Active phase separation in mixtures of chemically interacting particles

TL;DR

This paper presents a theoretical study of chemically-interacting particle mixtures, revealing a novel active phase separation driven by nonequilibrium chemical interactions that break action-reaction symmetry, with implications for biological and colloidal systems.

Contribution

It introduces a new class of active phase separation phenomena caused by nonequilibrium chemical interactions in mixtures of particles.

Findings

Chemically-interacting particles can undergo phase separation due to nonequilibrium interactions.

The model explains self-organization in biological and colloidal systems.

Action-reaction symmetry breaking is key to the observed phenomena.

Abstract

We theoretically study mixtures of chemically-interacting particles, which produce or consume a chemical to which they are attracted or repelled, in the most general case of many coexisting species. We find a new class of active phase separation phenomena in which the nonequilibrium chemical interactions between particles, which break action-reaction symmetry, can lead to separation into phases with distinct density and stoichiometry. Due to the generic nature of our minimal model, our results shed light on the underlying fundamental principles behind nonequilibrium self-organization of cells and bacteria, catalytic enzymes, or phoretic colloids.

Figures

Equations

$$\sum _i{\mu }_i{\alpha }_i{\rho }_{0i}<0.$$

$$(\delta {\rho }_1,\delta {\rho }_2,...,\delta {\rho }_M)=\left(1,\frac{{\mu }_2{\rho }_{02}}{{\mu }_1{\rho }_{01}},...,\frac{{\mu }_M{\rho }_{0M}}{{\mu }_1{\rho }_{01}}\right)\delta {\rho }_1.$$

$${\alpha }_1{N}_1^{\mathrm{clu}}+{\alpha }_2{N}_2^{\mathrm{clu}}=0,$$

Full text

Microorganisms and cells can chemotax in response to gradients of chemicals that they themselves produce or consume Wadhams and Armitage (2004); Van Haastert and Devreotes (2004). The same behaviour has been recently observed at the nanoscale for individual enzymes Sengupta et al. (2013); Jee et al. (2018); Agudo-Canalejo et al. (2018), and can be mimicked in synthetic systems using catalytically-active phoretic colloids Anderson (1989); Soto and Golestanian (2014, 2015); Niu et al. (2017, 2018); Yu et al. (2018); Varma et al. (2018); Rallabandi et al. (2019) . Importantly, when many such particles are placed in solution, they interact with each other through their influence on the chemical’s concentration field. Chemical interactions underlie a wide variety of phenomena such as self-organisation in heterogeneous populations of microorganisms and cells (e.g. quorum sensing Keller and Surette (2006) and competition for nutrients Hibbing et al. (2010) in bacterial ecosystems, or cell-cell communication via chemokines Friedl and Gilmour (2009)); aggregation of enzymes that participate in common catalytic pathways into a metabolon Sweetlove and Fernie (2018); Wu et al. (2015); Zhao et al. (2018), which could be harnessed in the design of better synthetic pathways Schwander et al. (2016); or the self-assembly of active materials from catalytic colloids Palacci et al. (2013); Massana-Cid et al. (2018).

A key feature of chemical interactions between two different species—whether they are synthetic catalytic colloids, biological enzymes, or whole cells or microorganisms—is that they are in general non-reciprocal Soto and Golestanian (2014, 2015). The concentration field of a fast-diffusing chemical around a chemically-active particle of species $i$ is, to lowest order, given by $c-c_{0}\propto\alpha_{i}/r$ where $\alpha_{i}$ is the activity of the species (positive and negative for producer and consumer species), $r$ is the distance to the particle’s centre, and $c_{0}$ is the reference concentration of the chemical at infinity. In turn, the motion of a particle of species $j$ in response to gradients of the chemical is given by a velocity $\boldsymbol{V}_{j}=-\mu_{j}\nabla c$ where $\mu_{j}$ is the mobility of the species (positive or negative if the species is directed towards regions of lower or higher concentration of the chemical). Combining these two expressions, one finds that the velocity of the $j$-species particle in response to the presence of the $i$-species particle is $\boldsymbol{V}_{ij}\propto\alpha_{i}\mu_{j}\boldsymbol{r}_{ij}/|\boldsymbol{r}_{ij}|^{3}$ with $\boldsymbol{r}_{ij}=\boldsymbol{r}_{j}-\boldsymbol{r}_{i}$, whereas the velocity of the latter in response to the presence of the former is $\boldsymbol{V}_{ji}\propto-\alpha_{j}\mu_{i}\boldsymbol{r}_{ij}/|\boldsymbol{r}_{ij}|^{3}$. Note that in general $\boldsymbol{V}_{ij}\neq-\boldsymbol{V}_{ji}$ for $i\neq j$ because $\alpha_{i}\mu_{j}\neq\alpha_{j}\mu_{i}$, implying a broken action-reaction symmetry for inter-species interactions, which would be impossible in a system at thermodynamic equilibrium Ivlev et al. (2015).

We have performed Brownian dynamics simulations Note (1) of the model just described for a wide range of mixtures of chemically-interacting species; see Fig. 1 and movies 1–12 in the Supplemental Material Note (1). For binary mixtures we find that, while in a large region of the parameter space the mixtures remain homogeneous, the homogeneous state can also become unstable leading to a great variety of phase separation phenomena. Here, phase separation is used in the sense of macroscopic (system-spanning) separation typically into a single large cluster (occasionally into two; see Fig. 1(a)) that coexists with a dilute (or empty) phase. The phase separation process may lead to aggregation of the two species into a single mixed cluster, or to separation of the two into either two distinct clusters or into a cluster of a given stoichiometry and a dilute phase. The resulting configurations are qualitatively distinct for mixtures of one chemical-producer and one chemical-consumer species, as opposed to mixtures of two producer (or consumer) species; compare panels (a) and (c) in Fig. 1. While the typical steady-state configurations are static, for mixtures of producer and consumer species we also find that static clusters can undergo a shape-instability that breaks their symmetry, leading to a self-propelling macrocluster (Fig. 1(b), movies 8 and 9). Randomly-generated highly-polydisperse mixtures of up to 20 species also show homogeneous as well as phase-separated states (Fig. 1(d), movies 11 and 12).

In the following, we will show how these results can be understood by means of a continuum theory, and how the observed phase separation behaviour is intimately related to the non-equilibrium and non-reciprocal character of the interactions. This represents a fundamentally new class of active phase separation, in which the activity arises from the non-equilibrium nature of the interactions between particles that are otherwise non-motile, rather than from the intrinsic activity of self-propelling particles as commonly studied Vicsek et al. (1995); Toner and Tu (1998); Golestanian (2012); Fily and Marchetti (2012); Redner et al. (2013); Saha et al. (2014); Cohen and Golestanian (2014); Zöttl and Stark (2014); Pohl and Stark (2014); Cates and Tailleur (2015); Liebchen et al. (2015, 2017).

We consider a system consisting of $M$ different species of chemically-interacting particles, with concentrations $\rho_{i}(\boldsymbol{r},t)$ for $i=1,...,M$; and a messenger chemical with concentration $c(\boldsymbol{r},t)$. The concentration of species $i$ is described by $\partial_{t}\rho_{i}(\boldsymbol{r},t)-\nabla\cdot[D_{\mathrm{p}}\nabla\rho_{i}+(\mu_{i}\nabla c)\rho_{i}]=0$ which includes a diffusive term with diffusion coefficient $D_{\mathrm{p}}$, which for simplicity is taken to be equal for all species (implying that all particles are of similar size or, in the case of microorganisms, all species show a similar baseline level of non-directed random motion); as well as an advective term describing motion in response to gradients of the chemical. The concentration of the chemical is described by $\partial_{t}c(\boldsymbol{r},t)-D\nabla^{2}c=\sum_{i}\alpha_{i}\rho_{i}$ which includes diffusion with coefficient $D$, and production or consumption of the chemical by all particle species. Performing a linear stability analysis Note (1) of this coupled system of $M+1$ equations around a spatially-homogeneous state with particle densities $\rho_{i}(\boldsymbol{r},t)=\rho_{0i}$, in the limit of a fast-diffusing chemical, we find that the homogeneous state becomes unstable towards a spatially-inhomogeneous state when the following condition holds

[TABLE]

The instability corresponds to macroscopic phase separation, in the sense that it occurs for perturbations of infinite wave length, specifically for perturbations with wave number $\boldsymbol{q}^{2}<-(DD_{\mathrm{p}})^{-1}\sum_{i}\mu_{i}\alpha_{i}\rho_{0i}$, with those having infinite wave length $\boldsymbol{q}\to 0$ being the first and most unstable. Importantly, the stability analysis also tells us about the stoichiometry of the different particle species at the onset of growth of the perturbation, which follows

[TABLE]

If only a single particle species is present ($M=1$), the instability criterion (1) describes the well-known Keller-Segel instability Keller and Segel (1970), which simply says that the homogeneous state is stable for particles that repel each other ($\mu_{1}\alpha_{1}>0$), whereas particles that attract each other ($\mu_{1}\alpha_{1}<0$) tend to aggregate, with the end state being a featureless macroscopic cluster containing all particles. In contrast, we will now show that as soon as we have mixtures of more than one species, the combination of the instability criterion (1) and the stoichiometric relation (2) predicts a wealth of new phase separation phenomena.

For binary mixtures ($M=2$), the instability condition (1) becomes $\mu_{1}\alpha_{1}\rho_{01}+\mu_{2}\alpha_{2}\rho_{02}<0$, and the stoichiometric constraint (2) implies that when $\mu_{1}$ and $\mu_{2}$ have equal or opposite sign, the instability will lead respectively to aggregation or separation of the two species. Combining these criteria we can construct a stability diagram for the binary mixture, although we must distinguish between two qualitatively-different kinds of mixtures: those of one producer and one consumer species, see Fig 2(a) where we have chosen $(\alpha_{1},\alpha_{2})=(+,-)$ without loss of generality; and those of two producer species, see Fig 2(b). The case of two consumer species is related to the latter by the symmetry $(\mu_{1},\mu_{2})\to-(\mu_{1},\mu_{2})$; see Fig. S1 in the Supplemental Material Note (1). In this way, the parameter space for each type of mixture can be divided into regions leading to homogeneous, aggregated, or separated states, which correspond directly to those observed in simulations; compare Fig 2(a,b) to Fig. 1(a,c). We note, however, that while for $(\alpha_{1},\alpha_{2})=(+,-)$ mixtures the simulations are always seen to match the predicted phase behaviour, for $(\alpha_{1},\alpha_{2})=(+,+)$ mixtures we have observed separation even when the continuum theory predicts the homogeneous state to be linearly stable, although proceeding much more slowly (see movie 7 in the Supplemental Material Note (1)), indicating that in this region separation may be occurring through a nucleation-and-growth process controlled by fluctuations. This is denoted as the shaded gray region extending past the instability line in Fig 2(b).

The wide variety of phase separation phenomena arising in these mixtures is intimately related to the active, non-reciprocal character of the chemical interactions. In particular, it is useful to consider the sign of both inter-species as well as intra-species interactions (as described above, species $i$ is attracted to or repelled from species $j$ when $\mu_{i}\alpha_{j}$ is negative or positive, respectively). In the stability diagrams in Fig. 2(a,b), we find that each quadrant corresponds to a distinct “interaction network” between species, as depicted in the boxed legends attached to every quadrant (as an example, the top-right interaction network in (a) can be read as “1 is attracted to 2, 2 is repelled from 1, 1 is repelled from 1, and 2 is attracted to 2”). We find that only three regions in the parameter space have passive analogs: (i) The bottom-right of (a) corresponds to electrostatics with opposite charges, where equals repel and opposites attract, allowing for the formation of small active molecules as studied in Refs. 7 and 8. (ii) The top-right of (b) corresponds to electrostatics with like charges, where all interactions are repulsive leading to a homogeneous state. (iii) The bottom-left of (b) corresponds to gravitation, where all interactions are attractive. The top-left of (a) can be thought of as the opposite of electrostatics (or as gravitation including a negative mass species), where equals attract and opposites repel. The remaining four quadrants involve intrinsically non-reciprocal interactions where one species chases after the other: in (a), a self-repelling species chases after a self-attracting species; whereas in (b), a self-attracting species chases after a self-repelling species. Importantly, we observe that the most non-trivial instances of phase separation, which are also those that can be triggered simply by density changes (e.g. by addition or removal of particles), occur in regions with such chasing interactions, which are in turn a direct signature of non-equilibrium activity.

Fourier analysis Note (1) of the simulation results (44 simulations with varying $N_{i}$, $\alpha_{i}$, and $\mu_{i}$; see Table S1 in the Supplemental Material Note (1)) agrees quantitatively with the theoretical prediction (2) for the stoichiometry at the onset of the instability; see Fig. 2(c). However, this initial value is not representative of the long-time stoichiometry of the phases. For $(\alpha_{1},\alpha_{2})=(+,+)$ mixtures, shown in Figs. 1(c) and 2(b), we always observe final configurations with either complete aggregation or separation of the two species. For $(\alpha_{1},\alpha_{2})=(+,-)$ mixtures, shown in Figs. 1(a) and 2(a), we always observe complete separation, but aggregation in this case leads to a cluster with non-trivial stoichiometry (Fig. 1(a), centre). Phenomenologically, we observe that the formation of such clusters proceeds by fast initial aggregation of the particles of the self-attractive species ($\alpha_{i}\mu_{i}<0$) followed by slower recruitment of particles of the self-repelling species ($\alpha_{i}\mu_{i}>0$) until the cluster is chemically “neutral”, in the sense that its net consumption or production of chemicals vanishes, namely

[TABLE]

where $N_{i}^{\mathrm{clu}}$ is the number of particles of species $i$ in the cluster. The long-time stoichiometry of the clusters thus depends on the activity of the species, but it is independent of their mobility; see Fig. 2(d). An intuitive explanation for this observation can be provided as follows: once the cluster becomes neutral, the remaining self-repelling particles will no longer “sense” its presence and stay in a dilute phase. However, at high values of activity and mobility for the self-attractive species, deep inside the instability region, these static neutral clusters can become unstable via shape-symmetry breaking towards a self-propelled asymmetric cluster (Fig. 1(b)), which also involves the “shedding” of some of the self-repelling particles; see Fig. S3 and movies 8 and 9 in the Supplemental Material Note (1). Finding a precise criterion for this symmetry-breaking to occur remains an open question, but we note that the existence of self-propelled clusters is a clear sign of non-equilibrium physics.

Going beyond binary mixtures ($M>2$), the phase separation phenomenology becomes even more complex due to the increasing number of parameter combinations, leading to a large variety of possible interaction networks between the different species. The instability condition (1) remains extremely useful, however. As a first example, in Fig. 3 we demonstrate how a small amount of a highly active “dopant” third species can be added to an otherwise homogeneous binary mixture in order to trigger macroscopic phase separation of the whole mixture on demand; see also movie 10 in the Supplemental Material Note (1). As a second example, we have simulated highly polydisperse mixtures made up of 20 different species with activities and mobilities randomly chosen in the intervals $-2\leq\tilde{\alpha}_{i},\tilde{\mu}_{i}\leq+2$ for each species; see Fig. 1(d) and movies 11 and 12. We find that the instability criterion (1) can rather reliably distinguish between phase-separating and homogeneous mixtures; see Fig. S4 in the Supplemental Material Note (1). We note that while all mixtures we predicted to phase-separate did so, some mixtures for which we predicted a linearly-stable homogeneous state were observed to phase-separate, albeit more slowly, once again pointing to a nucleation-and-growth mechanism rather than to a linear instability.

We have presented here a minimal model for phase separation in mixtures of chemically-interacting particles, and the generic phenomena that we predict should be applicable to a wide variety of systems. In the context of morphogenesis and collective migration in bacterial colonies and cells in tissues, the prediction of a transition between static and self-propelled clusters is particularly interesting. Here, it is important to take into account that what we call here “two species” may also represent a single species in two distinct states, each with different chemical activity or chemotactic behaviour. Regarding metabolon formation by enzymes in catalytic pathways, our prediction of “neutral” clusters (Eq. 3) is most intriguing, as it would correspond to a cluster in which one enzyme channels all of its product to be taken as substrate by the next enzyme, with no substrate missing or in excess. Finally, our predictions can be tested in detail in experiments using synthetic catalytic colloids, by systematically varying the sign and magnitude of the chemical activity, as well as the concentration of the different species. In future work, it will be interesting to characterize in more detail the non-equilibrium activity of the system by means of its energy dissipation or entropy production Sabass and Seifert (2012); Gaspard and Kapral (2018). Moreover, we note that in our simulations we have neglected hydrodynamic interactions between particles as well as near-field contributions in the chemical concentrations Note (1). While we we do not expect our results for the onset and stoichiometry of the instability to change, the detailed dynamics of aggregation and growth of the clusters as well as their internal dynamics will be affected by these additional effects.

This study was supported by the US National Science Foundation under MRSEC Grant number DMR-1420620.