# Weak, fast-off binding binding coupled with isomerization in fast exchange

Here we will compare outcomes of the four alternative models for conditions when binding is weak and the off-rate is fast. The isomerization is chosen to be slow here. I choose binding affinity of 106 M-1 and the off-rate constant of 500 s-1. R and RL are separated by 100 s-1 and R* or R*L - by 100 s-1 from corresponding species. Isomerization equilibrium is shifted 5:1 toward *-forms. The reverse isomerization rate is 170 s-1 to make kex =1000 s-1 .

Af_Bf/

Simulate setup U_Af

Simulate setup U_R_Af_Bf

Simulate setup U_L_Af_Bf

Simulate setup U_RL_Af_Bf

Forward and reverse titration is capable of picking up U_R/U_L character: there is remarkable shifting of the FWHH maximum to 0.6 in U_R while maximum in U_L stays about where U model has it 0.4 (in fact, a little earlier than that , at 0.3!)  Line broadening is most significant in U_RL model. However it's patter nicely matches the line broadening curve in U-model.

## Lets fit all these cases with the 2-site model

Optimum norm: 1.07e-03
[1] Kd: 5.93180e-07 +/- 1.6e-08       pre-set Kd=1e-6
[2] Koff: 5.17793e+02 +/- 5.6e+00  pre-set Koff=500
[3] Scale Factor: 2.04803e+00 +/- 1.9e-03
Optimum norm: 6.21e-02
[1] Kd: 1.37668e-06 +/- 1.6e-07
[2] Koff: 3.60079e+02 +/- 2.1e+01
[3] Scale Factor: 2.00497e+00 +/- 1.5e-02
Optimum norm: 8.81e-03
[1] Kd: 1.83048e-06 +/- 7.2e-08
[2] Koff: 4.94915e+02 +/- 1.5e+01
[3] Scale Factor: 2.04637e+00 +/- 5.5e-03
Optimum norm: 6.33e-03
[1] Kd: 1.23120e-07 +/- 1.8e-08
[2] Koff: 7.36641e+01 +/- 6.5e-01
[3] Scale Factor: 2.09303e+00 +/- 4.3e-03
Fitted parameters correspond well with the preset ones, though the confidence intervals are overly optimistic. Remarkable poor fit, characteristic of U_R model Accurate fit reproducing parameters of A transition Fit is very close but fails at the end of titration. Having more points at the end of titration should magnify this effect and help discern U_RL from U model.

## Conclusion

1. The four models give remarkably different fitting discrepancies with the 2-site model.
2. Forward and reverse titrations are able to tell apart U_R/U_L model from U and U_RL.
3. Additional points near equivalence region enhance discrepancy between the U and U_RL models, which should give similar pattern in forward and reverse titrations and enable distinguishing U_RL.

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