 # NMR Line Shapes: development of kinetic matrices

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## Introduction

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 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 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.

• LRcorrection is used to correct for the a possible error associated with initial concentration of the ligand stock. It is one global value applied for all datasets and all titration points. Therefore, it must be linked within the series (so all titration points have corrected ligand concentrations) and between the series (if they all originate from the same HSQC titration).

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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_T (temperature-dependent)

• Folder:  IDAP/Mathematical_models/NMR_line_shape_models/I_ab
• Derivation
• Testing
• MATLAB notebook:
• HTML report:

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

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### Models with ligand binding

#### U-L2

• Folder: U_L2
• Derivation
• HTML report:
• Testing
• MATLAB notebook:
• HTML report:

#### B  (R+L<=>RL2)

• Folder:  B
• This is a family of four models:
•  B_macro - the model using mAcroscopic constants. Binding sites are forced to be identical and statistical effects a factored into the binding and rate constants. Both single-bound species are lumped together: we cannot discriminate them. Use statistical relationships to obtain values of microscopic thermodynamic and kinetic constants. B_micro - the model using microscopic constants. For this model to be useful, the binding sites must have different affinity AND two single-bound species must produce DISTINCT SIGNALS in the spectrum. If this condition is not met, the model will be overparameterizing the dataset. Use B_macro instead. B_micro_rev and B_macro_rev are the same models as above but with labeled ligand instead of receptor (reverse labeling scheme).

##### B_macro

NOTE: I am assuming that two sites are identical and use their MICROscopic constants because they are more easily interpreted. I still allow for positive or negative cooperativity (K_a_A1 < K_a_A2 or K_a_A1 > K_a_A2). The microscopic constants are converted to their macroscopic counterparts inside B_macro_model_1D.m to input into the equilibrium populations model B_macro_model.m.

• Derivation
• Testing
• B_macro_reverse - Kinetic equations are derived in B-macro but resulted in non-linear system, which cannot be solved using linear matrix algebra. Temporarily, stopped here.

##### B_micro
•  (will be useful only if your data give you different chemical shifts for two single-bound species).
• Derivation
• HTML report:
• Testing
• MATLAB notebook:
• HTML report:

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### Multi-isomer models

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

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

• Derivation:  U_5R.mn  HTML
• NOTE: here I am using different order of species than U-R model used. Therefore, the single-isomer model in this family will be called U-1R.

#### 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

• Template (X<=>Y2
• Folder:
• Derivation
• HTML report:
• Testing
• MATLAB notebook:
• HTML report:

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