Promoting extinction or minimizing growth? The impact of treatment on trait trajectories in evolving populations

When cancers or bacterial infections establish, small populations of cells have to free themselves from homoeostatic regulations that prevent their expansion. Trait evolution allows these populations to evade this regulation, escape stochastic extinction and climb up the fitness landscape. In this study, we analyse this complex process and investigate the fate of a cell population that underlies the basic processes of birth, death and mutation. We find that the shape of the fitness landscape dictates a circular adaptation trajectory in the trait space spanned by birth and death rates. We show that successful adaptation is less likely for parental populations with higher turnover (higher birth and death rates). Including density- or trait-affecting treatment we find that these treatment types change the adaptation dynamics in agreement with a geometrical analysis of fitness gradients. Treatment strategies that simultaneously target birth and death rates are most effective, but also increase evolvability. By mapping physiological adaptation pathways and molecular drug mechanisms to traits and treatments with clear eco-evolutionary consequences, we can achieve a much better understanding of the adaptation dynamics and the eco-evolutionary mechanisms at play in the dynamics of cancer and bacterial infections.


Introduction
Cancer cells and bacterial pathogens show extensive adaptive potential, which helps them to establish 23 even in unfavourable conditions and outgrow competitors and external pressures, for example by the 24 immune system (Fridman et al., 2012;Winstanley et al., 2016). In healthy tissue or healthy micro-25 biomes, external regulation aims to maintain a constant population size, which together with stochastic 26 fluctuations in the population dynamics of individual subpopulations results in a constant turnover 27 characterized by the eventual stochastic extinction of a specific subpopulation and subsequent replace-28 ment by other subpopulations (Gallaher et al., 2019). This extinction can be prevented by adaptations 29 that give an emerging subpopulation of cells a fitness advantage over the remaining population. The 30 increased fitness reduces the subpopulation's risk of extinction in a process often termed evolutionary 31 rescue (Orr and Unckless, 2008;Alexander et al., 2014;Uecker et al., 2014;Marrec and Bitbol, 2020a). 32 Table 1 Reference parameter set. The parameters of the stochastic adaptive process are chosen such that without treatment about half of the replicate simulations show successful adaptation. The parameters of the deterministic model were set such that the time scales of the deterministic dynamics would match the time scales of the stochastic model. Deviations  δ. Focussing on the initial phases of adaptation, we assume that the carrying capacity K remains 88 constant. We will investigate treatment types that either target the density or the traits of the 89 evolving population (Fig. 1). Density-affecting treatment types are modelled as instantaneous density  (Cox and Hinman, 1961; 93 Sobel, 1997). Trait-affecting treatment types are implemented by prolonged additive changes to either 94 the birth or the death rates of the individual lineages. 'Static' drugs decrease the birth rate by ∆ β 95 (e.g. cytostatic chemotherapy or bacteriostatic antibiotics), 'toxic' drugs increase the death rate by 96 ∆ δ (e.g. cytotoxic chemotherapy, immunotherapy or bactericidal antibiotics). Different trait-affecting 97 treatment types can thus be represented by vectors (∆ β , ∆ δ ) in trait space (Fig. 1). Accounting for 98 treatment and logistic density dependence of birth rates the effective birth and death rates of lineage 99 i with population size N i are given by We ensure that effective birth rates are always greater than or equal to zero, setting them to zero if 101 they would be negative.

Stochastic model
We use these microprocesses of birth, death and mutation to construct a discrete-time stochastic model 104 (Eq. 2). We assume that the number of birth and death events per lineage i per time step dt, (B i (t+dt) 105 and D i (t + dt)) are Poisson-distributed around the expected numbers of birth events b i N i dt and death 106 events d i N i dt, given the effective birth and death rates b i and d i according to Eq. (1). The number 107 of mutants M i (t + dt) among the new-born cells is given by a binomial distribution with mutation 108 probability µ.  Defining the total population size as N (t) = i N i (t) and the population average traits as , we can construct a deterministic model from the above 123 microscopic model using a Quantitative Genetics approach (Lande, 1982), Here, the change in total population size is governed by the difference of logistic average birth rate and 125 average death rate. Treatment affects the effective birth and death rates as in Eq.
(1). The change in 126 the average birth and death rates are assumed to be proportional to the gradient of a function φ(t)

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(defined below) that describes the fitness of individuals with proportionality constants G β and G δ that 128 describe the additive genetic variance in the traits (Lande, 1982). The factors e −c/β(t) and e −c/δ(t) 129 ensure decelerating trait changes close to the trait axis, thus preventing negative trait values (Abrams 130 and Matsuda, 1997; Raatz et al., 2019). Note that also this deterministic model formulation assumes 131 independence of the two traits. The system of ordinary differential equations Eq. 3 is numerically 132 integrated using the LSODA implementation of the solve ivp function from the Scipy library (Virtanen 133 et al., 2020) in Python (version 3.8). Standard initial conditions are N (0) = 100, β(0) = 1, δ(0) = 1 134 (Tab. 1).

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Setting the temporal derivative of the population size to zero we can obtain the conditions for the 136 manifold where the population change equals zero. On this manifold, the population size is given by 137 the effective carrying capacity  (Doebeli et al., 2017;Kokko, 2021). One possible definition is lifetime-reproductive output, 144 which itself is a composite measure that includes net growth rate, but also the probability that newly 145 founded lineages survive stochastic population size fluctuations. Even in our simplified setting the 146 determinants of fitness are a priori not trivial, particularly in a regime of high rates of stochastic 147 extinction of lineages. An obvious choice may be the net growth of a lineage r, which determines how 148 quickly that lineage grows out of this regime of probable stochastic extinction and outcompetes other 149 lineages. Similarly, the survival probability of a newly founded lineage p may be selected for. Also, the 150 importance of these two fitness components may change with population size, with survival probability 151 being more important at small lineage size and net growth becoming more decisive for larger lineage 152 sizes. We define these two measures of fitness as Survival probability of newly founded lineage (6) The survival probability here follows from a simplified branching process under the assumption that 154 during the potential establishment of a mutant lineage, the population size of the remaining population 155 will stay approximately constant (see Supplementary Section A.1). Assuming a large carrying capacity 156 K, the density dependence vanishes and the survival probability becomes equal to one minus the 157 extinction probability for newly founded lineages as derived by others (Xue and Leibler, 2017;Coates 158 et al., 2018;Marrec and Bitbol, 2020b). 159 We numerically confirmed the agreement of the survival probability definition with simulations of our 160 model for the case of no mutation (µ = 0) (Fig. S1). Note that the fraction of birth rate over death rate 161 has also been proposed as a fitness measure for a model that is identical to ours, but lacks mutations 162 (Parsons and Quince, 2007).

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Adaptation will either be driven by selection for the fittest lineage in the stochastic model or determined In the deterministic model (Eq. 3) we explicitly prescribe whether adaptation should follow the net 169 growth or the survival probability fitness gradient and thus substitute φ(t) by r(t) or by p(t). If 170 adaptation is determined by net growth we obtain If adaptation is driven by survival probability we obtain  Figure 1 Different treatment types can either affect the cell density directly (left) or indirectly via changing the traits (right). Populations of cancer cells (yellow) or pathogenic bacteria (green) can be targeted with different mechanisms. Density-affecting treatment applies a bottleneck and reduces the population size instantaneously to a fraction f . Trait-affecting treatment, e.g. chemotherapy, alters the traits for a prolonged time period (the treatment duration) and displaces the population in trait space temporarily which results in population decline. Note that M = 1 + max(0, 1 − N (t)/K) 2 is a normalization factor.

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Treatment can either immediately kill part of the population or rig the chances of a population to 174 grow by decreasing birth rates or increasing death rates (Fig. 1). The first case, which affects density 175 directly, causes a direct, instantaneous population size reduction. The second case, which affects 176 traits, brings about an indirect, gradual population size decline where on average more death events 177 than birth events occur. These two treatment types thus differ in their temporal structure. Whereas 178 the first treatment occurs instantaneously, the latter treatment is applied for a defined time span, 179 during which the treatment alters the effective birth and death rates of cells, similar to (Marrec and 180 Bitbol, 2020b). We assume that the density-affecting treatment type targets all cells homogeneously, 181 irrespective of their traits. The additive trait changes during trait-affecting treatment are also equally 182 applied to all lineages, resulting in different relative trait changes, depending on the trait values of 183 each lineage. We represent different trait-affecting treatment types as vectors of length ∆ in trait 184 space with components given in Fig. 1. Besides the pure, static (affecting birth rates only, horizontal) 185 or toxic (affecting death rates only, vertical) treatments, we account for the fact that the boundaries 186 between static or toxic treatment are often blurred. The same drug can be static or toxic, depending 187 on the dose (Masuda et al., 1977), or treatment intentionally consists of two different drug types 188 that each act more static or toxic (Coates et al., 2018;Jaaks et al., 2022). Thus, we include a mixed 189 treatment where both treatment vector components ∆ β and ∆ δ have the same length. Additionally, we 190 propose two treatment types that also combine static and toxic components but additionally account 191 for the shape of the fitness landscape. The minimizing growth treatment counters the net growth rate 192 fitness gradient (Eq. 7) and has vector components (∆ β , ∆ δ ) ∝ ∇r(t) where r(t) is the average net   rubber band here that is extended by adaptive steps and contracts as growth closes the gap between 208 population size and effective carrying capacity. The adaptive steps form a trait space trajectory that 209 travels from the trait combination of the initial parental lineage to smaller death rates and larger birth 210 rates. 211 We hypothesize that this trajectory is the outcome of the stochastic exploration of trait space that 212 climbs up a fitness landscape, with fitter lineages out-competing less fit lineages. This fitness landscape 213 can be characterized by fitness gradients and we propose net growth rate and survival probability as 214 potential fitness components that generate these gradients. For our model, we see that the gradients

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The direction of the gradient of net growth ∇r is density-dependent, i.e. it changes with population 222 size (Eq. 7). The direction of the gradient of survival probability ∇p does not depend on population 223 size but is trait-dependent (Eq. 8). Interestingly, we find that both fitness gradients are parallel 224 as soon as the manifold of zero population size change is reached and the population size equals 225 the effective carrying capacity, N (t) = N * (Eq. 4, Fig. 3). Therefore, only in the initial phases of 226 adaptation ( Fig. 2a), or during and short after treatment when the population size deviates from N * 227 the two fitness components may have non-parallel directions and thus differently affect the direction 228 of adaptation steps. As soon as the total population size reaches N * , the effects of the two fitness 229 components cannot be disentangled, leaving us to conclude that they together dictate the trajectory 230 of trait adaptation.

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Successful adaptation in unfavourable conditions is a stochastic event. When starting with an initial 232 Figure 3 Predicted adaptation directions in trait space. (a) The direction of the net growth gradient is density-dependent, but trait-independent (Eq. (7)). (b) The direction of the survival probability gradient is density-independent, but trait-dependent and has a circular shape (Eq. (8)). At the effective carrying capacity N * , depicted by the red arrows in panel (a), the net growth fitness gradient is parallel to the survival probability fitness gradient. Note that the effective carrying capacity depends on the traits, this causes the apparent trait dependence of the net growth gradient at effective carrying capacity. Given these gradients and initial parental lineages starting from β 0 = δ 0 = 1 the trait trajectories are moving mainly within the region of trait space enclosed by the grey dashed rectangle. Therefore, we zoom in on this region when visualizing trait space trajectories such as in  Probability of evolutionary rescue. First parental populations with higher turnover as characterized by higher levels of equal birth and death rate are less likely to successfully adapt and escape extinction. Rescue probability is here defined as the fraction of non-extinct replicate populations after t = 500, which allows non-extinct populations to move far into trait space regions of high net growth rate and high survival probability (see for example Fig. 2). Simulations are started from the initial parental population size N 0 using 1000 replicates. and, if the population size equals the effective carrying capacity, the net growth fitness gradient.  (Fig. 6b). Similarly, the survival probability fitness component is independent 259 of population size and thus not affected by density changes (Fig. 6c). However, when the population 260 is displaced in trait space the circular shape of the survival probability fitness component changes the 261 predicted adaptation direction to become less vertical under trait-affecting treatment (Fig. 6d). Thus, 262 we hypothesize that both treatment types would drive less vertical adaptation trajectories. 263 We investigate the effect of treatment on the adaptation trajectory by periodically applying the dif- ferent treatment types on populations that grow from small population sizes and ascend the fitness 265 gradient (Fig. 7). If the replicate populations escape extinction, they increase in population size and 266 reach the carrying capacity K. The density-affecting treatment type reduces the population size of 267 each lineage by a bottleneck factor f . This decreases competition and allows surviving lineages to 268 achieve higher net growth rate. This competitive release causes the population size to recover to 269 higher levels after the first treatments than in the untreated control (Fig. 7a). However, newly estab-270 lished, fitter lineages are especially prone to extinction when the bottleneck treatment reduces lineage 271 sizes to small fractions, which limits the exploration of trait space and hinders a rapid adaptation 272 towards faster net growth rates and higher survival probabilities. Therefore, the populations that 273 undergo stronger bottleneck treatments approach the carrying capacity slower and have shorter trait 274 trajectories ( Fig. 7a,b). The trait-affecting treatment types also show the competitive release pattern 275 of recovery to population sizes higher than the untreated control. Here, the population sizes repeatedly 276 recover to higher values after treatment and the carrying capacity is approached faster than in the un-277 treated control (Fig. 7c,d). Similar to the untreated population size time series, also under treatment 278 the population size is tracking the effective carrying capacity N * . We find that the trait trajectories of 279 treated populations deviate from the untreated controls as predicted from our geometrical hypotheses 280 (Fig. 6). We observe that the deviations are caused by more horizontal adaptation steps right after which results in a ramp-like pattern of the traits over time Fig. 8b).

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We find that the dynamics of those trait-affecting treatment types that contain toxic components are 287 similar both in the population size and the trait dynamics. The purely static treatment, however, differs 288 considerably. As the population size approaches the carrying capacity, the effect of the static treat-289 ment is reduced as its net growth reduction is density dependent and proportional to 1 − N/K (Eq. 1).

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This manifests in decreasing density reductions during treatment phases (Fig. 7c). Accordingly, after 291 similar initial trajectories, the adaptation trajectory under purely static treatment later deviates from 292 the adaptation trajectories for the other treatment types that contain also density-independent toxic 293 components (Fig. 7d). We observe similar patterns also in the deterministic description of the adap-294 tive process using a quantitative genetics approach where we explicitly specify the gradient of trait  of the replicates go extinct without any treatment due to stochastic extinction in the initial phases 300 of adaptation. This pattern is caused by the initially equal birth and death rates. Equal birth and 301 death rates imply zero net growth and thus inevitable extinction due to stochastic population size 302 fluctuations. The adapting populations depart from this. Applying treatment increases the fraction 303 of extinct replicates, which we use as a measure to quantify the treatment success rate (Fig. 9). As the success rate of the density-affecting treatment type. Among the trait-affecting treatment types, 306 pure static and toxic treatments achieve a similar success rate. Interestingly, combining static and 307 toxic treatment components results in a considerably higher success rate. Here, the success rate of 308 treatment types that counter either the net growth fitness gradient or the survival probability fitness 309 gradient is slightly higher than the 'Mixed' treatment type that non-adaptively blends the static and 310 toxic components in equal proportion.

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An interesting pattern emerges for the overall number of lineages that are eventually created during the 312 adaptation from one parental lineage, which relates to the evolutionary potential of the population. 313 We find that treatments that particularly increase mortality while not decreasing birth rates lead 314 to a higher number of created lineages. The higher mortality decreases the density limitation of 315 birth rates, which enables high net birth rates and accordingly high mutation rates. Particularly the 316 stronger density-affecting treatments and the purely toxic treatment result in the creation of more In panels (c)-(f), the same lower-case letters above two treatments indicate that the two sets of data points could have been generated from the same underlying distribution. Differing lower-case letters thus indicate differences between treatments. Unique letters indicate treatments that are statistically different from all other treatments. The grouping into statistically different groups was determined using the Tukey's HSD implementation from the statsmodels module (v0.13.0) in Python 3.8 and assigned with the pairwisecomp letters function written by Philip Kirk (https://github.com/PhilPlantMan/Python-pairwise-comparison-letter-generator). A treatment can be part of multiple groups by being indifferent to each one of them and thus receive multiple letters. to be driven to population sizes below a single cell. During trait-affecting treatment, the relative effect 328 of treatment is smaller for smaller, but fitter lineages than for established, less fit lineages, whereas 329 the absolute effects are equal. This may explain the observed differences in the correlation of number 330 of lineages and evolved trait distance. It is interesting to note that treatments with higher success rate 331 were also found to induce faster trait changes (Fig. 8), pointing out a potential trade-off of treatment 332 success versus driving tolerance evolution. We found that treatment types that counter the potential fitness gradients achieve the highest success 335 rates. However, we have not conclusively answered whether the net growth fitness gradient or the 336 survival probability fitness gradient are more decisive for the eco-evolutionary dynamics in our model.

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To gather more evidence on this, we sampled the initial adaptation direction from different initial trait 338 combinations to visualize the realized fitness gradient that acts on the adapting populations in trait 339 space (Fig. S11). We indeed find that the realized fitness gradients are non-parallel in trait space,

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indicating that for larger birth rates and smaller death rates adaptation is driven by decreasing death 341 rate, and increasing birth rate becomes less important. The visual similarity of this pattern to the 342 survival probability fitness gradient hints at a larger importance of the survival probability fitness 343 gradient at first glance. However, also the net growth rate becomes larger for larger birth rates and 344 smaller death rates, which speeds up the population size increase during the short observation window 345 of initial adaptation. Because of the density-dependence, these larger population sizes turn the net 346 growth fitness gradient to be more vertical (see Fig. 3a). Also, we observe that the initial adaptation 347 direction is largely parallel along the diagonals in trait space, which correspond to the net growth 348 fitness isoclines for small population sizes, which favours the net growth fitness gradient to be more 349 important.

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To investigate whether the differences in initial adaptation direction are indeed caused by the density-351 dependence of the net growth fitness gradient, we again investigated the initial steps of adaptation 352 with parameters that minimize the density change within our observation window. We decreased the 353 initial population size and time span and increased the carrying capacity and find that the adaptation 354 direction indeed becomes more horizontal, indicating a larger importance of the net growth fitness 355 gradient than the survival probability fitness gradient. If the survival probability fitness gradient 356 would be predominantly driving the adaptation, we would expect that the initial steps of adaptation 357 change along the net growth fitness isoclines (except for the diagonal passing through the origin) and we would not expect a density dependence.

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In the deterministic model (Eq. 3), we are explicitly prescribing the fitness measure that determines 360 the direction of trait adaptation. If we choose the net growth as the determining fitness measure 361 we find trait trajectories that change with treatment and reproduce the trajectories obtained from 362 simulations (Fig. S9). However, if we set the survival probability as the determining fitness measure 363 in the deterministic model the trait trajectories under density-affecting treatment do not deviate from 364 the trajectories without treatment, thus contrasting the observation in the simulations (Fig. S10).

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Therefore, more evidence points towards net growth rate maximization as the determinant of trait space 366 adaptation trajectories in our simulations, even though we cannot falsify that the survival probability 367 fitness gradient could also play an important part. death rate (Fig. 3). Indeed, this circular trajectory is recovered in stochastic simulations (Fig. 5) and 379 altered by treatment in agreement with geometrically derived hypotheses (Figs. 6, 7). Interestingly, 380 we find that adaptive steps that maximize net growth rate or survival probability always have parallel 381 components, indicating no strong conflict between optimizing for either of the two plausible fitness 382 measures.

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In this study, we deliberately chose parameters that would result in occasional extinction of replicate 384 populations to represent the stochastic nature of the establishment of cancer or bacterial infections 385 and the stochasticity in treatment response (Coates et al., 2018;Alexander and MacLean, 2020).

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This results in a setting where evolutionary rescue is required for the populations to prevent their 387 extinction. In our model, the population dynamics are captured by the dynamics of the effective 388 carrying capacity which is the target population size that the total population size is tracking over 389 time. If birth rates and death rates are equal, the effective carrying capacity is zero and the population goes extinct deterministically. The effective carrying capacity becomes positive only if the death rate 391 becomes smaller than the birth rate by trait adaptation, thus also increasing the chances of population 392 establishment.

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The shape of the fitness landscape has important implications for the effect of turnover on the rescue 394 probability in the cancer or bacterial cell population, which we can again address using geometrical this pattern will change. Accordingly, we find that multiplicative mutational effects compensate for the 409 increasing distance of radial fitness isoclines at larger birth and death rates and the rescue probability 410 becomes largely independent of turnover.

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Besides the shape of the fitness landscape, the declining rescue probability for faster turnover may 412 also be explained with the higher rate at which the initial parental lineage declines. At equal birth 413 and death rate, the logistic competition term results in a deterministic rate of population decline of 414 −β 0 N 0 (t) 2 /K in our model, which increases proportional to the birth rate. As this initial parental

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Cancer cell populations as well as bacterial biofilms in chronic infections possess a considerable geno-424 typic and phenotypic heterogeneity (Caiado et al., 2016;Gay et al., 2016;Winstanley et al., 2016;425 Dhar et al., 2016). In a heterogeneous population consisting of lineages with different turnover but 426 individually equal birth and death rates our results imply that those lineages with smaller turnover 427 would persist longer. Evolutionary rescue would thus be achieved on average from those lower-turnover 428 lineages hinting at a selective advantage of low turnover in heterogeneous populations in challenging 429 environments, which may explain the therapeutic challenges posed by dormant subpopulations both 430 in cancer (Yeh and Ramaswamy, 2015;Ammerpohl et al., 2017) and bacterial infections (Wood et al.,431 2013). Birth (proliferation) and death (apoptosis) are partly interlinked in their regulation (Alenzi,432 2004) and measuring their rates in eukaryotic cells is possible in vitro and in vivo (Lyons and Parish,433 1994). Different tissue types were shown to have intrinsically different turnover rates (Sender and 434 Milo, 2021) and turnover can be altered experimentally (Casey et al., 2007). Several studies reported 435 a positive correlation of proliferation and apoptosis in breast cancer (de Jong et al., 2000;Liu et al., 436 2001;Archer et al., 2003), which suggests a positive correlation of birth and death rate. Prognosis was 437 found to be worse for higher birth rate (Liu et al., 2001). Our model proposes that such aggressive, 438 quickly growing tumours with a high cell death rate are actually less likely to persist than tumours 439 with lower turnover as the probability for evolutionary rescue decreases with turnover. This apparent 440 dichotomy indicates that the evolutionary rescue probability of a tumour not necessarily translates 441 into its prognosis and that clinically we tend to only observe the few high-turnover tumours that have 442 managed to escape homeostatic regulation, while remaining blind to those with lower turnover. Also in 443 the context of chronic bacterial infections there exist methods to assess turnover in bacterial pathogen 444 populations in vitro (Stewart et al., 2005;Wang et al., 2010). They are currently developed for in vivo 445 settings (Myhrvold et al., 2015; and will soon elucidate the different intrinsic 446 birth and death rates of bacterial strains and species, sometimes even working out spatial parameter 447 heterogeneity within the body . It will be interesting to see whether indeed 448 lower-turnover regions of the birth-death trait-space are found to be more populated and whether 449 trait evolution indeed proceeds along the circular trajectory predicted by our model.

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Fitness landscapes of mutational changes can be constructed from data (Watson et al., 2020) and used 451 in treatment via evolutionary steering (Nichol et al., 2015;Acar et al., 2020). Accounting for their 452 temporal variability (e.g. under the effect of treatment), then sometimes referring to them as fitness 453 seascapes, has important consequences for the understanding of adaptation, such as resistance evolution 454 (Lässig et al., 2017;King et al., 2022). For example, Hemez et al. (2020)  We found profound patterns of competitive release in the population dynamics of successfully adapting 462 populations (Wargo et al., 2007). In the off-treatment phases, the treated and non-extinct populations 463 quickly recover to population sizes up to twice as large as in the untreated reference. The competitive 464 release is particularly strong for the trait-affecting treatment types. This is in line with the fact that 465 the trait-affecting treatment exerts a higher relative penalty on less fit lineages than on fitter lineages 466 as we assumed additive treatment effects and thus the mortality during treatment is higher for less 467 fit lineages. In our model the effect of static drugs decreases as the population size approaches the 468 carrying capacity where the effective birth rate tends to zero even without treatment and thus can not subpopulations (Gatenby et al., 2009;Viossat and Noble, 2021).

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Time-resolved surveillance of treatment responses in both cancer and bacterial infections promises to 479 prevent resistance evolution, but is technically and practically challenging. Accordingly, the quest for 480 personalized, resistance-proof treatment approaches remains one to be fulfilled. In a recent paper, we 481 found that increasing the temporal frequency of surveillance has diminishing returns and also more 482 coarse-grained surveillance patterns could achieve large treatment improvements (Raatz et al., 2021).

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Interestingly, in the present study we find that the mixed treatment which is agnostic to real-time 484 information performs almost as good as the treatment types that counter the fitness gradient and thus 485 necessitate ongoing temporal information on the population trait average. This again suggests that 486 large treatment improvements can be achieved already with low surveillance effort. The high efficiency 487 of static and toxic treatment combinations is in agreement with theoretical predictions (Lorz et al.,488 2013) and recently explored approaches in cancer treatment, such as the combination of navitoclax, 489 a drug that increases the apoptosis rate, and cytostatics such as gemcitabine or brentuximab which 490 decrease the birth rate (Cleary et al., 2014;Ju et al., 2016;Montero and Letai, 2018). Also in bacteria, 491 recent findings suggest that a combination of bacteriostatic drugs (or nutrient deprivation) and bacte-492 ricidal drugs indeed increase the extinction probability of bacterial microcolonies (Coates et al., 2018). 493 However, awareness of the mechanisms of action and the interactive effects is essential, as treatment 494 efficiency can also be reduced in combination treatments, for example if the bactericidal drug relies 495 on cell growth that is reduced by the bacteriostatic drug (Bollenbach et al., 2009;Bollenbach, 2015;496 Coates et al., 2018). An additional advantage of combination therapies that was not considered in 497 our study is that resistance is less likely to evolve in parallel against two independently active drugs.

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Consequently, drug interactions have important consequences not only for treatment efficiency but also 499 for resistance evolution (Roemhild et al., 2018;  For a single initial individual with birth rate β and death rate δ they obtain a density-independent extinction risk of q = δ β from which the survival probability for a new lineage follows as Assuming that changes in the population size of the parental lineage are small on the time scale during which the fate of a mutant is decided, i.e. whether it escapes extinction from stochastic drift or not, allows us to fix the total population size to its value when the mutant occurred at time T . Thus, we can include the density dependence of our model in the survival probabilty (Eq. S1) by substituting This results in a density-dependent survival probability Including trait-affecting treatment effects and restricting the survival probability to the range between zero and one results in Eq. 6.
A similar derivation uses branching process techniques and arrives at an integral for the fixation probability of a mutant individual on the background of the parental population (Uecker and Hermisson, 2011) p fix (T ) = 2 Using the same assumption of N (t) = N (T ) = const. as above, this reduces to  Fig. S2 we tested the effect of multiplicative mutational effects on birth an death rates. The mutant lineages' birth rates here are determined by β mutant = β parental (1 + s), s ∼ N (0, σ), and death rates are independently determined as δ mutant = δ parental (1 + s), s ∼ N (0, σ). Under these assumptions, the rescue probability of initial parental populations is largely independent of turnover.  Figure S5 Exemplary dynamics for static treatment. Plot details and parameters as in Fig. 2. Black bars depict the times when ∆ β = 0.5. During treatment the effective carrying capacity can reduce to negative values. The population sizes, however, must be non-negative and thus approach zero when the effective carrying capacity becomes negative.  Figure S6 Exemplary dynamics for toxic treatment. Plot details and parameters as in Fig. 2. Black bars depict the times when ∆ δ = 0.5. During treatment the effective carrying capacity can reduce to negative values. The population sizes, however, must be non-negative values and thus approach zero when the effective carrying capacity becomes negative.    Figure S9 Deterministic adaptation dynamics under treatment -Net growth fitness gradient. Choosing the net growth gradient (Eq. (7)) as the fitness gradient in the deterministic model (Eq. 3) and parameter values from Tab. 1, we obtain adaptation dynamics that are similar to those presented for the stochastic model (Fig. 7).  8)) as the fitness gradient in the deterministic model (Eq. 3) and parameter values from Tab. 1, we obtain adaptation dynamics that are similar to those presented for the stochastic model (Fig. 7). However, the density-affecting treatment type has no effect on the trait trajectory as the survival probability fitness gradient is density-independent.  S11 Observed initial steps of adaptation. Shown is the average direction of the adaptation trajectories in trait space until time t f for different combinations of observation window t f , carrying capacity K and initial population size N 0 . Other parameters are chosen as given by Tab. 1. If the net growth was determining the adaptation trajectory, we expect adaptation steps that have a higher birth-rate component for decreasing density limitation (which can be realized by shorter observational window (blue arrows), higher carrying capacity (green arrows), smaller initial population size (yellow arrows) or all combined (red arrows)). If survival probability (grey arrows) was driving the adaptation we would expect the adaptation direction to not be affected by changes to t f , K or N 0 .