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B5. Analysis of the General Michaelis-Menten Equation

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  • The Michaelis-Menten equation can be simplified and studied under different conditions. First notice that \((k_2 + k_3)/k_1\) is a constant which is a function of relevant rate constants. This term is usually replaced by \(K_m\) which is called the Michaelis constant (which was used in the Mathematica graph above). Likewise, when \(S\) approaches infinity (i.e. \( S \gg K_m\), equation 5 becomes \(v = k_3[E_o]\) which is also a constant, called \(V_m\) for maximal velocity. Substituting \(V_m\) and \(K_m\) into equation 5 gives the simplified equation:

    \[v = \dfrac{V_m[S]}{K_m+ [S]} \label{10}\]

    It is extremely important to note that \(K_m\) in the general equation does not equal the \(K_s\), the dissociation constant used in the rapid equilibrium assumption! \(K_m\) and \(K_s\) have the same units of molarity, however. A closer examination of \(K_m\) shows that under the limiting case when \(k_2 \gg k_3\) (the rapid equilibrium assumption) then,

    \[K_m = \dfrac{k_2 + k_3}{k_1} \approx \dfrac{k_2}{k_1} = K_d = K_s. \label{11}\]

    If we examine Equations \ref{9} and \ref{10} under several different scenarios, we can better understand the equation and the kinetic parameters:

    • when \(S = 0\), \(v = 0\).
    • when \(S \gg K_m\), \(v = V_m = k_3E_o\). (i.e. \(v\) is zero order with respect to \(S\) and first order in E. Remember, \(k_3\) has units of s-1 since it is a first order rate constant. \(k_3\) is often called the turnover number, because it describes how many molecules of \(S\) "turn over" to product per second.
    • \(v = \dfrac{V_m}{2}\), when \(S = K_m\).
    • when \(S \ll K_m\), \(v = \dfrac{V_mS}{K_m} = \dfrac{k_3E_oS}{K_m}\) (i.e. the reaction is bimolecular, dependent on both on S and E. \(k_3/K_m\) has units of M-1s-1, the same as a second order rate constant.

    Notice that Equations \ref{9} and \ref{10} are exactly analysis to the previous equations we derived:

    • \(ML = \dfrac{M_oL}{K_d + L}\) for binding of \(L\) to \(M\)
    • \(J_o = \dfrac{J_m[A]}{K_d + [A]}\) for rapid equilibrium binding and facilitated transport of \(A\)
    • \(v_o = \dfrac{V_m[S]}{K_s + [S]}\) for rapid equilibrium binding and catalytic conversion of \(A\) to \(P\).
    • \(v_o = \dfrac{V_m[S]}{K_m+ [S]}\) for steady state binding and catalytic conversion of \(A\) to \(P\).

    Please notice that all these equations give hyperbolic dependencies of the y dependent variable (\(ML\), \(J_o\), and \(v_o\)) on the ligand, solute, or substrate concentration, respectively.

    Java Applet: Michaelis-Menten Plots