On Comparing Experimental and Calculated Structural Parameters

The tacit assumption underlying all science is that, of two competing theories, the one in closer agreement with experiment is the better one. In structural chemistry the same principle applies but, when calculated and experimental structures are compared, closer is not necessarily better. Structures from ab initio calculations, specifically, must not be the same as the experimental counterparts the way they are observed. This is so because ab initio geometries refer to nonexistent, vibrationless states at the minimum of potential energy, whereas structural observables represent specifically defined averages over distributions of vibrational states. In general, if one wants to make meaningful comparisons between calculated and experimental molecular structures, one must take recourse of statistical formalisms to describe the effects of vibration on the observed parameters. Among the parameters of interest to structural chemists, internuclear distances are especially important because other variables, such as bond angles, dihedral angles, and even crystal spacings, can be readily derived from them. However, how a rigid torsional angle derived from an ab initio calculation compares with the corresponding experimental value in a molecule subject to vibrational anharmonicity, is not so easy to determine. The same holds for the lattice parameters of a molecule in a dynamical crystal, and their temperature dependence as a function of the molecular potential energy surface. In contrast, vibrational effects are readily defined and best described for internuclear distances, bonded and non-bonded ones. In general, all observed internuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. To illustrate how the operations of an experimental technique affect the nature of its observables, gas electron diffraction shall be used as an example.

The Molecular Electrostatic Potential: A Tool for Understanding and Predicting Molecular Interactions

The quest for improved methods for elucidating and predicting the reactive behavior of molecules and other chemical species is a continuing theme of theoretical chemistry. This has led to the introduction of a variety of indices of reactivity; some are rather arbitrary, while others are more or less directly related to real physical properties. They have been designed and are used to provide some quantitative measure of the chemical activities of various sites and/or regions of the molecule. In this chapter our focus is on one of these indices, the electrostatic potential V(r) that is created in the space around a molecule by its nuclei and electrons. V(r) can be computed rigorously, given the electronic density function ρ(r), by Eq. (3.1).

Ab Initio Studies of Anti-Cancer Drugs

Cancer is an extraordinarily complicated group of diseases which are characterized by the loss of normal control of the maintenance of cellular organization in the tissues. It is still not completely understood how much of the disease is of genetic, viral, or environmental origin. The result, however, is that cancer cells possess growth advantages over normal cells, a reality which damages the host by local pressure effects, destruction of tissues, and secondary systemic effects. As such, a goal of cancer therapy is the destruction of cancer cells via chemotherapeutic agents or radiation. Since the late 1940s, when Farber treated leukemia with methotrexate, cancer therapy with cytotoxic drugs made enormous progress. Chemotherapy is usually integrated with other treatments such as surgery, radiotherapy, and immunotherapy, and it is clear that post-surgery, it is effective with solid tumors. This is due to the fact that only systemic therapy can attack micrometastases. The rationale for using chemotherapy is the control of tumor-cell populations via a killing mechanism. The major problem in this approach is the lack of selectivity of chemotherapeutic agents. Some agents indeed preferentially kill cancer cells, but no agents have been synthesized yet which kill only cancer cells and do not affect normal cells. Unfortunately, normal tissues are affected, giving rise to a multitude of side effects. In addition to drugs exhibiting cytotoxic activity, antiproliferative drugs are also formulated. According to their mode of action, anti-cancer drugs are divided into several classes. . . . alkylating agents antimetabolites DNA intercalators mitotic inhibitors lexitropsins drugs which bind covalently to DNA . . . Experimental studies of these molecules are complemented and enhanced by theoretical studies. Some of the theoretical studies use molecular mechanics methods while others apply ab initio or semi-empirical quantum-chemistry methods. Most of these molecules are large and besides their structures and properties it is important to investigate their interaction with DNA fragments (themselves large molecules). Ab initio calculations cannot always be applied to the whole system. Therefore, models are used and through a judicious choice of the entities investigated, the calculations can shed light on the problem and provide enough information to complement the experimental studies.

Ab Initio Calculations

Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its “time-independent” form is: ((−h2/8mπ2)⛛2+V)Ψ=E Ψ or, for short: HΨ = EΨ In this equation, H, the Hamiltonian operator, is defined by H = − ((h2/8mπ2)⛛2+V, where h is Planck’s constant (6.6 10−34 Joules), m is the particle’s mass, ⛛2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Ψ(nit) is in terms of probability: Ψ2(x,t) is the probability of finding the particle between x and x + dx, at time t. The Schrödinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Born-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. For a many-electron molecule, the Hamiltonian operator can thus be written as the sum of the electrons’ kinetic energy term, which in turn is the sum of individual electrons’ kinetic energies and the electronic and nuclear potential energy terms.

An Introduction to the Theoretical Basis of Semi-Empirical Quantum-Mechanical Methods for Biological Chemists

Computational methods that can be employed to investigate fundamental questions concerning the complex chemical and structural behavior of biological molecules such as proteins, carbohydrates, and nucleic acids have been traditionally limited by the large number of atoms that comprise even the simplest system of biochemical interest. As a consequence, highly parameterized, empirical force field methods have been developed that describe the energy of macromolecular structures as a function of the spatial locations of the atomic nuclei. In combination with algorithms for simulating molecular dynamics, these classical models allow relatively accurate calculations of the structural and thermodynamic properties associated with proteins and nucleic acids. On the other hand, empirical approaches cannot be used to model molecular behavior that is directly dependent on electrons and their energies. For example, no information can be obtained concerning the electronic spectra of macromolecule/ligand complexes, electron transfer reactions such as those that occur within the photosynthetic reaction center, nitrogenase, an enzyme involved in nitrogen fixation, or cytochrome c oxidase which catalyzes the reduction of oxygen in the last step of aerobic respiration. Accurate modeling of transition states, excited states, and intermediates in biological catalysis requires application of quantummechanical (QM) representations since all of these phenomena depend on the distribution and/or excitation of electrons. At present, the most accurate ab initio algorithms for calculating electronic structure cannot be applied to systems comprised of hundreds of atoms, as such calculations scale as N4–N7 on most workstations, where N is the number of functions used in constructing the many-electron, molecular wavefunction. Even with the implementation of ab initio codes optimized for use on parallel computing engines, and density functional approaches, it is likely that high-accuracy QM calculations in the near future will remain limited to systems that comprise tens, rather than hundreds, of nonhydrogen atoms. Semi-empirical quantum-mechanical methods combine fundamental theoretical treatments of electronic behavior with parameters obtained from experiment to obtain approximate wavefunctions for molecules composed of hundreds of atoms.

Ab Initio Calculations of Amino Acids and Peptides

In the late seventies and early eighties, a small number of researchers who had the requisite resources at their disposal made the first systematic attempts to employ the newly emerging quantum-chemical computational tools in experimental studies of molecular structures (for reviews see Boggs 1983G, 1988G; Schäfer et al. 1983G, 1987G, 1988aG; and Geise et al. 1988G). In this process, which provided an important testing ground for the evolution of computational chemistry, the ab initio geometric optimizations of glycine (Sellers et al. 1978AA) represent a special landmark. In a sequence of events unprecedented in conformational chemistry, the results of the optimizations first suggested the existence of a hidden conformation that had remained undetected in two independent microwave spectroscopic studies of the compound (Brown et al. 1978G; Suenram et al. 1978G), and then guided new experiments that led to the detection of the missing state (Suenram et al. 1980G; Schafer et al. 1980AA). In the first microwave investigations of glycine (Brown et al. 1978G; Guenram et al. 1978G)—experiments whose success represent a considerable achievement because glycine is difficult to work with in the vapor phase—the observed transitions were assigned to the cyclic form, C, of the compound. In both studies, it was emphasized that other conformers of glycine could have been present but were not detected in the microwave spectra because their line intensities were weaker than those of C. Nevertheless, since C was observed but not the stretched form, S, Brown et al. concluded that “the most likely conformation of glycine in the vapor state” is C and that the experimental result was in conflict with ab initio calculations of glycine by Vishveshwara and Pople (1977AA) in which S was found more stable than C. Brown et al. (1978G): “The microwave spectrum of glycine vapor has been measured and analyzed; it is in the molecular form with a dipole moment of 4.5–4.6D and probably having conformation (C), which is in conflict with a recent theoretical study that implies that conformation (S) is more stable.” In the sixties and seventies, theoretical chemistry had developed a somewhat uncertain reputation.

Applications of Density Functional Theory to Biological Systems

The term biological systems may be used in reference to a wide class of polyatomic systems. They can be defined as minimal functional units which perform specific biological functions: enzymatic reactions, transport across membranes, or photosynthesis. At present, such systems as a whole are not amenable to quantum-chemistry studies because of their large size. The smallest enzymes are built of few thousands of atoms (e.g., lysozyme consists of 129 amino-acid subunits), the smallest nucleic acids are of similar size (e.g., t-RNA molecules consist of about 80 nucleotide subunits), whereas biological membranes are even larger and include different biological macromolecules embedded in a phospholipide medium. On the other hand, a common-sense definition of the term biological systems refers to any chemical molecule or molecular complex which is involved in biological or biochemical processes. The latter definition, which will be used throughout this review, covers not only complete functional units performing biological functions but also fragments of such units. Theoretical studies have provided data on properties of such fragments and have helped understanding of the biological processes at the molecular level. Depending upon the size of such fragments, they can be studied by means of various quantum-chemical methods. Molecular systems of up to a few thousands of atoms can be studied using semi-empirical methods. For the Hartree-Fock or Kohn-Sham density functional theory (DFT) calculations, the current size limit is a few hundreds of atoms. (Throughout the text, Hartree-Fock refers to ab initio Self-Consistent Field calculations using the approximation of linear combination of atomic orbitals.) When the desired accuracy requires the calculation of electron correlation at the ab initio level, only systems containing no more than few tens of atoms can be treated. Therefore, a theoretician aiming at the elucidation of biological processes by quantum-mechanical calculations faces two crucial issues. The first one is the selection of a fragment for modeling at the quantum-mechanical level. The second one is the assessment of the effects associated with parts of the system which cannot be modeled at the quantum-mechanical level. In this review, the DFT studies of biological systems are divided into two groups corresponding to different ways of addressing the second aforementioned issue.