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

Effect of dose on riboflavin urinary recovery when given on an empty stomach

Effect of dose on ascorbic acid absorption. April 7, 2019

Steady-state Vitamin C plasma concentration as a function of dose in 13 female subjects receiving doses from 30 to 2,500 mg.

CAUSES OF DOSE DEPENDENT KINETICS Drug GI absorption Saturable transport in gut wall Riboflavin, L-dopa Intestinal metabolism Propanolol Distribution Saturable plasma protein binding Salicylic acid , phenytoin Cellular uptake 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 It assumes the formation of an enzyme- substrate complex It assumes that the ES complex is in rapid equilibrium with free enzyme 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 k-2 For the reaction E + S ES P + E the reverse reaction can be neglected at the beginning (initial rate with [P]=0) k-2

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 So: k1[E][S] = k-1[ES] + k2[ES]

[ES] = ([E][S])/KM 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

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

The "kinetic activator constant" Understanding Km The "kinetic activator constant" Km is a constant Km is a constant derived from rate constants Km is, under true Michaelis-Menten conditions, an estimate of the dissociation constant of E from S Small Km means tight binding; high Km means weak binding

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

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 (double-reciprocal)

Hanes–Woolf plot invert and multiply by [S]: Rearrange

b. In Vivo Determination Vmax Km K0 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 patient’s Km and Vmax for phenytoin.

Dose (mg/d) Css (mg/L) Dose Rate/Css (L/d) 100 3.7 27 300 47 6.4 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) Dose Rate/Css (L/d) 100 3.7 27 300 47 6.4 Vmax = 362 mg/d K0 (mg/d) Km = 9.7 mg/L K0/Css (L/d) April 7, 2019

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

Competitive inhibitor 2) Vmax unaltered, Km increased

Noncompetitive inhibitor 3) Km unaltered, Vmax decreased

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

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