First time posting questions here, so let me know if I need to edit anything.

I need to decide $dC/dt$ and $dP/dt$ when the reaction is like this, where I only know we must have 2 substrates $S$ to react with the enzyme $E$ at the same time:

$$2S + E \overset{k_a}{\longrightarrow} C \overset{k_c}{\longrightarrow} 2P + E, \quad C \overset{k_b}{\longrightarrow} 2S + E$$

Is $dC/dt = 2k_a SE - (k_b + k_c)C$? Is $dP/dt = 2k_cC$? Thanks in advance.

Edit: alright, so $dC/dt$ should be $k_a S^2 E - (k_b + k_c)C$, right? Now suppose I take $E_0 = E + C$ to be the total enzyme concentration, and transform $dC/dt$ as $$dC/dt = k_a E_0 S^2 - (k_a S^2 + k_b + k_c) C.$$ By quasi-equilibrium approximation, let $$dC/dt = 0 \Rightarrow C = (k_a E_0 S^2)/(k_a S^2 + k_b + k_c).$$ Then in this equilibrium, $$dP/dt = 2k_bC = (2k_a k_b E_0 S^2)/ (k_a S^2 + k_b + k_c).$$ So if we want to find out $V_m$ and $K_m$ s.t. $$\lim_{S \to \infty} dP/dt = V_m, \quad dP(S = K_m)/dt = V_m/2,$$ then we would need $V_m = 2 k_b E_0, K_m = \sqrt{(k_b + k_c)/(k_a)}$. Any mistakes so far?

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    $\begingroup$ This is very close to the Michaelis-Menten scheme. The rate equations are ok, you now need to assume that C is at steady state dC/dt=0 and solve the equation for C. Look up the MM scheme for how to do this if you are not sure. $\endgroup$ – porphyrin May 8 at 14:56
  • $\begingroup$ Sure thanks. May I also confirm that $V_m$ and $K_m$ in this case would be $2k_c E_0$ and $(k_b + k_c)/(2k_a)$ respectively? $E_0$ is the total enzyme concentration which is assumed to be constant. $\endgroup$ – Yuki.F May 8 at 15:00
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    $\begingroup$ IMO one of the best derivations of the Michaelis-Menten equation, together with a lucid explanation of why the steady-state hypothesis is superior to the 'equilibrium' assumption of Michaelis and Menten, is given by Haldane (1930) in his book 'Enzymes'. The first chapter of his book is available here, made available through DC's Improbable Science. (See also here). $\endgroup$ – tomd May 8 at 19:49
  • $\begingroup$ I did not notice before (sorry) but you should have $dC/dt=k_aS^2E-(k_b+k_c)C=0$ at steady state. $\endgroup$ – porphyrin May 9 at 6:46
  • $\begingroup$ @tomd I don't have any more info about any $k$ or any substances in this problem. How is it now? $\endgroup$ – Yuki.F May 9 at 7:37

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