Impact of currents on non-equilibrium coexistence in chemically driven mixtures

Abstract

Virtually every biological function emerges through the organization of molecules in time and space. Consequently, a major challenge in statistical physics is to uncover the universal principles governing macromolecular self-organization within the crowded, non-equilibrium environment of the cell. Here, we investigate a class of models where molecules maintain a conserved total concentration but can switch "identities", thereby modulating their intermolecular interactions. By enforcing thermodynamic consistency via the local detailed balance condition, we derive the steady-state criteria determining coexisting concentrations in a binary mixture. For non-constant transition rates and using a sharp-interface approximation, we obtain jump conditions that generalize Gibbs' coexistence criteria of equal pressure and chemical potential. We demonstrate that these jumps balance the chemical potential differences of individual species against their currents, which are confined to the interfacial region.

Figures (3)

• Figure 1: Chemically active binary mixture. (a) Sketch of macromolecules that can transition between two "species" (disk and oblique square). There are two reaction channels: spontaneous passive transitions and activated transitions involving the conversion of a molecular solute from substrate (S) to product (P) with rates $k^\nu_\pm$ ($\nu=\text{a,p}$). The concentrations of the molecular solutes are assumed to be fixed through reservoirs at constant chemical potentials. (b) Steady-state chemical fluxes at constant concentrations: Active transitions are biased towards species $2$ (for $\Delta\mu>0$), the chemical potential of which is increased. Consequently, passive transitions can decrease the free energy converting species 2 back to 1 so that overall the concentrations remain constant. (c) Chemical space spanned by the total concentration $\phi$ of macromolecules and their mole fraction $\alpha$. Along the nullcline $\bar{\alpha}(\phi)$ (solid black line) the net flux vanishes. In a uniform system the dynamics is confined to the reactive subspaces at constant $\phi$ (red lines). The blue line sketches the non-linear flux-balance space for an inhomogeneous system connecting both bulk phases with $\phi^\pm$ on the nullcline. The inset shows the corresponding profile $\phi(x)$ in real space. While the flux-balance space coincides with the nullcline in passive systems obeying detailed balance, driving the mixture away from equilibrium might displace the flux-balance space implying currents across the interface (thin red arrows).
• Figure 2: Active interface. (a) Sample integration region $\Omega$ for the divergence theorem. A non-zero "charge" $q$ at the interface enclosed by $\Omega$ would imply bulk currents $\bm j_k$. (b) Simplified interface constructed from the tangent at $\phi_\ast$, the intersection of which with the bulk concentrations $\phi^\pm$ determines the effective width $\ell$. (c) Sketch of the net flux $s(x)$ across a charge-free interface. We replace the smooth function $s(x)$ by two planar sheets separated by a small distance $\ell$ with surface charges $\pm\sigma$ adding to $q=0$. The charges drive lateral currents $\bm j_1=-\bm j_2=\sigma\bm e_x$ between the plates. Zooming out, the interface becomes a dipole sheet and the potential difference $\epsilon$ has to jump across the interface.
• Figure 3: Coexistence and phase diagram in a chemically active binary mixture with non-constant rates ($\zeta=0.1$, $\Delta f=0.35$, $k_0^\text{p}/k^\text{a}_0=1$). (a) Pressure $p(\phi)$ for three temperatures above, at, and below the critical temperature for $\Delta\mu=-0.05$. The dashed line indicates $p_\text{coex}$ for $T=0.2$. (b) Chemical potential at $T=0.2$ with the coexisting concentrations $\phi^\pm$. The insight shows the jump at $\phi_\ast$ between the chemical potentials in the dilute and dense regions. (c) Binodal curve together with nullclines for three selected temperatures. Dashed lines indicate the portion of the nullcline where the uniform system is unstable. Also shown is the position of the critical point and how it changes as the driving force $\Delta\mu$ is increased (solid gray line). (d) Binodal curve enclosing the two-phase region (shaded) between dilute and dense phase. Quenching a uniform system inside the two-phase region will lead to coexistence with the coexisting concentrations $\phi^\pm$ on the binodal (horizontal dashed tie lines). The gray dashed line shows the equilibrium reference phase diagram for $\Delta\mu=0$. The inset shows two binodals for stronger driving: $\Delta\mu=0.4$ (blue line) and $\Delta\mu=-0.2$ (green line). (e) Reactive flux through the active channel as a function of driving force $\Delta\mu$ in the dilute phase ($\phi^-$). The inset shows the chemical potentials $\mu^-_k$, which are equal in thermal equilibrium ($\Delta\mu=0$). (f) The gap $\Delta_0$ as a function of driving force.