Hierarchical microphase separation in non-conserved active mixtures

Yuting I Li,Michael E Cates

doi:10.1140/epje/s10189-021-00113-x

Yuting I Li, Michael E Cates

Open Access

https://doi.org/10.1140/epje/s10189-021-00113-x

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Abstract

Non-equilibrium phase separating systems with reactions, such as biomolecular condensates and bacteria colonies, can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative and non-conservative sectors of the dynamics can be governed by incompatible free energies; when both sectors are present, this is the leading-order TRS violation, captured in its simplest form by ‘Model AB’. Second, the diffusive dynamics can break TRS in its own right. This happens only at higher order in the gradient expansion (but is the leading behaviour without reactions present) and is captured by ‘Active Model B+’ (AMB+). Each of the two mechanisms can lead to microphase separation, by quite different routes. Here we introduce Model AB+, for which both mechanisms are simultaneously present, and show that for slow reaction rates the system can undergo a new type of hierarchical microphase separation, which we call ‘bubbly microphase separation’. In this state, small droplets of one fluid are continuously created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and diffusive dynamics.

Highlights

It has emerged that the non-membrane compartments of cells, known as biomolecular condensates, can be fruitfully viewed as phase-separated liquid–liquid mixtures undergoing chemical reactions [1,2,3,4,5,6,7]
The characteristics of phase separation in these biomolecular condensates are distinct from equilibrium counterparts, in that microphase separation is often observed without any energetic cause [4]. (In equilibrium, the latter requires long-range interactions mediated by charged species or block copolymers, for example. [8,9,10]) This echoes studies of bacteria colonies whose phase separation is arrested by birth-and-death dynamics to give patterns on a finite length scale [11]
We have furthered our investigation into scalar field theories for non-equilibrium phase separating systems

Summary

Introduction

It has emerged that the non-membrane compartments of cells, known as biomolecular condensates, can be fruitfully viewed as phase-separated liquid–liquid mixtures undergoing chemical reactions [1,2,3,4,5,6,7]. We know that the active currents represented by the φ-conserving AMB+ (which do not derive from any local chemical potential, of the form δF/δφ or otherwise), can independently cause phase separation via the reverse Ostwald process [14] Even if these are formally subdominant when expanding in (∇, φ), such terms can in principle interact in a strongly non-additive way with the microphase separation in Model AB, if parameters are chosen so that the two mechanisms act on well-separated length- and time-scales. These variables remain important even though, being higher order in the Landau–Ginzburg expansion, they are formally irrelevant in the neighbourhood of the Wilson– Fisher fixed point which controls the critical onset of bulk phase separation [23] This suggests that important physics could be lost by ignoring these non-integrable gradient terms in the conservative sector for systems with both conserved and non-conserved dynamics, whereas we did ignore them when constructing Model AB as a canonical model for that case [16].

Steady states

Conclusion