Molecular Concentration Gradient Design using Closed-Form Solutions for Concentration Field Steady-State
Year: 2025
Type: preprint
Abstract: Control over spatial concentration fields represents a fundamental challenge in designing synthetic biological systems and programmable soft materials. While nature creates morphogen gradients that orchestrate complex developmental processes, synthetic approaches have largely relied on empirical optimization and computationally intensive simulations. Here, we present a comprehensive analytical framework that transforms concentration field design for predictive engineering through closed-form mathematical solutions. Our approach focuses on localized source reaction-diffusion systems where localized production sources generate molecules that diffuse while being consumed by spatially distributed degradative catalysts, which is relevant for dissipative materials that require continuous energy input to maintain non-equilibrium organization. By deriving exact analytical solutions across one-, two-, and three-dimensional system configurations, we demonstrate that all concentration field characteristics are controlled by a single dimensionless parameter R' = R(k/D)0.5, representing the competition between reaction and diffusion length scales. We discover universal scaling behavior in the reaction-dominant regime (R' > 1), where dimensionless concentration decay becomes dimensionally independent, and power-law scaling in the diffusion-dominant regime (R' < 1) for designing gradients. This analytical framework eliminates computational overhead while providing generalizable design principles for applications ranging from biosensors to autonomous dissipative materials, establishing mathematical foundations for next-generation programmable matters.
Source: ChemRxiv
Authors: Dong Woo Kim, Alison Grinthal, Rebecca Schulman
Institution: Johns Hopkins University
Language: English
Subfield: Molecular Biology
Domain: Life Sciences
Open Access status: green
Award: G3535732435
OpenAlex ID: W4415605289