# 6.021/Notes/2006-09-29

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Revision as of 08:30, 11 October 2006 by Austin J. Che (talk | contribs)

## 4 state model

- Simplify the model with assumptions
- [math]\alpha_1=\alpha_3, \beta_1=\beta_3[/math] (binding same on inside and outside)
- [math]\alpha_2=\alpha_4, \beta_2=\beta_4[/math] (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 [math]\mathfrak{N}_E[/math] (per surface area) [math]\mathfrak{N}_E=c_E\cdot d[/math] where [math]d[/math] 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 [math]\alpha, \beta[/math]
- Then it is partitioned into whether has substrate bound by [math]K[/math] and [math]c_s[/math]
- 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:

[math]\mathfrak{N}^i_{ES}=\frac{\beta}{\alpha+\beta}\frac{c^i_s}{c^i_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^i_{E}=\frac{\beta}{\alpha+\beta}\frac{K}{c^i_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^o_{ES}=\frac{\alpha}{\alpha+\beta}\frac{c^o_s}{c^o_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^o_{E}=\frac{\alpha}{\alpha+\beta}\frac{K}{c^o_s+K}\mathfrak{N}_{ET}[/math]

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