Kernel Energy Method: Application to DNA

Massa

Karle

et al. 2009

Self Cite

Using the Kernel Energy Method we apply ab initio quantum mechanics to study the relative importance of weak and strong interactions (including hydrogen bonds) in the crystal structures of the title compounds TDA1 and RangDP52. Perhaps contrary to widespread belief, in these compounds the weak interaction energies, because of their large number and cooperativity, can be significant to the binding energetics of the crystal, and thus also to its other properties.CH-O bonds ͉ cooperativity of bonds ͉ cysteine macrocycles ͉ NH-CO bonds ͉ NH-S T he topic of weak interactions (including hydrogen bonds) is of increasing general interest because in large numbers and cooperativity they can be important to molecular properties. Desiraju and Steiner (1) discuss this aspect of weak hydrogen bonds and have published a collection of structures in which these bonds participate. The understanding of these weak interactions will likely be useful in future designs of complex selforganizing structures. It has been noted that cooperativity among weak interactions allows them to contribute effectively to tighten molecular packing in crystals to an extent beyond that associated with ordinary van der Waals forces. This can affect the thermal and electronic properties of molecules, including electron mobility, and physical properties such as hardness, melting point, and crystal stability (1-4). Because it is of increasing interest to recognize the significant and sometimes subtle effects on molecular properties caused by weak interactions, we have undertaken a quantum mechanical study of their energetics in crystals of 2 molecules that may stand as examples in addition to the compounds discussed by Desiraju and Steiner. These areIn the molecules to be studied the crystal structures and atomic coordinates are known (2, 3). This information is used to obtain the atomic coordinates of the near-neighbor molecules in the crystal to which they are related by operations of space group symmetry. In particular, it is the interaction energy between neighboring molecules of the crystal that is the object of our study. We wish to calculate the interaction energy between full neighboring molecules, and having that value, to calculate the components of such interactions that are contributed by the various ''pieces'' of the molecules through their strong and weak hydrogen bonds. Thus, the relative importance of the weak and strong hydrogen bonds would be an outcome of the study.Since the molecules to be studied may be thought of as having a variety of pieces, each contributing to the various strong and weak interactions, a natural method of quantum calculation from 1995 is that of kernel density matrices (5-7), which has evolved more recently to the Kernel Energy Method (KEM) of Quantum Crystallography (8)(9)(10)(11)(12)(13)(14). In the KEM, the results of X-ray crystallography are combined with those of quantum mechanics. It is assumed that the crystal structure is known for a molecule under study. With known atomic coordinates, the molecule is math...

mentioning

confidence: 99%

“…Of special relevance to this article, the KEM has been applied to the calculation of interaction energies between biological molecules and has been shown to be useful for that purpose (10,11). In this article we depend on the known ab initio accuracy of the KEM to show how it may be used to obtain strong and weak interaction energies.…”

mentioning

confidence: 99%

Calculation of strong and weak interactions in TDA1 and RangDP52 by the kernel energy method

Massa

Karle

et al. 2009

Self Cite

“…Initial studies (2)(3)(4)(5) showed that the KEM is capable of achieving good accuracy over a range of different molecules and model chemistries. The background for uniting quantum mechanics with crystallography (6-13) and a review of work related to ours (8) are available in the literature.…”

mentioning

confidence: 99%

The Kernel Energy Method: Application to a tRNA

Massa²,

Karle³

2006

Self Cite

The Kernel Energy Method (KEM) may be used to calculate quantum mechanical molecular energy by the use of several model chemistries. Simplification is obtained by mathematically breaking a large molecule into smaller parts, called kernels. The full molecule is reassembled from calculations carried out on the kernels. KEM is as yet untested for RNA, and such a test is the purpose here. The basic kernel for RNA is a nucleotide that in general may differ from those of DNA. RNA is a single strand rather than the double helix of DNA. KEM energy has been calculated for a tRNA, whose crystal structure is known, and which contains 2,565 atoms. The energy is calculated to be E ‫؍‬ ؊108,995.1668 (a.u.), in the Hartree-Fock approximation, using a limited basis. Interaction energies are found to be consistent with the hydrogen-bonding scheme previously found. In this paper, the range of biochemical molecules, susceptible of quantum studies by means of the KEM, have been broadened to include RNA. The tRNA molecule is designed to attach to a codon at one end and to deliver an amino acid at the other end. The amino acid methionine is the initiating codon in the protein production process in the ribosome. Such an initiator tRNA is the subject of the energy calculation in this paper.Given that the RNA molecules described above would be quite interesting to study by use of the techniques of quantum mechanics, the problem they present is their considerable size. That is the problem addressed here, by using the KEM approximation, whose main features are now reviewed.In the KEM, the results of x-ray crystallography are combined with those of quantum mechanics. The result is a reduction of computational effort and an extraction of quantum information from the crystallography, not otherwise available. Central to the KEM is the concept of the kernel. These are the quantum pieces into which the full molecule is mathematically broken. All quantum calculations are carried out on kernels and double kernels. Because the kernels are chosen to be smaller than a full biological molecule, the calculations are easily accomplished, and the computational time is much reduced. Subsequently, the properties of the full molecule are reconstructed from those of the kernels and double kernels.Initial studies (2-5) showed that the KEM is capable of achieving good accuracy over a range of different molecules and model chemistries. The background for uniting quantum mechanics with crystallography (6-13) and a review of work related to ours (8) are available in the literature.With known atomic coordinates, a molecule is mathematically broken into tractable pieces called kernels. The kernels are chosen such that each atom occurs in only one kernel. Schematically defined kernels and double kernels are shown in Fig. 2, and only these objects are used for all quantum calculations. The total molecular energy is reconstructed therefrom by summation over the contributions of double kernels reduced by those of any single kernels that have been overcounted. Two appr...

“…In the simplest terms, one may state that the complexity of an HF calculation is proportional to n 4 , where n is the number of atoms (and in a rough way this is proportional to the number of basis functions used). If a system of n atoms is broken into n K single kernels, then the number of double kernels, triple kernels, and quadruple kernels would be of the order of n K 2 , n K 3 , and n K 4 , respectively, and each individual multiple kernel would require computational effort of the order of (n/n K ) 4 . Thus, the total computational effort, in the case of using at most double, triple, or quadruple kernels, would be of the order of (n 4 /n K 2 ), (n 4 /n K ), and n 4 , respectively.…”

Section: Discussionmentioning

confidence: 99%

“…Most importantly, the use of quadruple kernels would allow the computation of extremely large molecules that would not otherwise be possible if the space required to store the large number of energy integrals were insufficient. For example, in an HF calculation where the number of integrals to be stored would grow as n 4 , the corresponding memory space required in the case of individual quadruple calculations would only grow as (n/n K ) 4 . There is a further point worth remembering.…”

Section: Discussionmentioning

confidence: 99%

The kernel energy method of quantum mechanical approximation carried to fourth-order terms

Massa

Karle

2008

Self Cite

It is now possible to calculate the ab initio quantum mechanics of very large biological molecules. Two things lead to this perspective, namely, (i) the advances of parallel supercomputers, and (ii) the discovery of a quantum formalism called quantum crystallography and the use of quantum kernels, a method that is well suited for parallel computation. The kernel energy method (KEM) carried to second order has been used to calculate the quantum mechanical ab initio molecular energy of peptides, protein (insulin and collagen), DNA, and RNA and the interaction of drugs with their biochemical molecular targets. The results were found to have good accuracy. In this article, the accuracy of the KEM is investigated up to an approximation including fourth-order interactions among kernels. Remarkable accuracy is achieved in the calculation of the energy of the ground state of the important biological molecule Leu 1 -zervamicin, whose crystal structure is known and used in the calculations. molecular energy ͉ n-body cluster ͉ quantum crystallography T he kernel energy method (KEM) combines structural crystallographic information with quantum mechanical theory. KEM may be described as the determination of the quantum mechanical molecular energy by the use of the parts of a whole molecule, which are referred to as kernels. Because the kernels are much smaller than a full biological molecule, the calculations of kernels and multiple kernels are practicable. Subsequently, kernel contributions are summed in a manner affording an estimate of the energy for the whole molecule. Thus, the task of obtaining a quantum mechanical energy has been simplified for biological molecules as large as those containing thousands of atoms. The computational time is much reduced by employing the KEM, and the accuracy obtained appears to be quite satisfactory, as shown in previous work.The first applications of the KEM (1) referred to above, involved a large number of peptides of various shapes and sizes, giving good accuracy, which was retained throughout a wide range of basis functions and computational methods (2). The KEM has been applied to the protein insulin (3), various A, B, and Z DNA (4), RNA (5), drug-target and ribosomal A-site RNA (6), and a collagen triple helix (7) with favorable results. The overall theoretical background for the application of quantum mechanics with crystallography may be found in refs. 8-15. References that review the quantum mechanical methods related to the KEM may be found in ref. 10. This article is devoted to calculating the KEM energy to an order of approximation including terms up to a fourth order of interaction among the kernels. Given that extending the accuracy of the KEM is a subject of interest, the problem that presents itself is the derivation of the approximation to higher orders and its application, to test whether an expected improvement in accuracy results from computation. That is the problem addressed here.To review, in the KEM, the use of atomic coordinates from x-ray crystallography lead...