Optimal Evolutionary Control for Artificial Selection on Molecular Phenotypes

Open Access

Optimal Evolutionary Control for Artificial Selection on Molecular Phenotypes

Armita Nourmohammad and Ceyhun Eksin

Phys. Rev. X 11, 011044 – Published 4 March 2021

Abstract

Controlling an evolving population is an important task in modern molecular genetics, including directed evolution for improving the activity of molecules and enzymes, in breeding experiments in animals and in plants, and in devising public health strategies to suppress evolving pathogens. An optimal intervention to direct evolution should be designed by considering its impact over an entire stochastic evolutionary trajectory that follows. As a result, a seemingly suboptimal intervention at a given time can be globally optimal as it can open opportunities for desirable actions in the future. Here, we propose a feedback control formalism to devise globally optimal artificial selection protocol to direct the evolution of molecular phenotypes. We show that artificial selection should be designed to counter evolutionary trade-offs among multivariate phenotypes to avoid undesirable outcomes in one phenotype by imposing selection on another. Control by artificial selection is challenged by our ability to predict molecular evolution. We develop an information theoretical framework and show that molecular timescales for evolution under natural selection can inform how to monitor a population in order to acquire sufficient predictive information for an effective intervention with artificial selection. Our formalism opens a new avenue for devising artificial selection methods for directed evolution of molecular functions.

Received 24 May 2020
Revised 2 December 2020
Accepted 11 January 2021

DOI:https://doi.org/10.1103/PhysRevX.11.011044

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Optimization problems Population genetics Stochastic processes

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Armita Nourmohammad ^*

Department of Physics, University of Washington, 3910 15th Ave Northeast, Seattle, Washington 98195, USA, Max Planck Institute for Dynamics and Self-organization, am Faßberg 17, 37077 Göttingen, Germany, and Fred Hutchinson Cancer Research Center, 1100 Fairview Ave North, Seattle, Washington 98109, USA

Ceyhun Eksin

Department of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77845, USA

^*Corresponding author. armita@uw.edu

Popular Summary

Controlling an evolving population is central in modern molecular genetics, including in directed evolution by artificial selection to improve function, and in devising public health strategies to suppress evolving pathogens. Current approaches in the lab are often ad hoc and, at best, use local evolutionary optimization to direct a population toward a desired target. However, local optimization can be ineffective, especially because of the ubiquity of evolutionary trade-offs in multivariate phenotypic traits. Thus, artificial selection should be devised by considering its impact over an entire evolutionary path to reach a target. Based on this principle of optimality, we introduce a control formalism to direct stochastic evolution of multivariate molecular phenotypes, such as the thermal stability and catalytic activity of an enzyme, to avoid undesirable outcomes in one phenotype by imposing selection on another.

Evolutionary control is challenged by the limited predictability of evolution due to stochastic effects at the molecular or environmental level. Predictability depends on the time span over which we aim to forecast and control evolution. To capture this, we introduce an information theoretical approach to gauge the efficacy of an evolutionary control based on the degree of predictive evolutionary information. Importantly, we show that the optimal schedule to monitor a population and to intervene with its evolution should be informed by the relevant timescales of molecular evolution under natural selection.

Our formalism brings together concepts from statistical physics and population genetics to introduce a new paradigm of optimal control for devising artificial selection schemes that control molecular evolution.

Key Image

Article Text

Click to Expand

Supplemental Material

Click to Expand

References

Click to Expand

Issue

Vol. 11, Iss. 1 — January - March 2021

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Artificial selection as an optimal stochastic adaptive control strategy. (a) Artificial selection is an external intervention to select for a desired trait (i.e., pinkness of cows) in a population, which is otherwise not favored by natural selection. Artificial selection should be viewed as an optimal feedback control, whereby monitoring a stochastically evolving population informs the intervention protocol to optimally bias breeding and reproduction over generations. (b) The schematic graph shows different paths with indicated costs for a system to evolve from a start to a target state. Bellman’s principle of optimality states that at each step an optimal decision is made, assuming that the following steps are also determined optimally. Although the first step (full line) of the blue path is more costly compared to the others (dotted lines), its cumulative cost is minimum, and hence, it should be chosen as the optimal path. This decision can be made best recursively, known algorithmically as dynamic programming.
Reuse & Permissions
Figure 2
Artificial selection with stochastic optimal annihilation. (a) Phenotypic trajectories starting from three distinct initial states (open circles) are shown for evolution under natural selection in a 1D quadratic adaptive landscape $F (x) = - c x^{2}$ , where $x$ is the phenotype centered around its optimum under natural selection. The trajectories are annihilated ( $†$ and low opacity lines) with a rate proportional to the cost of their deviation from target throughout evolution (dotted line), $V (| x - x^{*} |) / λ$ [Eq. (8)]. At each time point, the survived trajectories characterize the ensemble of evolutionary states for evolution under optimal artificial selection to reach the target at $x^{*} = 1$ . Control is designed under the assumption of infinite horizon. (b) The probability density function (PDF) shows the distribution of phenotypes for populations evolving subject to the annihilation protocol in (A). Populations start from an uncontrolled and natural state (blue distribution), and as a result of control annihilation, their distributions move toward the desired target at $x^{*} = 1$ (colors). (c) The expected control strategy $u^{*} (t) = - (1 / λ) k g (x - x^{*})$ (orange) and the cost of control $V (| x - x^{*} |) + \frac{1}{2} B u^{2}$ (blue) are shown as a function of time, as populations are driven from their natural state toward the target state. Time is measured in units of the characteristic time for natural evolution ( $1 / 2 k c$ ). Parameters are $k = 0.4$ , $c = 1$ , $λ = 0.01$ , $g = 4$ .
Reuse & Permissions
Figure 3
Artificial selection on covarying phenotypes. (a) Trajectories for evolution under natural (orange) and artificial (blue) selection are shown for a 2D phenotype $(x, y)$ , in a quadratic landscape. Parameters are $c_{x} = 2$ , $c_{y} = 4$ , $c_{x y} = 0$ ; $x^{*} = 1.2$ , $y^{*} = 0$ ; $k_{x} = 0.02$ ; $k_{y} = 0.05$ ; $k_{x y} = 0$ ; $g_{x} = g_{y} = 2$ ; $λ = 0.01$ . (b) The distribution of phenotypes at the end point of an artificial selection protocol (blue) is compared to the phenotypic distribution under natural selection (orange). Evolutionary parameters are the same as in (a). (c) The dynamics of the effective fitness peak is shown over time (colors) for 2D covarying phenotypes with correlations $ρ_{x y}$ indicated by the shape of the markers. From left to right, panels show increasing end point cost of deviation from the target along the $x$ phenotype, $g_{x} = 1$ , 2, 3, with $g_{y} = 2$ . Heat maps show the effective fitness landscapes associated with a specific fitness peak (indicated by the dotted arrows) for anticorrelated phenotypes at three different time points. The direction and length of the red arrows in each heat map indicate the direction and the strength of selection pressure toward the effective optimum. Parameters are $x^{*} = 3$ , $y^{*} = 0$ ; $c_{x} = c_{y} = 5$ , $c_{x y} = 0$ ; $k_{x} = k_{y} = 0.02$ ; $λ = 0.1$ .
Reuse & Permissions
Figure 4
Artificial selection with limited information. (a) Relative strength of artificial selection $α_{τ} / α_{0}$ [Eq. (11)] is shown as a function of the time interval for monitoring and intervention $τ$ , measured in units of the characteristic time for evolution under natural selection ( $1 / 2 k c$ ). The selection ratio is shown for various strengths of natural selection $c$ (top, with $x^{*} = 1$ ) and for various targets of artificial selection $x^{*}$ (bottom, with $c = 1$ ). (b) Power [Eq. (13)] is shown as a function of the time interval $τ$ for a range of parameters for the phenotypic diversity $k$ (full line) and the strength of natural selection $c$ (dotted line). The inset shows a collapse plot for power scaled with the expectation at the characteristic time for natural selection $power (τ) / power (τ = 1)$ . (c) Predictive mutual information $I (τ)$ [Eq. (14)] is shown to decay with the time interval $τ$ for a wide range of parameters ( $k$ , $c$ ). Inset shows an enlargement of a narrower range for the time interval $τ < 1$ . (d) Predictive information [Eq. (14)] is shown as a function of the scaled power for optimized artificial selection for a range of $τ$ values [Eq. (13)]. Each curve sets an information bound for artificial selection for a given set of evolutionary parameters ( $k$ , $c$ ). A nonoptimal selection intervention should lie below the information curve, shown by the gray shaded area as the accessible region associated with the dark blue curve. Color code in (c) and (d) is similar to (b). Other parameters are $λ = 0.6$ , $x^{*} = 3$ , $g = 2$ .
Reuse & Permissions

Physical Review X