Nonlinear Enzyme-Catalyzed Reaction

• Problem Description

  • Reaction involving substrate \((S)\) that is catalyzed by an enzyme \((E)\) leading to a product \((P)\)

  • Substrate combines with enzyme to form complex \((C)\)

  • Complex breaks down into product and enzyme

  • Product is mostly removed, so that reverse reaction doesn’t really occur

\[S \: + \: E \: \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} \: C \: {\stackrel{k_2}{\rightarrow}} \: P \: + \: E\]

• Model Descriptions

  • The differential equations model is given by the following nonlinear system

\[\begin{array}{rcl} {d S \over d t} & = & k_{-1} C \: - \: k_1 S E \\ {d E \over d t} & = & \left ( k_{-1} \: + \: k_2 \right ) C \: - \: k_1 S E \\ {d C \over d t} & = & k_1 S E \: - \: \left ( k_{-1} \: + \: k_2 \right ) C \\ {d P \over d t} & = & k_2 C \end{array}\]

• • A nonlinear response function is built to model the creation of product \((P)\) within a single system, where the inputs to the system are: initial substrate concentration \(S_0 =\) 20 mM, \(k_1\) = \(k_{-1}\) = \(k_2\) = 0.2, and 6 different initial enzyme concentrations \(E_0 =\) 2, 5, 10, 20, 50, and 100 mM.

• Results

  • Resulting \(P(t)\) curves are generated for the six \(E_0\) concentrations as well as two predicted \(E_0\) concentrations of 3 and 8 mM.

  • The \(P(t)\) curves are compared to “data,” where “data” represents the solution to the nonlinear system of differential equations

  • All of the resulting plots are shown here. Model curves are labeled M1, M2, M3, M4, M5, M6, and prediction curves are labeled P1, P2.

  • Fits to data are very accurate and require no other fitting of parameters.

  • In a Michaelis-Menten (MM) model, which is typically used to model enzyme kinetics, assumptions are made that limit its use, including only being applicable in describing the initial reaction velocity, assuming that the enzyme-substrate complex concentration \(C\) is constant (steady-state approximation), and assuming that the substrate concentration \(S\) is approximately equal to the total substrate concentration (free ligand approxmiation). The MM model could not be used for the full time-course response function model described here, due to the limiting assumptions, however the response function model does include the MM model as a subset and thus represents a valid general model of enzyme kinetics.

  • The nonlinear response function model is also able to accurately model non-MM kinetics, where Hill equations are typically used to describe quantitatively the degree of cooperativity. The Hill equation model requires fitting the Hill coefficient \(n\), which measures how much the binding of substrate to one active site affects the binding of substrate to the other active sites.

  • The nonlinear response function model automatically incorporates parameters that yield the same response as an optimized Hill equation with its optimized parameters.

• Conclusions

  • The nonlinear response function approach accurately models MM as well as non-MM enzyme kinetics, both of which represent solutions to a nonlinear system of differential equations.

  • The nonlinear response function model requires no additional fitting of parameters beyond those which are contained in the response function.

  • The nonlinear response function model represents a more universal formulation for use in modeling enzyme-catalyzed reactions.