# CHAPTER 6 - TRANSPORT AND KINETICS

## B:  Kinetics of Simple and Enzyme-Catalyzed Reactions

BIOCHEMISTRY - DR. JAKUBOWSKI

4/10/16

 Learning Goals/Objectives for Chapter 6B:  After class and this reading, students will be able to write appropriate chemical and differential equations for the rate of disappearance of reactants or appearance of products for 1st order, pseudo first order, second order, reversible first order reactions draw and interpret graphs for integrated rate equations (showing reactant or product concentrations as a function of time) and initial rate equations (showing the initial velocity vo as a function of reactant ; derive kinetic rate constants from data and graphs of integrated rate and initial rate equations; write appropriate chemical, differential equations, and initial rate equations for the rate of disappearance of reactants or appearance of products for simple enzyme catalyzed reaction; differentiate between rapid and steady state assumptions; simplify the initial rate equation containing rate constants for an enzyme catalyzed reactions to one replacing the rate constants with kcat and KM, and give operational and mathematical definitions of those constants;

# B5.  Analysis of the General Michaelis-Menten Equation

This equation can be simplified and studied under different conditions. First notice that (k2 + k3)/k1 is a constant which is a function of relevant rate constants. This term is usually replaced by Km which is called the Michaelis constant (which was used in the Mathematica graph above).  Likewise, when S approaches infinity (i.e. S >> Km, equation 5 becomes v = k3(Eo) which is also a constant, called Vm for maximal velocity. Substituting Vm and Km into equation 5 gives the simplified equation:

Equation 10)   v = Vm(S)/(Km+ S)

It is extremely important to note that Km in the general equation does not equal the Ks, the dissociation constant used in the rapid equilibrium assumption!    Km and Ks have the same units of molarity, however. A closer examination of Km shows that under the limiting case when k2 >> k3 (the rapid equilibrium assumption) then,

Equation 11)  Km = (k2 + k3)/k1 = k2/k1 = Kd = Ks.

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

• when S = 0, v = 0.
• when S >> Km, v = Vm = k3Eo. (i.e. v is zero order with respect to S and first order in E. Remember, k3 has units of s-1since it is a first order rate constant. k3 is often called the turnover number, because it describes how many molecules of S "turn over" to product per second.
• v = Vm/2, when S = Km.
• when S << Km, v = VmS/Km = k3EoS/Km (i.e. the reaction is bimolecular, dependent on both on S and E. k3/Km has units of M-1s-1, the same as a second order rate constant.

Notice that equations 9 and 10 are exactly analysis to the previous equations we derived:

• ML = MoL/(Kd + L) for binding of L to M
• Jo = JmA/(Kd + A) for rapid equilibrium binding and facilitated transport of A
• vo = VmS/(Ks + S) for rapid equilbirum binding and catalytic conversion of A to P.
• vo = Vm(S)/(Km+ 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, Jo, and vo) on the ligand, solute, or substrate concentration, respectively.

Java Applet:  Michaelis-Menten Plots