Jerome Karle (2006) - Kernel Energy Method Illustrated with Peptides

A Kernel Energy Method (KEM), for applying quantum crystallography to large molecules is described. Application has usually concerned the calculation of the molecular energy of peptides, is described. The computational difficulty of representing the system increases only modestly with the number of atoms

My collaborators in this work are Lulu Huang and Lou Massa. Slide 2. Listed here are the characteristics of the KEM, the Kernel Energy Method. Note especially that a large molecule is broken into pieces called kernels. And the kernels may be computed in parallel. The result is a large saving of computer time. Which nonetheless yields ab initio quantum accuracy. Slide 3. Here is the idea of KEM shown in an example with the use of a large molecule, a single strand of DNA. The large molecule is divided into kernels. The kernels are arbitrary, except that an atom is associated with only one kernel. At the places where bonds are broken, we mend them with hydrogen atoms. So that all the chemical bonds are saturated in the kernels. The single kernels can be combined into double kernels, the double kernels are of two types. A: double kernel in which the pair of single kernels are chemically bonded and B: double kernels in which the pair of single kernels are not chemically bonded. If we include only double kernels that are chemically bonded we find good accuracy combined with extremely fast calculations. However, for high accuracy we include the double kernels. In the calculations both single and double kernels play a role. Here are the equations which sum the energy contributions of the kernels and the double kernels. A double subscript refers to double kernels and a single subscript refers to single kernels. Thus EIJ is the energy of the double kernel composed of single kernels, INJ. And EJ is the energy of single kernels, the single kernel 1. These two equations sum the energy over all double kernels and subtract them from the energy of as many single kernels as have been thereby overcounted. The algebra of accounting differs depending on whether or not, 1: only the case of chemically bonded double kernels are included, or 2: all double kernels are included. Thus we have two equations to represent the energy sums depending on which approximation we wish to apply, we find the results are quite close using equation 1 and equation 2. These are the actual x-ray structures for each of the peptides. They have all been done at the Navel research lab, that's where we work, by - it says here 'Isabella', that happens to be my wife. Notice that these pictures represent a variety of peptide sizes and shapes. Therefore we expect such a variety of molecules to present a good sample for testing the accuracy and computational characteristics of the KEM, which is the kernel energy method, and that method is what we are using and talking about because it makes a very difficult computation, very simple and quick to carry out. Here we present the KEM, kernel energy method, that's what that KEM is. For eight different peptides that were pictured in the previous slide. In all cases shown we have chosen kernels such that each amino acid is a separate kernel. The type of calculation is that of Hartree-Fock, using a limited STO-3G basis. For each peptide the table shows the number of atoms, the number of kernels, the exact energy within the Hartree-Fock model, STO-3G basis. For the full molecule calculated as a whole and the energy calculated by the KEM based upon breaking the peptide into kernels. Also shown are the energy differences between the full molecule results and those from KEM. In this table only equation 1 is used to obtain the KEM result. That is only chemically bonded double kernels are used. Notice the molecular energies are listed in units, AU, but the differences are in the much smaller units kcal/mol. Thus the difference represented as a percentage of the exact energy, the differences represented are remarkably small. It may be concluded that the kernel energy method, including only the chemically bonded double kernels, is also very accurate as judged by these peptides. What this is all about is that this is a way to get quick answers but accurate ones. Saves a lot of time. And in fact the time, as you get the more complicated substances, gets to be so large that you just can't do it anymore. But this kernel energy method permits one to work with very, very large systems and get relatively quick answers that are also accurate. This is a graphic comparison which shows how much computing time is saved using the kernel energy method. The inset is a graph based upon the actual calculation times used. One sees here that KEM requires much less computing time than the conventional full molecule method. In order to project what would be the computed time saving in the case of much larger molecules, we look at the main graph. There we see two curves, the sharply rising red line is the fourth-order polynomial that has been fit to the actual calculation times of the full molecule result of the inset. The hardly rising blue line is the first-order polynomial, that was fit to the actual calculation time of the kernel energy method results of the inset. Notice, how dramatically the savings in computer time increases with the number of atoms in the molecule. Useful for very large biological molecules, which would not otherwise be calculable without the replacement of whole molecules, with the use of the procedure such as obtained by the kernel energy method. Here we see in tabulated form, results that are analogous to those of the previous graph. The table compares again the computation times for full molecules against the kernel energy method results. The table lists a few projected calculations, times from the previous graph. The molecules are of varying size. For just under 500 atoms the relatively comparison is 1.8 hours against .13 hours. This is already an order of magnitude difference in the time. Projecting to more than 10,000 atoms it is seen that the comparison involves an enormous saving of computer time. For example 162 days versus a few hours. In such a case, the full molecule calculation would be impractical while the kernel energy method calculation would be quite feasible. Thus the great advantage presented by the KEM, that is quantum mechanical results for very big molecules with high accuracy of results. Slide 9. We draw the preliminary conclusions of the talk. Thus the KEM makes possible true quantum mechanics of large biological molecules. Part 2. The point of part 2 is that, since it will be possible to describe large biological molecules by the KEM, this should make possible studies, useful in molecular medicine and drug design. As just one example, we discuss the calculation of the property we call the interaction energy. This slide defines the interaction energy between any pair of kernels. The pair can be within one molecule or between two molecules, as in the case of the drug and its target. A protein or a DNA, for example. I'll call this tall thing then 'I'. I is the interaction energy, E(IJ) is the energy of a double kernel, made of single kernels I and J. And E(I) and E(J) are the energies of the single kernels. The magnitude of I has the physical interpretation of measuring the strength of the interaction between kernels. But the sign determines whether the interaction is binding, negative or anti-binding, positive. The interaction example we choose is a tRNA molecule, it is called IYFG and is made of 75 nucleic acid residues. This is a very large molecule. However we have calculated the energy using the KEM. Here is the same tRNA molecule laid out as a single strand, folded back on itself. Note that in this slide the hydrogen bond interactions of various types between the various fragments and the molecules are shown. The hydrogen bonds have been determined crystallographically, that is they are based upon the distance between relevant hydrogen atom donors and acceptors. However using the kernel imaging method we can obtain directly the energetics of the imputed hydrogen bonds. Here we list a few of the interaction energies associated with the kernels. Which involve hydrogen bonding according to the independent crystallographic results based upon distances. Each of the crystallographic hydrogen bonds are confirmed as bonding in accordance with the negative sign for this large I. Interaction energies obtained in this case, thus the hydrogen bonding network of the molecule determined by experiment, is independently confirmed by the kernel energy method. Moreover, the strength of each of the interactions has also been obtained. It is to be realised in this case that the interaction energies obtained are all within 1 molecule. In the same way they could be obtained between molecules. As in the case of a drug and its target. Thus we conjecture that KEM will be useful for the study of interactions between drugs and their targets. Moreover, as the KEM is relatively new, we expect that in the future it is bound to be useful in many ways for the study of biological molecules. Just as quantum mechanics has proved to be invaluable for the study of the smaller organic molecules. Our final conclusion is that the KEM provides quantum interaction energies and these should be useful for the better understanding of biological molecules. We list publications, recent ones, where more can be learned about the KEM. E-N-D, end.

Jerome Karle (2006)

Kernel Energy Method Illustrated with Peptides

Jerome Karle (2006)

Kernel Energy Method Illustrated with Peptides

Abstract

A Kernel Energy Method (KEM), for applying quantum crystallography to large molecules is described. Application has usually concerned the calculation of the molecular energy of peptides, is described. The computational difficulty of representing the system increases only modestly with the number of atoms. The calculations are carried out on modern parallel supercomputers. By adopting the approximation that a full biological molecule can be represented by smaller "kernels" of atoms, the calculations are greatly simplified.

Collections of kernels are, from a computational point of view, well suited for parallel computation. The result is a modest increase in computational time as the number of atoms increases, while retaining the ab-initio character of the calculations. A test of the method and its accuracy is illustrated with the use of 15 different peptides of biological interest.

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