Past exam of the mathematical physiology course of the University of Oxford / 2023 / 3 / b / Solution

Choose positive reference concentrations $C,C_s$ and a reference flux $J_0$, and defineOverdots now mean derivatives with respect to the dimensionless time $\tau$. The store balance becomesThe total balance becomes $\dot{u}+(C_s/C)\dot{v}=r/(kC)-u$. ThereforeandThe reference quantities are scale choices, not additional kinetic variables; typically the response functions supply natural saturation concentrations and a maximal uptake flux. The identity $\delta /\epsilon =k_s/k$ checks the nondimensionalization. A small $\epsilon$ means store exchange is fast compared with external cytosolic removal.