Trait-space patterning and the role of feedback in antigen-immunity coevolution

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Trait-space patterning and the role of feedback in antigen-immunity coevolution

Hongda Jiang and Shenshen Wang

Phys. Rev. Research 1, 033164 – Published 10 December 2019

Abstract

Coevolutionary arms races form between interacting populations that constitute each other's environment and respond to mutual changes. This inherently far-from-equilibrium process finds striking manifestations in the adaptive immune system, where highly variable antigens and a finite repertoire of immune receptors coevolve on comparable timescales. This unique challenge to the immune system motivates general questions: How do ecological and evolutionary processes interplay to shape diversity? What determines the endurance and fate of coevolution? Here, we take the perspective of responsive environments and develop a phenotypic model of coevolution between receptors and antigens that both exhibit cross reactivity (one-to-many responses). The theory predicts that the extent of asymmetry in cross reactivity is a key determinant of repertoire composition: small asymmetry supports persistent large diversity, whereas strong asymmetry yields long-lived transients of quasispecies in both populations. The latter represents a different type of Turing mechanism. More surprisingly, patterning in the trait space feeds back on population dynamics: spatial resonance between the Turing modes breaks the dynamic balance, leading to antigen extinction or unrestrained growth. Model predictions can be tested via combined genomic and phenotypic measurements. Our work identifies cross reactivity as an important regulator of diversity and coevolutionary outcome, and reveals the remarkable effect of ecological feedback in pattern-forming systems, which drives evolution toward nonsteady states different than the Red Queen persistent cycles.

Received 13 June 2019

DOI:https://doi.org/10.1103/PhysRevResearch.1.033164

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.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Organismal, population, evolutionary & ecological systems

Populations

Physics of Living Systems

Authors & Affiliations

Hongda Jiang and Shenshen Wang ^*

Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA

^*shenshen@physics.ucla.edu

Article Text

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References

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Issue

Vol. 1, Iss. 3 — December - December 2019

Subject Areas

Biological Physics

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Images

Figure 1
Schematic of antigen-receptor interaction with asymmetric range of inhibition and activation in the phenotypic space. (a) $R_{inh} > R_{act}$ : the receptor (blue Y shape) is not activated by the antigen (red flower shape) but nevertheless inhibits it. (b) $R_{act} > R_{inh}$ : the antigen activates the receptor but is not subject to its inhibition. Lower row: in addition to predation (black arrows; blunt for inhibition, acute for activation), antigens self-replicate (red arrow) whereas receptor-expressing cells spontaneously decay (blue arrow pointing to an empty set symbol) in the absence of stimulation. If a finite carrying capacity of receptors $θ_{2}$ is explicitly considered, self-inhibition will also be present [Figs. 4, 6, and 6].
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Figure 2
Phases in a 1D reaction-diffusion system under local predator-prey interactions. (a) Phase diagram on the plane spanned by the ratio between diffusion constants $D_{1} / D_{2}$ and that between birth and death rates $λ_{1} / λ_{2}$ of antigens (activators) and receptors (inhibitors). Dynamics start from a local dose of antigens and uniform receptors. The early extinction phase is color coded for the logarithm of the inverse time to antigen extinction. The persistence phase (blank) divides into a propagating wave state (upper) and a uniform coexistence state (lower). Insets show typical kymographs in each subphase, red for antigen and blue for receptor; the upper pair corresponds to the filled circle at $λ_{1} / λ_{2} = 200, D_{1} / D_{2} = 10^{- 2}$ , and the lower one corresponds to the open circle at $λ_{1} / λ_{2} = 10, D_{1} / D_{2} = 10^{- 2}$ . (b) Representative abundance trajectories. Top: $λ_{1} / λ_{2} = 20, D_{1} / D_{2} = 10^{- 3}$ [red dot in (a)]; bottom: $λ_{1} / λ_{2} = 10, D_{1} / D_{2} = 10^{- 2}$ [white dot in (a)]. Corresponding phase plots are shown on the right; vertical dashed lines indicate the extinction threshold. $B_{in} = 10, α_{1} = 10^{- 3}, α_{2} = 10^{- 4}$ .
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Figure 3
Asymmetric cross-reactive interactions simultaneously organize receptor and antigen distributions. (a), (b) The pattern wavelength $λ$ , identical for both populations, is symmetric under the interchange of the interaction ranges $R_{inh}$ and $R_{act}$ . (a) The scaled wavelength increases with the extent of asymmetry $γ \equiv (R_{inh} - R_{act}) / (R_{inh} + R_{act})$ ; $R_{inh} + R_{act} = 0.015, 0.02, 0.03$ from top to bottom. (b) Pattern diagram in the ( $R_{act}, R_{inh}$ ) plane. The white region corresponds to stable behavior, whereas patterning occurs in the colored areas. Solid lines indicate the instability onset [Eq. (4)]. The color bar shows the values of the wavelength determined from the critical mode. (c) Typical mutual distributions of receptor (blue) and antigen (red) in a 1D trait space with coordinate $x$ . The actual (solid line) and effective (dashed line) population densities (scaled by total abundance) show mismatch for receptors (antigens) when $R_{inh} > R_{act}$ ( $R_{inh} < R_{act}$ ), leading to colocalized (alternate) density peaks between two populations, as indicated by the yellow bars. Shaded are the effective density fields $A_{eff} (x)$ and $B_{eff} (x)$ . These two examples correspond to the open circle ( $R_{inh} = 0.025, R_{act} = 0.005$ ) and the filled circle ( $R_{inh} = 0.005, R_{act} = 0.025$ ) in (b). $λ_{1} = 10, λ_{2} = 1, α_{1} = 10^{- 3}, α_{2} = 10^{- 4}, D_{1} = 10^{- 6}, D_{2} = 10^{- 4}$ .
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Figure 4
Distinct regimes of coevolutionary dynamics. Population trajectories (top row) and concomitant pattern evolution (lower rows) of antigen (red) and receptor (blue) are shown for late antigen extinction (a), persistent coexistence (b), and antigen escape (c), which are realized by varying the range of cross reactivity and the size of carrying capacity. Concentration changes progress via three distinct stages: uniform steady state, stationary pattern, and oscillatory pattern. An extinction threshold is crucial for the termination of branches (a), (b) and the formation of forks (b) shown in the evolutionary kymographs. Color bars code for population densities. (a) $R_{inh} = 0.025, R_{act} = 0.005, θ_{2} = \infty$ ; (b) $R_{inh} = 0.005, R_{act} = 0.025, θ_{2} = \infty$ ; (c) $R_{inh} = 0.005, R_{act} = 0.025, θ_{2} = 3.5 \times 10^{5}$ . Other parameters are identical to those in Fig. 3.
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Figure 5
Theory predicts pattern amplitudes and abundance shift induced by coupling between Turing modes. Shown are scaled first- (a) and second- (c) order pattern amplitudes and abundance shift (b) as a function of $ε$ , the dimensionless deviation from $D_{1}^{*}$ . Lines are analytical predictions; symbols are numerical solutions. Solid line and filled symbol: $R_{inh} = 0.025, R_{act} = 0.005$ ; dashed line and open symbol: $R_{inh} = 0.005, R_{act} = 0.025$ . Red (blue) for antigen (receptor). $θ_{2} = \infty$ .
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Figure 6
Asymmetric cross reactivity yields diverse phases. (a) Without homeostatic constraints on lymphocyte counts ( $θ_{2} = \infty$ ), above the critical asymmetry (beyond the light blue region), patterns form. The pattern-forming boundaries are symmetric about the diagonal. The boundary between the late antigen extinction phase (blue) and the persistent patterned phase (yellow) is determined by tracking the prevalence trajectories until $t = 100$ . (b) Under a finite carrying capacity ( $θ_{2} = 3 \times 10^{5}$ ), the pattern-forming region is no longer symmetric and an antigen escape phase (red) emerges at the small- $R_{inh}$ large- $R_{act}$ corner, where the phase boundary corresponds to the transition between supercritical and subcritical bifurcations. (c) First-order pattern amplitudes as a function of carrying capacity $θ_{2}$ . Lines are analytical solutions of amplitude equations, and symbols are numerical values extracted from Fourier spectrum of stationary patterns right after abundance shift. Insets show examples of population dynamics in escape (subcritical) and persistence with pattern (supercritical) phases; pattern amplitudes diverge near the transition. Red (blue) for antigen (receptor).
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