Pseudocontact shift analysis

This tutorial shows how to use Spinach for analysis of paramagnetic NMR data. The examples are designed in a way to familiarize users with Spinach functions and basic theory of paramagnetic shift.

Paramagnetic chemical shift consists of two contributions: Fermi-contact (FC) and pseudocontact shift (PCS). It can be expressed via trace of the product of hyperfine (A) and susceptibility (χ) tensors. There are four different approaches for PCS analysis currently presented in Spinach. Difference in the presented methods appears only at the evaluation of the hyperfine tensor while susceptibility is either defined by user or optimized. Note that PCS depends only on the traceless part of χ, whereas Fermi contact part of the paramagnetic shift depends on the Tr(χ).

Point approximation

The simplest approach is based on the point-dipole approximation where dipolar hyperfine tensor depends on the position of a nucleus relative to the paramagnetic center \(\left(\vec r\right)\) as \(\frac{1}{{4\pi {r^3}}}\left( {\frac{{\vec r \cdot {{\vec r}^{\rm{T}}}}}{{{r^2}}} - \frac{1}{3}} \right)\). Therefore PCS is defined as \(\sigma = \frac{1}{{4\pi {r^3}}}{\rm{Tr}}\left( {\left( {\frac{{\vec r \cdot {\vec r^{\rm{T}}}}}{{{r^2}}} - \frac{1}{3}} \right) \cdot {\bf{\chi }}} \right)\). This approach is valid at the distance where paramagnetic center can be viewed as a point. To solve a direct problem use the function ppcs.m. This function takes structure of a molecule and susceptibility tensor as an input and gives PCS at the nuclei as an output. To solve an inverse problem use ippcs.m. The function evaluates susceptibility tensor and position of the paramagnetic center for provided PCS data with corresponding nuclear coordinates.

Example 1a. Direct problem for PCS using the point-dipole approximation: Metal porphyrin proton PCS.

Structure of metal porphyrin. Color code: metal - orange, nitrogen - blue, carbon - grey, hydrogen - light grey

In this example we will compute PCS on protons of the Co(II)/Cu(II) porphyrin and see how magnetic anisotropy affects PCS. The structure of the molecule in proper coordinate system is given at M_porphyrin.txt.

• Enter the coordinates of protons:

     nxyz=[  
           4.551635888      2.658552774      0.000000000
           2.658552774      4.551635889      0.000000000
          -2.658552774      4.551635888      0.000000000
          -4.551635889      2.658552774      0.000000000
          -4.551635888     -2.658552774      0.000000000
          -2.658552774     -4.551635889      0.000000000
           2.658552774     -4.551635888      0.000000000
           4.551635889     -2.658552774      0.000000000
           4.533874147      0.000000000      0.000000000
           0.000000000     -4.533874147      0.000000000
          -4.533874147      0.000000000      0.000000000
           0.000000000      4.533874147      0.000000000];

• Enter the susceptibility tensor in cubic Angstrom in SI. To compute susceptibility tensor from g-tensor for S=1/2 at room temperature use the formula \({\chi _i} = {\mu _0}\mu _B^2\frac[[:Template:G i^2]]{{4{k_B}T}} \approx 1.9571\frac[[:Template:G i^2]]{{T[K]}}[{{Å}^3}]\). Note that the coordinate system for tensor is defined by molecular frame.

     % g-tensor
     g_Co=[3.0 3.0 2.0];
     g_Cu=[2.0 2.0 2.2];
     % susceptibility tensor in A^3 in SI
     T=298; %K
     chi_Co=1.9571/T*diag(g_Co.^2);
     chi_Cu=1.9571/T*diag(g_Cu.^2);

• Define position of paramagnetic center

     mxyz=[0 0 0];

• Compute proton PCS with different susceptibility tensors

     pcs_Co=ppcs(nxyz,mxyz,chi_Co)
     pcs_Cu=ppcs(nxyz,mxyz,chi_Cu)

In the output you see that there are two groups of protons: four with larger absolute shift and eight with smaller shift. These correspond to two symmetry unique protons in the D4h symmetric molecule. The shifts (in ppm) are

     Co     Cu
    5.95  -1.00
    9.35  -1.57

In the case of easy plane anisotropy like in Co the PCS in equatorial plane is positive, and in the case of easy axis anisotropy - negative. In both cases g_x and g_y are the same so the PCS field has a shape of second rank spherical harmonic Y₂₀. In the case of non-zero rhombisity the shape of PCS field is a linear combinations of several second rank spherical harmonics.

File:PCS axial rh chi.png

PCS field in the case of axial susceptibility tensor (left) and in the case of limiting rhombisity.

Example 1b. Inverse problem for PCS using point-dipole approximation. Calbidin D9K PCS.

In this example we will process the experimental PCS data for calbindin D9K in which one of the two calcium ions has been replaced by a thulium ion from [Balayssac, S.; Jiménez, B.; Piccioli, M. Journal of Biomolecular NMR 2006, 34, 63]. The matrix of coordinates from PDB file 1IGV atoms with corresponding PCS from the paper is stored in calbidin_xyz_pcs.txt

Comparison of the point model PCS with experiment

• Import the data from calbidin_xyz_pcs.txt to x y z and expt_pcs
• Solve the inverse problem with initial guess of the Tm position [-5 5 -15]. One may need several tries with different initial guesses to converge the solution.

     mguess=[-5 5 -15];
     [mxyz,chi,pred_pcs]=ippcs([x y z],mguess,expt_pcs);

• Plot experimental vs predicted PCS

     plot(expt_pcs,pred_pcs,'bo');
     xlabel('Experimental PCS, ppm');
     ylabel('Predicted PCS, ppm');
     axis equal; axis square;

• Print the position of Tm and traceless part of suceptibility tensor in molecular frame

     disp('Susceptibility tensor:'); disp(chi);
     disp('Point electron location:'); disp(mxyz);

You can see that the fit with the point model is quite good but there are several outliers, which we will talk about later.

Corrections to PCS from non-point source

The next level of theory that corrects the point approximation is based on the multipole expansion of a spin distribution. It works for nuclei outside the spin density bounding sphere and accounts for its anisotropy that affect PCS (see paper). It should be noted that any isotropic distribution of the spin gives the same PCS as the point source. Direct problem is solved using lpcs.m and inverse problem is solved using ilpcs.m. This method requires from user list of the ranks of spherical harmonics (L). If L=0 the density is isotropic and result is identical to point approximation. Normally L=[0 1 2] accounts for the major effects of the anisotropy in the spin density distribution.

Example 2a. Correction to point PCS from spin density delocalization. Cobalt porphyrin

Spin density in cobalt porphyrin is clearly not a point object. In fact the delocalized spin density over nitrogen leads to the detectable superhyperfine signal in EPR. Let's assume that 10% of spin density is localized on each nitrogen atom. How that affects proton PCS? A simple way to to account for spin delocalization is to compute multipole moments of the distribution using the function points2mult.m.

• To compute multipole moments we have to specify nuclear coordinates with spin density

     xyz=[
      0.000000000      0.000000000      0.000000000   
      1.452280156      1.452280157      0.000000000
     -1.452280156     -1.452280157      0.000000000
     -1.452280157      1.452280156      0.000000000
      1.452280157     -1.452280156      0.000000000];
     % spin population
     rho=[0.6 0.1 0.1 0.1 0.1]'; 

• Compute multipole moments with ranks from 0 to 14

     L=[0:14];
     % multipole moments
     Ilm=points2mult(xyz,mxyz,rho,L,'points')

• Compute PCS with account of spin delocalization

     theor_pcs=lpcs(nxyz,mxyz,L,Ilm,chi_Co)

The resulting PCS shows that effect of the spin delocalization on PCS is significant. Point model predicts shifts at 5.95 and 9.35 ppm and non-point model - at 7.18 and 10.76 ppm.

Example 2b. Account for spin tag mobility. Calbidin D9K PCS

To account for possible mobility of thulium ion in the coordination cites we solve the inverse problem with additional non-point contributions using ilpcs.m. Using the experimental data from example 1b we solve the inverse problem with L=[0 1 2]:

     [mxyz,Ilm,pred_pcs]=ilpcs(nxyz,expt_pcs,L,mguess)

Fit of the experimental values improves significantly with account for spin delocalization. Standard deviation drops from 1.26 to 0.67 ppm.