\ No. rotatable bonds angle increment
There are three common quantitative ways to obtain estimations of preferred molecular shapes required for a good fit at the receptor. The first, which is the oldest and considered the most accurate, is x-ray crystallography. When properly done, resolution down to a few angstrom units can be obtained. This permits an accurate mathematical description of the molecule, providing atomic coordinates in 3D space that can be drawn by using a chemical graphics program. A serious limitation of this technique is the requirement for a carefully grown crystal. Some chemicals will not form crystals. Others form crystals with mixed symmetries. Nevertheless, with the newer computational techniques, including high-speed computers, large databases of x-ray crystallographic data are now available. These databases can be searched for structures, including substructures, similar to the molecule of interest. Depending on how close is the match, it is possible to obtain a pretty good idea of the low-energy conformation of the drug molecule. This is a common procedure for proteins and nucleic acids after obtaining the amino acid and nucleotide sequences, respectively. Obtaining these sequences is now largely an automated process.
There also is the debate that asks if the conformation found in the crystal represents the conformation seen by the receptor. For rigid molecules, it probably is. The question is very difficult to answer for flexible molecules. A common technique is to determine the crystal structure of a protein accurately and then soak the crystal in a nonaqueous solution of the drug. This allows the drug molecules to diffuse into the active site. The resulting crystal is reanalyzed using different techniques, and the bound conformation of the drug can be determined rapidly without redoing the entire protein. Often, the structure of a bound drug can be determined in a day or less.
Because of the drawbacks to x-ray crystallography, two purely computational methods that require only a knowledge of the molecular structure are used. The two approaches are known as quantum mechanics and molecular mechanics. Both are based on assumptions that (a) a molecule's 3D geometry is a function of the forces acting on the molecule and (b) these forces can be expressed by a set of equations that pertain to all molecules. For the most part, both computational techniques assume that the molecule is in an isolated system. Solvation effects from water, which are common to any biological system, tend to be ignored, although this is changing with increased computational power. Calculations now can include limited numbers of water molecules, where the number depends on the amount of available computer time. Interestingly, many crystals grown for x-ray analysis can contain water in the crystal lattice. High-resolution nuclear magnetic resonance (NMR) provides another means of obtaining the structures of macromolecules and drugs in solution.
There are fundamental differences between the quantum and molecular mechanics approaches, and they illustrate the dilemma that can confront the medicinal chemist. Quantum mechanics is derived from basic theoretical principles at the atomic level. The model itself is exact, but the equations used in the technique are only approximate. The molecular properties are derived from the electronic structure of the molecule. The assumption is made that the distribution of electrons within a molecule can be described by a linear sum of functions that represent an atomic orbital. (For carbon, this would be s, px, py, etc.) Quantum mechanics is compu tation intensive, with the calculation time for obtaining an approximate solution increasing by approximately N4 times, where N is the number of such functions. Until the advent of the high-speed supercomputers, quantum mechanics in its pure form was restricted to small molecules. In other words, it was not practical to conduct a quantum mechanical analysis of a drug molecule.
To make this technique more practical, simplifying techniques have been developed. Whereas the computing time is decreased, the accuracy of the outcome is also lessened. In general, use of calculations of the quantum mechanics type in medicinal chemistry is a method that is still waiting to happen. It is being used by laboratories with access to large-scale computing, but there is considerable debate about its utility, because so many simplifying approximations must be made for larger molecules.
To overcome the limitation of quantum mechanics, there has been motivation to develop alternative approaches to calculation conformations of flexible molecules. The reason is that manipulation of computer models is much superior to the use of traditional physical models. Mathematical models using quantum mechanics or the more common force field methods (see later) better account for the inherent flexibility of molecules than do hard sphere physical models. In addition, it is easy to superimpose one or more molecular models on a computer and to color each structure separately for ease of viewing. Medicinal chemists use the superimposed structures to identify the necessary structural features and the 3D orientation (pharmacophore) responsible for the observed biological activity. The display of the multiple conformations available to a single molecule can provide valuable information about the conformational space available to druglike molecules. Rather than measuring bond distances with a ruler, as was done years ago with handheld models, it is relatively easy to query a computergenerated molecular display. Because the coordinates for each atom are stored in computer memory, rapid data retrieval is achieved. Moreover, the shape and size of a molecular system can be visualized and quantified, unlike the situation with handheld models, when only visual inspections are possible. Exactly how much energy does it cost to rotate torsion angles from one position to the next? Understanding drug volumes and molecular shapes is critically important when defining the complementary (negative volume image) receptor sites needed to accommodate the drug molecule.
High-resolution computer graphics that accompany the conformation calculations have revolutionized the way drug design is carried out. Once a molecular structure has been entered into a molecular modeling software program, the structure can be viewed from any desired perspective. The dihedral angles can be rotated to generate new conformations, and functional groups can be eliminated or modified almost effortlessly. As indicated previously, the molecular features (bond lengths, bond angles, nonbonded distances, etc.) can be calculated readily from the stored 3D coordinates.
Because of ease of calculations relative to quantum mechanics, medicinal chemists are embracing molecular mechanics. Force field calculations rest on the fundamental concept that a ball-and-spring model may be used to approximate a molecule.20,21 That is, the stable relative positions of the atoms in a molecule are a function of through-bond and through-space interactions, which may be described by relatively simple mathematical relationships. The complexity of the mathematical equations used to describe the ball-and-spring model is a function of the nature, size, and shape of the structures. Moreover, the fundamental equations used in force fields are much less complicated than those found in quantum mechanics. For example, small strained organic molecules require greater detail than less strained systems such as peptides and proteins. Furthermore, it is assumed that the total energy of the molecule is a summation of the individual energy components, as outlined in Equation 2.18. In other words, the total energy (Etotal) is divided into energy components, which are attributed to bond stretching (£stretching), angle bending (Ebending), nonbonded interactions (£nonbonded), torsion interactions (Etorsion), and coupled energy terms (Ecross-teims). The cross-terms combine two interrelated motions (bend-stretch, stretch-stretch, torsion-stretch, etc.). The division of the total energy into terms associated with distortions from equilibrium values is the way most chemists and biological scientists tend to think about molecules.
Etotal ^bondsEstretching ^ ^anglesEbending
^ ^nonbonded(EVDW ^ Eelectrostatics) ^ ^dihedralsEtorsion ^ ^Ecross-terms (Eq. 2.18)
Each atom is defined (parameterized) in terms of these energy terms. What this means is that the validity of molecular mechanics depends on the accuracy of the parameterization process. Historically, saturated hydrocarbons have proved easy to parameterize, followed by selective heteroatoms such as ether oxygens and amines. Unsaturated systems, including aromaticity, caused problems because of the delocalization of the electrons, but this seems to have been solved. Charged atoms such as the carboxylate anion and protonated amine can prove to be a real problem, particularly if the charge is delocalized. Nevertheless, molecular mechanics is being used increasingly by medicinal chemists to gain a better understanding of the preferred conformation of drug molecules and the macromolecules that compose a receptor. The computer programs are readily available and run on relatively inexpensive, but powerful, desktop computers.
The only way to test the validity of the outcome from either quantum or molecular mechanics calculations is to compare the calculated structure or property with actual experimental data. Obviously, crystallographic data provide a reliable measure of the accuracy of at least one of the low-energy conformers. Because that is not always feasible, other physical chemical measurements are used for comparison. These include comparing calculated vibrational energies, heats of formation, dipole moments, and relative conformational energies with measured values. When results are inconsistent, the parameter values are adjusted. This readjustment of the parameters is analogous to the fragment approach for calculating octanol/water partition coefficients. The values for the fragments and the accompanying correction factors are determined by comparing calculated partition coefficients with a large population of experimentally determined partition coefficients.
Figure 2.19 • Potential energy for butane. The energy (kcal/mol) is plotted on the y-axis versus the torsion angle Csp3-Csp3-Csp3-Csp3, which is plotted on the x-axis. There are three minima. The two gauche conformers are higher in energy that the anti conformer by approximately 0.9 kcal/mol.
A simplified energy diagram for the hydrocarbon butane is shown in Figure 2.19. It illustrates that the energy rises and falls during rotation around the central Csp3-Csp3 bond as a function of the relative positions of the methyl groups. The peaks on the curve correspond to energy maxima, whereas the valleys correspond to energy minima. For butane, there are two different types of minima: one is for the anti butane conformation and the other two correspond to the gauche butane conformations. The anti conformation is the global minimum, meaning it has the absolute lowest energy of the three possible low-energy conformations. The differences in the conformational energies cannot be attributed to steric interactions alone. Structures with more than one rotatable bond have multiple minima available. Knowing the permissible conformations available to druglike molecules is important for design purposes.
A more typical energy diagram is shown in Figure 2.20. Notice that some of the minima are nearly equivalent, and it is easy to move from one minimum to another. From energy diagrams alone, it is difficult to answer the question, which of the ligand's low or moderately low conformations fits onto the receptor? This question can be answered partially by assuming that lower-energy conformations are more highly populated and thus more likely to interact with the receptor. Nevertheless, specific interactions such as hydrogen bond formation and dipole-dipole interactions can affect the energy levels of different conformations. Therefore, the bound conformation of a drug is seldom its lowest-energy conformation.
Because looking solely at the molecule can lead to ambiguous conclusions regarding the conformation when docking at the receptor, this has lead to calculating conformations of the macromolecule that is the receptor and visualizing the results. A capability that today's graphics software is representing molecular structures in many
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