Example. Fitting a kinetic rate law to time-dependent data

Example. Fitting a kinetic rate law to time-dependent data 

There need not always be an analytical expression for the cost function. Often, the cost function itself is computed by a numerical calculation. For example, let us say that we are studying the enzymatic conversion of a substrate S to a product P in a batch bioreactor. We expect the rate of conversion, in units of micromoles converted per minute per milligram of enzyme, to be described by Michaelis-Menten kinetics, with possibly substrate inhibition, 

For a bioreactor of volume Fr, tiie number of micromoles of substrate. Ns, is related to the substrate molar concentration [S] by 

For a reactor of volume 100 ml containing 10 mg of enzyme. Table 5.1 records the substrate concentration as a function of time, starting from an initial concentration of 2 M. We wish to fit 0 = [ Fm Kja Ib minimizing the cost function 

In the preceding sections, we considered only unconstrained optimization problems in which X may take any value. Here, we extend these methods to constrained minimization problems, where to be acceptable (or feasible), x must satisfy a number e of equality constraints gi (x) = 0 and a number n of inequality constraints hj x) 0, where each g, (x) and hj(x) are assumed to be differentiable nonlinear functions. This constrained optimization problem 

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