Michaelis-Menten Kinetics Presented by Avinash Kodoori M.Pharm I semester Dept. of Pharmaceutics, Univ. College of Pharmaceutical Sciences, Kakatiya University, Warangal.

Contents Linear and Non linear kinetics Causes The Michelis Menten kinetics Introduction Assumptions Derivation Important conclusions

Understanding Km and Vmax Determination of Km and Vmax In vitro , Lineweaver-Burk plot In vivo Applications % of Dose Recovered in Urine

Effect of dose on riboflavin urinary recovery when given on an empty stomach 60 50 40 30 20 10 0 0

5 10 15 20 25 30 35 Dose (mg) 4 % of Dose Absorbed 100

80 60 40 20 0 100 1000 10000

Daily Dose (mg) Effect of dose on ascorbic acid absorption. 2/6/20 5 Steady-state Vitamin C plasma concentration as a function of dose in 13 female subjects receiving doses from 30 to 2,500 mg. 6

CAUSES OF DOSE DEPENDENT KINETICS Cause Drug GI absorption Saturable transport in gut wall Riboflavin, L-dopa

Intestinal metabolism Propanolol Distribution Saturable plasma protein binding Cellular uptake Salicylic acid , phenytoin Methicillin

Renal elimination Active secretion p-amino hippuric acid Tubular reabsorption Riboflavin, ascorbic acid Metabolism

Saturable metabolism Phenytoin, valproic acid Metabolite inhibition Diazepam The Michaelis-Menten Kinetics

Louis Michaelis and Maude Menten's theory (1913) Applies to Enzymes - Carriers - proteins Related terms - Dose dependent kinetics - Saturation Kinetics

- Capacity limited kinetics Saturation Kinetics/ Capacity limited kinetics Assumptions 1. It assumes the formation of an enzymesubstrate complex 2.

It assumes that the ES complex is in rapid equilibrium with free enzyme 3. Breakdown of ES to form products is assumed to be slower than 1) formation of ES and 2) breakdown of ES to re-form E and S Derivation

For the reaction E+S ES P+E k-2 the reverse reaction can be neglected at the beginning (initial rate with [P]=0)

1) The rate of formation of the product v = k2[ES], 2)[Enzyme]total = [E]t = [E] + [ES] 3) At steady state: d[ES]/dt = 0 4) KM = (k-1 + k2)/k1 The rate of formation of ES is given as Assume steady state:

d[ES]/dt = 0 0 k1 [ E ][ S ] k 1 [ ES ] k 2 [ ES ] So: k1[E][S] = k-1[ES] + k2[ES] Rearranging we get [ES] = (k1/(k-1 + k2))[E][S]

Substituting (KM = (k-1 + k2)/k1): [ES] = ([E][S])/KM KM[ES] = [E][S] Substituting ([E] = [E]t - [ES]): KM[ES] = [E]t[S] - [ES][S]

Rearranging: [ES](KM + [S]) = [E]t[S] So:

The rate of formation of the product is given as [ ET ][ S ] [ ES ] (Km S ) the maximum rate is given as hence The Michaelis Mentens Equation

At [S] Km, Vo is proportional to [S] At [S] Km, Vo = Vmax Important Conclusions of Michaels Menten Kinetics when [S]= KM, the equation reduces to

when [S] >> KM, the equation reduces to when [S] << KM, the equation reduces to Understanding Km The "kinetic activator constant" K is a constant m K is a constant derived from rate

m constants K is, under true Michaelis-Menten m conditions, an estimate of the dissociation constant of E from S Small K means tight binding; high K m m means weak binding

Understanding Vmax The theoretical maximal velocity V max is a constant V max is the theoretical maximal rate of the reaction - but it is NEVER achieved in reality To reach V max would require that ALL enzyme molecules are tightly bound with substrate V

max is asymptotically approached as substrate is increased The dual nature of the Michaelis-Menten equation Combination of 0-order and 1st-order kinetics When S is low, the equation for rate is 1st order in S When S is high, the equation for rate is 0-order in S

The Michaelis-Menten equation describes a rectangular hyperbolic dependence of v on S a)Invitro determination Lineweaver Burk Double Reciprocal Plots It is difficult to determine Vmax experimentally

The equation for a hyperbola can be transformed into the equation for a straight line by taking the reciprocal of each side Vmax C v Km C 1 Km C

v Vmax C Km 1 1 v Vmax C Vmax The formula for a straight line is y = mx + b

A plot of 1/V versus 1/[S] will give a straight line with slope of KM/Vmax and y intercept of 1/Vmax Such a plot is known as a Lineweaver-Burk double reciprocal plot

Lineweaver-Burk plot (doublereciprocal) 29 HanesWoolf plot invert and multiply by [S]: Rearrange

b. In Vivo Determination Vmax Vmax Css v K m C ss If K 0 input rate Vmax C ss K0 K m C ss

Km K0 K 0 ( K m C ss ) Vmax Css K 0 K m K 0Css Vmax C ss K 0Css (Vmax Css ) K 0 K m K 0 Vmax K0

K m Css K0/Css JB is an 18 year male receiving phenytoin for prophylaxis of post-traumatic head injury seizures. The following steady state concentrations were obtained at the indicated doses: Dose (mg/d) Css (mg/L) 100

3.7 300 47 From this data, determine this patients Km and Vmax for phenytoin. JB is an 18 year male receiving phenytoin for prophylaxis of post-traumatic head injury seizures. The following steady state concentrations were obtained at the indicated doses: Dose (mg/d) Css (mg/L)

100 3.7 27 300 47 6.4 Dose Rate/Css (L/d) Vmax = 362 mg/d

K0 (mg/d) Km = 9.7 mg/L K0/Css (L/d) 2/6/20 34 Significance of Km Km is a constant Small Km means tight binding; high Km means weak

binding Useful to compare Km for different substrates for one enzyme Hexokinase : D-fructose 1.5 mM D-glucose 0.15 mM Useful to compare Km for a common substrate used by several enzymes Hexokinase: D-glucose 0.15 mM Glucokinase: D-glucose 20 mM

Other applications 1)The Catalytic Efficiency kcat, the turnover number, is the number of substrate molecules converted to product per enzyme molecule per unit of time, when E is saturated with substrate. kcat/Km is an apparent second-order rate constant which measures how the enzyme performs when S is low 2) Competitive inhibitor

Vmax unaltered, Km increased 39 3) Noncompetitive inhibitor Km unaltered, Vmax decreased 40

References Leon Shargel, A Text book of Applied Bio pharmaceutics and pharmacokinetics,5th edition Milo Gibaldi , Donald Perrier, Pharmacokinetics,2nd edition V.Venkateshwarlu, Biopharmaceutics and pharmacokinetics,(2004) Garret & Grisham , Biochemistry, 2nd edition http://en.wikipedia.org/wiki/Michaelis-Menten_kinetics German biochemist and physician

Canadian medical scientist Thank You