Molecular concentration field design using closed-form steady-state solutions.

Area of Science:

• Synthetic Biology
• Materials Science
• Biophysics

Background:

• Controlling spatial concentration fields is crucial for synthetic biology and soft materials.
• Nature uses morphogen gradients for development, but synthetic methods rely on empirical or simulation-based approaches.

Purpose of the Study:

• To present an analytical framework for steady-state concentration fields from localized sources in diffusion-degradation systems.
• To derive closed-form solutions for 1D, 2D, and 3D geometries.
• To establish a quantitative design strategy for programming concentration ranges.

Main Methods:

• Developed an analytical framework for diffusion-degradation systems.
• Derived closed-form solutions for concentration fields.
• Expressed solutions in dimensionless form using the Thiele modulus.
• Introduced a design strategy based on threshold criteria.

Main Results:

• Derived dimensionless solutions showing gradient steepness depends on the Thiele modulus.
• Identified distinct design regimes: dimension-independent exponential decay in degradation-dominated systems and dimension-dependent power-law scaling in diffusion-dominated systems.
• Demonstrated consistency with numerical solutions and experimental data.

Conclusions:

• The framework provides a generalizable and physically transparent method for designing steady-state concentration fields.
• Enables predictive control of gradient-mediated organization in synthetic biological and soft matter systems.
• Reduces reliance on computationally intensive simulations for gradient design.