Tuning environmental timescales to evolve and maintain generalists

TL;DR

This paper demonstrates that intermediate-timescale environmental changes can effectively promote the evolution and maintenance of generalist genotypes in populations, offering new insights into adaptive dynamics.

Contribution

It reveals that tuning environmental change timescales can reliably evolve generalists, providing design principles for dynamic fitness landscapes.

Findings

Intermediate environmental change timescales enhance generalist evolution.

Dynamic environments with increasing frequency further increase generalist yield.

Changing environments on the right timescale funnels populations to otherwise inaccessible genotypes.

Abstract

Natural environments can present diverse challenges, but some genotypes remain fit across many environments. Such `generalists' can be hard to evolve, out-competed by specialists fitter in any particular environment. Here, inspired by the search for broadly-neutralising antibodies during B-cell affinity maturation, we demonstrate that environmental changes on an intermediate timescale can reliably evolve generalists, even when faster or slower environmental changes are unable to do so. We find that changing environments on timescales comparable to evolutionary transients in a population enhances the rate of evolving generalists from specialists, without enhancing the reverse process. The yield of generalists is further increased in more complex dynamic environments, such as a `chirp' of increasing frequency. Our work offers design principles for how non-equilibrium fitness `seascapes' can dynamically funnel populations to genotypes unobtainable in static environments.

Equations

$${\tau }_{\text{min}}\sim \frac{1}{\mu }{C}_{\text{init}}(L_c,L)<{\tau }_{\text{epoch}}<{\tau }_{\text{max}}\sim \frac{1}{\mu }D(L_c,L)\log N$$

$${\tau }_{\text{min}}\sim d_{12}/\mu <{\tau }_{\text{epoch}}<{\tau }_{\text{max}}\sim (1/\mu )e^{\Delta F_{g}N}$$

$$⟨F^{(1)}|F^{(2)}{⟩}_s\equiv \frac{c_{1,2}}{\sqrt{c_{1,1}{c}_{2,2}}}$$

Full text

00footnotetext: Correspondence: shenshen@physics.ucla.edu, amurugan@uchicago.edu

Evolutionary outcomes are driven by environmental pressures, but environments are rarely staticLevins1968-qz . In a changing environment, some genotypes – termed generalists – maintain a uniformly high fitness over time, even if they are not globally fit at any particular instant. A striking example is that of broadly-neutralizing antibodies (bnAbs) against HIV and other viruses – these antibodies maintain potency against the large diversity of viral strains that may arise in an infected individual over time Burton2004-jw ; Burton2012-zd ; Wu2010-xh . It is desirable for the immune system to select for generalist antibodies during B-cell affinity maturation, a rapid evolutionary processCobey_Sarah2015-tb , but generalists are often out-competed by specialists that only bind particular viral strains.

Recent work has suggested that sequential vaccination with different viral antigens, rather than a single cocktail of those antigens, can better select for generalist antibodies during affinity maturation Pissani2012-to ; Malherbe2011-cg ; Wang2017-tx ; Wang2015-jg . This result is consistent with the broader idea that a time-varying environment can drive evolution out of equilibrium and into genotypes unevolvable in static environments Mustonen2010-kf ; Mustonen2009-fu ; Arndt2004-yq ; Kussell2014-mg ; Goldenfeld2011-pm . However, the broader principles underlying generalist selection by dynamic environments remain unknown. In particular, the interplay of environmental and evolutionary timescales and choices of correlated antigens generates a high-dimensional space of possible vaccination protocols. Hence guiding principles are needed to find optimal protocols for evolving generalist genotypes.

Here, we take a phenomenological approach to design dynamic environments that select generalists. We analyze situations in which generalists are entropically disfavoured or isolated by fitness valleys, and thus unevolvable in a static environment. We find that a dynamic environmental protocol can maximize the yield of generalists if the environment changes on the same timescale as the evolutionary transients of the population, i.e., on the timescale for allele frequencies to reach steady state. Consequently, switching antigens before antibody populations have evolved to a steady state can dynamically funnel finite populations from specialists to generalists, even when faster or slower switching is unable to do so.

We understand these results in terms of a kinetic asymmetry between generalists and specialists. Environmental dynamics at the right timescale perturbs specialist populations while leaving generalists relatively undisturbed. This asymmetry favours evolution from specialists to generalists without enhancing the time-reversed process. In contrast, faster or slower environmental dynamics may be cast into effective static fitness landscapes Cvijovic2015-xy , and are thus unable to maintain a strong kinetic asymmetry between specialists and generalists. In this sense, the intermediate cycling mechanism studied here exploits a truly non-equilibrium evolutionary ‘seascape’ Kussell2014-mg ; Mustonen2009-fu with no static analog.

Our framework proposes novel protocols for evolving generalists, such as a ‘chirp’ where the environment is cycled at an increasing frequency, and predicts optimal correlations between antigens to be used. Since we use a sufficiently abstracted model of B-cell affinity maturation, our analysis might be adapted for other temporal evolution protocols, e.g., to avoid antibiotic resistanceToprak2011-ae ; Marrec2018-vv ; De_Jong2018-ao and for cancer treatments Gatenby2009-hp ; Katouli2011-uk .

Numerous works have studied evolution in time-varying environments, including in the context of evolving generalists Kassen2002-co ; Desponds2016-ko ; Uecker2011-kn ; Hemery2015-hv ; Kashtan2005-zp ; Lipson2002-un ; Xue2016-al ; Raman2016-bj . Relatively fewer worksCvijovic2015-xy ; Mustonen2008-im ; Kussell2006-ch ; Mayer2017-gl have analyzed the case of intermediate timescales where the environment changes before populations reach steady state. In this broader sense, our work is a step towards a theory of evolution in time-varying environments with no separation of timescale between the evolutionary response of populations and environmental changes.

Results

We present our results in two broad models of how specialists and generalists can be distributed in sequence space, paralleling different assumptions about antigen-antibody binding. In both cases, we model populations (e.g., the population of B-cells across all germinal centers in an organism). We explain our results in terms of the rate at which a population of specialists evolves generalists in time-varying environments relative to the rate of the time-reversed processes from generalists to specialists.

.1 Entropically disfavored generalists

A basic difficulty in evolving generalists is that generalists are often far fewer in number than specialists. This is schematically shown in Fig. 2a, where specialists in each environment form a connected set of genotypes of similar fitness. The relatively few generalists, found at the intersection of such sets, can easily mutate into the more numerous specialists in any fixed environment.

We study the problem quantitatively in a simplified molecular model of antigen-antibody binding, as used for affinity maturation against HIV antigens. Antibodies bind to a single epitope, partially conserved across antigens $\eta=1$, $2$. A (binary) antibody sequence $\mathbf{x}$ binds to an epitope sequence $\mathbf{h}^{\eta}$ with an affinity given by an additive sum-of-sites model: $\mathbf{x}\cdot\mathbf{h}^{\eta}$. Antibodies that bind above a threshold $T$ are assigned fitness $s(\epsilon-1)>0$, while those that bind weaker have fitness $-s<0$. We take $1<\epsilon<2$, such that the average fitness of an antibody across antigens is negative.

Since the epitope is relatively but not entirely, conserved across antigens, $\mathbf{h}^{\eta}$ for different antigens are assumed to share a conserved region of length $L_{c}=12$ but have a variable region of length $L_{v}=7$ Wang2015-jg (see SI for other choices). While based on a simple model of molecular binding, our results below apply broadly to the phenomenological description of specialists as connected islands of relatively uniform fitness, with no fitness barriers separating the generalists.

We simulate a finite population ($N\sim 500$) of antibodies in an environment that switches between antigens 1 and 2 on a timescale $\tau_{\text{epoch}}$ using a birth-death model (see SI), working in the limit of frequent mutations ($\mu N>1$). Initializing a monoclonal population in a random specialist state for antigen $\eta=1$, we monitor the fraction of generalists in the populations at late times (Fig. 2d), systematically varying the timescale of switching $\tau_{\text{epoch}}$. Averaging over many simulation, we find that neither fast nor slow cycling is able to reliably elicit generalists in the population; however, an intermediate timescale of switching is able to do so (Fig.2b).

We sought to understand the origin of this non-monotonic behaviour by examining population dynamics in the limits of fast and slow cycling. For fast cycling, (i.e., small $\tau_{\text{epoch}}$), the initial specialist population is repeatedly confronted with an antigen it cannot bind to. Without enough time to mutate into a generalist, purifying selection drives the population to extinction (Fig. 2d,i). Consequently, the fraction of trials in which specialists evolve into generalists, $\chi_{\text{s}\to\text{g}}$, is low (Fig. 2c).

In fact, in this limit the dynamics of the population are effectively described by a static, average landscape, where the specialist has fitness $s(\epsilon-2)<0$. In this regime, we find that purifying selection drives the population to extinction when $s>\mu\log N$; see SI for derivation and discussion of alternative cases.

On the other hand, for very slow cycling (large $\tau_{\text{epoch}}$), any generalists that arise have enough time to specialize again by mutational drift (Fig. 2d,iii). As a result, the fraction of an initially-generalist population that stay generalists over an environmental cycle, $\chi_{\text{g}\to\text{g}}$, falls with $\tau_{\text{epoch}}$, as seen in Fig.2c.

Consequently, we find that intermediate timescale cycling strikes a balance: providing enough time to for specialists to evolve into generalists (high $\chi_{\text{s}\to\text{g}}$), but not enough time for generalists to switch back to specialists again (high $\chi_{\text{g}\to\text{g}}$). In the SI, we determine this regime to be,

[TABLE]

where $C_{\text{init}}$ and $D$ are combinatorial factors that account for the mutational distance of the initial naive repertoire from generalists and the number of generalist genotypes, respectively; see SI.

Note that if population sizes are small and sequence space is large, we find $\tau_{\text{min}}>\tau_{\text{max}}$, i.e., it takes longer for generalists to evolve from specialists than to specialize again. In this regime, the entropic bias in sequence space driving generalists to specialists is large; even fixed frequency cycling may not produce generalists.

Hence, we propose a new dynamic protocol - a ‘chirp’ - that can alleviate this tension between evolving generalists from specialists ($\chi_{\text{s}\to\text{g}}$), which requires slower cycling, and the ability to maintain a population of generalists ($\chi_{\text{g}\to\text{g}}$), which requires faster cycling. A chirp, shown in Fig. 3, starts with slow cycling and increases the cycling frequency over time. Such highly dynamic ‘chirp’ protocols outperform any fixed frequency cycling protocol; see Fig. 3c.

.2 Generalists isolated by fitness valleys

We now consider a more general case where fitness valleys separate viable genotypes, and specialists and generalists form disconnected sets in sequence space. Such models have been used to describe antibodies for influenza and malaria Munoz2005-am ; Chaudhury2014-tq ; Deem2003-yu ; Childs_Lauren_M2015-zn , as well as describing RNA molecular fitness landscapes Pressman2019-at ; Blanco2019-to . In the affinity maturation context, such a model naturally arises if each antigen has multiple epitopes, with one epitope shared across antigens Childs_Lauren_M2015-zn . Epistasis in antigen-antibody binding interactions, quantified recentlyAdams2019-eb , can also give rise to such disconnected sets.

Here, we take a phenomenological approach that is agnostic to molecular details. Exploiting Hopfield’s Hopfield1982-fb (or more generally, Gardner’s Gardner1987-op ) construction, we construct fitness landscapes for each antigen with fitness islands around sequences corresponding to each epitope. In particular, consider $P$ epitopes on each antigen $\eta=1$, $2$, that bind to antibody sequences $\mathbf{h}^{\eta}_{\alpha}$ ($\alpha=1,\ldots P$). The fitness of an antibody with sequence $\mathbf{x}$ confronted by antigen $\eta$ is chosen to be $F^{\eta}\propto\sum_{\alpha}(\mathbf{x}\cdot\mathbf{h}^{\eta}_{\alpha})^{2}$. This minimal construction produces fitness landscape with peaks at the specified epitopes $\mathbf{h}^{\eta}_{\alpha}$, provided $P$ is sufficiently small compared to sequence length $L$ Amit1985-as .

By making different choices for the epitopes $\mathbf{h}^{\eta}_{\alpha}$, we may construct fitness landscapes with arbitrary amounts of correlation between them. We begin by studying the minimal case where $1$ epitope is shared between the two antigens, $\mathbf{h}^{1}_{1}=\mathbf{h}^{2}_{1}$, with the others epitopes being uncorrelated. Later, we relax this assumption. For our theoretical analysis, we assume selection is strong and beneficial mutations are rapidly fixed, $sN\gg\mu N$, $sN\gg 1$; hence fitness valleys between islands play a significant role.

We simulate a finite population of antibodies evolving via Moran dynamics. Initialising a monoclonal population at a specialist, we once again carried out simulations at different antigen switching times, $\tau_{\text{epoch}}$, and quantified the fraction of generalists in the population at long times. As seen in Fig.4b, an intermediate timescale of switching elicits generalists in the population. This is reminiscent of the entropic model above, but for different underlying reasons.

Here, fast switching fails to produce generalists because populations stay confined to their initial positionWeissman2009-uj (Fig.4d). In fact, as $\tau_{\text{epoch}}\to 0$, the fitness landscape is effectively static, $F\sim F^{(1)}+F^{(2)}$, corresponding to the time-averaged fitness. This static landscape $F$ generally inherits the attractors of $F^{(1)}$ and $F^{(2)}$, as well as potentially numerous additional ‘spurious’ attractors Amit1985-as . Consequently, the population remains segregated away from the generalist genotypes by valleys of low-fitness, and generalist acquisition, $\chi_{\text{s}\to\text{g}}$, is small. In practice, such populations stuck in a specialist genotype for extended time can go extinct in the presence of multiple antigens Wang2015-jg .

In contrast, at slower switching times, evolution in each environment can shift the population away from its initial position in the prior environment (Fig.4d). As shown in the SI, this requires at least time $\tau_{\text{min}}\sim d_{12}/\mu$, where $d_{12}$ is the typical mutational distance separating specialists across environments. Consequently, the population is forced to continually traverse genotype space. This continual evolution is by necessity stochastic (Fig.4f), contingent on the random order of mutations that arise, as well as on any potential population variance. This cycling-induced mobility, augmented by stochasticity, allows the population to widely explore genotype space and find the generalist, and hence $\chi_{\text{s}\to\text{g}}$ rises (Fig.4d).

Importantly, upon evolving into generalists, environmental cycling no longer disturbs the population, as the fitness of generalist sequences does not appreciably change over time. Thus cycling breaks the symmetry between specialists and generalists and enhances $\chi_{\text{s}\to\text{g}}$ without enhancing $\chi_{\text{g}\to\text{s}}$. Intuitively, intermediate cycling selectively ‘warms up’ (i.e., increases stochasticity) specialist parts of sequence space, naturally leading the population to collect in ‘cooler’ generalist sequences.

Cycling significantly slower than $\tau_{\text{min}}$ is counterproductive. The cycling-induced leaks from specialists to generalists only occur due to environmental switches; hence unnecessarily long $\tau_{\text{epoch}}$ only adds dead time with no additional population divergence.

In the meantime, as shown in the SI, escape from generalists to specialists becomes significant on timescales of $(1/\mu)e^{\Delta F_{g}N}$ where $\Delta F_{g}$ is the fitness of the generalist relative to the fitness valley separating it from specialists; $N$ is the population size. See Jain2007-tl ; Van_Nimwegen2000-qh ; Weissman2009-uj for calculations of valley crossing rates in other parameter regimes. These considerations limit intermediate timescales favorable for evolving generalists:

[TABLE]

Correlation between specialists

The effectiveness of this theoretical cycling mechanism depends on the correlation between specialists of $F^{(1)}$ and $F^{(2)}$ Wang2019-al . For example, if specialists of $F^{(1)}$ and $F^{(2)}$ are similar or are carefully arranged as to be well within each other’s attractors, cycling will primarily cycle the population between specialists with minimal divergence into generalists. As shown in the SI, we can quantify relevant correlations by

[TABLE]

where $c_{\eta,\gamma}=\frac{1}{LP}\sum_{\alpha,\beta\neq 1}\mathbf{h}^{\eta}_{\alpha}\cdot\mathbf{h}^{\gamma}_{\beta}$ excludes the generalist pattern $\mathbf{h}^{1}_{1}=\mathbf{h}^{2}_{1}$. When $\langle F^{(1)}|F^{(2)}\rangle_{s}$ is high, cycling-induced variance is low; see Fig.4g. Consequently, the small asymmetry between $\chi_{\text{s}\to\text{g}}$ and $\chi_{\text{g}\to\text{s}}$ created by a single environmental cycle must be compounded by cycling multiple times; however, in practice, other considerations might limit the number of such cycles. Hence, our proposal requires the specialists of $F^{(1)}$ and $F^{(2)}$ to be sufficiently uncorrelated (low $\langle F^{(1)}|F^{(2)}\rangle_{s}$).

Is cycling a practical strategy given physiological parameters for population dynamics and the correlations between specialists antibodies found during HIV infection? We analyzed specialist and generalist antibody sequences collected from an HIV patient Liao2013-xu ; Gao2014-uy ; Bonsignori2016-du ; see Fig.5a. We constructed landscapes $F^{(1)},F^{(2)}$ with fitness peaks at these observed specialist and generalist sequences following Gardner’s construction Gardner1987-op ; see SI for details.

Simulations of cycling environments $F^{(1)},F^{(2)}$ constructed from the above sequence data evolved generalist antibodies, while simultaneous presentation of both antigens, a practical alternative to fast cyclingWang2015-jg , fails to produce such generalists; see Fig.5b. We then artificially shuffled antigen labels for antibodies, so that CH105 was considered a Ag2 specialist and CH186, an Ag1 specialist and reconstructed $F^{(1)},F^{(2)}$. This artificial shuffling significantly increased the correlation $\langle F^{(1)}|F^{(2)}\rangle_{s}=0.78$ compared to the real data ($\langle F^{(1)}|F^{(2)}\rangle_{s}=0.43$). Cycling is now less effective in evolving generalists. We conclude that the low correlation between specialists in the real data is crucial for time-varying selection of generalists.

While our model here did not explicitly account for extinction, simultaneous presentation or fast cycling can cause most specialist B-cells to perish, especially if many distinct antigens are use (see SI). In this more realistic case, ‘chirped’ cycling at increasing frequency as in Fig.3 will provide the dual advantage of evolving generalists from specialists through slow cycling and then enhance generalist yield by removing specialists through fast cycling.

Discussion

We have shown that environmental changes on intermediate timescales can dynamically funnel populations from specialists to generalists. Our quantitative framework suggests broad new classes of experimental protocols such as ‘chirped cycling’ that further enhance the evolution of generalists.

The relevant intermediate timescale here is that of evolutionary transients in a population - the environment must change slow enough for significant changes to accumulate but fast enough to prevent the population from settling to a steady state distribution. This intermediate regime creates a highly dynamic fitness ‘seascape’ with no effective static descriptionMustonen2009-fu . This dynamic regime has been relatively less exploredCvijovic2015-xy than limits where the environment changes much faster or slower than evolutionary transients and which can be understood using effective static environments.

The simple models studied here ignore many ingredients present in B-cell affinity maturation and other evolutionary processes in the natural world. For instance, affinity maturation starts from a specific naive antibody repertoireElhanati_Yuval2015-xh , population response timescales can vary widely Hallatschek2018-dq ; our results require an ensemble of lineages to participateWang2015-jg and ignores clonal interference Park2007-ke .

Nonetheless, our analysis has broad applicability, since it relies only on a simple phenomenological characterization of how specialist and generalist genotypes are organized in sequence space. Further, our results are fundamentally linked to the fact that generalists experience less time variation of fitness than specialists, leading, e.g., to higher stochasticity and mobility for the specialist parts of sequence space but not for the generalists. In this sense, the dynamic strategies presented here represent a broader class of non-equilibrium evolutionary strategiesMustonen2009-fu ; Kussell2014-mg that can enhance the rate of transitions from specialists to generalists without enhancing the time-reversed processes.

Dynamic protocols have been investigated recently in other evolutionary contexts, e.g., in antibiotic resistance, where correlations in response to different antibiotics have been exploited to maximize cross-vulnerabilityMarrec2018-vv ; Munck2014-vr ; Chait2007-bb ; Nichol2015-zg . While such cross-vulnerabilities have been primarily studied in the slow switching limit, switching antibiotics after a partial evolutionary response like that explored here might open a larger space of strategies.

While we have discussed dynamic environments as a prescriptive mechanism, natural environments are also dynamic Levins1968-qz .For example, co-evolution of pathogens Papkou2019-ur ; Nourmohammad2016-jm , movement through spatially heterogeneous environments Zhang2011-wa and ecological changes Pelletier_F2009-zg ; Roxburgh2004-wl ) can naturally create the intermediate timescale variations discussed here. The quantitative principles developed here suggest new experiments to both understand and exploit this understudied regime of evolution with no separation of timescales between perturbation and response.

Acknowledgements

We thank Sarah Cobey, Aaron Dinner, Allan Drummond, Muhittin Mungan, Sidney Nagel, Stephanie Palmer, David Pincus, Rama Ranganathan, Olivier Rivoire, Thomas Witten for useful discussions. VS thanks the NIH for support through NIBIB T32 EB009412. KBH and AM thank the James S. McDonnell Foundation and the Simons Foundation respectively for support. SW is grateful to funding from UCLA.