Cellular turnover can increase or decrease the mutant burden in expanding cell population

Cellular turnover can increase or decrease the mutant burden in expanding cell population | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Cellular turnover can increase or decrease the mutant burden in expanding cell population Samrat S. Mondal , Natalia L. Komarova , Dominik Wodarz doi: https://doi.org/10.1101/2025.11.17.688895 Samrat S. Mondal 1 Department of Ecology, Behavior and Evolution, University of California San Diego , La Jolla, CA 92093, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: s5mondal{at}UCSD.edu Natalia L. Komarova 2 Department of Mathematics, University of California San Diego , La Jolla, CA 92093, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Dominik Wodarz 1 Department of Ecology, Behavior and Evolution, University of California San Diego , La Jolla, CA 92093, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract Expanding cell populations, such as bacterial and tumor colonies, continuously accumulate mutations as they grow. However, how mutational burden depends on cell turnover, i.e. the ratio of birth and death rates, remains poorly understood. Elucidating this relationship is crucial for predicting how populations adapt to changing environments, including during evolutionary rescue and resistance evolution. Previous theory suggested that higher turnover increases mutant abundance at a given population size, since more cell divisions are required to reach that size. Using well-mixed and spatial stochastic models, we find the relationship is more nuanced. For advantageous mutants, higher turnover does increase mutant numbers. For disadvantageous mutants, however, mutant abundance can actually decreases with turnover or exhibit non-monotonic behavior, with these somewhat counterintuitive patterns being more pronounced in spatially expanding populations. We derive explicit analytical boundaries separating different regimes and quantify the contribution of stochastic “jackpot” events to mutant burden. In spatial models, we show that the interplay between two timescales is critical: the time to reach the target population size versus the time for wild-type cells to erode disadvantageous mutant clusters. Our results reveal how basic demographic parameters influence the ability of cell populations to overcome selective barriers during growth, with implications for understanding both evolution in natural settings and disease progression. 1 Introduction The accumulation of mutations in expanding cell populations is a fundamental process that determines their potential to adapt to changing environments. When a population experiences environmental stress, whether bacteria facing nutrient limitation in natural ecosystems, pathogens encountering antibiotic therapy, or cancer cells subjected to therapeutic intervention, the number of mutants present becomes a critical determinant of survival [ 1 ]. Populations with larger mutant reservoirs are more likely to harbor variants capable of withstanding the new selective pressure, a phenomenon central to the concept of evolutionary rescue [ 2 ]. The distribution of mutant numbers when a population reaches a given size is a classic problem in mutation theory, famously characterized by the Luria-Delbrück distribution for exponentially growing populations without cell death [ 3 ]. The most fundamental parameters underlying cell growth are the rate of cell division and cell death, and the balance between these processes can influence the mutant burden in a population that has grown to a given size. We refer to the death-to-birth ratio as cellular turnover, with higher turnover indicating a higher rate of cell death compared to the rate of cell division. The turnover varies widely across biological contexts. In bacteria such as E. coli , growth under nutrient-rich conditions is characterized by rapid division and relatively low death rates, resulting in low turnover [ 4 ]. Under sub-inhibitory stress, however, elevated death rates increase turnover substantially [ 5 ]. Even under favorable growth conditions, however, high turnover can occur [ 6 ]. During phytoplankton blooms in the North Sea, several bacterial taxa exhibited death rates that almost matched their division rates, brought about by mortality from grazing and viral lysis, and leading to an overall slow rate of population growth [ 6 ]. Turnover also varies considerably across cancers: pancreatic ductal adenocarcinoma exhibits a relatively high turnover, with substantial cell death balanced by continuous proliferation [ 7 ], while other cancers, such as chronic lymphocytic leukemia, have been reported to display lower turnover due to disrupted apopotosis and extended survival of the tumor cells [ 8 ]. Within specific malignancies, the balance between apoptosis and proliferation can vary, and this variation has been shown to correlate with clinical aggressiveness in complex ways, with both higher and lower turnover linked to poorer outcomes in different contexts [ 9 , 10 , 11 ]. Elucidating how turnover shapes mutant burden in growing populations is therefore crucial for understanding their evolutionary potential and, ultimately, their capacity to escape selective barriers or to resist therapeutic intervention. Mutations in expanding populations arise during cell division. When cell death is rare, population expansion proceeds rapidly, and mutants have limited time to accumulate before a target population size is reached. Conversely, when death events are frequent, more birth–death cycles are required for the population to achieve the same net size, increasing the number of mutational opportunities. One might therefore expect that higher turnover should increase the number of mutants at a given population size. This effect was reported under various assumptions in [ 12 , 13 , 14 ]. These studies showed that the average number of mutant cells at a threshold population size grows with the number of divisions in the tumor’s history, which in turn correlates with larger cell death relative to division rates (higher turnover). More recently, analysis of stochastic colony growth revealed more complex patterns in mutant accumulation, see e.g. [ 15 , 16 , 17 ]. It became apparent that studying mutant burden accumulated by a given time is different from requiring a fixed tumor size [ 18 ]. Further, significant differences between mutant accumulation in spatially growing colonies compared to well-mixed systems have been discovered [ 19 , 20 , 21 , 22 ]. In spatially expanding populations such as growing tumors or bacterial biofilms, cells can divide primarily at the periphery while the interior remains near constant cell densities [ 20 , 23 ]. This coupling between local birth and death constrains the effective rate of reproduction and modulates the interplay between mutation and selection. In such systems, mutant accumulation may differ markedly from that in well-mixed populations. For example, in two-dimensional tumor growth, mutant clones that arise in the interior may be quickly lost, while those generated at the expanding edge can persist and spread along the frontier [ 20 , 23 ]. Hence, the spatial pattern of turnover can fundamentally change how mutants accumulate during expansion. In this study, we investigate comprehensively how the average number of mutants at a fixed population size depends on the degree of turnover. We analyze stochastic models of both well-mixed populations and spatially explicit two-dimensional populations. Our results reveal that while higher turnover correlates with a larger burden of advantageous mutants, the relationship between turnover and mutant accumulation is quite complex for deleterious mutants: mutant numbers may increase or decrease with higher turnover, or exhibit a peak at intermediate values. Unexpectedly, we find that stochasticity and spatial structure strengthen these so far unobserved dependencies, effectively amplifying the disadvantage of mutants as turnover increases. These findings have important implications for understanding how demographic parameters influence the extent of standing genetic variation in cell populations, with special relevance to the evolution of drug resistance in bacteria and tumors, since such mutations are often associated with a fitness cost [ 24 , 25 ]. 2 Results We consider a number of models where a population of individuals undergoes unlimited growth through births and deaths, and deleterious mutants are produced by mutations that happen with a certain probability during cell divisions. Let us denote by r w and d w the basic, per capita division and death rates of the wild-type cells. The quantity that is, the death-to-birth ratio of the wild-type cells, characterizes the cellular turnover. The mutant kinetic parameters are given by r m = (1 + s 1 ) r w and d m = (1 − s 2 ) d w , such that positive values of the selection factors s 1 and s 2 generally align with advantageous, and negative values with disadvantageous mutants. A deterministic, well-mixed colony The simplest model of deterministic, exponential growth of a well-mixed system is given by the following ordinary differential equations: where x ( t ) and y ( t ) measure the populations of wild-type and mutant cells respectively. Under the initial condition x (0) > 0, y (0) = 0, we let the population grow to the total size N , and evaluate the mutant burden, M N , as a function of the turnover, τ . First, we note that advantageous mutant burden always increases with the turnover (see SI.1.2). Next, suppose first that mutants are deleterious because their division rate is smaller than that of the wild types ( s 1 < 0), while their death rate is the same as that of the wild-type cells ( s 2 = 0). In this case, again the mutant burden M N always increases with turnover (see SI.1.3 for a detailed proof). This intuitive result is usually explained by noting that in a higher-turnover colony, more cell divisions are required for the population to reach the target size N , and therefore there are more opportunities for mutant lineages to arise. This reasoning is not entirely correct and, in fact, breaks down when mutants are disadvantageous in terms of death. Figure 1 shows three possible types of dependence on the death-to-birth ratio: The number of mutants can increase with τ , decrease, or vary in a non-monotonic way. For example, in the special case s 1 = 0 and s 2 < 0, where mutants are disadvantaged only in terms of death, the mutant burden as a function of turnover, M N ( τ ), first increases and then decreases with τ for small values of | s 2 | . Moreover, as | s 2 | exceeds a threshold (see SI.1.4 for a detailed proof), the function becomes monotonically decreasing in τ . A significant decay of M N with τ is observed when the selection coefficient | s 2 | exceeds ∼1 / ln N (see SI.1.5 for a detailed proof). Download figure Open in new tab Figure 1: Dependence of mutant burden on the death-to-birth ratio in well mixed systems. (a)–(d) represent outcomes from the deterministic model, and (e)–(h) those from the stochastic model (using equation (2)). (a) and (e) are phase diagrams showing the qualitative behavior of mutant accumulation across the ( s 1 , s 2 ) parameter space for the deterministic and stochastic models, respectively: Blue indicates a monotonic increase; green denotes non-monotonic behavior with a decrease in mutant accumulation at high death-to-birth ratios; gray indicates a monotonic decrease; and teal represents non-monotonic behavior with an increase at large death-to-birth ratios. The line (see SI.1.4 for a derivation) separates the green and gray regions in the deterministic case. (b)–(d) show illustrative mutant accumulation curves as functions of the death-to-birth ratio τ , corresponding to the white dots marked in (a), which lie in the blue, green, and gray regions, respectively. These points correspond to ( s 1 , s 2 ) = (0.14, 0.14), (−0.14, −0.14), and (−0.42, −0.42). Similarly, (f)–(h) display mutant accumulation curves as functions of τ , corresponding to the white dots marked in (e), located in the blue, green, and teal regions, respectively, at ( s 1 , s 2 ) = (0.14, 0.14), (−0.35, −0.20), and (−0.50, 0.25). The rest of the parameters are r w = 1, µ = 2 × 10 −5 , and N = 10 3 . Even though the deterministic well-mixed system is too simplistic to provide quantitative predictions about mutant burden in real colonies, it still reveals qualitative patterns that also hold in more realistic settings. For example, The deleterious mutant burden can increase, decrease, or vary in a non-monotonic fashion with the death-to-birth ratio. We will see later in this text that this behavior also occurs in other, more realistic models, such as models incorporating finiteness of population and spatial structure. In the deterministic well-mixed system, a decrease in M N with τ requires mutants to be disadvantaged in death, with the strength of the disadvantage on the order of 1 / ln N , which is unrealistically large. As we will show, although this behavior is qualitatively true, this stringent condition is much more relaxed in the stochastic and spatial systems. Thus, while the deterministic well-mixed system captures useful qualitative patterns, it cannot account for the inherent randomness of reproduction, mutation, and death. In the next step, we incorporate stochasticity, which makes the model more realistic and can lead to outcomes that differ drastically from the deterministic predictions. A stochastic, well-mixed colony In the well-mixed stochastic model, wild-type individuals reproduce at rate r w (1 − µ ) giving rise to wild-type offspring, or a mutation occurs during reproduction at rate r w µ giving rise to mutant offspring. Wild-type individuals die at rate d w . Mutant individuals reproduce at rate r m and die at rate d m . These events occur probabilistically, so that at any time t the (integer) numbers of wild types and mutants, X ( t ) and Y ( t ), evolve as two coupled stochastic processes. The system is initiated from a single wild type individual and no mutants, ( X (0), Y (0)) = (1, 0), and trajectories are followed until the total population first reaches the target size X + Y = N ; runs that end up in extinction at ( X, Y ) = (0, 0) are excluded. For a detailed description of the model, see Methods 4.1. Figure 1(e) gives a summary of the mutant burden dependency on τ . While advantageous mutant behavior does not change qualitatively, it turns out that in some cases, the deleterious mutant burden in the stochastic system behaves differently compared to the deterministic model. In Figure 2 we illustrate this by considering the case where mutants do not differ from wild-type cells in their death rate, only in their reproduction rate, i.e., s 2 = 0. When s 1 < 0 (mutants disadvantageous in birth), the well-mixed deterministic system predicts monotonic growth of mutant burden with increasing death-to-birth ratio (the blue line, Figure 2 ). In the well-mixed stochastic system, however, the outcome is substantially different and exhibits non-monotonic behavior, see the green symbols. Download figure Open in new tab Figure 2: Dependence of mutant burden on the death-to-birth ratio in a well-mixed stochastic system. The blue line represents the deterministic model prediction, the green symbols with error bars (standard error) are stochastic simulations, and the green line is a theoretical approximation, see SI.2.1 for details. For τ (1 + s 1 ), mutants become subcritical and the burden declines toward the deterministic limit. Each point is computed from 2 × 10 7 non-extinct runs. The parameters are r w = 0.5, µ = 10 −4 , N = 10 4 , and ( s 1 , s 2 ) = (−0.3, 0.0). The reason for the difference in behavior is so-called “jackpot events”. In a stochastic system, there exists the possibility of an extreme event where the final state is ( X, Y ) = (0, N ). This corresponds to the extinction of the wild-type lineage while the mutant seed lineage, generated by the wild type just before extinction, reaches the target population size. We theoretically compute (see SI.2.1) the probability of jackpot events in the limit of rare mutation as follows: where R m = [ τ (1 − s 2 )] / (1 + s 1 ). Because this probability allows the mutant population to occasionally reach the target size N , we correct the well-mixed prediction by adding the occurrence probability multiplied by the target size, to obtain the corrected stochastic mutant burden, : which generally deviates from the deterministic well-mixed prediction (see the green solid line in Figure 2 ). The mutant burden exceeds the deterministic prediction for lower τ due to takeovers of mutants with r m > d m (supercritical mutants), but as the death-to-birth ratio increases further, such that r m < d m (subcritical mutants), the mutant burden decreases to coincide with the deterministic prediction, since mutant takeovers become nearly impossible. Figure 1(e-h) illustrates the differences between deterministic and stochastic models. Generally, jackpot events result in an increase of mutant burden compared to the deterministic system. As τ increases from zero, the super-criticality of mutants diminishes while the total production of mutants by wild-types rises. At an intermediate turnover, the stochastic system generates sufficient numbers of subcritical mutants to offset their extinction, producing a peak in mutant accumulation (as evident from the humps of the mutant accumulation curves in Figure 1(g, h) and Figure 2 ). At this point, the elevated probability of jackpot events elevates the mutant burden, relative to the deterministic prediction M N , by an amount on the order of ∼ µN . While the stochastic “hump” is drowned out by a rapidly increasing mutant population in the case of advantageous mutants ( Figure 1(e) ), for deleterious mutants, jackpot events can turn a monotonically increasing curve into a non-monotonic dependence on τ , see the teal region in Figure 1(e) and panel (h). Moreover, even when the deterministic curve decreases monotonically but at an exceedingly slow rate, jackpot effects can also be significant, transforming an otherwise monotonic decline into a non-monotonic profile with a single maximum in the mutant burden. The curve in Figure 1(g) corresponds to the point in the phase diagram (panel (e)) below the dashed line. In the deterministic system it was inside the “gray” region (monotonic decline); in the stochastic system we observe non-monotonic behavior. This type of contributions from jackpot events explains why the green region in Figure 1(e) extends beyond the deterministic analytical separator between the green and gray regions in Figure 1(a) . Next, we incorporate spatial structure into our model, making it more realistic. A stochastic, spatial colony In the spatial stochastic model, individuals occupy sites on a two-dimensional L × L grid, with each site containing at most one individual. Wild types divide at rate r w and mutants at rate r m = r w (1 + s 1 ), while death occurs at rates d w and d m = d w (1 − s 2 ), respectively, leaving the site empty. Division requires an empty neighboring site, chosen uniformly from the Moore neighborhood, and mutation arises only during wild-type division: with probability µ the offspring is mutant, otherwise it is wild type. These events occur probabilistically, see Section 4.2 for details. The system is initiated from a single wild type at the center of the grid and trajectories are followed until the total population size first reaches N ; the grid size is set large enough that the expanding population never reaches the boundary of the space. Again, runs that end in extinction are excluded. Stochastic simulations of spatially expanding colonies demonstrate the following patterns. Similar to the other models described above, advantageous mutants always show an increase in numbers as the death-to-birth ratio increases (see Figure. SI.1(a) and compare with panels (a), (b), (e), (f) in Figure 1 ). For disadvantageous mutants, the overall result is that the mutant load again is not necessarily an increasing function of the death-to-birth ratio. In fact, the reversal of this relationship is seen even more often than in well-mixed stochastic systems. Figure 3 illustrates this by showcasing results for a mutant that has a two percent disadvantage in birth rate compared to the wild type while having identical death rates (see Figure 3(a) ), as well as a two percent disadvantage in death and no change in birth (see Figure 3(b) ). For these parameters, the deterministic and stochastic well-mixed models predict a monotonic increase in mutant burden with increasing cellular turnover for both cases. In contrast, the spatial stochastic model predicts a much higher mutant burden than the deterministic one, particularly within the range of turnover rates over which the spatial simulations are evaluated. In addition, in the spatial model, we observe a clear decrease in mutant burden as turnover increases. Similarly, mutants that are overall disadvantageous but have an advantage in deaths (that is, s 1 + s 2 0) also exhibit a decline in numbers as τ increases, see Figure SI.1(b). Download figure Open in new tab Figure 3: Dependence of mutant burden on the death-to-birth ratio in a spatial stochastic system. (a) Results for a mutant that is disadvantageous in birth but not in death, i.e., ( s 1 , s 2 ) = (−0.02, 0.0). (b) Results for a mutant that is disadvantageous in death but not in birth, i.e., ( s 1 , s 2 ) = (0.0, −0.02). Green symbols with error bars (standard error) represent data from spatial stochastic simulations averaged over 1.6 10 4 non-extinct runs. Blue solid lines correspond to the well-mixed deterministic system, and blue symbols with error bars (standard error) represent results from well-mixed stochastic simulations averaged over 2 × 10 6 non-extinct runs. Other parameters are set to r w = 0.5, µ = × 10 −6 , and N = 4 × 10 6 . To explain these observations, consider mutant and wild-type co-dynamics in a two-dimensional expanding colony. Growth is sustained by cell divisions that occur within a narrow proliferative layer at the advancing edge of the colony. Random birth and death events within this layer continually reshape the boundaries between neighboring segregated lineages of wild-type and mutant individuals, causing them to drift and occasionally merge as the colony expands [ 19 ]. When a mutant cell arises near the advancing front, its descendants form a compact, wedge-like patch whose boundaries move laterally due to stochastic fluctuations and any difference in uneven selective forces between mutant and wild-type cells [ 26 ]. Under conditions of low turnover, when deaths are rare and this selective bias is slightly negative (i.e., when mutants are mildly disadvantageous) they can still expand sideways along the frontier and form extended, temporary sectors. Even though the mutant sectors eventually get surrounded by (advantageous) wild-types, the presence of resulting mutant “bubbles” leads to a higher overall mutant burden compared with a well-mixed population (see Figure 4(a) ). Similar frontier-driven amplification and boundary dynamics have been discussed in the context of expanding cellular populations [ 27 ]. Download figure Open in new tab Figure 4: Dependence of the density of disadvantageous mutants bubble on increasing turnover. A mutant type with disadvantage in birth only is considered, with ( s 1 , s 2 ) = (−0.05, 0.0). Other parameters are r w = 0.5, µ = 4 × 10 −4 , and N = 5 × 10 6 . Panels (a) and (b) show final population snapshots for turnovers τ = 0 and τ = 0.4, respectively. For large turnover, frequent death events make the frontier more diffuse and render the interior of the colony less densely occupied. The effective expansion speed of the colony slows down, and random fluctuations at the frontier begin to dominate. Under these conditions, disadvantageous mutants fail to maintain coherent bubbles and are gradually overtaken by wild-type lineages (see Figure 4(b) ). The number of surviving mutants therefore declines, approaching the well-mixed expectation that scales with the total population size and the mutation rate. This crossover reflects a competition between frontier-driven lateral expansion, which dominates when turnover is low, and turnover-driven dilution and interior mixing, which dominate when turnover is high. When do we expect to see the most pronounced decline of the mutant burden with the death-to-birth ratio? To answer this question on a qualitative level, it is useful to consider low and high turnover situations separately. At low turnover, advantageous mutants create large mutant sub-colonies, mildly disadvantageous mutants form intermediate-sized bubbles, and strongly disadvantageous mutants generate only small bubbles. At high turnover, a substantial reduction in mutant burden occurs only when mutants are disadvantageous. Even mildly disadvantageous mutants are unlikely to persist, as high-turnover populations take longer to reach the target size (provided it is large enough), giving such mutants more opportunities to be lost before the colony hits the target size. By connecting these two conditions, we see that mildly disadvantageous mutants are the ideal candidates to show both sufficiently elevated and decreased mutant burdens at low and high turnovers, respectively, therefore exhibiting the sharpest decline in mutant burden as turnover increases. 3 Discussion A long-standing paradigm in evolutionary dynamics and tumor modeling holds that increased cellular turnover necessarily elevates the mutant burden, since additional deaths require compensatory divisions to reach a given final size, creating more mutational opportunities. This principle underpins a number of studies, from the early stochastic frameworks of clonal evolution [ 28 ] and multistage cancer progression [ 29 ] to generalized birth–death and branching-process models of mutation accumulation [ 12 ]. Collectively, these works form the prevailing view that mutant accumulation monotonically increases with cellular turnover. Our results demonstrate that this expectation, while valid under specific conditions, does not represent a universal principle. Specifically, we found that an increase of mutant burden with the death-to-birth ratio is always observed for advantageous mutants, in all the three models. For disadvantageous mutants, however, the relationship between turnover and mutant burden is more complex and depends critically on the mechanistic basis of mutant disadvantage, the presence of demographic stochasticity, and spatial population structure. Here we show that for disadvantageous mutants, even for the deterministic well-mixed model, the canonical expectation (that the mutant burden increases with the turnover) holds only under a limited range of conditions, see Figure 1(a) and Table 1 . The mutant burden may decrease with turnover when mutants experience disadvantage in their death rates ( s 2 < 0), although this requires significant levels of disadvantage: if s 1 = 0, | s 2 | must exceed 1 / ln N for the mutant burden curve to decline with turnover. View this table: View inline View popup Download powerpoint Table 1: Conditions, under which a decay in disadvantageous mutant burden with death-to-birth ratio is observed in different models. Including stochasticity and spatial constrains facilitates this counterintuitive behavior of deleterious mutants. The well-mixed stochastic model can exhibit a decline in disadvantageous mutant burden for a wider range of parameters, see Figure 1(b) and Table 1 . The difference arises due to occasional “jackpot” events, where mutants that are generated early on, take over the whole final population. For example, assuming s 1 = 0, when mutants are supercritical (| s 2 | > (1 − τ ) /τ ), these rare events lead to a higher average mutant burden than predicted by the deterministic model. As turnover increases and mutants become subcritical ( | s 2 | < (1 − τ ) /τ ), takeovers become unlikely, and the average mutant burden decreases to the well-mixed deterministic values. As with a deterministic system, to observe a decrease in the mutant burden as a function of the death-to-birth ratio, the disadvantage of the mutants (whether it is in divisions, death, or both) must be relatively high. It is only in the spatial stochastic colony that we observe a mutant burden decline with the turnover rate even for very small levels of disadvantages, making this widely applicable to a range of biological systems. Spatial structure introduces a distinct mechanism through which turnover modulates mutant burden, which operates even for weakly disadvantageous mutants that would show monotonically increasing burden in both deterministic and stochastic well-mixed models. This mechanism relies on the strictly spatial feature of mutant dynamics, namely, the formation of “bubbles”, compact spatial mutant regions that first form at the advancing colony front [ 20 , 30 ], and whose lateral boundaries undergo stochastic fluctuations and slow selective drift [ 21 , 22 ]. At low turnover, these bubbles contribute disproportionately to final mutant burden because spatial segregation partially insulates disadvantageous mutant lineages from direct competition with wild-types, allowing them to accumulate numbers far exceeding well-mixed predictions. As turnover increases, this protective mechanism breaks down through two related processes. First, frequent death events render the frontier more diffuse and reduce cell density in the colony interior, slowing effective expansion velocity and allowing wild-type lineages more opportunities to overtake disadvantageous mutant sectors through lateral invasion. Second, high turnover increases the time (measured in generations) required to reach target size N , providing extended opportunity for disadvantageous mutant sectors to be eliminated by boundary fluctuations and selective pressure. The combination produces a crossover: at low turnover, frontier-driven sector formation dominates and elevates mutant burden; at high turnover, sector dissolution and mixing dominate, reducing burden toward well-mixed expectations. Importantly, spatial models exhibit burden reduction even when | s 1 | ≪ 1, | s 2 | ≪ 1, that is, under a wider sets parameters compared to the non-spatial models, increasing the applicability of the mechanism to a wider variety of biological systems. These findings have direct implications for interpreting the ability of growing cell populations to overcome selection barriers or to survive in the face of changed and adverse conditions. This is described generally by the concept of evolutionary rescue [ 2 ], which investigates how populations escape extinctions during adverse conditions through evolution. The mutant burden at the time at which the environment changes is a crucial component of this, and we have shown that the simple heuristic “more divisions equals more mutations” requires substantial qualification: the relationship between cellular turnover and mutational burden depends on the nature of the mutation (difference in division vs. death rates), how demographic fluctuations propagate (jackpot events), and whether populations expand spatially (frontier dynamics). Our results are especially relevant for understanding resistance evolution, since drug resistance frequently carries a fitness cost, both in bacterial and in cancer cells [ 24 , 25 ], and our reported complex relationship between mutant burden and turnover applies to deleterious mutations. In the context of cancer, our results allow us to interpret the contradictory relationships that have been reported between turnover and outcome. While some studies have shown that higher turnover correlates with poor outcomes [ 31 , 32 , 33 , 34 ], other studies have demonstrated the opposite [ 9 , 10 ]. Our analysis has shown that either outcome is possible, depending on the parameter regions and particulars about the population structure. Prognosis and survival in cancer likely correlate with the evolutionary potential of the tumors (i.e. the mutant burden that accumulates during growth), and our analysis has shown that both low turnover or high turnover, or even an intermediate turnover can maximize the mutant burden. The dynamics analyzed here have further implications for adaptive cancer therapy approaches, where treatment phases alternate with “off” phases, during which the tumor cell population is allowed to expand to a certain extent in an attempt to contain drug-resistant mutants at low levels [ 35 ]. The effect of cellular turnover on adaptive therapy approaches has been investigated in a different context [ 36 ], and the results reported here can further inform such investigations. In summary, incorporating stochasticity and spatial structure fundamentally alters the classical picture of mutation accumulation. While deterministic models predict monotonic increases in mutant burden with turnover, realistic stochastic and spatial effects may introduce non-monotonicity in the relationship between mutant burden and turnover, and the number of mutants in the population can decline with increased turnover. The interplay between turnover, stochastic survival, and spatial effects determine how an expanding colony accumulates mutant burden. 4 Methods 4.1 A stochastic, well-mixed system Within the well-mixed stochastic model, let X ( t ) denote the number of wild types at time t , and Y ( t ) the number of mutants. We use capital letters, in contrast to the deterministic well-mixed system, to emphasize the stochasticity and discreteness of these variables. The stochastic system evolves via five distinct types of reactions. A wild-type individual reproduces at a rate proportional to r w . Each reproduction event yields either a wild-type offspring, with probability 1 − µ , or a mutant offspring, with probability µ . Wild-type individuals die at a rate proportional to d w . Mutants reproduce at a rate proportional to r m and die at a rate proportional to d m . The dynamics can be summarized by the following five reactions: Equivalently, the dynamics of the stochastic variables X ( t ) and Y ( t ) can be written as We simulate the stochastic dynamics starting from the initial condition ( X (0), Y (0)) = (1, 0). The system admits an absorbing state at (0, 0), which we exclude from our analysis. Thus, all results for the stochastic well-mixed system are obtained from trajectories that terminate upon first hitting the target size X + Y = N . Specifically, for each independent realization, the number of mutants M N at the hitting time is recorded, and the ensemble-averaged mutant burden ⟨ ⟩ is computed. 4.2 A stochastic, spatial system To simulate the stochastic dynamics of the spatial colony, we implemented a discrete event-based algorithm on a two-dimensional lattice, where each lattice site can either be empty, occupied by a wild-type cell, or occupied by a mutant cell. The lattice has size L × L , large enough to ensure that the boundary of the expanding population never reaches the boundary of the lattice, and initially only the central site is occupied by a single wild-type cell. At each update, a randomly chosen occupied site ( i, j ) is selected from the list of all occupied positions. If the selected cell is wild type, it dies with probability d w or attempts to divide with probability r w ; if it divides, one of the eight neighboring sites (Moore neighborhood) is chosen uniformly at random. If the target site is empty, the offspring is placed there; otherwise, the division is aborted. Mutation occurs only during the division of a wild-type cell: With probability µ the offspring becomes mutant, and with probability 1 − µ it remains wild type, corresponding to the reactions If the selected cell is mutant, it dies with probability d m = d w (1 − s 2 ) or divides with probability r m = r w (1 + s 1 ), producing a mutant offspring when an empty neighboring site is available and chosen uniformly and randomly to place the offspring, according to Each event updates the grid and the population counts ( X, Y ), where X and Y denote the numbers of wild-type and mutant cells, respectively. The simulation proceeds asynchronously, with on average one update attempt per occupied site per iteration, until a stopping condition is reached. 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