A partial differential equation for pseudocontact shift

Abstract

It is demonstrated that pseudocontact shift (PCS), viewed as a scalar or a tensor field in three dimensions, obeys an elliptic partial differential equation with a source term that depends on the Hessian of the unpaired electron probability density. The equation enables straightforward PCS prediction and analysis in systems with delocalized unpaired electrons, particularly for the nuclei located in their immediate vicinity. It is also shown that the probability density of the unpaired electron may be extracted, using a regularization procedure, from PCS data.

Model-free extraction of spin label position distributions from pseudocontact shift data

“…If the paramagnetic centre is distributed with some probability density ρ ( r ), the convolution of this density with eqn (1) obeys 21 the partial differential equation that we have recently derived: 20 where ∇ is the gradient operator, H ρ ( r ) is the Hessian of ρ ( r ) and the denominator is to be understood as the inverse Laplacian. The best way to solve this equation is to use fast Fourier transforms: 21 …”

“…20 We demonstrate here that it may be used to recover the spin label position distribution from the experimental PCS data. This creates a new window into protein structure and dynamics.…”

“…The conformational mobility of lanthanide tags is visible in the distance distributions measured by double electron resonance, 17 and in molecular dynamics simulations. 18 In this situation the commonly used point paramagnetic centre approximation 3,19 for PCS is not expected to be valid, 14,20,21 but quantum chemical calculations 6,7 are prohibitively expensive.…”

Theoretical studies of the active-site structure, spectroscopic and thermodynamic properties, and reaction mechanism of multicopper oxidases

“…Another important source of experimental information concerning the structure of the metal sites in proteins is EXAFS spectroscopy [60,61]. As mentioned above, it is known from EXAFS studies that the PI has a Cu-Cu distance of ~3.4 Å [35] and that the NI has at least two Cu-Cu distances of 3.3 Å [15].…”

“…Considering that the overall accuracy of the EXAFS geometries (~0.01 Å [60,61]) is better than that of QM calculations (~0.05 Å [69,70]), w EXAFS is normally increased until the geometry reaches a state when χ 2 does not change. Calibration studies showed that this EXAFS/QM approach worked excellently and actually could improve the EXAFS structures of small inorganic complexes (i.e.…”