill functions H are identical. This is because those functions also involve the parameter w (maximum growth rate), which will differ between genotypes when there are costs of resistance. Very low and very high drug concentrations In the simplest case where A1, A2 ? 0, epistasis is determined only by the maximum growth rates of the four genotype (E = wSS + wRR wSR wRS). If we denote by cSR, cRS and cRR the costs of resistance for the two resistance mutations (i.e. cSR = wSS wSR, cRS = wSS wRS and cRR = wSS wRR), we get E = (cRR cSR cRS), which is the negative of the deviation from additivity in fitness costs of the doubly resistant genotype. Epistasis in fitness is thus equal to the negative of epistasis in fitness costs of resistance. When one drug concentration is very high and the other very low, for example A1 ? ∞, A2 ? 0, we have RR SR RS (7) E ¼ wSS þ wRR wSR wRS uSS 1 þ u1 u1 u1

Table 1 Parameters and variables used in the model. Parameters w u f j Amax r T Variables A H w E N

Growth rate in the absence of antibiotics Maximum reduction in growth caused by antibiotic Minimum inhibitory concentration (MIC) of drug Hill coefficient; determines steepness of Hill function Maximum antibiotic concentration in pharmacokinetics Pharmacokinetic rate of decay of antibiotic Periodicity in pharmacokinetics Antibiotic concentration Reduction in growth caused by antibiotics Fitness, or net growth rate, of bacteria, depending on antibiotic concentrations Epistasis Population size

Where applicable, subscripts on each parameter or variable denote antibiotics (1 or 2) and superscripts denote genotypes (SS, SR, RS or RR).

Thus, epistasis in this case is the difference between the negative of epistasis in fitness costs of resistance as described in the previous paragraph, and epistasis in the maximum reduction in fitness caused by drug 1. The latter type of epistasis will be zero when the two resistance mutations are drug-specific (as defined in eqn 6), so that epistasis in overall fitness reduces to the negative of epistasis in fitness costs. Finally, when both drugs are present at very high concentrations (A1, A2 ? ∞), epistasis computes to RR SR RS E ¼ wSS þ wRR wSR wRS uSS 1 þ u1 u1 u1 (8) RR SR RS uSS 2 þ u2 u2 u2 ; which is the difference between the negative of epistasis in costs of resistance and the sum of epistases in the maximum reduction in fitness caused by drugs 1 and 2. Again, when there is drug specificity of the resistance



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mutations, epistasis in fitness reduces to the negative of epistasis in fitness costs. Special cases with nonepistatic fitness landscapes In two special cases, the pharmacodynamic fitness landscapes were found to be nonepistatic for all (or almost all) drug concentrations. Let us first assume that there are no fitness costs associated with either of the two resistance mutations, so that the growth rates of all four bacterial genotypes are the same in the absence of any antibiotics (wSS ¼ wRS ¼ wSR ¼ wRR ). Under this assumption, we get EðA2 ; A2 Þ ¼ wSS H1SS ðA1 Þ H2SS ðA2 Þ þ wRR H1RR ðA1 Þ H2RR ðA2 Þ SR w H1SR ðA1 Þ H2SR ðA2 Þ wRS H1RS ðA1 Þ H2RS ðA2 Þ (9) ¼ wSS þ wRR wSR wRS SS SR H1 ðA1 Þ H1 ðA1 Þ H1RR ðA1 Þ H1RS ðA1 Þ H2SS ðA2 Þ H2RS ðA2 Þ RR H2 ðA2 Þ H2SR ðA2 Þ Hence, if both resistance mutations are drug-specific (as defined in eqn 6), all terms in brackets are zero and there is no epistasis at any combination of antibiotic concentrations (EðA2 ; A2 Þ 0). A second special case is the limiting case where jij ! 1 for all genotypes i and both drugs j so that all Hill functions become infinitely steep. Each Hill function (1) can then be simplified to the step function

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8 wRS, wSR > wRR and dark blue: wRR > wRS, wSR > wSS), whereas orange colours denote regions where the single-resistant genotypes take extreme fitness values (dark orange: wSR > wSS, wRR > wRS and light orange: wRS > wSS, wRR > wSR); grey colours denote other fitness orderings that imply sign epistasis. Plot (b) shows mutant selection windows, that is, combinations of drug concentrations where the double-resistant (lilac), one of the single-resistant (blue and red) or the susceptible genotype (green) can grow and outcompete all other genotypes. Black lines indicate MICs for the four genotypes. Plot (c) gives the magnitude and sign of epistasis, calculated from eqn (7). All parameters as in Fig. 1b. ª 2014 THE AUTHOR. J. EVOL. BIOL. 27 (2014) 840–853 JOURNAL OF EVOLUTIONARY BIOLOGY ª 2014 EUROPEAN SOCIETY FOR EVOLUTIONARY BIOLOGY


genotypes can be the fittest genotypes with appropriate antibiotic concentrations. Moreover, for most combinations of antibiotic concentrations, the two singly resistant genotypes SR and RS take either both intermediate fitness values (blue regions) or take the highest and lowest fitness value (orange regions). However, there are also regions (shown in grey) where this is not the case, indicating the presence of sign epistasis. Figure 2b shows the mutation selection windows for the different strains, defined as combinations A1 and A2 of drug concentrations for which a given genotype i can grow (wi(A1, A2) > 0) and has a higher fitness than the 0 other genotypes (wi(A1, A2) > wi (A1, A2)∀i0 6¼ i). This plot shows that mutation selection windows may extend to drug concentrations below the MICs of the susceptible strains. In other words, a resistant strain may be selected for even with drug concentrations where susceptible genotypes are still able to grow. As in the case of single drug treatment (Gullberg et al., 2011), defining mutation selection windows as extending from the MIC of the susceptible strains to the MIC of the resistant strains may therefore be misleading. Finally, the magnitude and direction of epistasis is shown in Fig. 2c. When drug concentrations are very low or high, there is no epistasis. This result can be understood from the previous section, which showed that with drug-specific resistance mutations and additive fitness costs, epistasis occurs only around the slopes of the Hill functions. For intermediate drug concentrations, there are both positive and negative epistases. However, in the region of the A1 9 A2 plane that falls below the MICs of the RR genotype and that therefore is arguably the most relevant region for drug resistance evolution, negative epistasis predominates. By contrast, positive epistasis is found almost exclusively at drug concentrations above the MICs of the doubly resistant strain, that is, at drug concentrations where none of the genotypes can grow. As shown in Section 1 of the Supporting Information (SI), this overall picture of emerging epistasis appears to be robust with regard to the assumption that Hill functions underlie the pharmacodynamics in the model. Moreover, I demonstrate in SI Section 2 that the reason why epistasis emerges from fitness costs is due to the assumption that a costly mutation conferring resistance to one drug does not affect the MIC for the other drug. In particular, around the MICs of the resistant genotypes, two of the four genotypes (e.g. SS and SR with only drug 1 present) will both have zero fitness whereas the other two genotypes (RS and RR) will differ in fitness by as much as the cost of resistance, so that negative epistasis of magnitude as high as the cost of resistance can arise. If the above assumption is relaxed, fitness costs per se may not result in epistasis (Fig. S2, see also Discussion). Figure 3 shows more examples for epistasis in the pharmacodynamics fitness landscapes when resistance

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mutations are drug-specific, illustrating general effects of the different parameters. An overall, genotypeunspecific increase in the maximum reduction in fitness caused by the antibiotics leads to an increase in the magnitude of both negative and positive epistases around the areas when either of the two antibiotic concentrations equals one of the two MICs for that drug (Fig. 3a). In accord with the previous section, decreasing the Hill coefficient j causes epistasis to take less extreme values but to spread over a wider range of antibiotic concentrations (Fig. 3b). Higher fitness costs, as incorporated through reduced maximum growth rates w for resistant genotypes, increase the overall amount of epistasis in a similar way as increasing u does (Fig. 3c). Increasing the MIC conferred to by the two resistance mutations leads to increasing separation of the areas in the A1 9 A2 plane where epistasis is negative and positive, respectively (Fig. 3d). Figure 3e, f illustrates cases where one or two groups of parameters are asymmetric with respect to the two drugs. Finally, Fig. 3g–i gives examples for cases with epistasis in the fitness costs. This induces another source of epistasis in fitness, distinct from but acting in addition to epistasis caused by the interplay between fitness costs and the Hill functions described above. As can be seen in Fig. 3g, h, negative or positive magnitude epistasis in fitness costs leads to an overall shift in epistasis in the respective direction, affecting especially areas of low and high drug concentrations where epistasis is close to zero in the absence of such epistasis in fitness costs. Figure 3i shows an example where there is sign epistasis in the fitness costs: the resistance mutation for drug 2 compensates for the fitness cost of resistance mutation for drug 1 but does not itself impose a fitness cost. This is a situation similar to the one reported by Trindade et al. (2009) and produces sign epistasis in fitness for low concentrations of drug 1 (not shown). Cross-resistance Many resistance mutations confer simultaneous resistance to several different drugs. Thus, it is important to relax the assumption of drug specificity (eqn 6) and also investigate cases where one or both resistance mutations affect parameters associated with both drug 1 and 2. I will focus on simultaneous increases in the MICs for both drugs caused by one resistance mutation. In order to isolate epistatic effects arising from cross-resistance, I will assume in this section that there are no fitness costs associated with either resistance mutation. Figure 4 shows the emerging epistasis in different scenarios of cross-resistance. In Fig. 4a,b, cross-resistance is complete: both resistance mutations increase the MIC for both drugs by exactly the same amount. In Fig. 4a, increases in MICs are multiplicative (additive on a log-scale), implying that there is no intrinsic epistasis in the f parameters. Despite this, there is pervasive


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Fig. 3 Epistasis in example pharmacodynamic fitness landscapes with drug-specific mutations. Each plot shows the magnitude and sign of epistasis for a particular pharmacodynamic fitness landscape that has been derived through modification of one or few parameters from the standard fitness landscape shown in Figs 1b and 2. Altered parameters in these fitness landscapes are (a) uij ¼ 0:5 for all i, j, (b) jij ¼ 0:5 for SR RR RR SS SR i i all i, j, (c) wSR = wRS = 0.08, wRR = 0.06, (d) fRS 1 ¼ f2 ¼ f1 ¼ f2 ¼ 1000, (e) u2 ¼ 0:5 for all i, (f) j2 ¼ 0:5 for all i and f2 ¼ f2 ¼ 0:1, (g) wRR = 0.0875, (h) wRR = 0.0925, (i) wSR = wRR = 0.1.

positive and negative epistasis in fitness that is very similar to the epistatic distribution found in the presence of costs of resistance (compare Fig. 2b). Note, however, that a major difference in the fitness landscape is that unlike in Fig. 2b, the order of fitness values for all fitness landscapes considered in this section is strictly wRR > wRS, wSR > wSS for all A1, A2 > 0. In Fig. 4b, there is intrinsic negative epistasis in the MICs, SR SS RR that is, lnðfRS i Þ þ lnðfi Þ [ lnðfi Þ þ lnðfi Þ for both drugs i. As a consequence, the fitness landscape is now characterized by strong negative epistasis, and some areas of positive epistasis in Fig. 4a have vanished.

Figure 4c shows an example for when cross-resistance is only partial. Here, each resistance mutation strongly increases the MIC for one drug, but also slightly increases the MIC for the respective other drug. This leads to the same pattern of epistasis as with complete cross-resistance in Fig. 4a, but the magnitude of epistasis is lower. Finally, Fig. 4d assumes that one resistance mutation increases the MIC for both drugs, whereas the other resistance mutation is specific to drug 2. Epistasis in this case takes negative and positive values below and above the MIC for drug 1, respectively, but is independent of the concentration of



Fig. 4 Epistasis in example pharmacodynamic fitness landscapes with cross-resistance. Maximum growth rates are assumed to be the same for all genotypes (no costs of resistance): wSS = wRS = wSR = wRR = 0.1. MICs in genotypes with a single resistance mutation take the values (a) RS SR SR fRS 1 ¼ f2 ¼ f1 ¼ f2 ¼ 10, (b) RS RS SR f1 ¼ f2 ¼ f1 ¼ fSR 2 ¼ 20, (c) SR SR RS fRS 1 ¼ f2 ¼ 20, f1 ¼ f2 ¼ 5, (d) SR RS ¼ f ¼ 10; f ¼ 1; fSR fRS 1 1 2 2 ¼ 100. All other parameters as in Fig. 1b.

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drug 2 (because all Hill functions H2i cancel out in the expression for epistasis, see eqn 9). Drug interactions So far, I have assumed throughout that the two drugs act independently from each other. More precisely, a form of Bliss independence was assumed in which fitness reductions caused by the two drugs are additive (see eqn 3 and see also the Discussion where the difference to Loewe additivity is examined). In this section, I will consider the case where there are interactions between the two drugs, which can be captured by replacing the fitness function (3) with the following function: wi ðA2 ; A2 Þ ¼ wi H1i ðA1 Þ H2i ðA2 Þ I i ðA1 ; A2 Þ

(11)

Here, i stands for any of the four genotypes and the Hill functions Hji are defined as above. It is clear from eqns (9) and (11) that in the absence of costs of resistance and with drug specificity of the two resistance mutations, epistasis in fitness reduces to the negative of epistasis in the interaction terms (E = (ISS + IRR IRS ISR)). Let us now consider one specific out of many possible sets of functions Ii, namely functions of the b form I i ðA1 ; A2 Þ ¼ a H1i ðA1 ÞH2i ðA2 Þ=ui1 ui2 . Here, the parameter a describes the strength and direction of drug interactions, and the terms in the bracket are normalized such that with very high concentrations of the two drugs (A1, A2 ? ∞), a gives the deviation in

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net growth rate from the expectation of additivity of the two Hill functions. Positive values of a imply synergistic drug interactions (stronger reduction in growth than expected based on the reduction in growth caused by each drug individually), whereas negative values of a imply antagonistic drug interactions. The parameter b determines how fast the functions Ii approach the maximum value of a with increasing values of H1i H2i . Some insights into the fitness landscapes resulting from this form of drug interactions can be gained by again considering the case of jij ! 1 for all genotypes i and both drugs j (Hill functions converge to step-functions) and assuming no fitness costs of resistance and drug specificity. Ignoring drug concentrations at exactly the MICs of the two resistance mutations, we then get the simple result RR if fSS j \Aj \fj for jf1; 2g E ¼ a 0 otherwise. This means that there will be a constant level of epistasis for combinations of drug concentrations spanning the area between the four MICs of the susceptible and resistant allele for both drugs, and this epistasis will be negative for synergistic and positive for antagonistic drug interactions. For drug concentrations outside of this area spanning the MICs, there will be no epistasis. Figure 5 explores two pharmacodynamic fitness landscapes that emerge with smaller values of j, resulting in less steep Hill functions. As before, when the two


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drugs interact in a synergistic way, this produces negative epistasis, but now epistasis varies on magnitude and also extends beyond the MICs (Fig. 5a,b). In Fig. 5c,d, antagonistic drug interactions are assumed resulting in positive epistasis. More precisely, the drug interaction assumed in Fig. 5c,d is one of reciprocal suppression of the two drugs (an extreme case of drug antagonism; see Yeh et al., 2009); here, the two drugs suppress each other when combined so that the greatest reduction in growth is achieved when each drug is used on its own. For some combinations of drug concentrations, this results in reciprocal sign epistasis where the fitness of the two singly resistant genotypes (RS, SR) is lower than those of either the susceptible or the doubly resistant genotype (wRS, wSR < wSS, wRR) so that with these drug concentrations, double resistance is difficult to evolve.

Evolutionary dynamics In this section, I will briefly investigate how epistasis in the pharmacodynamic fitness landscapes influences the evolutionary dynamics of adaptation to the presence of two drugs. Rather than a realistic exploration of within-host evolution of a population of bacterial pathogens, this is meant as a proof of principle of how standard population genetic results concerning the impact of epistasis on the rate of adaptation play out in the context of the present model. Consequently, I will consider a very simple, deterministic model. Time is continuous in this model, and the population is assumed to consist of the four genotypes SS, SR, RS and RR. The model incorporates only different net

Fig. 5 Epistasis in two example pharmacodynamic fitness landscapes with synergistic (plots a and b) and antagonistic (panels c and d) drug interaction. Plots (a) and (c) show net growth rates of the fully susceptible genotype, whereas plots (b) and (d) show the magnitude of epistasis. The shaded area in plot (d) indicates combinations of drug concentrations with sign epistasis in which wRR > wSS > wRS, wSR. All parameters as in Fig. 1b except jij ¼ 2 for all i, j, (a,b) a = 0.3, (c,d) a = 0.25, and b = 0.5.

growth rates (fitnesses) of these genotypes but ignores mutation and recombination. Denoting the number of bacteria with genotype i by Ni, the dynamics of the population are described by the following system of ordinary differential equations: d i N ðtÞ ¼ wi ðA1 ðtÞ;A2 ðtÞÞN i ðtÞ;withifSS, SR, RS, RRg (12) dt The population is initialized with predominantly the SS genotype, but with a small proportion of single-resistant genotypes and an even smaller proportion of double-resistant genotypes in a way that the genotype frequencies are in linkage equilibrium. Figure 6a shows the dynamics in a case where the drug concentrations are constant. These drug concentrations take values above the combined MIC for the fully susceptible genotype SS, but below the combined MIC of the doubly resistant genotype, thus ensuring the growth of the latter bacterial genotype. With the standard pharmacodynamic fitness landscape studied in Figs 1b and 2, these drug concentrations fall in the region of the pharmacodynamic landscape where there is negative epistasis. In agreement with population genetic theory, this results in a reduced rate of fixation of the RR genotype compared with the case where epistasis was ‘artificially’ set to zero (compare bold and dashed lines in Fig. 6a). Next, I investigated the evolutionary dynamics when drug concentrations vary through time according to a simplified form of pharmacokinetics. Specifically, the concentration of drug j is assumed to be present at an initial concentration Amax , decline exponentially at rate j



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Fig. 6 Evolutionary dynamics on pharmacodynamic fitness landscapes, for (a) constant and (b) varying drug concentrations. For both plots, the underlying pharmacodynamic fitness landscape is the same as in Figs 1b and 2. Bold lines show the frequencies of the four genotypes RR (green), SR (red), RS (blue) and RR (lilac). Dashed lines of the same colours show the corresponding dynamics in the complete absence of epistasis; to obtain these latter dynamics, the fitness of the RR genotype was set to wSR + wRS wSS for all antibiotic concentrations. In (a), drug concentrations are A1 = 1, A2 = 2. In (b), the pharmacokinetic parameters are Amax ¼ 5, r1 = 0.1, Amax ¼ 15, 1 2 r2 = 0.25, T1 = T2 = 24.

rj, and returns to the original concentration after a time interval Tj: Aj ðtÞ ¼ Amax exp rj ðt%Tj Þ ; (13) j where % denotes the modulo operator (remainder of division). When drug concentrations change over time, fitness values of all four genotypes will also change. As a consequence, epistasis will also vary through time and may take both positive and negative values. Moreover, depending on the degree of variation in drug concentrations, even the order of fitness values may change so that different genotypes are selected for at different time points. Figure 6b shows example evolutionary dynamics under pharmacokinetics, on the same pharmacodynamic fitness landscape as in Fig. 5a. It can be seen that the rate of increase in the frequency of the RR genotype varies over time according to drug concentrations. Epistasis takes both positive and negative values in this example, but the decelerating effect of negative epistasis clearly outweighs the accelerating effect of positive epistasis. Note that with both increasing and decreasing time scales of the pharmacokinetics relative to the time scale at which natural selection takes place (i.e. when Tj is varied but rjTj is kept constant), the dynamics become increasingly similar to those with constant drug concentrations (results not shown).

Discussion The relationship between antibiotic drug concentrations and bacterial fitness (net growth) is often described by pharmacodynamic functions. In the case of combina-

tion treatment with two distinct drugs, these functions give the fitness for each combination of drug concentrations (e.g. Ankomah & Levin, 2012). I have constructed and analysed fitness landscapes that emerge when generic multidrug pharmacodynamic functions are assumed to determine fitness of both susceptible and resistance genotypes. This analysis showed that these fitness landscapes are generally characterized by pervasive epistasis over a wide range of drug concentrations. Epistasis emerges due to three factors in the model presented here: (i) fitness costs of resistance that lead to reduced maximum growth, (ii) cross-resistance so that one resistance mutation provides partial or complete resistance to both drugs and (iii) drug interactions that produce nonindependent fitness reductions in the two drugs. These three factors will now be discussed in more detail. Fitness costs of resistance Drug resistance mutations usually incur fitness costs in the absence of drugs (Andersson et al., 2007; Andersson & Hughes, 2010). If there is epistasis in the fitness costs of resistance – as has been observed in several cases (Trindade et al., 2009; Hall & MacLean, 2011; Silva et al., 2011) – this will naturally lead to a certain level of overall epistasis in fitness also in the presence of antibiotics on the pharmacodynamic fitness landscapes. However, epistasis also emerges (at drug concentrations around the MICs for sensitive and resistant genotypes) even if fitness costs are additive. This counterintuitive result is due to the assumption that two genotypes with the same allele at a particular resistance locus are characterized by the same MIC for the corresponding drug. In reality, it is not clear how fitness costs imposed by a


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mutation conferring resistance to drug 1 will affect growth in the presence of another drug 2 to which this mutation provides no resistance. In principle, four simple situations are conceivable that differ in how the MIC and the maximum reduction in growth are influenced by the cost of resistance and that are decisive for epistasis in the resulting fitness landscapes. The first situation, assumed throughout this paper, is when fitness costs only affect maximum growth, but do not affect the MIC or the maximum reduction in growth caused by the drug. This has the conceptual advantage that the costs of resistance can be captured by only a single parameter (w). On the other hand, this option may not be the most realistic because net growth is reduced both at very low and very high drug concentrations by the cost of resistance, but not at intermediate concentrations around the MIC. Second, the resistance mutation to drug 1 may reduce net growth by the same amount for all concentrations of drug 2. Although this seems an intuitive assumption, it has the potentially problematic property that the resistance mutation for drug 1 will reduce the MIC for drug 2, and vice versa. Another consequence of this assumption is that fitness costs per se will not induce any epistasis in the resulting pharmacodynamic fitness landscape. Third, costs of resistance may only affect maximum growth, but not the MIC or minimum growth. This means that the maximum reduction in growth caused by drug 2 will be smaller in genotypes carrying a costly resistance mutation to drug 1. Finally, costs of resistance to drug 1 may reduce maximum growth and the MIC for drug 2, but not the minimum net growth rate at high concentrations of drug 2. In Section 2 of the SI, these four situations and their consequences for the emerging fitness landscapes are explored in more detail; it is also shown how they can all be accommodated within the framework of the model proposed here. An interesting extension of the model proposed here would be to also incorporate compensatory mutations, that is, mutations that reduce the costs of antibiotic resistance and therefore may delay or prevent reversion to susceptibility once treatment is ceased (Bjorkman et al., 2000; Levin et al., 2000; Maisnier-Patin & Andersson, 2004; Besier et al., 2005). Fitness effects of compensatory mutations are usually only determined in the absence and sometimes, the presence of drugs at a given antibiotic concentration. Examining the alleviation of fitness costs associated with resistance to mupirocin in Salmonella typhimurium, Paulander et al. (2007) showed that some compensatory mutations compensate only in absence and others in both the presence and absence of drugs (see also Schulz zur Wiesch et al., 2010). However, we still remain largely ignorant of the fitness effects of compensatory mutations for intermediate drug concentrations, epistatic interactions between different compensatory mutations or with other resis-

tance mutations, and the evolutionary implications of the resulting fitness landscapes. Cross-resistance The second factor that produces epistasis in the pharmacodynamic fitness landscapes analysed here is nonspecificity in the two drug resistance mutations. This means that at least one of the two resistance mutations alters the fitness effect of not just one, but of both drugs simultaneously. In particular, epistasis will readily emerge in the presence of cross-resistance, that is, when one resistance mutation increases the MIC for both drugs. Cross-resistance is a common situation for many resistance mutations and drugs. For example, the newly emerged New Delhi Metallo-betalactamase 1 (NDM-1) is an enzyme capable of inactivating many betalactam antibiotics, including penicillins, cephalosporins and carbapenems (Yong et al., 2009; Moellering, 2010). Similarly, resistance genes coding or efflux pumps often impart cross-resistance, as many of them are capable of pumping out many different and chemically distinct antibiotic compounds (reviewed in Nikaido & Pages, 2012). Epistasis emerges from cross-resistance through two factors: deviations from independent effects of the resistance mutations on the MICs on the one hand and the nonlinearity in the pharmacodynamic functions on the other hand. To illustrate this point, it is sufficient to focus on a single drug. For simplicity, let us also assume that cross-resistance is complete, that is, both mutations confer the same level of resistance to the drug. If the two resistance mutations increase the MIC only slightly individually but substantially when combined, this will produce positive epistasis. Conversely, if both mutations when combined increase the MIC only slightly more or not at all compared with their individual effects, negative epistasis will ensue. Roughly speaking, epistasis in the MICs produces epistasis in fitness of the same sign. However, even if the MICs of the two partially resistant genotypes is the (geometric or arithmetic) mean of the fully susceptible and fully resistant genotype, there will still be negative epistasis below and positive epistasis above the MIC. This is because the pharmacodynamic functions are sigmoidal and not linear and clearly shows that the absence of epistasis in some or even all of the parameters describing the pharmacodynamic fitness landscape does not imply the absence of epistasis in fitness. Although the model presented here focuses on multidrug resistance evolution, the argument above illustrates that by setting one of the drug concentrations to zero, the model contains as a submodel the important case of pharmacodynamic fitness landscapes involving two mutations that confer resistance to a single drug. Such fitness landscapes have been studied more extensively experimentally than multidrug-dependent fitness



landscapes (e.g. Weinreich et al., 2006; Goulart et al., 2013; Schenk et al., 2013), but generally without incorporating pharmacodynamic considerations. What the present model shows is that for a certain range of drug concentrations, epistasis in fitness between several mutations that increase resistance to a drug may be a necessary consequence of the nonlinear relationship between drug concentration and fitness. Drug interactions Antibiotics and other drugs often interact in a synergistic (more reduction in fitness when combined than expected from their individuals effects) or antagonistic manner (less reduction in fitness when combined). There are two main approaches to defining the absence of such drug interactions (with long-standing controversy as to which is more appropriate, reviewed in Greco et al., 1995). Loewe additivity, on the one hand, means that when two drugs in concentrations A1 and A2 produce the same fitness reduction when applied individually, all combinations of the two drugs with concentrations xA1 and (1–x)A2 (where x e [0,1]) will also produce the same fitness reduction. Bliss independence, on the other hand, is given when the fitness reduction caused by several drugs simultaneously is equal to the sum of the individual fitness reductions caused by the drugs of the same concentrations (where fitness may be measured on a logarithmic scale). The present model assumes additive fitness reductions caused by the two drugs as the null model for the absence of drug interactions (as in the model by Ankomah & Levin, 2012); this can be regarded as a form of Bliss independence. In the absence of any other factors generating epistasis, deviations from Bliss independence produce epistasis equal to the negative of epistasis in the drug interaction terms. Using drug interaction functions based on the product of the reductions in fitness caused by the two drugs, synergistic drug interactions lead to negative epistasis, whereas antagonistic drug interactions lead to positive epistasis. It is expected that with pharmacodynamic functions based on Loewe additivity as the null model for the absence of drug interactions, the resulting fitness landscapes would always exhibit epistasis for most combinations of drug concentrations. Whereas with Bliss independence, different components of fitness reduction act additively and therefore may readily cancel out in certain simple cases so that no epistasis ensues, this is never the case with Loewe additivity. Preliminary results (see Section 3 of the Supporting Information) indicate that indeed, even comparatively simple fitness landscapes characterized by Loewe additivity and drugspecific resistance mutations exhibit positive epistasis in the absence of fitness costs, and both positive and negative epistasis in the presence of additive fitness costs. However, the fitness landscapes emerging from phar-

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macodynamic functions based on Loewe additivity remain to be fully investigated in future studies. An interesting case occurs when the two drugs suppress each other’s activity to the extent that a combination of both drugs reduces fitness less than each drug individually. This situation has been dubbed ‘reciprocally suppressive drug interaction’, and it has been argued that this extreme form of antagonistic drug interaction can lead to a situation where the fitness of two singly resistant genotypes is lower than that of both the fully susceptible and the doubly resistant genotype (Yeh et al., 2009). The pharmacodynamic fitness landscapes constructed here indeed show this effect: for some parameters and combinations of drug concentrations, reciprocally suppressive drug interactions produces sign epistasis, with wRR > wSS > wRS, wSR. Such sign epistasis implies a ‘rugged’ fitness landscape with two fitness peaks. Importantly, the fact that the fully susceptible genotype occupies a local fitness peak means that resistance evolution may be slowed down or prevented entirely, opening up the possibility of ‘resistance-proof’ drug combination therapy (Yeh et al., 2009). Although reciprocally suppressive antibiotic combinations remain to be discovered, unidirectional suppression (where one drug suppresses the action of the other but not vice versa) has been described and has also been shown to delay the spread of resistance (Chait et al., 2007). Concluding remarks The fitness landscapes analysed here, based on additively combined Hill functions, represent the perhaps simplest generic pharmacodynamic fitness landscapes with two resistance mutations for varying concentrations of two drugs. The fact that despite this simplicity, these fitness landscapes are generally characterized by pervasive epistasis indicates that real multidrug resistance fitness landscapes should also exhibit epistasis for most drug concentrations. For example, more complicated pharmacodynamic functions that incorporate nonmonotonic effects (such as the Eagle effect; Eagle & Musselman, 1948) or more complex ways in which drug interactions materialize are expected to only further increase the overall level of epistasis. Given that epistasis is well known to affect the speed of adaptive evolution, this result points to the importance of empirically determining multidrug resistance fitness landscapes for a wide range of drug concentrations in order to optimize treatment regimens with respect to resistance evolution. This is particularly important for subMIC concentrations of drugs, which mirroring previous results for single drugs (Gullberg et al., 2011), may still select for multidrug resistance (see Fig. 2b). The mathematical framework of a pharmacodynamic fitness landscape presented here may serve as a model amenable for parametrization or as a null model


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against which deviations can be tested. To this end, extension of the model to more than two loci and more than two drugs is in principle straightforward, although graphical representation of the resulting multidimensional epistasis or combinations of antibiotics will be challenging. The model may also be useful to conceptually disentangle different types of agent interaction involved in multidrug resistance, in particular different forms of epistasis (in overall fitness, in fitness costs of resistance, in MICs etc.), cross-resistance and drug interactions.

Acknowledgments I thank Pia Abel zur Wiesch and two anonymous reviewers for helpful comments on the manuscript.

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Supporting information Additional Supporting Information may be found in the online version of this article: Figure S1 (A) Comparison of the original function H (black) and the two alternative functions Halt1 (red) and Halt2 (blue), as well as the resulting epistasis with (B) the original Hill function, (C) Halt1 and (D) Halt2. All parameters as in Fig. 2 in the main text. Figure S2 Four ways to incorporate costs of drug resistance. Figure S3 (A) Pharmacodynamic fitness landscape with Loewe additivity, with parameters matching those in SR SS RS RS Fig. 1 in the main text fSS 1 ¼ f1 ¼ f2 ¼ f2 ¼ 1; f1 RR SR RR i i ¼ f1 ¼ f2 ¼ f2 ¼ 100, u ¼ 0:2 and j ¼ 1 for all i. (B) Resulting epistasis in this fitness landscape. (C) Epistasis in the same fitness landscape but without fitness costs of resistance mutations (wi = 0.1 for all genotypes i). Received 25 November 2013; revised 17 January 2014; accepted 10 February 2014


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