Question

Derive the Michaelis-Menten equation. Convert the derived equation to a linear form.

Answer 1

Michaelis - Menten equation can be derived as follows.

Michaelis - Menten equation Mechanism for the kinetics of enzyme catalysed reactions were proposed by Michaelis and Menten. Enzyme catalysed reactions take place in two steps. 1 Enzyme-Substrate complex is fogmed. E+S > ES (complex) © Enzyme- Substrate complex is decomposed. Es ke PTE (product) To derive the equation for kinetics of enzyme catalysed reaction, we use steady state approximation, for the intermediate, Es. d [ES] - k, EJ[S] - ky[es] -k, [ES] = 0 - ☺ dt Here [] cannot be measured experimentally. This can be calculated from the enzyme conservation equation.

E1. – [E] + [ES] - © >>[E] = [E]. - [ES] where [[] is the total enzyme concentration [E] the free enzyme concentration ES] is the botand enzyme concentration

Substituting eqn. in equation , we have. d[es] . k, (E).- [ES]) 6J - K [es] -k, [ES] = 0 dt dt → d[es] . k, 11.S] - [ES][S] - [es] -k, [ES] - O => K, (E).E] = (k, [S] + k, + k) [ES] => [ES] Ek, EL [S] k[s] tk, tk, Rate of the reaction is determined by the slowest step of the readion. This slowest stepis given by decomposition of enzyme substrate complex, Hence the rate of the reaction would be r - d[s] = + d[p] =k, [ES] - dt dt 0 mes Substituting egn. ④ in equ. we have war k k, E]. S] K [s]+k, tk / 7 kg k, [c]. [s] meleg k[s] tk, tko no Dividing the fraction by k, we have 4= K[C].[s] [s] +((k_, +km)/k) 21 = k, E). [s] [s]+ km

where km = ky tkz and is called Michaelis constant. Equation is the michaelis-menten equation, a r ): 19. Conversion into linear form The reaction occurs at maximum rate when all of the enzyme has reacted with the Substrate at high concentration. In this ease no free enzyme remains. Hence, [J. = [es] (From equation © og 35 W => nate (1) = "max = Vmax =k, Eb - (from equ. NOTE: may is a notation for maximum rate. COM Now substituting egn. (8 in ean 6 we have, HN max

a = Vmax [s] [s] + km Taking reciprocal on both sides, 1 = [s] + km max [s] 4 I Vmax [5]. Vmax Rearranging we have Vmax IST max is the linear foam of Michaelis - Menten Equ. ean. Islope (m) = km Vmax 1/a Vmax [S]