| Abstract: |
| We study a PDE-constrained variational problem modeling spatial distributions of enzymes that maximize their reaction with substrates in biological cells. With a given enzyme concentration, the substrate concentration is determined by a reaction-diffusion equation. The reaction functional of enzyme concentrations is defined to be the total amount of reaction flux that is a nonlinear form of product of enzyme and substrate concentrations. We construct reaction-maximizing sequences through localization of enzymes, calculate the first and second variations of the reaction functional, and show the nonexistence of local and global maximizers. We further propose regularized reaction functionals and study their variational properties. Applications to biological cells are discussed. This is joint work with Yasir Khan. |
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