This document summarizes my development of different models for 1D NMR line shapes. General discussion of Bloch-McConnell equations and their use to account for conformational exchange and binding is summarized in my lectures here.
Equilibrium thermodynamic models for corresponding line shapes were developed in Mathematical_models/Equilibrium_thermodynamic_models/.
The critical component of math required to compute line shapes is an appropriate kinetic matrix for the exchanging system accounting for multiplicity of observed spins in particular species. For all models where spins "gather" in some species, such as R<=>R2 (so not simply transfered in one-to-one fashion as in R<=>RL) I perform explicit derivation of the kinetic matrix using MuPad to document the derivation. The simplest example of such derivation is R2 model (R<=>R2 process) developed in My_lectures (see R2.html and R2.mn).
NOTE: I am not normalizing spin concentrations to range from 0 to 1 (to become populations). This creates opportunity to fit concentration-dependent signals if raw spectral data is imported. One needs to make sure that no normalization is applied to the data upon import. Important: the current version (as of 6/23/2011) of BiophysicsLab_1D_NMR module DOES normalize the data!
IMPORTANT: I have switched signs of the rates in derivation of the kinetic matrices starting from U-R-RL. This has no effect on the resulting matrices but is more logical.
NOTE 1: The most recent workflow for testing of the model operation in BiophysicsLab is in Tutorial_6.Testing_a_new_model.
NOTE 2: For models that are solved numerically---keep L/R>=0.01, otherwise numeric solutions are unstable! Same applies to values of constants if you want to 'turn off' a specific transition: make them small but not too small!