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Scientist: Examples: Implicit Equations

This example shows how Scientist easily handles implicit equations.

The rates of many enzymatic reactions can be described by an equation of the following form:

Rate(C) = (VMAX * C)/(Km + C)

The change in concentration with time is therefore given by

dC/dt = -Rate(C) = -(VMAX * C)/(Km + C)

This differential equation has no explicit analytical solution for C as a function of t. Nonetheless, Scientist provides several options for working with it. The above equation could be entered in its differential form, we could define a model in terms of the INTEGRAL operator, then solve the equation implicitly, or we could integrate the equation by hand and come up with an implicitly defined analytic solution. The most straightforward method would be to use the above equation in differential form and let Scientist do all of the work. The above equation leads to the following Scientist model.

Michaelis-Menten Implicit Equation

After entering the Michaelis-Menten model, we enter parameter values as follows:

Michaelis-Menten Implicit Equations Parameters

We ask Scientist to do a simulation with T going from 0 to 120 hours in 10 increments. After a brief calculation, we get the following output:

Michaelis-Menten Implicit Equations Spreadsheet

We then choose the Quick Chart function from the function toolbar to display the model graphically.

Michaelis-Menten Implicit Equations C vs. T

Note that the curve has the characteristic "hockey stick" shape and is nearly linear for concentrations greater than Km.