Drug-target binding quantitatively predicts optimal antibiotic dose levels

Combatting antibiotic resistance will require both new antibiotics and strategies to preserve the effectiveness of existing drugs. Both approaches would benefit from predicting optimal dosing of antibiotics based on drug-target binding parameters that can be measured early in drug development and that can change when bacteria become resistant. This would avoid the currently frequently employed trial-and-error approaches and might reduce the number of antibiotic candidates that fail late in drug development.Here, we describe a computational model (COMBAT-COmputational Model of Bacterial Antibiotic Target-binding) that leverages accessible biochemical parameters to quantitatively predict antibiotic dose-response relationships. We validate our model with MICs of a range of quinolone antibiotics in clinical isolates demonstrating that antibiotic efficacy can be predicted from drug-target binding (R2 > 0.9). To further challenge our approach, we do not only predict antibiotic efficacy from biochemical parameters, but also do the reverse: estimate the magnitude of changes in drug-target binding based on antibiotic dose-response curves. We experimentally demonstrate that changes in drug-target binding can be predicted from antibiotic dose-response curves with 92-94 % accuracy by exposing bacteria overexpressing target molecules to ciprofloxacin. To test the generality of COMBAT, we apply it to a different antibiotic class, the beta-lactam ampicillin, and can again predict binding parameters from dose-response curves with 90 % accuracy. We then apply COMBAT to predict antibiotic concentrations that can select for resistance due to novel resistance mutations.Our goal here is dual: First, we address a fundamental biological question and demonstrate that drug-target binding determines bacterial response to antibiotics, although antibiotic action involves many additional effects downstream of drug-target binding. Second, we create a tool that can help accelerate drug development by predicting optimal dosing and preserve the efficacy of existing antibiotics by predicting optimal treatment for possible resistant mutants.

It is even more challenging to optimize dose levels to minimize the emergence of antibiotic 60 resistance, both for existing and novel antibiotics. There remains substantial debate about which 61 dosing strategies best prevent the emergence of resistant mutants during treatment [7][8][9]. In this 62 context, a useful concept that links antibiotic concentrations with resistance evolution is the 63 resistance selection window (mutant selection window) that ranges from the lowest 64 concentration at which the resistant strain grows faster than the wild-type, usually well below the 65 wild-type minimum inhibitory concentration (MIC), to the MIC of the resistant strain [10][11][12]. 66 Antibiotic concentrations above the resistance selection window safeguard against de novo 67 resistance emergence. Antibiotic concentrations below the resistance selection window do not 68 kill the susceptible strain, but also do not favor the resistant strain and therefore do not promote 69 emergence of resistance. The latter may be preferable if one cannot dose above the MIC of the 70 resistant strain due to toxicity or solubility limits. To limit resistance emergence, it is therefore 71 important to identify the resistance selection window and optimize dosing accordingly. 72 73 Limitations in our knowledge of how antibiotic treatment regimens affect bacterial populations 74 contribute to the need for lengthy and expensive trial-and-error approaches, with the sheer 75 number of possible dosing regimens making it difficult to identify an optimal regimen. We argue 76 that this knowledge gap is a major limitation for the improvement of dosing regimens of existing 77 drugs and a real obstacle for the development of new antibiotics [13,14]. 78 79 Pharmacodynamic models that can make predictions of bacterial killing and selection on the 80 basis of drug-target interactions offer new promise to inform rational antibiotic dosing 81 practices [15]. Recently described models that include drug-target binding have been useful in 82 gaining a better qualitative understanding of complicated drug effects, such as post-antibiotic 83 effects, inoculum effects, and bacterial persistence [15][16][17][18]. However, to speed the development 84 of new antibiotics or to inform practices which minimize resistance, we require quantitative 85 predictions for antibiotics or resistant bacterial strains that do not exist yet. Models which permit 86 quantitative predictions of changes in drug efficacy as a function of modification of antibiotic 87 molecules (i.e. new drugs) or novel resistance mutations would be invaluable. Such tools would 88 advance our general mechanistic understanding of antibiotic action, could guide dosing trials of 89 new drugs, and suggest better dosing of existing drugs. 90

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In this report, we describe a mechanistic computational modeling framework  COmputational Model of Bacterial Antibiotic Target-binding) that allows us to predict drug 93 effects based solely on accessible biochemical parameters describing drug-target interaction. 94 These parameters can be determined early in drug development. We use this framework to 95 investigate how changes in drug target binding, either due to improvements in existing 96 antibiotics or due to resistance mutations in bacteria, affect antibiotic efficacy. We first show that 97 COMBAT accurately predicts bacterial susceptibility as a function of drug-target binding and, 98 conversely, allows inference of these biochemical parameters on the basis of observed patterns of 99 bacterial growth suppression or killing. We then use COMBAT to predict the susceptibility of 100 newly arising resistant variants based on the molecular mechanism of resistance and determine 101 the resistance selection window. 102 To investigate how biochemical changes in antibiotic action modifies bacterial susceptibility, we 106 explored how the affinity of antibiotics to their target affects the MIC. We compared the MICs of 107 quinolones, an antibiotic class in which individual antibiotics have a wide range of affinities to 108 their target, gyrase (KD ~10 -4 -10 -7 M) but are of similar molecular sizes and have a similar mode 109 of action [19]. This choice allowed us to isolate the effects of differences in drug-target affinity 110 on the MIC. 111 112 We obtained binding affinities of quinolones to their gyrase target in Escherichia coli from 113 previous studies [20][21][22][23][24]. We then retrieved MIC data for several quinolones from clinical 114 Enterobacteriaceae isolates collected before 1990 [25], i.e., before the widespread emergence of 115 quinolone resistance [19]. We assume that quinolone affinities obtained from clinical 116 Enterobacteriaceae isolates collected before the emergence of resistance correspond to those 117 measured in wild-type E. coli. 118

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To make qualitative predictions of MICs, we employed a simplified model based on the 120 assumptions that i) drug-target binding occurs much more quickly than bacterial replication, ii) 121 the antibiotic concentration remains constant and iii) that during the 18 hours of an MIC assay, 122 the concentration gradient of the drug inside and outside the cell has equilibrated. Under these 123 assumptions, the MIC can be expressed as 124 describes the binding and unbinding of antibiotics to their targets and predicts how such binding 143 dynamics affects bacterial replication and death (Fig. 2a). In previous work linking drug-target 144 binding kinetics with bacterial replication[18 ], we described a population of bacteria with 145 target molecules per cell with a system of + 1 (bacteria with 0, 1, …, bound target 146 molecules) ordinary differential equations (ODEs). This system increases in complexity with the 147 number of target molecules and makes fitting the model to data computationally too demanding 148 for most settings. To simplify this prior approach, we developed new mathematical models based 149 on partial differential equations (PDEs), where a single equation describes all bacteria 150 simultaneously. The sum of bacteria within all target occupancy states over time can be 151 described by a time kill curve (Fig. 2b), during which the bacterial population is characterized by 152 the distribution of bacterial cells with different levels of target occupancies at each time-step 153 ( Fig. 2c). This curve can be visualized as a two-dimensional surface in a three-dimensional 154 coordinate system where the number of bacteria is represented on the z-axis, the percent of 155 bacteria with the fraction of bound target molecules on the x-axis, and time on the y-axis (Fig.  156 2d). 157 158 Antibiotic action is described by rates of binding (kf) and unbinding (kr) to bacterial target 159 molecules (Fig. 2a, e). The binding of an antibiotic to a target results in the formation of an (2) 165

166
The term for binding kinetics is given in brown, the term for replication in blue and the term for 167 death in red. 168 The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Equation 4 (part of the replication term in equation 2) describes how daughter cells inherit bound 174 target molecules from the mother cell during replication: 175

Model fit to ciprofloxacin time-kill data 182
We used the quinolone ciprofloxacin to quantitatively fit bacterial time-kill curves, since this is a 183 commonly used antibiotic for which binding parameters have been directly measured. 184 Supplementary Tab. S1 gives an overview of the known parameters used for fitting; 185 Supplementary Tab. S2 gives the parameters resulting from our fit. 186

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The functional relationship between the levels of bacterial replication and death on the fraction 188 of bound target molecules is extremely hard to obtain experimentally. We therefore treated the 189 relationships between the fraction of bound target and bacterial replication and death as free The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Overall, we found that COMBAT could fit the time-kill curves well (R 2 = 0.93, Fig. 3a).

Accurate prediction of target overexpression from time-kill data 204
Having shown that COMBAT can quantitatively fit experimental data on antibiotic action within 205 biologically plausible parameters, we continued to test the predictive ability of the model. Given 206 our hypothesis that modifications in antibiotic-target interactions lead to predictable changes in 207 bacterial susceptibility, we experimentally induced changes in the antibiotic-target interaction of 208 ciprofloxacin in E. coli. We then quantified these biochemical changes by fitting COMBAT to 209 corresponding time-kill curves and compared them to the experimental results. Ciprofloxacin 210 acts on gyrase A2B2 tetramers [19]. We used an E. coli strain for which both gyrase A and gyrase 211 B are under the control of a single inducible promoter (PlacZ), such that the amount of gyrase 212 A2B2 tetramer can be experimentally manipulated [32]. We measured net growth rates for this 213 strain at different ciprofloxacin concentrations in the presence of 10 µM isopropyl β-D-1-214 thiogalactopyranoside (IPTG; mild overexpression) and 100 µM IPTG (strong overexpression) 215 and compared it to the wild-type in the absence of the inducer (Fig. 4a). 216 217 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Like previously reported, we find that increasing gyrase content makes E. coli more susceptible 218 to ciprofloxacin [32]. We fitted net growth rates allowing the target molecule content, i.e. gyrase 219 A2B2, to vary. We assumed that the only change between the different conditions was the amount 220 of target. We further assumed that the relationship between bound target and bacterial replication 221 or death did not differ between the control strain containing a mock plasmid (no IPTG) and the 222 experiments with overexpression ( Fig. 4b, between 0 % and 100 %). Finally, we assumed that 223 the maximal kill rate at very high antibiotic concentrations was accurately measured in our 224 experiments and forced the function describing bacterial death through the measured value when 225 all target molecules are bound. We found the best fit for a 1.31x increase in GyrA2B2 target 226 molecule content for bacteria grown in the presence of 10 µM IPTG and a 2.02x increase in 227 GyrA2B2 target molecule content for those grown in the presence of 100 µM IPTG. 228

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We subsequently tested these predictions experimentally by analyzing Gyrase A and B content 230 by western blot Fig. 4c; Supplementary Fig. S2). Using realistic association and dissociation 231 rates for biological complexes[33], we predicted a range of functional tetramers based on the 232 relative amount of Gyrase A and B proteins (Fig. 4d) Next, we tested whether COMBAT can be applied to the action of the beta-lactam ampicillin, a 241 very different antibiotic with a distinct mode of action from quinolones. Using published 242 pharmacodynamic data of E. coli exposed to ampicillin[31] also allowed us to compare 243 COMBAT predictions to established pharmacodynamic approaches. Most of the biochemical 244 parameters for ampicillin binding to its target, penicillin-binding proteins (PBPs), have been 245 determined experimentally (Supplementary Tab. S1). Ampicillin is believed to act as a 246 bactericidal drug [34], and this mode of action is supported by findings from single-cell 247 microscopy [26]. We therefore assume that ampicillin binding does not affect bacterial 248 replication. In order to model the consumption of beta-lactams at target inhibition and eventual 249 target recovery, we made small adjustments to equation 13 (see Methods, description of beta-250 lactam action). 251

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We fitted COMBAT to published time-kill curves of E. coli exposed to ampicillin (Fig. 5a). 253 Again, COMBAT provides a good fit to the experimental data between 0 min and 40-60 min. 254 After that time, observed bacterial killing showed a characteristic slowdown at high ampicillin 255 concentrations which is often attributed to persistence[18] (Fig. 5a). For the sake of simplicity, 256 we chose to omit bacterial population heterogeneity in this work and therefore cannot describe 257 persistence, even though COMBAT can be adapted to capture this phenomenon [18]. Because 258 ampicillin acts in an entirely bactericidal manner, we assume a constant replication rate (see 259 Methods & Supplementary Fig. S1) and fitted bacterial death as a function of target binding, 260 ( ) (Fig. 5b, fitted parameters in Tab. S4). Fig. 5c shows the predicted net growth rate over a 261 range of drug concentrations. We estimated a MIC of 2.6 mg/L. This MIC is based on the 262 Clinical & Laboratory Standards Institute definition of the MIC determined at 18 h. The original 263 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder.

Sensitivity of antibiotic efficacy to parameters of drug-target binding 274
It is possible to vary all parameters in COMBAT and explore their effect. We used this to test 275 how hypothetical chemical changes to ampicillin or ciprofloxacin would affect antibiotic 276 efficacy (Supplementary Fig. S3-S11). These changes could reflect either bacterial resistance 277 mutations or modifications of the antibiotics themselves. We predict that changes in drug-target 278 affinity, KD, have more profound effects than changes in target molecule content, bacterial 279 reaction to increasingly bound target (i.e. d(x) and r(x)), or changes in target molecule content.

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We also predict that the individual binding rates kr and kf, and not just the ratio of these terms, 281 the KD, are important factors in efficiency. The faster a drug binds, the more efficient we 282 predicted it will be. One intuitive explanation for the observation that kf drives efficacy is that a 283 slow binding fails to rapidly interfere with bacterial replication, which may allow for the 284 production of additional target molecules and thereby reduce the ratio of free antibiotic to target 285 molecules. 286 All rights reserved. No reuse allowed without permission.

Forecasting the resistance selection window 288
Finally, we illustrate how COMBAT can be used to explore how the molecular mechanisms of 289 resistance mutations affect antibiotic concentrations at which resistance can emerge, i.e., the 290 resistance selection window. We compared predicted net growth rates as a function of 291 ciprofloxacin concentrations for a wild-type strain and an archetypal resistant strain. For this 292 analysis, we assumed that the resistant strain has a 100x slower drug-target binding rate (i.e. 293 ~100x increased MIC, realistic for novel point mutations [36]) and that the maximum replication 294 rate of the resistant strain is 85 % of the wild type strain [37]. We then predicted the antibiotic 295 concentrations at which resistance would be selected. Interestingly, when comparing COMBAT 296 to previous pharmacodynamics models (Fig. 5), we observed that estimates of replication rates 297 depend on the selected time frame (Fig. 6a). When the timeframe for MIC determination is set to 298 18 h as defined by CLSI[38], the "competitive resistance selection window", i.e., the 299 concentration range below the MIC of both strains where the resistant strain is fitter than the wild 300 type, ranges from 0.002 mg/L to 0.014 mg/L for ciprofloxacin (Fig. 6a) and 1 mg/L to 2.6 mg/L 301 for ampicillin ( Supplementary Fig. S12), respectively. This corresponds well with previous 302 observations that ciprofloxacin resistance is selected for well below MIC [11]. However, when 303 measuring after 15 min or 45 min, the results are substantially different. The reason for this is 304 illustrated in Fig. 6b. COMBAT reproduces non-linear time kill curves where bacterial 305 replication continues until sufficient target is bound to result in a negative net growth rate. This 306 compares well with experimental data around MIC in Fig. 3a and 5a. In Fig. 6b Optimizing dosing levels of antibiotics is important for maximizing drug efficacy against wild-318 type strains as well as for minimizing the rise of resistant mutants. The determination of optimal 319 dosing strategies typically requires expensive empirical studies; the need for such studies arises 320 in part from our currently limited capacity to predict how antibiotics will affect bacteria at a 321 given concentration. In fact, drug attrition is mainly due to insufficient predictions of efficacy 322 (pharmacodynamics) rather than pharmacokinetics [6]. For optimizing drug development and for 323 minimizing resistance, we need quantitative predictions for antibiotics or resistant bacterial 324 strains that do not exist yet. The ability to accurately predict MICs on the basis of biochemical 325 parameters and, more generally, to define antibacterial activity across a range of drug 326 concentrations, would allow us to estimate antibiotic efficacy for novel compounds or against not 327 yet emerged resistant strains [15,39]. Recent studies have reported methods to predict MICs from 328 whole genome sequencing data [40,41]. However, these methods require transfer of prior 329 knowledge on how the resistance mutations affect MICs in other organisms. There are no 330 methods that could predict a priori how chemical changes to an antibiotic structure or novel 331 resistance mutations affect bacterial growth at a given antibiotic concentration. 332 333 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Here, we accurately predict antibiotic action on the basis of accessible biochemical parameters of 334 drug-target interaction. Our computational model, COMBAT provides a framework to predict 335 the efficacy of compounds based on drug-target affinity, target number, and target occupancy. 336 These parameters may change both when improving antibiotic lead structures as well as when 337 bacteria evolve resistance. Importantly, they can be measured early in drug development and 338 may even be a by-product of target-based drug discovery [42]. When these data are available, 339 COMBAT makes only one assumption: that the rate of bacterial replication decreases and/or the 340 rate of killing increases with successive target binding. While fitting, we allow this relationship 341 to be gradual or abrupt and select the best fit. This means we do not model specific molecular 342 mechanisms down-stream of drug-target binding, but their effects are subsumed in the functions 343 that connect the kinetic of drug-target binding to bacterial replication and death. 344

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In previous work, for example on antipsychotics [16], antivirals[17] and antibiotics [15,18], 346 models of drug-target binding kinetics have been used to improve our qualitative understanding 347 of pharmacodynamics. Our study substantially advances this work by making accurate 348 quantitative predictions across antibiotics and bacterial strains when measurable biochemical 349 characteristics change. This is possible because COMBAT employs an elegant mathematical 350 approach, based on partial differential equations, that makes it computationally feasible to fit the 351 model to a large range of data. Importantly, we are not only able to predict antibiotic action from 352 biochemical parameters, but can also vice versa use COMBAT to accurately predict biochemical 353 changes from observed patterns of antibiotic action. We have confirmed the excellent predictive 354 power of COMBAT with clinical data as well as experiments with antibiotics with very different 355 mechanisms of action. This gives us confidence that biochemical parameters are major 356 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint determinants of antibiotic action in bacteria and that COMBAT helps to make rational decisions 357 about antibiotic dosing. 358

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In drug development, our mechanistic modeling approach provides insight into which chemical 360 characteristics of drugs may be useful targets for modification. For example, our sensitivity 361 analyses indicate that antibiotics with a similar affinity but faster binding inactivate bacteria 362 more quickly and therefore prevent replication and production of more target molecules, which 363 would change the ratio of antibiotic to target. Furthermore, because e.g. antibiotic binding and 364 unbinding rates can be determined early in the drug development process, such insight can help 365 the transition to preclinical and clinical dosing trials. This may contribute to reducing bottlenecks 366 between these phases of drug development and thereby save money and time. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Our approach also offers insight into determinants of the resistance selection window. Rather 379 than determining the resistance selection window for a comprehensive collection of possibly 380 arising resistance mutations in each bacteria-drug pair, it would be attractive to build 381 transferrable knowledge that allows estimating the resistance selection window. In concordance 382 with a recent meta-analysis of experimental data[43], our sensitivity analyses predict that 383 changes in drug target binding and unbinding have a greater impact on the MIC than changes in 384 target molecule content or down-stream processes. Thus, a more comprehensive characterization 385 of the binding parameters of spontaneous resistant mutants would allow an overview of the 386 maximal biologically plausible levels of resistance that can arise with one mutation. Dosing 387 above this level should then safeguard against resistance. This is especially useful for compounds 388 for which it is difficult to saturate the mutational target for resistance, or for safeguarding against 389 resistance to newly introduced antibiotics for which we do not yet have a good overview of 390 resistance conferring mutations. If toxicity, solubility or other constraints do not allow dosing 391 above the MIC of expected resistant strains, COMBAT can predict the concentration range at 392 which resistance is less strongly selected. This could guide decisions on treating with low versus 393 high doses, which is currently controversially debated [7,8]. Good quantitative estimates on the 394 dose-response relationship of new drugs would also help defining the therapeutic window, i.e. 395 the range of drug concentrations at which the drug is effective but not yet toxic. 396 397 Our quantitative work can help to identify optimal dosing strategies at constant antibiotic 398 concentrations for homogeneous bacterial populations. These measures are commonly used to 399 assess antibiotic efficacy. In addition, previous work has demonstrated that drug-target binding 400 models can qualitatively describe antibiotic efficacy over the fluctuating concentrations that 401 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint actually occur in patients [26,44]. They can also explain complicated phenomena such as 402 biphasic kill curves, the post-antibiotic effect, or the inoculum effect [15,18,45]   In the most basic version of COMBAT, we ignored differences between extracellular and 417 intracellular antibiotic concentrations and only followed the total antibiotic concentration A, 418 assuming that the time needed for drug molecules to enter bacterial cells is negligible. We model 419 ciprofloxacin (to which there is a limited diffusion barrier [46]) and ampicillin (where the target 420 is not in the cytosol, even though the external membrane in gram negatives has to be crossed to 421 reach PBPs). We therefore believe that this assumption is justified in wild-type E. coli. This 422 basic version of COMBAT is therefore more accurate for describing antibiotic action where the 423 diffusion barrier to the target is weak. 424 All rights reserved. No reuse allowed without permission.

Binding kinetics 426
We describe the action of antibiotics as a binding and unbinding process to bacterial target 427 molecules [18]. For simplicity, we assume a constant number of available target molecules . The 428 binding process is defined by the formula + ⇌ , where the intracellular antibiotic 429 molecules A react with target molecules T at a rate k f and form an antibiotic-target molecule 430 complex x, where values for x range between 0 and . If the reaction is reversible, the complex 431 dissociates with a rate k r . 432 In [18], the association and dissociation terms are described by the following terms 433 , kf is the association rate, Vtot is the volume in which the experiment is 437 performed, nA is Avogadro's number, kr is the dissociation rate, Bi is the number of bacteria with 438 i bound targets, and is the total number of targets. Green denotes the association term, while 439 the dissociation term is in orange. 440 This approach requires the use of a large number of ordinary differential equations, ( + 1) for 441 the bacterial population and one for the antibiotic concentration. To generalize this approach, we 442 assume that the variable of bound targets is a real number ∈ ℛ. Under this continuity 443 assumption, we consider the bacterial cells as a function of x and the time t, thereby reducing the 444 total number of equations to two. 445 Under the continuity approximation ( ∈ ℛ), we can rewrite the binding kinetics in the form 446 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. We assume that the total number of target molecules doubles at replication, such that each 472 daughter cell has the same number as the mother cell. We also assume that the total number of 473 drug-target complexes is preserved in the replication and that the distribution of x bound target 474 molecules of the mother cell to its progeny is described by a hypergeometric sampling of n where is a complex number. In this way, the distribution can be expressed as a probability 483 density function of continuous variables. The amount of newborn bacteria is given by the term 484 ( ) ( , ) H<I ( ). We assume that bound target molecules are distributed randomly between 485 mother and daughter cells, with each of them inheriting 50% upon division on average. This 486 means that twice the amount of newborn cells must be redistributed along x to account for the 487 random distribution process. For example, if a mother cell with 4 bound targets divides, we have 488 two daughter cells, each with a number of bound targets between 0 and 4 (their sum has to be 4), 489 following the generalized hypergeometric distribution. For simplicity, we define S(x,t) to be a 490 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. We consider several potential functional forms of the relationship between the percentage of 503 bound targets and replication and death rates, because the exact mechanisms how target 504 occupancy affects bacteria is unknown (Supplementary Fig. S1). We use a sigmoidal function 505 that can cover cases ranging from a linear relationship to a step function. When the inflection 506 point of a sigmoidal function is at 0 % or 100 % target occupancy, the relationship can also be 507 described by an exponential function. We assume that replication in bactericidal and death in 508 bacteriostatic drugs is independent of the amount of bound target. With sufficient experimental 509 data, the replication rate r(x) and/or the death rate ( ) can be obtained by fitting COMBAT to 510 time-kill curves of bacterial populations after antibiotic exposure. The sigmoidal shape of r(x) 511 and d(x) can be written as: 512 ( ) = Z g )oB • ‚ lƒ"ƒ ‚s… n ; ( ) = > † ‡ƒ )oB "•ˆlƒ"ƒˆs … n (12) 513 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint 514 where xrth is the replication rate threshold, xdth is the death rate threshold, and both represent the 515 point where the sigmoidal function reaches ½ of its maximum. gr and gd represent the shape 516 parameters of the replication and death rate functions, respectively. These factors determine the 517 steepness around the inflection point. When they are extreme, the relationship approaches a 518 linear or a step function. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint Beta-lactams acetylate their target molecules (PBPs) and thereby inhibit cell wall synthesis. The 536 acetylation of PBPs consumes beta-lactams. However, PBPs can recover through deacetylation. 537 We modified the term of drug-target dissociation in the equation describing antibiotic 538 concentrations (equation 3), and set the unbinding rate kr = 0. To reflect the recovery of target 539 molecules, we substituted the dissociation rate kr in the equation describing the bacterial 540 population with the deacetylation rate ka, as described in [26]. 541 542

Initial and boundary conditions 543
At t = 0, we assume that all bacteria have zero bound targets ( = 0), and the initial 544 concentration of bacteria is ( , 0) = 0, > 0, and (0,0) = Y . 545 At the boundaries of the partial differential equation ( = 0, = ), we specify that the outgoing 546 velocities are zero. For = 0, i.e. no bound target molecules, the unbinding velocity Z (0, ) = 547 0, and in = , i.e. all targets are bound, the binding velocity ' ( , ) = 0. When the 548 replication term at = 0 and the death term at = are known, we can solve the partial 549 differential equation with two ordinary differential equations at the boundaries. They are similar 550 to the equations at = 0 and at = described by Abel zur Wiesch et al. [18], but taking into 551 account that x is a continuous variable instead of a natural number. 552 553 Numerical schemes 554 To solve our system of differential equations, we used a first-order upwind scheme. Specifically, 555 we used the spatial approximation * ' = Š(<)*Š(<*)) ∆/ for the binding term (vf > 0) and the spatial 556 for the unbinding term (vr < 0). For the time approximation of 557 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint both the PDEs and the ODEs, we used the forward approximation [47]. We also 558 verified that the Courant-Friedrichs-Lewy condition is satisfied. For fitting the experimental data 559 of bacteria exposed to ciprofloxacin and ampicillin, we used the particle swarm method 560 ("particleswarm" function in Matlab, MathWorks software). 561 562

Concentrations of gyrase A2B2 tetramers 563
We assumed that gyrases A and B first homo-dimerize to A2 and B2, respectively, which in turn 564 bind to each other to form the tetramer TR Latin hypercube approach from a biologically plausible range where the association rates are 574 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint between 10 7 -10 9 M -1 s -1 and the dissociation rates between 10 -3 -10 -1 s -1 [33]. This results in 10 4 575 estimates for each of the six experimental replicates quantifying gyrase A and B (Fig. 4 correlation are given. In cases where there was more than one KD value reported in the literature, 828 All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint we used the mean for this analysis. The tested MIC values are the median of several clinical 829 isolates described previously [25]. 830 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint The values of the fitted parameters are listed in Supplementary Tab. S2. c, The net growth rate as 854 determined by the slope of a line connecting the initial bacterial density and the final bacterial 855 density of a time-kill curve at 18 h on a logarithmic scale, is given as function of the drug 856 concentration (blue). The dotted horizontal line indicates zero net growth, and the intersection 857 with the blue line predicts the MIC (0.0139 mg/mL). 858 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint GyrAB expression conditions as in (a). The x-axis shows the percentage of bound antibiotic 870 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint targets predicted by the model fit in (a). The black line indicates the predicted distribution of 883 target occupancies in a bacterial population (both living and dead cells) exposed to ampicillin at 884 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint the MIC for 18 h. c, The net growth rate, as determined by the slope of a line connecting the 885 initial bacterial density and the bacterial density at 18 h on a logarithmic scale predicted from the 886 model fit in (a), is shown as function of the drug concentration (blue). The dotted horizontal line 887 indicates zero net growth, and the intersection with the blue line predicts the MIC (2.6 mg/mL). 888 All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint The copyright holder for this preprint (which was not peer-reviewed) is the author/funder. . https://doi.org/10.1101/369975 doi: bioRxiv preprint window depends on the time at which bacterial growth is observed. The x-axis shows the 903 observed time at which replication rates were determined, the y-axis shows ciprofloxacin 904 concentrations. The dotted curve shows the ciprofloxacin concentration at which the resistant 905 becomes fitter than the WT (FitnessResistant > FitnessWT), the solid line the MIC of the WT 906 (MICWT), and the dashed line the MIC of the resistant strain (MICResistant). The area between the 907 dotted and dashed line indicates the competitive resistance selection window. 908 All rights reserved. No reuse allowed without permission.