Resolve ambiguity in evaluating reactant concentration differences at sharp interfaces

Determine a theoretically justified rule for evaluating the concentration differences Δs(v) and Δp(v) across the droplet interface in the sharp-interface limit of the self-consistent theory of active condensates. Specifically, specify whether inner-side values, outer-side values, or an appropriate interpolation at the discontinuity should be used when computing ∫_D ∂_z s(z) dz and ∫_D ∂_z p(z) dz, and derive this rule from a consistent continuum limit (e.g., by introducing a smooth or kinked interface profile) rather than relying on heuristic comparison to simulations.

Background

In the sharp-interface formulation, enzyme concentration is piecewise constant, which induces discontinuities (jumps) in substrate and product concentrations at the droplet boundary due to reciprocal interactions. The self-consistency relations for 1D motion involve Δs(v) and Δp(v), defined by integrals of spatial derivatives across the droplet domain. Because of the discontinuities at the interfaces, these integrals become ambiguous depending on whether one uses inner- or outer-side boundary values or an interpolation.

The authors highlight that their current treatment chooses the inner-side values heuristically by comparison with full FEM simulations. A principled theoretical resolution is needed to eliminate this ambiguity and to make the sharp-interface theory internally consistent without reliance on simulation-based calibration.