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4 state model

  • Simplify the model with assumptions
    • [math]\displaystyle{ \alpha_1=\alpha_3, \beta_1=\beta_3 }[/math] (binding same on inside and outside)
    • [math]\displaystyle{ \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]\displaystyle{ \mathfrak{N}_E }[/math] (per surface area) [math]\displaystyle{ \mathfrak{N}_E=c_E\cdot d }[/math] where [math]\displaystyle{ 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]\displaystyle{ \alpha, \beta }[/math]
    • Then it is partitioned into whether has substrate bound by [math]\displaystyle{ K }[/math] and [math]\displaystyle{ 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]\displaystyle{ \mathfrak{N}^i_{ES}=\frac{\beta}{\alpha+\beta}\frac{c^i_s}{c^i_s+K}\mathfrak{N}_{ET} }[/math]

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

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

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

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