BL482
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BASIC BIOCHEMICAL TECHNIQUES
"Scoop" on Enzyme Kinetics
Dr. Bill Campbell
I. Introduction
Enzymes are biocatalysts. To serve this role in living systems, enzymes must have a specific site for binding the substrates of the reaction they catalyze and the enzyme must form a complex with its substrates. In honor of the first biochemists to describe enzyme kinetic phenomena, the complex of enzyme and substrate is called the Michaelis complex:
Figure 1. Formation of ES complex.
This complex can breakdown to form the free enzyme and substrate or go on to form product, which we will treat as an irreversible reaction:
Figure 2. Production of product and regeneration of Enzyme from ES complex.
We assume also that the formation of the ES complex is very rapid and that [ES] reaches a "steady state" or constant level in a few millisec. Hence, this type of enzyme kinetic analysis is know as steady state kinetic analysis.
II. Substrate Saturation of the Enzyme
The definition of a catalyst is something which accelerates the rate of a reaction without being changed itself. For a biocatalyst or enzyme to be most effective, it should be able to do its job with a much lower concentration of itself as compared to the substrate(s) it processes. This leads to the concept that there are a limited number of catalytic sites in a solution containing enzyme and substrate(s): [E] much less than [S]. Consequently, the enzyme will become "saturated" with substrate and the initial velocity (v) of the enzyme catalyzed reaction will react a limiting value called the maximum velocity (Vmax). This is most easily observed by plotting initial velocity versus substrate concentration.
Graph 1. v vs. [S] plot.
This is response of the enzyme catalyzed reaction can be described by the Michaelis-Menton equation:
Figure 3. Michaelis-Menton Equation.
This equation defines the shape of a square hyperbola, which is the shape shown for the plot of v vs. [S]. The constants -- Vmax and Km --- can be obtained from a set of experimental data where v is measured at different [S]. You are doing this experiment this week in lab for the acid phosphatase. And while you need to make a plot of v vs. [S] and turn it in as part of your lab report to establish that the phosphatase is obeying the Michaelis-Menton equation, it is best to use other methods to obtain values for Vmax and Km. But before we get to that stuff, let us make a couple of definitions a little clearer. Vmax is the asymptote of the graph above which v approaches at very high [S]. The units of Vmax are the same as those of v which should be in terms of the quantity of product formed per unit time (i.e. for the acid phosphatase v and Vmax should be in µmol or nmol of pNitrophenolate formed/min or hr). Operationally, the Km is defined as the substrate concentration which gives 0.5 Vmax (verify this for yourself by substituting [S] = Km in the Michaelis-Menton equation shown above). So Km values are in the same units as substrate concentration (i.e. for the acid phosphatase if [pNPP] is in µM then Km must also be in µM). However, more importantly, the Km is a measure of the strength of the Michaelis complex between enzyme and substrate. Let us make a clear distinction between the binding constant of substrate to the enzyme (Ks) and the Michaelis Constant (Km) -- it is often stated that the Km measures the "affinity" of the enzyme for its substrate. But this is only true in special cases because the Km also includes a measure of how well the enzyme catalyzes the formation of product after the enzyme substrate complex has formed. As it turns out the ratio of Vmax/Km is the best way to compare enzymes. The "V/K" constant as it is sometimes called is a measure of the "efficiency of catalysis" -- if the V/K is large then the enzyme is very efficient. However, there is an upper limit to efficiency which is determined by how fast the substrate can diffuse to the enzyme in solution. Some enzymes have attained this limit over evlutionary time, but most have not and are limited by chemistry that takes place during catalysis.
III. Calculating Km and Vmax from Experimental Data
The best solutions to the problem of calculating Km and Vmax are to use a computer program. I have a computer program available in my lab that runs on a PC in DOS (called EnzPlot) which you can use to obtain these kinetic constants for your data (It also has a tutorial on enzyme kinetics). But even if you use the computer, you must also make a graphical representation of these results using the "Lineweaver-Burk" plot, which is a linear transformation of the Michaelis-Menton equation generated by taking the reciprocal of both sides of the equation:
Figure 4. Equation for Lineweaver-Burk Plot (Linear Transformation of M&M Equation).
To use this equation and make the graph required for your lab report, you take the reciprocal of each of your datum points or in other words for each v you calculate 1/v and for each [S] you calculate 1/[S], then you plot 1/v versus 1/[S]:
Graph 2. Double Reciprocal Plot of 1/v versus 1/[S].
You get the Km and Vmax from the graph as shown on it. Since the above equation is the one for a line (i.e. y = a + bx), you can make a linear regression or "least-squares fit" of your data for this graph and calculate Km and Vmax from the equation of the line.
IV. Simple Inhibitors of Enzymes and their Kinetic Analysis
Inhibitors of enzymes have a long history of usefulness in gaining understanding of how enzymes work and unraveling metabolic pathways. You will use two simple inhibitors in the lab this week: 1) the first is inorganic phosphate, Pi, which is the product of the phosphatase reaction and acts as a classical competitive inhibitor of the enzyme; and 2) fluoride, F-, which is an anion often found to inhibit phosphate metabolizing enzymes and acts as a classical noncompetitive inhibitor of the acid phosphatase. Well, what are competitive and non-competitive inhibitors of an enzyme. A competitive inhibitor is a substance which chemically resembles the substrate and its inhibition can be overcome by high concentrations of the substrate. A noncompetitive inhibitor is a substance with no chemical similarity to substrate and its inhibition can not be overcome by high concentration of substrate. Let us go back to our model of the kinetics of the phosphatase and see how these inhibitors act:
Competitive Inhibition:
Figure 5. Model for Competitive Inhibitor.
The competitive inhibitor, Ic, forms a complex with the enzyme at the substrate binding site and prevents the substrate from getting in. But the EIc complex is reversible and so as more substrate is added, the substrate manages to get in and overcomes the effects of the inhibitor. Consequently, the overall effect on the kinetics is to increase the Km and we call this new kinetic constant, the apparent Km or Km* (called Km prime). The maximum velocity which is attained is the same in the presence and absence of the competitive inhibitor, so Vmax is the same for both the uninhibited and competitively inhibited reaction. This is illustrated below using graphes for the response of an enzyme to competitive and noncompetitive inhibitors.
NonCompetitive Inhibition:
Figure 6. Model for Noncompetitive Inhibitor.
The noncompetitive inhibitor, Inc, forms a complex with enzyme, which is unaffected by the substrate concentration, such that it does not matter if substrate is already bound to the enzyme when the noncompetitive inhibitor binds or not. In this classical type of noncompetitive inhibition, the Km is not altered and only the Vmax is decreased. Thus, the real Vmax remains the same and you obtain an apparent Vmax or Vmax* (called Vmax prime).
Graph 3 and Graph 4 illustrate the expected results for a competitive and noncompetitive inhibitor.
Graph 3: v vs. [S] plot for competive and noncompetitive inhibitors.
Graph 4: Double Reciprocal plot for competitive and noncompetitive inhibitors.
V. Calculation of the Ki for Competitive and NonCompetitive Inhibitors
The strength of the complex formed between the enzyme and the inhibitor can be calculated from the kinetics constants obtained in the analysis of steady state kinetics:
Figure 7. Table of Equations for Calculating Inhibitor Binding Constants (Ki values).
The symbols Km and Vmax have the same meaning here as in the standard kinetic analysis, while in the inhibited reactions, Km* represents the "Km" obtained in the graphical solution of the data collected in the presence of a competitive inhibitor and Km* will always be greater than Km. Vmax* is obtained from the graphical solution of the data collected in the presence of the noncompetitive inhibitor and will always be less than Vmax. The inhibitors of both types are used at known concentrations ([I]) and these must be used to find the strength of the binding of the inhibitor to the enzyme (Ki) and the Ki has the same units of concentration as does the inhibitor. So if [I] = 1 mM, then the units of Ki will be mM. So the degree to which the inhibitor effects the kinetics of the enzyme depends on both the strength of its complex with the enzyme, with a low Ki indicating strong binding of the inhibitor to the enzyme, and the concentration of the inhibitor added to the kinetic analysis. When you make your lab report, calculate the Ki constants for Pi and F- and compare them. The one with the lower Ki is the stronger inhibitor.
Which is the stronger inhibitor of acid phosphatase, Pi or F-, according to your kinetic data?
You can in another sense compare the Ki constants to the Km for substrate, which is also expressed as a concentration, and make a general judgment as to whether the strength of the Michaelis complex (sort of the affinity of the enzyme for the substrate) is greater than the strength of the complex of inhibitor and enzyme. Since the competitive inhibitor binds at the same site as the substrate (especially in the case of the phosphatase you are studying since the competitive inhibitor being used is the product of the reaction), how does the Ki for Pi compare to the Km for pNPP? In another sense, the noncompetitive inhibitor, which apparently binds to the enzyme at a place other than the substrate binding site, will generally have a Ki with less meaningful relationship to the Km of the enzyme.
©Wilbur H. Campbell, 1995, 1996; wcampbel@mtu.edu