**Nutrient Uptake with Michaelis-Menten Dynamics**

by **Mike Martin**

You can use this page to investigate the nutrient uptake of a cell. Let *n* (*t*) be the nutrient concentration used by a cell. Nutrient molecules (or substrate) enter the cell by attaching to membrane-bound receptors or enzymes *x*. Let *p* denote the product formed when a nutrient molecule is taken in by the cell. *x*_{0} denotes a receptor (or enzyme) not occupied by a nutrient and *x*_{1} denotes a receptor that is occupied by a nutrient. This may be symbolized as

The *k*_{i}'s are the rate constants for the reaction and various directions. The number of receptors, *x*_{0} + *x*_{1}, is constant and we let *r* equal that quantity. Also, it is convenient to introduce two new constants (that involve *r* and the rate constants) to simplify the equations.

The nutrient concentration is usually much higher than that of the receptors (or enzymes), so it is reasonable to make a quasi-equilibrium assumption. It can then be shown that the nutrient concentration then obeys

It follows, too, that the rate of nutrient uptake is the opposite of the rate of product formation.

On this page the nutrient uptake dynamics are investigated and plotted. The parameter *k*_{max} can be interpreted as the maximum rate of nutrient uptake by the cell. The parameter *k*_{n} is the half-saturation constant. The initial nutrient concentration is *n* (0) and the simulation is only valid for the entered time interval. The parameters for these functions can be entered below.