The Laplacian - the 2nd derivative

The Laplacian is an important concept in charge density analysis. Pinoneered by Bader (1994), in his Theory of Atoms in Molecules the topological analysis of electron density through the Laplacian remains a popular analysis in ultrahigh resolution structure determination.

The laplacian of electron density is the sum of the diagonal elements of the second derivative matrix, or the trace of the hessian.

There is a nice document here about charge density analysis from Glasgow University, Topological Analysis of Electron Density (Faurrugia, 2007).

My Laplacian is simply a 2nd partial derivative, and is not very illuminating because it so closely matches the density itself. I use a hue that has 0 as a middle and red and blue on the extremes (as I do with difference density) so that the features of the 2nd derivative as -ve or +ve are clear.

The laplacian of the disulfide bonds

If I look at the 2nd partial derivatives of the disulfide bonds in 1ejg, and compare them to the density, it would be hard not to be concerned about the rings.

And yet, can we dismiss it entirely? There are clear volumes of deeper density in specific locations around the atoms - is it the product of overlapping rings or real information?

A challenge would be to remove the rings - what has held me back so far is the fear of adding layer upon layer of artifical interpretation and having no idea what is really left at the end.

The laplacian of an electron crystallograhy structure

If we add the laplacian to the density and radient of the 7uly peptide bonds we have calculated I am a little intrigued:

Now we have no rings and no model bias, and distinctive differences in the character of electron distribution in the peptide plane.

Discussions about Gaussian interference are coming up ahead, however.

Critical point definitions

The laplacian is the diagonal of the Hessian, which is the sum of the 2nd partial derivatives. For all critical points, the signature of the critical point is the number of -ve or +ve 2nd derivatives. If it is a stable critical point it will have a rank of 3, that is 3 real eigenvalues.
The signs of the eigenvalues define it's signature, where:
- (3,-3) All negative eigenvalues - a local maxima
- (3,-1) 2 negative, 1 positive - a bond cp
- (3,+1) 1 negative, 2 positive - a ring cp
- (3,+3) All positive, a local minimum - a cage cp

We can use a simplified numerical scheme to itemise these as our values. I have called this "derivative" the criticalpoint - it involves both the radient and the laplacian for calculation so is more time-consuming. Where the (g)radient is not close to 0 the colour is 0, so there is a 5 colour scheme:
- -3 -> (3,-3) = dark red
- -1 -> (3,-1) = pink
- 0 -> NA = black
- +1 -> (3,+1) = pale blue
- +3 -> (3,+3) = dark blue

The image is greyed out where there is not a critical point. However this can lead to a very sparse image, so the arbitrary decision is made to categorise <0.75 as a critical point for the purpose of visualsation. As most of an atom is not at a critical point, this leads largely to the atoms themselves being greyed out - with the background a bit more noisy. Within the atoms you can see clearly the crtitical points and their signatures. Quite satisfactory for this structure:
- all the atom positions are at maxima
- 2 of the C=O bonds have a bond critical point
- 2 of the C=O bonds have a ring critical point

A tyrosine ring

Because it is always nice to have the large plane of a tyrosine ring, I have the 4 derivatives for residue 44 in the crambin structure 3nir. 3nir is even higher resoluton than 1ejg at 0.48Å (Schmidt et al, 2011). I left the atom positions on the density image but removed them from the others so they don't obscure anything.

I increased the samples to 120 to get a clear image as the width is 9Å.

Orientation of above is:

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Colab Page to replicate

All the plots can be run from the colab example found here: colab-maps-laplacian