Chapter Three: Part Two

Uses of Enzymes Food Production Cheese Beverage brewing Meat tenderizers Tofu production Wine-making Lactose removal in milk

Uses of Enzymes Chemical Industries Detergents Fructose production from glucose Amino acid production Vitamin production Glucose production from starch Aspartic acid production Urocanic acid production Ethanol production

Medical Uses Medical Applications Antibiotic production Blood purification Treatment of metabolic disorders Wound healing and antibacterial agents Treatment of stomach problems Blood clot removal Anticancer medicines

Michaelis-Menten Kinetics

Conservation of enzyme Enzyme Kinetics Enzymatic reaction E + S ES E + P k1 k-1 k2 Rate expression for product formation v = dP/dt = k2(ES) d(ES)/dt = k1(E)(S)-k-1(ES)-k2(ES) Conservation of enzyme (E) = (E0) – (ES)

K’m = k-1/k1 = [E][S]/[ES] Two Methods to Proceed Rapid equilibrium assumption: define equilibrium coefficient K’m = k-1/k1 = [E][S]/[ES] [ES]= [E][S]/ K’m Quasi-steady state assumption [ES] = k1[E][S]/(k-1+k2) [ES] = [E][S]/ Km Km = (k-1+k2)/k1

Michaelis-Menten Kinetics When v= 1/2 Vm, [S]= Km so Km is sometimes called the “half-saturation constant” and sometimes the “Michaelis constant”

Michaelis-Menten Kinetics unit on k2 is the inverse of time unit (k2 is also called the “turnover number”). Units on E0 are amount of enzyme (moles, grams, units, etc.) per unit volume Km has the same units as [S] (mole/liter, etc.)

Experimentally Determining Rate Parameters for Michaelis-Menten Kinetics Lineweaver-Burk Eadie-Hofstee Hanes- Woolf Batch Kinetics

Determining Parameters Rearrange the equation into a linear form. Plot the data. What kind of data do we have for an experiment examining enzyme kinetics (i.e., v, 1/v, [S], 1/[S], units)? The intercept and slope are related to the parameter values.

Enzyme Kinetics Experiment Place enzyme and substrate (reactants) in a constant temperature, well stirred vessel. Measure disappearance of reactant or formation of product with time. Why constant temperature? Why well stirred? What about the medium? Buffer?

Lineweaver-Burk Plot (double reciprocal plot) Rewrite Michaelis-Menten rate expression Plot 1/v versus 1/[S]. Slope is Km/Vm, intercept is 1/Vm

Graphical Solution y-intercept Slope 1/ V Slope = Km/ Vm 1/ Vm -1/ Km

Eadie-Hofstee Plot Rearrangement of Eq. 3.12b yields; V 15

Hanes-Woolf Plot Rearrangement of Eq. 3.12b yields; 16

Example Find Vm and Km using Lineweaver-Burk plot Eadie-Hofstee plot V(M/s) [S], (M) 0.0000117 0.00001 0.000015 0.0000175 0.00002 0.0000194 0.000025 0.000021 0.00003 0.0000223 0.000035 0.0000233 0.00004 0.0000242 0.000045 0.00005 Find Vm and Km using Lineweaver-Burk plot Eadie-Hofstee plot Hanes-Woolf plot 17

Lineweaver-Burk Slope=Km/Vm= 0.5686 y-intercept=1/Vm= 28687 Vm(M/s)= V(M/s) [S], (M) 1/V 1/S 0.0000117 0.00001 85470.09 100000 0.000015 66666.67 0.0000175 0.00002 57142.86 50000 0.0000194 0.000025 51546.39 40000 0.000021 0.00003 47619.05 33333.33 0.0000223 0.000035 44843.05 28571.43 0.0000233 0.00004 42918.45 25000 0.0000242 0.000045 41322.31 22222.22 0.00005 20000 Slope=Km/Vm= 0.5686 y-intercept=1/Vm= 28687 Vm(M/s)= 3.4859E-05 Km (M)=Slope*Vm= 1.98208E-05

Eadie-Hofstee Plot Slope=-Km= -0.00002 y-intercept=Vm= 0.000035 V(M/s) [S], (M) V/S 0.0000117 0.00001 1.17 0.000015 1 0.0000175 0.00002 0.875 0.0000194 0.000025 0.776 0.000021 0.00003 0.7 0.0000223 0.000035 0.63714 0.0000233 0.00004 0.5825 0.0000242 0.000045 0.53778 0.00005 0.5 Slope=-Km= -0.00002 y-intercept=Vm= 0.000035 Vm(M/s)= Km (M)= 0.00002

Hanes-Woolf Plot Slope=1/Vm= 28628 y-intercept=Km/Vm= 0.5701 Vm(M/s)= V(M/s) [S], (M) S/V 0.0000117 0.00001 0.854701 0.000015 1 0.0000175 0.00002 1.142857 0.0000194 0.000025 1.28866 0.000021 0.00003 1.428571 0.0000223 0.000035 1.569507 0.0000233 0.00004 1.716738 0.0000242 0.000045 1.859504 0.00005 2 Slope=1/Vm= 28628 y-intercept=Km/Vm= 0.5701 Vm(M/s)= 3.49308E-05 Km (M)= 1.99141E-05

Comparison of Methods   Lineweaver-Burk Eadie-Hofstee Hanes-Woolf Km 1.98208E-05 2.0E-5 1.99141E-05 Vm 3.4859E-05 3.5E-5 3.49308E-05 Lineweaver-Burk: supposedly gives good estimate for Vm, error is not symmetric about data points, low [S] values get more weight Eadie-Hofstee: less bias at low [S] Hanes-Woolf: more accurate for Vm.

Assignment # One Problem 3.1 (Textbook) Problem 3.2 (Textbook) Due date: Sunday, 16/10/2011 (during lecture). Late submission will not be marked and student will get zero.

Batch Kinetics The time course of variation of [S] in a batch enzymatic reaction can be determined from; By integration yield; Km and Vm from plotting (on y-axis) vs (on x-axis) using the following equation: Slope=Km Y-intercept=-Vm Or by plotting (on y-axis) vs (on x-axis) using the following equation: Slope= Y-intercept= 23

General Form E + S ES E + P k1 k2 k-1 The following set of ordinary differential equations (ODEs) must be solved simultaneously in order to obtain more general and more accurate parameter values (1) (2) (3) (4)

Allosteric Enzyme Kinetics In an enzyme with more than one substrate binding site, binding of one substrate molecule affects the binding of another. The cooperativity coefficient (n) can be determined from: n>1, cooperation; n<1, interference

Allosteric Enzymes Shape of rate curve is sigmoidal Michaelis-Menten