Michaelis menten kinetics Nonlinear Pharmacokinetics

The saturable concentration when the velocity of the reaction is equal to half of the maximum velocity or 0.5 V max

1. M ichaelis M enten kinetic characteristics Syed Umar Farooq M pharm ( Ph.D ) Asst Professor Care College of Pharmacy

2. contents Introduction Michaelis Menten characteristics' Assumptions and limitation Derivation Applications References

3. Definition The saturable concentration when the velocity of the reaction is equal to half of the maximum velocity or 0.5 V max

4. Nonlinear pharmacokinetics In some cases the rate process of drug depends on carrier or enzyme that are substrate specific ,have definite capacities and susceptible to saturation at high concentration In such cases first order transforms in to mixture of first order and zero order rate process and pharmacokinetic parameters change with dose Pharmacokinetics of such drugs are said to be dose-dependent, or mixed-order ,nonlinear and capacity limited kinetics

5. Test to detect non linear pharmacokinetics Determination of steady state plasma concentration at different doses If steady state plasma concentration is directly proportional to dose then linearity in kinetics exist. Determination of some of the important pharmacokinetics parameter such as fractional bio available, elimination half life or total systemic clearance at different doses of drug any change in these parameters indicate non linearity

6. Causes of non-linearity Occurs in Absorption When absorption involves carrier mediate transport system eg . Absorption of riboflavin, ascorbic acid When absorption is solubility or dissolution rate limited e.g griseofulvin When pre systemic gut wall or hepatic metabolism attains saturation

7. In drug distribution Non linearity in drug distribution of drugs administered at high doses may be due to Saturation of binding sites on plasma proteins e.g naproxen Saturation of tissue binding sites E,g thiopental and fentanyl

8. In metabolism Capacity limited metabolism due to enzyme and co factor saturation E.g phenytoin alcohol Enzyme induction E.g carbamazepine In drug excretion Active tubular secretion e.g penciline g After saturation of carrier system a decrease in renal claerence occur Active tubular re absorption e.g water soluble vitamins

9. Change in concentrations over time for enzyme E, substrate S, complex ES and product P

10. … -dc/ dt = VmC /Km +C Where –dc/ dt = rate of decline of drug concentration at the time t, Vm = theoretical maximum rate of the process. Km = michaelis menten constant.

11. characteristics Three situation can be considered depending up on values of Km and C When Km= C -dc/ dt = Vmax /2 i.e rate of process is equal to one half its maximum rate When Km= C

12. When Km = C A plot of Michaelis Mentens equation

13. When Km>>C Here Km + C = Km and equation reduces to The above equation is identical to one that describes to first order elimination of a drug i.e drug concentration in the body is below Km PHENYTOIN and ALCOHOL are exceptions

14. When Km << C Under this condition Km + C = C And equation will become the above equation is identical to that of zero order process i.e. rate process occurs a constant rate Vmax and is independent of drug concentration E.g metabolism of ethanol

15. Assumptions and limitations The first step in the derivation applies the law of mass action, which is reliant on free  diffusion . However, in the environment of a living cell where there is a high concentration of protein, the cytoplasm often behaves more like a gel than a liquid, limiting molecular movements and altering reaction rates

16. Equilibrium approximation In their original analysis, Michaelis and Menten assumed that the substrate is in instantaneous  chemical equilibrium  with the complex, and thus  k f [ E ][ S ] =  k r [ ES ].  Combining this relationship with the enzyme conservation law, the concentration of complex is

17. Contd … where  K d  =  k r  /  k f  is the  dissociation constant  for the enzyme-substrate complex. Hence the velocity  v  of the reaction – the rate at which P is formed – is where  V max    =  k cat [ E ]  is the maximum reaction velocity.

18. Derivation Applying the  law of mass action , which states that the rate of a reaction is proportional to the product of the concentrations of the reactants, gives a system of four non-linear  ordinary differential equations  that define the rate of change of reactants with time  t

19. In this mechanism, the enzyme E is a  catalyst , which only facilitates the reaction, so its total concentration, free plus combined, [ E ] + [ ES ] = [ E ]  is a constant. This conservation law can also be obtained by adding the second and third equations above

20. Applications Enzyme K m  (M) k cat  (1/s) k cat  /  K m  (1/M.s) Chymotrypsin 1.5 × 10 -2 0.14 9.3 Pepsin 3.0 × 10 -4 0.50 1.7 × 10 3 Tyrosyl-tRNA synthetase 9.0 × 10 -4 7.6 8.4 × 10 3 Ribonuclease 7.9 × 10 -3 7.9 × 10 2 1.0 × 10 5 Carbonic anhydrase 2.6 × 10 -2 4.0 × 10 5 1.5 × 10 7 Fumarase 5.0 × 10 -6 8.0 × 10 2 1.6 × 10 8

21. … The constant  k cat  /  K m  is a measure of how efficiently an enzyme converts a substrate into product. It has a theoretical upper limit of 10 8  – 10 10  / M.s ; enzymes working close to this, such as fumarase, are termed superficies

22. Michaelis- Menten kinetics have also been applied to a variety of spheres outside of biochemical reactions alveolar  clearance of dusts the  richness of species  pools clearance of  blood alcohol the  photosynthesis-irradiance  relationship and bacterial  phage  infection

23. References Milo Gibaldi , pharmacokinetics second edition, page no 271-278 http://www.ncbi.nlm.nih.gov/books/NBK22430/figure/A1052/?report=objectohttp://en . wikipedia.org/wiki/File:Michaelis-Menten_saturation_curve_of_an_enzyme_reaction_LARGE.svg nly

24. Thank u………..