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NMR Line Shapes: development of kinetic matrices




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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. An overview of behavior of NMR line shapes of multi-state systems is given in my paper: Kovrigin EL NMR line shapes and multi-state binding equilibria (2012) JBNMR 53, 257-270  (available upon request).

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

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 are 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 IDAP_1D_NMR module DOES normalize the data!

NOTE 2: IDAP models used in TITAN do not normalize the data.


Dimensionality definitions

Generally, line shapes in NMR spectra are multidimensional. For example, the spectrum may have one frequency axis, two or more. This gives rise to two-, and three-dimensional data sets, etc. because of addition of intensity axis. However, by convention, we indicate dimensionality of the datasets only by number of frequency axes. Therefore, the traditional CW NMR spectrum is one-dimensional (1D), a heteronuclear experiment with 1H-15N correlation is 2D, and so on.

NOTE: Development of kinetic matrices for particular models was first done for analysis of one-dimensional line shapes. In August 2017, I developed interface for using IDAP models in TITAN for 2D line shape analysis. However, both one-dimensional and two-dimensional IDAP models will use the same kinetic matrices. Therefore, I am continuing their development in the same document.

In case of one-dimensional models, their testing was performed here as well as in IDAP/IDAP_Online_Documentation/NMRLineShapes1D/NMRLineShapes1D_index.htm. Starting in August 2017, I only use this document for derivation of kinetic matrices. Their testing is performed in IDAP/IDAP_Online_Documentation/NMRLineShapes2D_TITAN/NMRLineShapes2D_TITAN_index.htm.


Model parameters

Typically, model parameters are described in the model definitions in the class definition IDAP/code/@NMRLineShapes1D/NMRLineShapes1D.m. The self.add_model() sections contain (1) name of the  model, (2) a list of parameter names, usually, self-descriptive. The utilization of the parameters may be checked in  model files  in IDAP/code/+line_shape_equations_1D.

This following list describes some of the commonly used parameters.


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     Model listing

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 IDAP 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!

Most recent derivation: nU-R-RL (n=1..5)


Models without binding of ligands

I-ab (Isomerization, Ra<=>Rb

I-ab_T (temperature-dependent) 

R2 (R<=>R2)

I-abcd (Isomerization, equilibrium between four isomers, a "box"


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Models with ligand binding









B  (R+L<=>RL2

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Multi-isomer models


U-nR (n=1..5)

path: /IDAP/Mathematical_models/NMR_line_shape_models/2D/U_5R



U-nR-RL (n=1..5)

path: IDAP/Mathematical_models/NMR_line_shape_models/2D/U_5R_RL


nU-R-RL (n=1..5)

path: /IDAP/Mathematical_models/NMR_line_shape_models/2D/U_5R




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Models still in development


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