6.021/Notes/2006-09-29

4 state model

 * Simplify the model with assumptions
 * $$\alpha_1=\alpha_3, \beta_1=\beta_3$$ (binding same on inside and outside)
 * $$\alpha_2=\alpha_4, \beta_2=\beta_4$$ (ability for protein to translocate/flip is independent of solute)
 * Binding fast relative to translocation
 * Only care about the dissociation constant as it will always be in steady state
 * Instead of concentrations (which is per volume), it is easier to think about $$\mathfrak{N}_E$$ (per surface area) $$\mathfrak{N}_E=c_E\cdot d$$ where $$d$$ is the membrane thickness
 * This leads to the simple symmetric four state carrier model
 * The solution can be interpreted intuitively
 * The enzyme is first partitioned into facing in or out depending on $$\alpha, \beta$$
 * Then it is partitioned into whether has substrate bound by $$K$$ and $$c_s$$
 * The concentration difference between inside and outside is not important. All that matters is the concentration relative to K.

Solution to simple symmetric 4-state carrier model:

$$\mathfrak{N}^i_{ES}=\frac{\beta}{\alpha+\beta}\frac{c^i_s}{c^i_s+K}\mathfrak{N}_{ET}$$

$$\mathfrak{N}^i_{E}=\frac{\beta}{\alpha+\beta}\frac{K}{c^i_s+K}\mathfrak{N}_{ET}$$

$$\mathfrak{N}^o_{ES}=\frac{\alpha}{\alpha+\beta}\frac{c^o_s}{c^o_s+K}\mathfrak{N}_{ET}$$

$$\mathfrak{N}^o_{E}=\frac{\alpha}{\alpha+\beta}\frac{K}{c^o_s+K}\mathfrak{N}_{ET}$$

$$\phi_s=(\phi_s)_{max}(\frac{c^i_s}{c^i_s+K}-\frac{c^o_s}{c^o_s+K})$$; $$(\phi_s)_{max}=\frac{\alpha\beta}{\alpha+\beta}\mathfrak{N}_{ET}$$