Enhanced stability and chaotic condensates in multi-species non-reciprocal mixtures

Abstract

Random non-reciprocal interactions between a large number of conserved densities are shown to enhance the stability of the system towards pattern formation. The enhanced stability is an exact result when the number of species approaches infinity and is confirmed numerically by simulations of the multi-species non-reciprocal Cahn-Hilliard model. Furthermore, the diversity in dynamical patterns increases with increasing number of components and novel steady states such as pulsating or spatiotemporally chaotic condensates are observed. Our results may help to unravel the mechanisms by which living systems self-organise via metabolism.

Figures (4)

• Figure 1: (a) Schematic representation of a minimal living cell, which we define as a collection of a large number of bio-molecular building blocks with random interactions. (b) Non-reciprocity is a reliable way to control LLPS, as it always lowers the phase separation temperature $\Theta$. (c) Activity stirs the mixture while simultaneously stabilising it, leading to chaotic dynamics as shown in the representative kymographs.
• Figure 2: The eigenvalue of the interaction matrix $M_{ab}$ [see Eq. \ref{['eq:M']}] with the smallest real part, $\lambda_0$, determines the linear stability of the mixed state. (a) Scatter plots of the mean $\bar{\lambda}_0$ in the complex plane versus $\alpha$ for $N=10^3$ (error-bars show the variance). Inset: The same in log-log scale. (b) Solid lines are spinodal curves in the space spanned by temperature $\Theta$ and average composition $\phi_0$ at selected values of activity $\alpha$. The curves shift towards lower $\Theta$ at high $\alpha$ as $\hbox{Re}(\lambda_0)$ decreases showing that non-reciprocity stabilises the mixed state. Markers highlight the order parameter $\Delta$ [see Eq. \ref{['eq:OrderParameter']}], which vanishes outside the spinodal and changes discontinuously to a finite value inside it. The dashed line at maximum activity $\alpha \sim 1$ is the lower bound on the stable region.
• Figure 3: Statistics of steady states close to the spinodal. (a) Probability of finding patterns and waves in the steady state. Simulations results match well with the predictions of the linear theory for $\Theta(\alpha,0) <\approx \Theta_{\rm{sp}}(\alpha,0)$. The round markers and the blue line are results from an ensemble of size $300$ and $3\times10^5$ showing that the results converge very quickly with size of the ensemble. (b) The composition of the bulk phase separated final states as determined from simulations is compared with predictions from the eigenvector associated with $\lambda_0$.
• Figure 4: Pattern formation with decreasing $\Theta$. Panels (a-c) and (d-f) illustrate the dynamics in the steady state for two realisations of $M_{ab}$ ($\alpha = 0.89$), with real and complex $\lambda_0$, respectively. (a) For $\Theta<\Theta_{\rm sp}$ the system with vanishing $\phi_0$ shows bulk phase separation where roughly half of the species are enhanced in one of the phases. As $\Theta$ is lowered, the condensates pulsate with a finite number of distinct frequencies in the steady state as seen in panel (b). Upon lowering $\Theta$ further the oscillations become incoherent and the system starts to exhibit chaotic behaviour. In panel (e) waves (in which all species travel with fixed phase differences) give way to more complex chaotic dynamics. We find a zoo of complex dynamics that can be broadly categorised as intermittent bands and condensates in panel (e) and simply chaotic condensates in (f). (g) At sufficiently low $\Theta$ we only find chaotic condensates as characterised by the Lyapunov spectrum with at least one positive index.