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# Eco-evolutionary control of pathogens

Edited by Arup K. Chakraborty, Massachusetts Institute of Technology, Cambridge, MA, and approved June 28, 2020 (received for review November 19, 2019)

## Significance

Vaccinations and therapies targeting evolving pathogens aim to curb the pathogen and to steer it toward a controlled evolutionary state. Control is leveraged against the pathogen’s intrinsic evolutionary forces, which in turn, can drive an escape from control. Here, we analyze a simple model of control, in which a host produces antibodies that bind the pathogen. We show that the leverages of host (or external intervention) and pathogen are often highly imbalanced: an error threshold separates parameter regions of efficient control from regions of compromised control, where the pathogen retains the upper hand. Because control efficiency can be predicted from few measurable fitness parameters, our results establish a proof of principle how control theory can guide interventions against evolving pathogens.

## Abstract

Control can alter the eco-evolutionary dynamics of a target pathogen in two ways, by changing its population size and by directed evolution of new functions. Here, we develop a payoff model of eco-evolutionary control based on strategies of evolution, regulation, and computational forecasting. We apply this model to pathogen control by molecular antibody–antigen binding with a tunable dosage of antibodies. By analytical solution, we obtain optimal dosage protocols and establish a phase diagram with an error threshold delineating parameter regimes of successful and compromised control. The solution identifies few independently measurable fitness parameters that predict the outcome of control. Our analysis shows how optimal control strategies depend on mutation rate and population size of the pathogen, and how monitoring and computational forecasting affect protocols and efficiency of control. We argue that these results carry over to more general systems and are elements of an emerging eco-evolutionary control theory.

Control of human pathogens is a central goal of medicine. Important examples are antimicrobial and antiviral therapies and vaccinations; similarly, cancer therapies aim to control tumor cell populations. Biological hosts, notably the human immune system, face related issues of pathogen control. In most cases, control targets pathogen populations with fast-paced replication and evolution. Its goal is to alter these dynamics: to prevent or elicit an evolutionary process of the pathogen or to curb the pathogen population by reducing its ecological niche. Pathogen control has seen spectacular successes (e.g., in the eradication of smallpox and in HIV combination therapies) (1). However, control is often compromised by escape evolution of the pathogen, highlighting the importance to factor pathogen evolution into control protocols (2, 3). Promising evolutionary avenues include adaptive pathogen control and cancer therapy (4⇓–6), vaccination, drug development and immunotherapy strategies based on evolutionary predictions (7⇓⇓–10), and controlled evolution of immune antibodies (11⇓–13). However, we need quantitative relations between leverage and cost of control in order to generate optimization criteria and protocols that are comparable across systems. These are central elements of an eco-evolutionary control theory.

Because population dynamics and evolution are stochastic processes, any eco-evolutionary control operates on the likelihood of future states. Successful control turns a likely process into an unlikely one (e.g., the evolution of antibiotic resistance) or vice versa (e.g., the evolution of a broadly neutralizing antibody). In a broader scientific context, directing a stochastic process toward a future objective is a classic subject of control theory (14, 15). There is a well-established conceptual and computational framework to optimize control protocols, given complete knowledge of the dynamical rules and the ability to forecast likely future outcomes. However, the swords of eco-evolutionary control are blunter, and establishing an appropriate control theory faces new challenges. First, the control of an evolving population is based, at best, on limited dynamical information and forecasting capabilities. Here, we compare three modes of control update dynamics: by Darwinian evolution of a biotic host system, by regulation (which requires sensing of the current pathogen state as input), and by computation (which requires sensing and forecasting). For human interventions, optimizing control is inextricably linked to predictive evolutionary analysis, which is a topic of high current interest but far from a comprehensive understanding (16). Second, control theory has to factor in the underlying biological mechanism of control. Host–pathogen interactions are often based on biomolecular interactions, such as drug–target or antibody–antigen binding (17). The form of these interactions imposes specific constraints on control forces and their leverage on the pathogen system, which are discussed below. Third, developing an appropriate dynamical model of control update and pathogen response calls for a merger of control theory with ecological dynamics and population genetics. These questions are the topic of the present paper.

In the first part, we develop dynamical principles of eco-evolutionary control. The evolution and population dynamics of the pathogen are governed by intrinsic forces, including fitness and entropy of pathogen traits, and by the additional selective force imposed by control. We derive general minimum-leverage relations that specify the strength of control needed to alter the evolution of the pathogen toward the host’s control objective. The control force has a payoff function in the host system, which includes the pathogen load and the cost of control, and is updated in response to the pathogen. For control by evolution or regulation, host and pathogen follow similar dynamical rules. The fixed points of the coupled dynamics, which are relevant for long-term control, are coevolutionary (Nash) equilibrium points. Through control by computation, however, a host can globally maximize its integrated payoff over an extended control period, often trading an initial dip for a later gain. We show that computational control defines a class of dynamical fixed points, called computational equilibria, which differ from Nash equilibria and reflect the added value of computation.

The second part focuses on applications to biomolecular control of pathogens. We analyze a minimal model of control, in which the host produces antibodies that bind to the pathogen. This model captures two complementary control modes, which are associated with different host–pathogen interactions. For ecological control, bound antibodies impede pathogen growth. The control objective is to reduce the pathogen’s carrying capacity; a deleterious collateral is the evolution of resistance. For evolutionary control, bound antibodies reduce pathogenicity. The control objective is the adaptive evolution of an antibody binding site (epitope) in the pathogen population; a collateral is the concurrent increase of the carrying capacity. We develop an analytical solution of the minimal model for ecological and evolutionary control. The solution includes maximum-payoff stationary and time-dependent protocols for antibody dosage, and it maps efficiency phase diagrams that delineate parameter regimes of efficient and compromised control.

The control theory of the minimal model has a number of key characteristics we argue to be general features of biomolecular control. First, control phase diagrams contain an error threshold, marking a switch in molecular antibody–antigen recognition and a rapid change of control efficiency. Second, control by computation shows striking differences to evolutionary and regulatory host protocols: computational equilibrium points can reach higher payoff than Nash equilibria and in turn, be outcompeted by protocols with intermittent time dependence. Third, protocols and success of control depend on the pathogen’s mutation rate and population size, highlighting how control depends on the underlying population genetics and ecology. We discuss implications of these results for biomedical applications: how information processing impacts mode and efficiency of control and how measurements of core pathogen and host data can be used to predict control outcomes.

## Eco-evolutionary Control Theory

### Eco-evolutionary Dynamics of Pathogens.

Consider a population of pathogens with a quantitative, heritable trait G that is a target of host–pathogen interactions, including control. Selection on the trait is described by a fitness landscape *Methods* and *SI Appendix*.

First, common mutations with individually small trait effects **17**] in *Methods*, this Darwinian process generates a trait distribution peaked at its mean *Methods*).

Second, mutations of large effect

Third, the pathogen population dynamics follows a minimal ecological model, which is given by [**18**] in *Methods*. This model describes a stochastic birth–death process with basic reproductive rate

In an individual pathogen population, the controlled evolutionary dynamics defines a control path *SI Appendix*, building on previous results in stochastic thermodynamics and evolutionary statistics (20, 21). Pathogen evolution often involves large flux amplitudes (

### Minimum-Leverage Relations for Pathogen Evolution.

Eco-evolutionary control is exerted by altering selection: the controlled system is governed by a free fitness landscape **3**] to determine lower bounds on

A control protocol ζ can elicit a trait value **3**] for any segment covered in an arbitrary subperiod **1**] (20). This amounts to the local minimum-leverage condition**4**] is valid up to small fitness troughs that can be crossed by common mutations at low effective population size.

If the population states **4**] takes the simpler form*SI Appendix*) (20, 22). In the case of a constant **5**] says that escape mutations from

Fig. 1 illustrates the minimum-leverage relation for a minimal model of control with a time-independent amplitude ζ. The pathogen free fitness landscape *A*) and evolutionary control aimed at maintaining an evolved equilibrium *B*). In both cases, the minimum-leverage relation [**5**] is seen to delineate a strong control (SC) regime, where the control objective is achieved, and a weak control (WC) regime, where the objective is missed. These regimes and their dependence on the control amplitude ζ will be further discussed below.

The inequalities [**4** and **5**] specify the minimum leverage that a controlling host system must exert on the controlled pathogen system in order to elicit a feature that would not evolve spontaneously and to maintain this feature against reverse evolution toward the wild type. These relations are formally related to the maximum-work theorem of thermodynamics, which specifies the minimum-work uptake (or maximum-work release) associated with a given free energy change of a thermodynamic system. Unlike in thermodynamics, however, the minimum-leverage relations say nothing about cost and benefit of control for the controlling system. This requires explicit modeling of the host’s control dynamics and of the host–pathogen interactions, to which we now turn.

### Control by Darwinian Evolution or Regulation.

How can pathogen evolution under control be optimized for the controlling host system? To address this question, we have to specify a host payoff function and the resulting dynamics of the control amplitude **19** in *Methods* specifies a stochastic update rule: *Methods*).

Control by local update rules can readily lead to a local maximum of the host’s payoff **20**] in *Methods*. In a complex payoff landscape, this involves bridging payoff valleys and requires large host populations that harbor large-effect control amplitude changes in their standing variation. Otherwise, evolutionary update of the control amplitude becomes mutation limited and often prohibitively slow. In appropriate host systems, as in the example given below, global payoff maximization can be implemented more rapidly by regulation. Given a recurrent pathogen, a trained regulatory network can sense the instantaneous pathogen load and generate an approximately optimal control amplitude as output. Because regulation is faster than the eco-evolutionary pathogen dynamics, it produces a greedy maximization of the instantaneous host payoff. These dynamics can again be modeled as a stochastic or deterministic process in the payoff landscape *Methods*).

Given a control path **2**]. Under a joint stochastic process of pathogen evolution and (gradient or greedy) instantaneous control, these fluxes take symmetric roles. The total flux *SI Appendix*). In the deterministic limit, instantaneous-update protocols have a nonnegative flux,

A stable fixed point

### Control by Computation.

A more ambitious goal, and the subject of control theory, is to optimize protocols toward an objective defined over an extended period of the dynamics, including the future. Here, we use a scoring function of the form*Methods*, we solve the HJB equation analytically for deterministic control with scoring functions of the form [**9**]. Computational protocols often steer through payoff valleys, trading transient periods with **7**]; hence, they cannot be realized by any instantaneous-update rule in the payoff landscape

A stationary state **10**] can lead to different fixed points than the Nash equilibrium condition [**8**]. Specifically, computational control can reach higher stationary payoff

## Pathogen Control by Antibodies

### Antibody–Antigen Interactions.

We now focus on a specific control scenario, in which a host exerts control by producing antibodies that bind to a pathogen (also referred to as antigen in this context). The probability that the pathogen is bound,

### Pathogen Fitness and Host Payoff Landscapes.

We assume that host and pathogen live in coupled landscapes of the form**18**] and measured in units of c. In the case of evolutionary control, we use a load function **12**] for ecological and evolutionary control. In both cases, the pathogen has two local fitness maxima (solid and dashed lines) with a rank order depending on the antibody dosage.

### Optimal Stationary Control.

We now focus on pathogens that can be contained by sustained treatment but cannot be eradicated by a short-time protocol (this is, currently, an appropriate assumption for HIV and some cancers). First, we compute the maximum-payoff stationary control protocol, as given by the computational equilibrium condition [**10**], which is relevant for long-term treatment. Given a large pathogen population under stationary control, the trait evolves to a dosage-dependent fitness maximum, **12**], the resulting optimal control **27**–**30**] in *Methods*). In the case of ecological control, we find two control regimes, SC and WC, as shown schematically in Fig. 1. These regimes are separated by a transition at*A*). At the transition, the resistance trait switches from *SI Appendix*, Fig. S1). Antibody–antigen binding switches from *A* shows the efficiency as a function of the cost parameters

For stationary evolutionary control, we find a similar emergence of two control regimes with an error threshold*SI Appendix*, Fig. S1). Fig. 3*B* shows the control efficiency

### Instantaneous-Update Control.

Next, we consider the dynamical accessibility of the computational equilibrium point. In the case of ecological control, **8**] (*Methods*). Hence, this point can be reached by a control dynamics based on host evolution or regulation; examples of such control paths are shown in Fig. 4*A*. In the SC regime, the pathogen wild type is stable (*B*), reflecting the mutually deleterious effects of host and pathogen in the landscapes of [**12**]. The host and pathogen fluxes, however, are positive throughout in accordance with the flux conditions [**3**, **4**, and **7**] (Fig. 4*C*). In this case, computational update generates control paths with a similar convergence to the fixed point *SI Appendix*, Fig. S2). We conclude that the ecological control equilibrium can be dynamically reached and maintained in a robust way.

### Computational Control of Adaptive Trait Formation.

In contrast, the optimal stationary (computational equilibrium) protocol **12**] can be evaluated analytically (Fig. 4*D* and *Methods*). These protocols have two phases. Starting from a wild-type pathogen, we apply a high initial dosage *E*). In a second “breeding” phase, small-effect trait and dosage changes increase host payoff and steer the control path toward the stationary point **3** and **4**]. The host flux has negative increments in the initial phase and final phase, which violate the condition [**3**] and mark strong deviations of the computational protocol from evolutionary or regulatory protocols (Fig. 4*F*). These instantaneous-update protocols converge to a Nash equilibrium point *SI Appendix*, Fig. S2).

Computational protocols of adaptive trait formation maximize the score Ω by jointly tuning payoff and duration of the initial and the breeding phase. The optimal protocol depends on the speed scoring parameter λ; protocols with large λ steer the pathogen along paths of near-maximal speed of trait evolution (*SI Appendix*, Fig. S3). Importantly, the optimal protocol also depends on the pathogen population size, which affects the mutational supply in the adaptive process (*SI Appendix*, Fig. S3). To map these effects, we evaluate the score of a time-dependent path relative to the optimal stationary protocol, *Methods*, we solve an extended HJB equation for *Methods*). In small populations (*Methods*). That is, the optimal protocol of adaptive trait formation eliminates the bottleneck of waiting for de novo gain-of-function mutations by circumnavigating the pathogen fitness valley (*SI Appendix*, Fig. S4). Larger values of *G*). We conclude that in small pathogen populations, adaptive trait formation requires higher control effort and is more costly.

### Metastable Computational Control.

Strikingly, optimal computational protocols can be time-dependent even for long-term control with the stationary objective of maximizing the host payoff integral *A*). In smaller populations, such mutations become rare, and the pathogen will stay in a metastable state (here, the wild type) over extended periods. Computational control can reinforce metastability by deepening the pathogen fitness valley for escape mutations; we refer to such protocols as metastable control. Provided we can detect escape mutants at low frequency, we can keep the bulk pathogen population permanently in the metastable state by a two-state protocol (Fig. 4*H*). As long as no escape mutant is detected, we apply a baseline antibody dosage *I*) and flux (Fig. 4*J*), which mark strong deviations from instantaneous-update protocols.

For metastable control over a long period T, the average score relative to the optimal stationary state per unit of time, **37** in *Methods* and *SI Appendix*, Fig. S4). For *K*, which compares the efficiency of maximum-score metastable and stationary protocols,

For evolutionary control in the WC regime, a similar protocol can stabilize a metastable evolved trait

## Discussion

In this paper, we solve the optimization problem for a minimal model of pathogen control operating on realistic biomolecular host–pathogen interactions. Specific features of host and pathogen biology enter the eco-evolutionary control theory of this system at several stages. In particular, control mechanisms are based on physiological effects of host–pathogen interactions—here, antibody–antigen binding—and the objective functions of control, Eq. **12**, account for the fitness and payoff effects of these interactions. As we have shown, these biological features strongly impact the eco-evolutionary control dynamics and the efficiency of optimized protocols.

A salient feature of eco-evolutionary control is the emergence of high- and low-efficiency parameter regimes separated by an error threshold of molecular recognition (Fig. 3). This behavior can be traced to the nonlinearities in the Hill function of antibody–antigen binding, Eq. **11**. First, binding-mediated control has a diminishing-return leverage, which is bounded by the pathogen selection coefficient **5**] has a moderate cost in the SC regime but is too expensive or mechanistically impossible in the WC regime. Because such nonlinearities are a generic feature of biomolecular interactions, we expect error thresholds to emerge also in more complex models of pathogen control.

In biomedical applications of pathogen control, it is crucial to predict the likely efficiency of control prior to any intervention. Our results show that few independently measurable cost parameters can inform such estimates. The pathogen cost parameters (

Optimal control protocols and their efficiency also depend on mutation rate and population size of the pathogen. Eqs. **15** and **16** describe opposing effects: adaptive formation of pathogen traits becomes less costly in large populations; maintenance of traits by metastable control is more efficient in small populations. These effects reflect differences between control strategies. Evolutionary control aimed at eliciting a pathogen trait has to circumnavigate valleys of the pathogen’s fitness and entropy landscape in order to catalyze its adaptive dynamics. In contrast, metastable control has to broaden fitness valleys surrounding a metastable state in order to suppress pathogen adaptation by escape mutations. In both cases, we can compute optimized control paths by combining probabilistic, discrete jumps across fitness valleys with continuous dynamics on smooth flanks in between. These navigation principles and their computational implementation are expected to extend to strong-selection control in more complex landscapes.

An equally important issue for control by humans and by natural hosts is optimizing the control dynamics in tune with the monitoring and computation capabilities of the host system (29). Instantaneous-update dynamics following local payoff gradients can often be realized efficiently by Darwinian evolution in the host (i.e., by variation of and selection on antibody levels). Protocols based on regulation can, in principle, circumnavigate payoff valleys and implement an approximate maximization of the instantaneous host payoff. However, signaling and regulatory networks require prior training by learning or by evolution, and they generate an additional cost to the host. Optimization of control toward future objectives depends, to various degree, on computation. The minimal model displays the relative efficiency of these control modes. For ecological control, the computational equilibrium point (i.e., the stationary state of maximal host payoff) is also a Nash equilibrium and can be reached by instantaneous-update protocols; for evolutionary control, this point can only be reached by computation. In more complex systems, the success of human computational control is limited by the ability to predict pathogen evolution. An example is vaccine selection for human influenza based on predictive analysis, where current methods have a prediction and control horizon of about one year (8, 16). For biotic, nonhuman host systems, it remains a fascinating question how far evolutionary and regulatory mechanisms can emulate control by computation.

A case in point is metastable control, which realizes the time-independent objective of maintaining a controlled pathogen state by a time-dependent protocol responding to recurrent pathogen attacks. In the control theory literature, this class of protocols is known as closed loop control (14). Here, we have solved a minimal model of metastable control, which maintains a metastable pathogen state against escape mutations by a two-state protocol of baseline control and rescue boosts. We find that metastable control can outperform the computational equilibrium protocol **[5]**), and escape mutants have a substantial intrinsic cost (generating positive selection for the metastable state during rescue boosts). In the minimal model, an optimized resource allocation between baseline and rescue protocols can only be achieved by computation. We can compare this model with the adaptive immune system of vertebrates, which controls multiple pathogens by a complex pattern of resource allocation (30). From the perspective of control theory, the immune system mounts a bilayer baseline protocol of naive and memory B cells; control boosts involving the evolution of target-specific high-affinity antibodies (affinity maturation) respond to acute infections by specific antigens (akin to escape mutations in the minimal model). Remarkably, affinity maturation continues for several weeks after an infection, which is an investment toward future infections by similar antigens. Together, adaptive immunity may be regarded as an instance of metastable control by nested circuits of evolution (of antibody–antigen affinities) and regulation (of antibody levels).

In summary, this paper establishes a conceptual framework and infers navigation principles for eco-evolutionary control based on a minimal model of host–pathogen interactions with stationary control objectives. Several complexities of biomedical control problems are not captured by the minimal model. For example, the human immune system has a complex antibody repertoire, and pathogens often have multiple antigenic binding sites (microbial and viral epitopes, cancer neoantigens). This generates multiple antibody–antigen interactions (i.e., multiple potential control channels with independent control amplitudes, leverage, and cost parameters). Conversely, pathogens with a high mutation rate often have multiple channels of escape mutations from a given antibody (31). Consequently, the fitness and payoff landscapes of host and pathogen live in a multidimensional parameter space; an appropriate control dynamics is to navigate this space toward high-efficiency protocols. An important example is the competitive selective dynamics of broadly neutralizing and specific antibodies for HIV (11, 13). Another layer of complexity arises for pathogens embedded in microbiota, which are multispecies systems with a tightly connected ecology. The control of a given pathogen can perturb the entire microbiota, generating cross-resistance of multiple pathogens and complex collateral effects for the host (32). Gearing up eco-evolutionary control theory to these systems is an important avenue for future work.

## Methods

### Stochastic Pathogen Dynamics.

We consider a population of pathogens with a quantitative trait G, which has a peaked trait distribution **1**], the trait dynamics of Eq. **17** depend on the entropy landscape **18**] depend on the fitness *SI Appendix*.

### Instantaneous-Update Control Dynamics.

The diffusive update rule for the control amplitude takes the form**17**]. Greedy update is defined by*A*; stochastic control paths are shown in *SI Appendix*, Fig. S2.

### Computational Control Dynamics.

Given deterministic pathogen evolution and landscapes **10**, and we define **17**]. The maximum payoff *SI Appendix*.

### Pathogen Equilibrium States.

In the parameter regime of interest, the pathogen fitness landscape of [**12**] has two local fitness maxima. 1) The wild type

### Computational Control Equilibria.

We compute the point of optimal stationary control, **25**]. We obtain the equilibrium point*SI Appendix*, Fig. S1). The transition between the SC and WC control regimes is determined by the condition **13**.

For evolutionary control, the pathogen has an evolved pathogen trait **26**], where the pathogen has the maximum stable **24**], the host payoff *SI Appendix*, Fig. S1). The transition between the SC and WC control regimes is determined by the condition **14**.

### Dosage Protocols for Adaptive Trait Formation.

In the protocols described above, an initial-phase dosage ζ elicits a gain-of-function mutation to a trait value G, which is the starting point for the subsequent breeding phase. The extended HJB equation for these two-phase protocols determines the relative score**23**]. To compute **12**. Prior to the start of control, all mutants with **31**–**34** displays the scaling form of the score, Eq. **15**, where the scaling function ω describes the cross-over in prevalence between de novo mutations (*SI Appendix*). Using Eqs. **23** and **34** together with a maximum-likelihood approximation for the gain-of-function amplitude,**31** determines the global maximum-score protocol (*SI Appendix*, Fig. S4). This protocol has a gain-of-function dynamics**22**] (Fig. 4 *D*–*G* and *SI Appendix*, Figs. S2 and S3).

### Dosage Protocols for Metastable Control.

For the two-state protocol, we evaluate the relative score per unit of time as a function of the escape amplitude G and baseline dosage ζ for **32**], **37** (*SI Appendix*, Fig. S4). This protocol has a baseline dosage*H*–*J*). The maximum score per unit of time, *K*).

## Data Availability.

Analysis notebooks are available at Open Science Framework (OSF), https://osf.io/6nakg/.

## Acknowledgments

We thank Armita Nourmohammad and Johannes Cairns for discussions. This work has been supported by Deutsche Forschungsgemeinschaft Grants SFB 680 (to M.L.) and SFB 1310 (to M.L.). We acknowledge the CSC–IT Center for Science, Finland, for computational resources.

## Footnotes

↵

^{1}M.L. and V.M. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: lassig{at}thp.uni-koeln.de or v.mustonen{at}helsinki.fi.

Author contributions: M.L. and V.M. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: Analysis notebooks are available at Open Science Framework (OSF), https://osf.io/6nakg/.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1920263117/-/DCSupplemental.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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