Effective models and the search for quantitative principles in microbial evolution

Abstract

Microbes evolve rapidly. Yet they do so in idiosyncratic ways, which depend on the specific mutations that are beneficial or deleterious in a given situation. At the same time, some population-level patterns of adaptation are strikingly similar across different microbial systems, suggesting that there may also be simple, quantitative principles that unite these diverse scenarios. We review the search for simple principles in microbial evolution, ranging from the biophysical level to emergent evolutionary dynamics. A key theme has been the use of effective models, which coarse-grain over molecular and cellular details to obtain a simpler description in terms of a few effective parameters. Collectively, these theoretical approaches provide a set of quantitative principles that facilitate understanding, prediction, and potentially control of evolutionary phenomena, though formidable challenges remain due to the ecological complexity of natural populations.

With their small size and rapid generation times, microbes have a unique ability to evolve on human-relevant timescales. Recent years have seen an explosion of data, first from laboratory experiments, but increasingly from pathogens and other natural populations, that document these short-term evolutionary changes at ever increasing breadth and precision [1]. This glut of quantitative data poses a challenge for theorists and experimentalists alike: are there simple quantitative principles of microbial evolution that can be extracted from these measurements? And if so, can we leverage these principles to better predict or control evolution in the future? We highlight recent progress in this direction, and we comment on the further challenges imposed by the ecological complexity of natural populations.

The simplest principle of microbial evolution: microbes adapt rapidly

One of the most robust empirical findings in microbial evolution is that adaptive evolution is common. Even in the simplest laboratory settings, microbes will readily uncover mutations that can displace their parent strain. A canonical example is Lenski’s long-term evolution experiment in E. coli (the LTEE [2]), whose 12 replicate populations are still accumulating adaptive mutations after 70,000 generations of evolution [3, 4]. The fitness benefits of each mutation are often small by physiological standards (i.e., s ≲ 1%), and may not be apparent in spot assays or other traditional growth rate measurements. But compounded over time, these small benefits can still drive mutants to high frequency over the hundreds of thousands of generations that take place in a year.

The high rates of adaptive evolution can also be traced to the large number of cells in a typical microbial population. Even with a small target size for weakly beneficial mutations, a population of a few hundred million cells will produce multiple adaptive variants in each generation (NU_b ≳ 1). Natural microbial populations like the gut microbiome can grow to be even larger, with an estimated ~ 10⁹ mutations produced in a single day [5].

Together, these order-of-magnitude estimates and empirical examples suggest a very simple principle of microbial evolution: microbes will readily and rapidly adapt to a variety of environmental conditions. But beyond this simple rule of thumb, it is far more di cult to predict the phenotypic or genetic changes that will take place, even in relatively simple settings like the LTEE. Studies have shown that evolution can often be repeatable, at least at a statistical level, when populations are exposed to the same conditions [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 3, 4]. But it is much more challenging to predict the range of responses ahead of time, or to extrapolate the predictions of one experiment to slightly different conditions.

At some level, these difficulties are not surprising, since ab initio predictions of evolution would require a detailed understanding of the mapping between genotype, phenotype, and fitness. To make progress, microbial evolutionists have focused on two broad approaches, which we describe in more detail below. The first is a reductionist (or bottom-up) approach that builds on familiar concepts from biophysics and systems biology. The second is a top-down approach that works backward from the emergent dynamical processes of population genetics.

Biophysical models of evolution from the bottom up

A common strategy for dealing with the complexity of evolution is to focus on a narrowly defined trait or pathway, where biophysical models can be used to populate the mapping between genotype, phenotype, and fitness. This approach is often based on metabolic control analysis, or its close relatives, which relate microbial fitness in chemostat culture to the flux through a growth-limiting linear pathway (Fig. 1a) [17, 18]. Fitness can then be connected to the expression and activity of the relevant enzymes using standard biophysical models, which can include varying degrees of molecular detail [18, 19, 20, 21, 22]. Figure 1 shows a typical example of this approach for the lactose pathway in E. coli.

Figure 1: The bottom up approach to microbial evolution, illustrated using the lactose pathway in E. coli.

(a) A model of growth in lactose-limited conditions connects fitness (i.e., biomass production) to flux through the lac pathway. (b) This model can be used to predict fitness as a function of lacZ expression [20]. The circles illustrate how a mutation with a constant effect on expression can be both beneficial or deleterious, depending on the genetic background [21]. (c) Deep mutational scanning and (d) thermodynamic models of transcription provide a map between the DNA sequence of the lac promoter and lacZ expression [23]. In (c), the colored grid denotes the inferred change in binding energy of the CRP activator (green), RNA polymerase (blue), and the Lac repressor (red) when each of the indicated bases is mutated to the corresponding value. Together, the different biophysical layers in (a-d) enable the prediction of an organismal fitness effect s for each mutation in the lac promoter. Panels (c) and (d) are reproduced with permission from Ref. [23]. ©IOP Publishing. All Rights Reserved.

A major advantage of the reductionist approach is its conceptual simplicity. Previously puzzling evolutionary patterns can be explained using existing biophysical models, where the basic mechanisms are often well understood. For example, an experimental study inM. extorquens showed that mutations that produce a constant change in gene expression can be beneficial in one genetic background but deleterious in other [21]. The authors showed that this complex pattern of epistasis could be explained simply by overshooting the optimal expression level predicted by metabolic control analysis [20] (Fig. 1b). Another recent study showed that the order of the mutations that accumulated in the influenza nucleoprotein in last 40 years was strongly constrained by the “fitness cli ” predicted from the thermodynamics of protein stability [24].

In addition to conceptual insights, reductionist approaches offer clear practical advantages for obtaining quantitative predictions. In simple cases, computational methods can be used to predict binding energies [25] and protein stability [26] from the underlying DNA sequences, though in general, these molecular traits must still be determined experimentally. Recent hybrid approaches, which combine high-throughput functional assays of mutagenized libraries with statistical models of the DNA-phenotype map [27, 28, 29] make it increasingly easy to perform such measurements. Variations of this technique have started to provide insight into the sequence landscape of promoter function (Fig. 1c,d) [23], drug resistance activity [30], and antibody binding [31]. Even without a specific fitness model, these laboratory measurements can provide model-free predictions for the mutations that are tolerated in natural populations [23, 32].

By definition, the bottom-up approach is most easily applied to small functional units. Yet when left to their own devices, laboratory populations often acquire mutations in a number of different genes, which influence multiple biomolecular processes within a cell [33, 11, 3, 13, 34, 35]. In these cases, it is much more challenging to predict the targets of selection from biophysical principles, since they depend on organism-wide networks of metabolic and regulatory interactions. Genome-scale models like flux balance analysis [36], which combine networks inferred from omics data with optimality principles for choosing unconstrained parameters, represent one potential solution to this problem, along with recent experimental advances in assaying fitness in large gene knockout libraries [37, 38, 39]. So far, however, these methods have been most successful in predicting extreme phenotypes (e.g. nutrient auxotrophies [40] or synthetic lethality [38]) or in predicting adaptation to severe metabolic limitations [41]. It remains to be seen whether these methods can scale down to the moderate fitness benefits (s ~ 1%) observed in more permissive conditions.

Insights into evolutionary dynamics from effective models

Despite the successes of the bottom up approach, it remains far from being able to predict evolution even in simple laboratory experiments like the LTEE, let alone the more complex environments encountered in nature. Nevertheless, the statistical regularities observed in laboratory evolution experiments have inspired a second body of research, which seeks to characterize principles of microbial evolution that emerge at higher levels of description.

Much of this work has focused on questions in evolutionary dynamics. That is, given a known genotype-to-fitness map, how does evolution determine which combinations of mutations ultimately reach observable frequencies, where they could be detected in sequencing or phenotyping assays. These dynamical questions are implicit in the “bottom-up” approach as well, though they are often studied only after the genotype-to-fitness map has been determined. By switching the order of emphasis, the “top-down” approach to microbial evolution tries to make progress by asking a slightly different set of questions: if we can understand which aspects of the genotype-to-phenotype map influence evolutionary dynamics, it might be possible to infer these properties directly from population-level measurements (e.g. genetic diversity, rates of change, etc).

Given its inherent complexity, the study of evolutionary dynamics has been greatly aided by the use of effective models. These are simple mathematical objects that capture the limiting behavior of a much wider range of models (known as a universality class) which may differ in their underlying details. Since the members of the universality class behave similarly, mathematical statements about effective models can translate into general quantitative principles, which can apply even in messy biological settings.

One of the best-known effective models in evolutionary biology is the Wright-Fisher process (also known as the single-locus diffusion model), which describes the competition between two genetic variants in a well-mixed population [45] (Fig. 2). In this model, the frequency of “mutant” individuals is controlled by just two numbers: an effective fitness advantage s_e that describes the deterministic effect of selection, and an effective population size N_e that controls the strength of random genetic drift (Fig. 2, inset). Powerful limit theorems in population genetics have shown that this model captures the long-time behavior of a large class of models in the limit that population sizes are large and the relevant frequency changes occur over many generations [46, 45]. Of particular relevance to microbial evolution, this domain of attraction includes chemostat growth [47], batch culture [43], and even some scenarios with rapid temporal variation [42] or spatial structure with frequent mixing [48] (Fig. 2a).

Figure 2: An example of an effective model of evolutionary dynamics: the Wright-Fisher diffusion process.

(a) Different microscopic dynamics that converge to the Wright-Fisher diffusion (inset) on long timescales. From left to right: microscopic Wright-Fisher diffusion, an environment that rapidly fluctuates between two states [42], serial passaging [43], and spatially segregated passaging with frequent mixing [44]. (b) Emergent dynamics of the mutation frequency f at the population level, described by the effective parameters s_e and N_e. Genetic drift is dominant when N_es_ef(1 − f) « 1, so mutations with N_es_ef « 1will have a high probability of drifting to extinction (grey line). Once N_es_ef » 1, natural selection will deterministically drive the mutation to fixation. If $s\ll s_{\text{min}}\sim N_e^{-1}$, genetic drift will dominate over the entire frequency range, and the mutation will be effectively neutral.

In simple cases like batch culture, N_e and s_e can be related to the census population size and growth rate differences between the two strains [43]. But in most practical settings, these are simply effective parameters that must be self-consistently fit from the data. However, even in these cases, the Wright-Fisher model already reveals a useful general principle relating the magnitudes of N_e and s_e. In particular, the model predicts that natural selection will be unable to distinguish between mutants with a fitness difference much less than the drift barrier, s_min ~ 1/N_e, where fluctuations from genetic drift start to dominate. This limiton the efficiency of selection, which arises purely from population genetic considerations, has been conjectured to set bounds on the intrinsic mutation rates of different microorganisms [49] and the stability of different proteins [50].

When reasoning with effective models, it is important to apply them only in situations where their limiting behavior is expected to apply. For example, convergence to the Wright-Fisher model will eventually break down on short timescales (Fig. 2a), when life-history details and other deviations from the model start to become important. With the increasing density of temporal sampling [51], and the low frequencies (and recent times) probed by large cohorts [52], these deviations are more likely to be encountered in real data. Other limits to convergence may be more difficult to anticipate. In the case of the Wright-Fisher model, seemingly innocuous changes in assumptions or parameter values can sometimes drive convergence to an entirely different class of effective models, which cannot be captured by adjusting N_e or s_e. Classic examples include temporal variation in N_e or s_e, which become relevant when the variation is slow compared to drift or selection [53, 42]. Another relevant example is spatial structure, which can be important in crowded [54, 55, 56, 57] or otherwise poorly mixed populations [58, 59].

Some of these complications can be avoided in carefully controlled experiments like the LTEE. However, other deviations from the single-locus diffusion model arise due to genetic linkage, which cannot be switched off so easily. By definition, the single-locus model describes the competition between a single mutant type and its ancestor. Yet as we saw above, large microbial populations often harbor many adaptive variants at different locations (or sites) along the genome. Since these different sites are inherited together on the same physical genome (Fig. 3a), the fate of a mutation at a particular site will depend on all of the other genetic variants present in the population, a phenomenon is known as clonal interference [60] or interference selection [61].

Figure 3: Evolutionary dynamics in rapidly adapting populations with many linked mutations.

(a) Many limiting behaviors are captured by the staircase model of adaptation on an asexual genome. A target mutation (s_e, blue) resides on a genetic background that can acquire further beneficial mutations (s₀, red) at a large number of other sites (white) at total rate U₀. Mutations on the same genetic background are inherited together and their fates become coupled. (b) Macroscopic epistasis influences long-term evolutionary dynamics by changing the effective parameters. Left: an initial distribution of fitness effects, U_ᑭ (s), and the corresponding effective parameters s₀ and U₀ [69]. After several substitutions, epistasis leads to a shifted DFE (right) and a new set effective parameters. A similar effect occurs during the acquisition of a mutator phenotype, which increases the genome-wide mutation rate by a factor r [83]. (c) Diverse genetic backgrounds can be coarse-grained into a population fitness distribution (green), which translates to higher fitness at a constant rate. Black lines illustrate a stereotypical genealogy constructed from samples at the two timepoints. Rapidly branching internal nodes indicate a high-fitness lineage, which is likely to dominate the population in the future [84, 85].

Early work [62] had suggested that these effects could be absorbed within the single-locus diffusion model by rescaling N_e. However, more recent theoretical analyses have shown that this is true only in the limit of rapid recombination, when correlations between mutations can be neglected [63, 61, 64, 65]. For large microbial populations, by contrast, theory and experiment both suggest that clonal interference is likely to be a generic feature of short-term evolution [66].

At first glance, linkage would seem to make the evolutionary dynamics hopelessly complex. To model the fate of a given mutation, one would need to know the rates and fitness effects of all other mutations across the genome. Fortunately, simpler effective models emerge in the limit that there are many competing mutations. In the simplest cases, these models augment the single-locus diffusion by assuming that an infinite “staircase” of linked beneficial mutations can occur at other loci (Fig. 3a). This requires at least two additional parameters: a typical fitness benefit s₀ for the additional linked mutations, and a total mutation rate U₀ for these mutations to occur. The staircase model is clearly an abstraction, and would be a poor approximation to the genetics of any real population. Yet in terms of evolutionary dynamics, the staircase model has been shown to capture the limiting behavior in a variety of more realistic scenarios, including distributions of fitness effects [67, 68, 69, 70, 71] (Fig. 3b) and moderate amounts of recombination [72, 61, 65, 64]. The main advantage of the staircase model is that it can be characterized analytically in different limits [73, 74, 75, 68, 76, 72, 77, 78, 70, 79, 80], often by recasting the population as a traveling wave in fitness space (Fig. 3b). Together, these theoretical studies have revealed some general principles that apply to microbes evolving under clonal interference.

First, a generalization of the drift-barrier (s_min ~ 1/N_e) still applies, provided that we replace N_e with the coalescence time T_c at a neutral locus (Fig. 3c). This reduces to the familiar drift barrier in the absence of linkage, where T_c ~ N_e. But in the presence ofclonal interference, T_c is more strongly influenced by the rates and fitness effects of linked mutations (genetic draft [81]), and only weakly dependent on N_e. This draft barrier imposes fundamental limits on the ability of microbes to optimize certain functions when faced with other opportunities for adaptation [82, 78]. In the rush to run uphill, many mistakes and missed opportunities can be tolerated, potentially fueling further bouts of adaptation in the future.

Second, recent work has shown that the genealogies of populations under strong clonal interference approach a universal form, which is qualitatively different from the genealogies that arise in the Wright-Fisher model [86, 87, 88, 89]. The new genealogies are distinguished by their skewed branching structure and multiple merger events, which arise when multiple lineages are spawned by the same highly fit ancestor (Fig. 3c). Forward in time, the coarse-grained dynamics of a target mutation can be still mapped to an effective single locus model, which does not explicitly keep track of linked mutations [90, 88, 91, 92]. In this case, however, the slow diffusion of the Wright-Fisher process must be supplemented by sudden jumps in allele frequency [93], which reflect lucky mutational “jackpots” with anomalously high fitness [92].

These universal signals, which arise independently of what trait is being selected for, can be used to identify exceptionally fit lineages that may come to dominate the population in the future [84, 85] (Fig. 3c). A recent study in influenza has demonstrated the utility of this approach, in the context of forecasting which strains are likely to dominate the next flu season [85]. By mapping the observed influenza genealogy to a simplified version of the staircase model, the authors showed that one can make predictions that rival more complex methods that incorporate functional information about the influenza genome [94]. This study highlights the power of combining effective models with highly quantitative evolutionary data, in order to make testable predictions about the future.

Working in the opposite direction, dynamical models have also shed light on the structure of microbial fitness landscapes, complementing the traditional biophysical approaches above. For example, by analyzing an effective model for the emergence of beneficial mutations in barcoded subpopulations [95], a recent study surveyed hundreds of the most important beneficial mutations in the S. cerevesiae genome and mapped them onto key nutrient-responsive pathways [96]. Other studies have investigated how the fitness landscape shifts over time, as different mutations are added to the genetic background [97, 14, 98, 99, 100, 101, 16]. In the context of the staircase model, these changes manifest as differences in the effective parameters U₀ and s₀ that control the overall speed of adaptation (Fig. 3b). An active area of research [97, 102, 103, 101, 104, 105] seeks to understand how these macroscopic measures of epistasis are connected to the microscopic or biophysical epistasis probed by traditional methods [106, 101, 107, 39]. Simpler effective models of physiology [108, 109, 110] may be useful for investigating these connections.

Emergence and evolution of ecology

With the ability to track mutations over longer times, laboratory experiments have started to uncover more fundamental deviations from the simple evolutionary models above. In many cases, mutations are observed to neither fix or go extinct, but instead persist at intermediate frequencies for long periods of time (Fig. 4). This constitutes a primitive form of ecology, in which subsets of the population have evolved to colonize different ecological niches.

Figure 4: Spontaneous ecological diversification during microbial evolution.

(a) A canonical microscopic example of coexistence through crossfeeding. The wildtype strain metabolizes a nutrient (light blue) and excretes a metabolic by-product (dark blue). A mutation occurs that enables growth on the by-product, which allows the mutant and wildtype to stably coexist. Further beneficial mutations can occur in this background, leading to the replacement of the mutant strain while maintaining coexistence with the wildtype. (b) A schematic illustration of the mutation frequencies for the hypothetical scenario in (a). The diversifying mutation rapidly invades, but stabilizes at an intermediate frequency. A second beneficial mutation sweeps through the mutant background, leading to a shift in this ecological equilibrium [111].

Such diversification was once thought to be rare, emerging only under special conditions. However, evidence for coexistence continues to arise in a variety of laboratory settings [112, 113, 114, 115, 116, 117, 118, 119, 120], even in experiments that were designed to avoid this behavior, like the LTEE [113, 121, 4]. This suggests that emergent ecological interactions cannot be easily controlled or switched off in the lab. Moreover, many natural microbial populations are embedded in larger microbial communities, and recent evidence suggests that short-term evolution can occur in these settings as well [122, 5, 123, 124]. This suggests that diversification and community structure may be a generic feature of microbial evolution, similar to the rapid adaptation described above. Will similar simple principles emerge when ecology and evolution are intertwined?

At first glance this seems like a di cult problem, since there are many potential ways for microbes to interact. We can again gain some insight by focusing on simple systems, where the ecological interactions can be understood at a biophysical level. In microbial populations, these interactions often involve the consumption or exchange of metabolites (Fig. 4a). In a few cases, recent multi-strain extensions of flux balance analysis have been shown to predict coexistence from individual metabolic profiles [125, 126, 127], providing an intriguing route for investigating evolutionary perturbations. In an even simpler example in yeast, a recent study fully characterized the ecological landscape of mutants that adhere to the walls of their culture vessel [115]. This enabled the authors to predict when evolution would disrupt coexistence, based on fitness measurements in non-adherent conditions.

To complement these bottom-up approaches, we would also like to know whether there are simpler effective models that describe some generic features of evolving ecosystems. But in contrast to the purely evolutionary case, there is much less consensus about what the relevant effective models will look like. Recent studies have started to explore the range of potential behaviors using computer simulations, in which the interaction parameters in existing ecological models are allowed to mutate in pre-defined aways [128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140]. The results can vary depending on the detailed assumptions of the model. For example, some studies suggest that evolution generally acts to disrupt coexistence [133], while others suggest that evolution promotes coexistence [130, 136]. Without a corresponding set of effective models, it can be di cult to draw connections between these studies, or to understand how they map on to experiments.

Drawing inspiration from the evolutionary studies above, we believe that progress will depend on the development of simple effective models, in which the limiting behavior can be investigated analytically. Understanding these limits is crucial for determining whether two models share the same limiting behavior. Is it reasonable to expect an ecological analogue of the staircase model, which proved so useful for understanding clonal interference? If the previous literature is any guide, it may not be obvious which effective models will turn out to be the most relevant, and a certain amount of theoretical exploration will be required. Simple biophysical systems like Refs. [115, 141] or the abstract trait models of adaptive dynamics [142, 143] may both prove to be useful starting points. In Ref. [111], we investigate another promising candidate that emerges as the generalization of the Wright-Fisher model when there are multiple fitness dimensions.

In addition, theoretical ecology has a long tradition of seeking limiting behaviors that arise in ecosystems with a large number of species or niches [144, 145]. An exciting body of recent work has leveraged this approach, in combination with techniques from statistical physics, to analytically characterize the steady states and dynamical properties of several ecosystem models [146, 147, 148, 149, 150, 151]. Extending this approach to allow for heritable evolutionary changes may provide an additional route for understanding evolution in large microbial communities. Of course, natural communities possess additional factors that may influence their short-term evolution, including their nontrivial spatial organization and the ability to exchange genetic information across species boundaries. Understanding how these complexities map onto simpler effective models is a key challenge for future studies.