Analytical field theory for the onset of homochirality in the coupled reaction–flocking model of Hoffmann and Giomi

Hoffmann and Giomi (arXiv:2307.14360; published as Phys. Rev. E 111, 015427 (2025)) coupled the Frank–Jafarpour stochastic chiral reactions (autocatalytic replication of left- and right-chiral cells, death, and influx from an external reservoir) to Vicsek flocking dynamics and showed numerically that a globally homochiral state is reached in finite time only when the influx rate vanishes, with a fat-tailed transition-time distribution and power-law dependence of the transition time on noise, speed, and reaction rate. All of these findings are purely numerical: the analytical treatment that Jafarpour et al. developed for passively diffusing molecules breaks down here because the spatial dynamics is anisotropic and far from equilibrium. The open problem is to derive a closed analytical (field-level or stochastic) theory for this coupled reaction–flocking model, valid inside the flocking phase and extendable to chirality-dependent rotation, and to obtain from it the conditions, statistics, and parameter scalings of the transition to homochirality. This analytical theory has not been constructed; the problem remains open.

Background

Chirality is the property of an object that differs from its mirror image; a system is homochiral when only one of the two mirror-related states is present even though its microscopic dynamics treats both equally. Biology is homochiral at many scales: amino acids are almost exclusively left-handed, sugars right-handed, and individual eukaryotic cells can show a consistent left–right bias in their crawling and internal organization. Frank (1953) proposed that autocatalysis plus mutual antagonism between the two chiral states could amplify a tiny imbalance into complete homochirality. Jafarpour, Biancalani, and Goldenfeld (Phys. Rev. Lett. 115, 158101 (2015); Phys. Rev. E 95, 032407 (2017)) turned this into a stochastic theory for replicating molecules that diffuse passively: for the reactions L+S→2L and R+S→2R at rate $k_m$, death L,R→S at rate $k_d$, and influx S→L,R at rate $k_c$ (with S a solvent/nutrient species), the chirality order parameter $\omega=N_L/(N_L+N_R)\in[0,1]$, with the racemic state at $\omega=\tfrac12$ and the homochiral states at $\omega=0$ and $\omega=1$, obeys, in the well-mixed and diffusive limits, a stochastic differential equation whose multiplicative reaction noise has fixed points at $\omega=0$ and $\omega=1$: noise itself drives the system to a homochiral state, but only if the influx rate $k_c$ vanishes, i.e. only in a closed system.

Hoffmann and Giomi (arXiv:2307.14360; published retitled as Phys. Rev. E 111, 015427 (2025)) asked what happens to this mechanism when the replicating entities are cells that move collectively rather than diffuse. Cells in a monolayer are well described at minimal level by the Vicsek model (Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995)): each self-propelled particle moves at constant speed $v_0$ and aligns its heading with neighbors within a radius $\mathcal{R}$, perturbed by angular noise of strength $\eta$; for small noise and high density the particles undergo a transition to a flocking state with polar order. The model of Hoffmann and Giomi runs the Frank–Jafarpour reactions via the Gillespie algorithm (an exact stochastic simulation method that draws each reaction event from the rates $k_m$, $k_d$, $k_c$ with their correct probabilities) inside boxes of side $2\mathcal{R}$ ($m$ reaction steps per Vicsek time step), with an optional chirality-dependent deterministic rotation $\lambda\Omega$ ($\lambda=\pm1$ for left/right cells) in the heading update. Numerically they found: (i) homochirality is still reached in finite time only for $k_c=0$; (ii) the transition time $\tau$ has a fat-tailed distribution with mean and standard deviation of the same order; (iii) $\tau$ decreases with orientational noise $\eta$, speed $v_0$, and number of reaction cycles $m$, approximately as $\eta^{-0.16}$ and $v_0^{-0.25}$; (iv) like-chiral cells are more strongly spatially correlated than opposite-chiral cells, although chirality enters neither the reactions nor (for $\Omega=0$) the equations of motion; (v) deep in the flocking phase, where reduced mixing slows convergence, the phenomenology persists; (vi) for $\Omega\neq0$, simulations show two chirality-separated clusters moving on a circular trajectory, stable for long times.

The analytical gap is explicit in the source. Jafarpour et al.'s derivation yields a reaction–diffusion equation $\partial_t\omega=-\frac{2Vk_ck_d}{Nk_m}(\omega-\tfrac12)+\mathcal{D}\nabla^2\omega+\sqrt{\frac{2k_d}{N}\omega(1-\omega)}\,\zeta$ for passively diffusing particles (with $V$ the system volume, $N$ the particle number, and $\zeta$ unit-variance Gaussian white noise); near the isotropic–flocking transition the Vicsek dynamics can still be mimicked by an anisotropic diffusion tensor, which leaves the fixed-point structure intact. But well inside the flocking state this reasoning fails: the dynamics is anisotropic, out of equilibrium, and exhibits the giant density fluctuations characteristic of Vicsek-type flocks, so no diffusion-like closure is available. The authors note that a stochastic field equation for the order parameter 'could be developed following Jafarpour et al.', that the 'main difficulty ... consists in including the anisotropic, non-equilibrium spatial dynamics', and that the theory of Malthusian flocks — polar active matter whose particle number is not conserved because particles are born and die (Toner, Phys. Rev. Lett. 108, 088102 (2012)) — is a possible starting point. The Malthusian-flock hydrodynamics has since been developed further, including two-dimensional chiral Malthusian flocks whose fluctuations fall in the Kardar–Parisi–Zhang universality class (the universal scaling class of rough interface growth and one-dimensional driven diffusive systems) (Chen, Lee, Toner, arXiv:2506.03488 and arXiv:2507.17762) and a nonperturbative renormalization-group analysis of their ordering (Jentsch and Erzberger, arXiv:2608.05851), but none of these works includes the two-state autocatalytic chiral reactions or addresses the onset of homochirality.

Problem Statement

Derive an analytical theory for the coupled reaction–flocking model of Hoffmann and Giomi, and use it to establish the onset of homochirality analytically. Concretely: starting from the microscopic model — Vicsek dynamics (speed $v_0$, alignment within radius $\mathcal{R}$, angular noise $\eta$) with the Frank–Jafarpour reactions L+S→2L, R+S→2R (rate $k_m$), L,R→S (rate $k_d$), and S→L,R (rate $k_c$) advanced by $m$ Gillespie steps per Vicsek step in boxes of side $2\mathcal{R}$, and with optional chirality-dependent rotation $\lambda\Omega$ — construct a closed coarse-grained description (for example a stochastic partial differential equation for the chirality order parameter $\omega(\mathbf{x},t)$, together with any density and velocity fields it couples to) that remains valid for the anisotropic, out-of-equilibrium spatial dynamics of the flocking phase, where the reaction–diffusion treatment of Jafarpour et al. and its anisotropic-diffusion generalization break down, including deep in the strong-flocking regime. The theory should analytically reproduce or correct the numerically observed phenomenology; all four items that follow are required: the guarantee of homochirality in finite time only for $k_c=0$ and its prevention by influx; the fat-tailed distribution of the transition time $\tau$ with mean and standard deviation of the same order; the enhanced spatial correlation of like-chiral cells; and the power-law dependence of $\tau$ on $\eta$, $v_0$, and $m$. A numerical observation counts as corrected only where the theory supplies a definite replacement prediction together with a concrete argument for why the simulation measures otherwise (for example a finite-size or finite-time measurement artifact); a bare discrepancy is a failure to reproduce, not a correction. It should further characterize the regime of chirality-dependent rotation, $\Omega\neq0$, in which left- and right-chiral cells have divergent trajectories and simulations show long-lived, chirality-separated clusters moving on circular trajectories: determine the resulting phase behavior and its consequences for whether and how a globally homochiral state is reached. The requested result is one coherent analytical account of the model — derivation, predictions, and regime analysis — not a collection of separate studies; Malthusian-flock hydrodynamics is a suggested starting point, not a prescribed method.

References

1. L. A. Hoffmann and L. Giomi, 'Theory of cellular homochirality and trait evolution in flocking systems', arXiv:2307.14360 (2023), DOI 10.48550/arXiv.2307.14360, https://arxiv.org/abs/2307.14360
2. L. A. Hoffmann and L. Giomi, 'Homochirality in the Vicsek model: Fluctuations and potential implications for cellular flocks', Physical Review E 111, 015427 (2025), DOI 10.1103/PhysRevE.111.015427, https://doi.org/10.1103/PhysRevE.111.015427 (journal version of the preceding reference)
3. F. Jafarpour, T. Biancalani, and N. Goldenfeld, 'Noise-Induced Mechanism for Biological Homochirality of Early Life Self-Replicators', Physical Review Letters 115, 158101 (2015), DOI 10.1103/PhysRevLett.115.158101, https://doi.org/10.1103/PhysRevLett.115.158101
4. F. Jafarpour, T. Biancalani, and N. Goldenfeld, 'Noise-induced symmetry breaking far from equilibrium and the emergence of biological homochirality', Physical Review E 95, 032407 (2017), DOI 10.1103/PhysRevE.95.032407, https://doi.org/10.1103/PhysRevE.95.032407
5. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, 'Novel Type of Phase Transition in a System of Self-Driven Particles', Physical Review Letters 75, 1226 (1995), DOI 10.1103/PhysRevLett.75.1226, https://doi.org/10.1103/PhysRevLett.75.1226
6. J. Toner, 'Birth, Death, and Flight: A Theory of Malthusian Flocks', Physical Review Letters 108, 088102 (2012), DOI 10.1103/PhysRevLett.108.088102, https://doi.org/10.1103/PhysRevLett.108.088102
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