and K, which make them less efficient at taking up a given nutrient, but allows them to take up more of that nutrient. These phytoplankton tend to be large (e.g. diatoms) with a smaller area to volume ratio. Their larger enables them to become larger. This relationship is generally applied to nitrate concentrations, as that is the limiting nutrient in much of the ocean, but other nutrients—for example, iron and phosphate—might also be limiting. Derivation Applying the law of mass action, which states that the rate of a reaction is proportional to the product of the concentrations of the reactants (i.e. ), gives a system of four non-linear ordinary differential equations that define the rate of change of reactants with time In this mechanism, the enzyme E is a catalyst, which only facilitates the reaction, so that its total concentration, free plus combined, is a constant (i.e. ). This conservation law can also be observed by adding the first and third equations above. Equilibrium approximation In their original analysis, Michaelis and Menten assumed that the substrate is in instantaneous chemical equilibrium with the complex, which implies From the enzyme conservation law, we obtain Combining the two expressions above, gives us Upon simplification, we get where is the dissociation constant for the enzyme-substrate complex. Hence the velocity of the reaction – the rate at which P is formed – is where is the maximum reaction velocity. Quasi-steady-state approximation An alternative analysis of the system was undertaken by British botanist G. E. Briggs and British geneticist J. B. S. Haldane in 1925. They assumed that the concentration of the intermediate complex does not change on the time-scale of product formation – known as the quasi-steady-state assumption or pseudo-steady-state-hypothesis. Mathematically, this assumption means . This is mathematically the same as the previous equation, with replaced by . Hence, following the same steps as above, the velocity of the reaction is where is known as the Michaelis constant. Assumptions and limitations The first step in the derivation applies the law of mass action, which is reliant on free diffusion. However, in the environment of a living cell where there is a high concentration of proteins, the cytoplasm often behaves more like a viscous gel than a free-flowing liquid, limiting molecular movements by diffusion and altering reaction rates. Although the law of mass action can be valid in heterogeneous environments, it is more appropriate to