Progress in Medicinal Chemistry - Vol. 11, edited by G.P. Ellis and G.B. West 0 1975 - North-Holland Publishing Company
2 Quantum Chemistry in Drug Research W.G. RICHARDS, M.A., D.Phi1. and MOIRA E. BLACK, B.A. Physical Chemistry Laboratory, University of Oxford INTRODUCTION
68
QUANTUM CHEMICAL METHODS General considerations Extended Huckel theory Iterative extended Huckel theory Complete neglect of differential overlap Intermediate neglect of differential overlap Perturbative configuration interaction using localised orbitals A b initio molecular orbital methods
69 72 72 73 73 74 74 75
THE CALCULATION O F GEOMETRIES
75
THE CALCULATION O F CONFORMATION
76
CALCULATED PARAMETERS Charge density Orbital energies Frontier electron density
79 79 80 81
THEORIES OF DRUG ACTIVITY
82
ACTIVITY AND BINDING TO RECEFTORS
84
APPLICATIONS T O SERIES O F COMPOUNDS
85
APPLICATIONS T O SINGLE COMPOUNDS
a7
DISCUSSION
88
REFERENCES
89
61
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QUANTUM CHEMISTRY IN DRUG RESEARCH
INTRODUCTION A good guideline for medicinal chemists who are confronted with the increasing volume of literature on applications of quantum chemistry t o drug research is a quotation due to A.N. Whitehead: ‘There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain’. Drug research involves complicated biological problems where the relatively pure techniques of quantum chemistry might seem totally out of place. However, provided that a sufficiently critical attitude is maintained, much useful work can be done. The wealth of recent literature on the subject, which includes two books [1,2] provides examples both of the utility, and the dangers of naivety in the approach. Using the largest of electronic computers, good electronic wavefunctions can be computed for molecules, with up to about twenty atoms. In the realm of isolated gaseous molecules many properties can be calculated as accurately, or even on occasion more accurately than they can be measured. Examples include geometries and barriers to internal rotations. Physico-chemical experiments such as e.s.r. spectra or X-ray photoelectron spectra can be rationalised by means of theoretical calculation. Chemical reactivity remains a serious problem, but drug activity is in a sense a problem intrinsically more tractable. The activities of drugs fall into two main groups. There are those whose activities appear t o be related to their macroscopic properties, such as lipid solubility, and which are non-specific. Secondly, there are specific drugs where activity is closely allied to fine structural properties of the molecule. In such cases, minor alterations in structure can alter the activity of the drug by several orders of magnitude. We will only consider the latter type. The active molecule is envisaged as interacting with a receptor molecule in order to produce the response. This interaction is frequently considered as analogous to a ‘lock and key’ mechanism, although the search for receptors has not yet resulted in any very detailed understanding of their activity in terms of structure. For drugs which are highly specific, the classical methodology of pharmacology is to find an active compound; similar molecules are then synthesised by the organic chemist and tested for activity similar t o that of the lead compound (agonist activity), or for possible blocking (antagonist activity). The choice of molecules t o be prepared and screened may perhaps be made on the basis of the ease of synthesis. More logically, some relationship between structure and activity is preferred as a basis for testing. Measured physical properties of the
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W.G. RICHARDS A N D MOIRA E. BLACK
molecule may be correlated with activity. In particular, Hansch has used a type of linear free energy relationship with some notable successes in such correlations. Theoretical calculations have two potential advantages over empirical correlations. Firstly, submolecular properties such as local charge densities or polarisabilities are easily calculated and may be more directly related to binding than overall molecular or bulk properties like partition. Secondly, and even more attractive, is the fact that the calculations may be performed without synthesising the molecule first. Hence, provided properties computed from wave functions correlate with activity, then predictions may be made about the activity of similar molecules and used as a guide to synthesis and screening. The organic chemistry and pharmacology will be as important as ever, but a little guidance could reduce the number of molecules which have to be tested. If care is taken in the calculation of the wavefunction, and series of related compounds are compared, then meaningful correlations can and have been obtained; geometries of active molecules may be predicted; tautomer and conformer populations can be determined and detailed charge distributions in the active molecules can be given. Perhaps most tempting of all, from the detailed charge distribution of the active molecule, something of the complementary nature of the receptor may be inferred.
QUANTUM CHEMICAL METHODS It is well known that all the problems of chemistry are in principlt solved in the Schrodinger equation, H+ = E $ . Were it possible to solve this equation for a system containing many electrons, then the resulting wave function could be used to calculate any property of the system. It is equally well known that beyond the example of the hydrogen molecule ion, H i , the equation cannot be solved exactly. However approximate solutions are possible and if computers are used, then these approximate wave functions do provide a basis for calculating observable properties to an almost arbitrary level of accuracy. The only problem is that to obtain very accurate results a considerable amount of expensive computer time must be employed. All the common approximate methods for obtaining wave functions express the wave function of the whole molecule, $, as a product of orbitals, @,which individually represent functions which deal with one single electron, i.e.
+
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QUANTUM CHEMISTRY IN DRUG RESEARCH
Each @ is further expressed as a sum of atomic orbitals, appropriate coefficients, i.e. $1 = C,Xl -t c2x2 -t
x, multiplied
by
... .
This expression is often referred to as the linear combination of atomic orbital (LC AO) expansion. As a simple example we can consider the molecule LiH which has four electrons in two doubly occupied sigma-type molecular orbitals
This product, which has to be antisymmetrised to take account of the Pauli Principle, indicates the occupied molecular orbitals @ j . Each 4 can be expressed as a linear combination, for example, as
~ ~ , = ~ ~ x ~ ~ ~ , ~ ~ ) + c ~ X ( l s , N ) The two atomic orbitals, x, are known from work on atoms and the whole problem for the quantum chemist is to determine the coefficients Ci.Once the coefficients are known, the $ j and hence the overall wave function, $, are specified. One of the irritations of quantum chemistry is that it is bedevilled with a plethora of acronyms. Each refers to one of the methods of determining the coefficients. The acronyms and names of the methods most commonly applied to pharmacological problems are indicated in Figure2.1. In this figure the methods produce increasingly accurate answers as the table is read from the bottom to the top and from left to right. Generally the more accurate the method the more costly in terms of computer time, which also mounts rapidly as the number of electrons in the molecule under consideration increases. Roughly this increase in time is proportional to the fourth power of the number of electrons in the molecule. For molecules of interest to the medicinal chemist this means that only the relatively crude methods are employed. To assess the literature, a medicinal chemist need only know of the strengths and weaknesses of the particular techniques. An excellent review of some of the methods has been published by Hoyland [2] and below we summarise some of the salient points of the various molecular orbital methods. Before doing so one further important point must be stressed. The starting data for a particular calculation are merely the Cartesian coordinates of the various atoms in the molecule. These may be calculated directly
W.G. RICHARDS AND MOIRA E. BLACK
%
HARTREE MCK UMlT
t
I
INITIQ, EXTENW) BAYS
t
AB INITIO, MlNlMUM W S
I
INDO, MIND0
I -
AMXlRATE RWLTS
CI
CI
CI
CNDO
IEHT
71
pc'Lo
-
t
EHT
Figure 2. I . A classification of Molecular Orbital Methods. Accuracy improves from bottom to top of the dingram and left to right.
from the bond lengths and bond angles. The resulting wave function and energy of the molecule for this particular geometry is thus strictly appropriate only to the lone gas phase molecule. There is no simple way of including any solvent effects. Thus, applying the results to a biological situation where the molecule will be in a solution and possibly in an environment of uncertain pH requires an act of faith. It is an act of faith which is scarcely justified if one is only considering a single molecule. There may be more justification if one is concerned with a series of similar molecules, where solvation may be a constant factor, or different geometrical arrangements of the atoms in a single species where again differences in energies may be realistically computed. It is not too harsh a judgement to say that quantum mechanical calculations on drug molecules should only be given much credence if they are performed on a series of similar compounds or geometries and preferably if also there are some physico-chemical measurements which lend support to the values of the wave functions. We now give a brief resume of the molecular orbital methods commonly applied to biologically active molecules.
12
QUANTUM CHEMISTRY IN DRUG RESEARCH GENERAL CONSIDERATIONS
If we use our orbital expansion in the Schrodinger equation, it becomes, for each one electron orbital @,
Here H is a hamiltonian operator and expansion then we have
E
the orbital energy. In the LCAO
These equations can conveniently be transformed into the Roothaan equations
where
A set of such equations is at the heart of all molecular orbital methods. In principle a knowledge of all integrals of the types H l k and S Ik enable both the coefficient Clk (and hence the orbitals @)and the orbital energy E to be calculated. The various approximations differ in the methods used to deduce appropriate values for the integrals H l k and S l k which are frequently referred to as matrix elements. EXTENDED HUCKEL THEORY [3] (EHT)
This method ignores core Is electrons on all atoms except hydrogen but is capable of dealing with molecular orbitals of both u- and n-types. Integrals of the type Hll are replaced by experimental ionisation potentials and those of type Hlk deduced from ionisation potentials using the prescription
K is an empirical constant usually set equal to 1.75. All the overlap integrals S l k are actually calculated using so-called Slater atomic orbital functions for the X I .
W.G. RICHARDS AND MOIRA E. BLACK
73
The energy of the molecule is assumed to be a sum of the energies, E , of all the occupied molecular orbitals. This implies a neglect both of electron-electron and nuclear-nuclear repulsions. The theoretical basis of this method is very shaky indeed. However SO much experimental data is put into the calculation that little actual calculation remains and EHT can almost be considered as an interpolation formula connecting data. Despite its simplicity, it does appear to give sensible indications of the preferred geometries and conformations of molecules, particularly aliphatic hydrocarbons, although charge distributions are exaggerated and less reliable. Its greatest utility is the prediction of conformation but not barriers t o conformational change. ITERATIVE EXTENDED HUCKEL THEORY [ 4 ] (IEHT)
Simple EHT uses ionisation potentials for H 11 and also shows that two carbon atoms in a molecule may have different charges. Thus it would be more realistic to have slightly different ionisation potentials for these two carbon atoms since the ease of removing an electron will depend on the electronic charge surrounding the atom as well as its nuclear charge. IEHT achieves this by doing an EHT calculation, then using the charges produced t o modify the values of H 11 and repeating this cycle until the charges are self-consistent. The charge distiibutions are improved and predictions of dipole moment are reasonable. The method is as good as EHT for conformational prediction. COMPLETE NEGLECT OF DIFFERENTIAL OVERLAP [ 5 ] (CNDO)
This method is less empirical than Huckel approximations. Integrals of the type Hlk are no longer treated simply as empirical parameters. The actual nature of the operator H is considered and each matrix element H l k is expanded as an
appropriate set of integrals over atomic orbitals x, multiplied by products of the coefficients C. Since determining the coefficients C is part of the exercise, the method is again an iterative self-consistent field technique. Values of C in the expansion
are guessed, the integrals over functions x deduced and ultimately the Roothaan equations are solved giving new values of C and this is repeated until convergence is achieved.
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QUANTUM CHEMISTRY IN DRUG RESEARCH
Only valence electrons are considered and the integrals over the x functions are calculated in some simple cases, ignored, taken from experiment or treated as empirical parameters. In particular, many integrals involving x on two separate atoms are put equal to zero which is the origin of the name of the method. Computer programmes of this type can deal with molecules with up to about twenty atoms before the computing time becomes a serious problem. Generally this type of method gives good geometries but is unreliable for conformational studies. Charge distributions on the other hand do agree with experimental observation from dipole moment or e.s.r. studies. INTERMEDIATE NEGLECT OF DIFFERENTIAL OVERLAP [ 6 - 9 ] (INDO AND MINDO)
INDO differs from CNDO by the inclusion of some electron repulsion integrals which are neglected in the simpler method. One advantage is that the method does distinguish between singlet and triplet electronic states and further, e.s.r. hyperfine coupling constants can be predicted. MINDO differs from INDO in the procedures used to estimate the various integrals. It provides very good agreement with heats of formation of many compounds, but fails t o predict the relative stabilities of rotational isomers and gives poor dipole moments. The electron distribution in molecules containing hetero atoms is exaggerated. PERTURBATIVE CONFIGURATION INTERACTIONS USING LOCALISED ORBITALS LOCALISED ORBITALS [ 101 (PCILO)
This is a method based on the expansion of molecular orbitals in terms of bond orbitals rather than atomic orbitals. The main hypotheses of the CNDO approximation are maintained and the status of this method is roughly parallel to the CNDO calculations. Some improvement is obtained by including configuration interaction which is a technique of improving a wave function and hence an energy calculation by allowing the wave function to take some account of excited states of the molecule. There are an infinite number of excited states which could be included but the PCILO method restricts attention to the few closest in energy to the ground state of the molecule. In terms of predictions of geometry or conformation the results of the PCILO method are broadly similar to those of the CNDO method but they are obtained much more quickly. This is an important consideration since, if calculations can be performed using only small amounts of computer time, then it becomes possible to do many calculations on a molecule rather than just a few. PCILO
W.G. RICHARDS A N D MOIRA E. BLACK
15
has in this way been responsible for a significant advance in the application of quantum chemical methods to drug research: using PCILO methods it is now possible t o produce complete energy maps with contours when considering conformational energy. These give considerably more information than the earlier results which were normally presented as graphs of energy against some conformational twist. AB INITIO MOLECULAR ORBITAL METHODS [ 11,121
In ab initio methods all electrons including the core electrons are considered. No integrals are ignored or replaced by empirical parameters. This means that many millions of integrals have to be computed for a molecule containing more than about twenty atoms and the computer time needed which is roughly proportional to the fourth power of the number of electrons, mounts prohibitively. These rigorous techniques are improving steadily in both speed and accuracy but drug molecules are still beyond their practical capabilities especially when it is borne in mind that not one but perhaps several hundred calculations need to be performed on each molecular species if a thorough study is to be made. The only serious impact which ab initio methods have had on calculations on pharmacologically interesting molecules is that they provide bench-mark calculations on small molecules against which any approximation should be compared before applying i't to complicated species. This obvious test of reliability has not always been performed. THE CALCULATION OF GEOMETRIES The method which has to be followed to determine geometries by theoretical calculation is logical. We perform a calculation of the energy of the molecule for a given geometrical arrangement of the constituent atoms. One by one all bond lengths and bond angles are varied by small increments and the predicted geometry will be that structure which is calculated t o be the most stable. This logical procedure would involve many thousands of separate molecular orbital calculations and even in principle is limited t o calculations of the ab initio type since the empirical parameters used in the approximate work are only suitable for standard bond lengths. Thus an approximation to the idealised scheme is favoured. This involves starting with known bond distances and angles (perhaps from published crystal data) or using standard values. There is an invaluable set of data published by Sutton [ 131. Bond angles may be found by studying the variation of calculated
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QUANTUM CHEMISTRY IN DRUG RESEARCH
energy with bond angle and the results are very satisfactory for most of the molecular orbital methods discussed above. Table 2.1 gives some typical results for the calculation of bond angles and lends confidence to the view that rigid geometries can be calculated theoretically.
THE CALCULATION OF CONFORMATION The theoretical calculation of geometry gives some indication of the reliability of the various molecular orbital methods but is of no real value to the medicinal chemist. Satisfactory physical methods may be used to determine geometries and even the use of molecular models is normally sufficient. The same cannot be said of conformation. Molecular models of most drug molecules show considerable flexibility due to the rotation of groups about single bonds and it is of vital importance to have some idea of the conformation of an active species as this may lead to possible inferences about the nature of the receptor. In some simple cases nuclear magnetic resonance techniques may give some information about conformation and X-ray crystallographic studies can be made on crystalline phases, but in general the problem of determining conformation and relative conformational preferences remains a difficult one. Quantum chemical calculations should be able t o treat conformational studies in precisely the same manner as investigations of geometries. A calculation is performed for a series of positions of one part of the molecule with respect to another and the energies are then compared. If there is only one bond about which rotation can occur the results can be presented simply as a graph of energy against angle. If, as is frequently the case Table 2.1. Predicted Bond Angles 114, 15, 161 Molecule
Experiment
Ab initio
H20 CHI NH3 MeOH
104.5 103.2 106.6 (IICH) 109.5 (HCO) 109.5 (COH) 105.9 (HCH) 108.8 (HCC) 110.1
110
C2H6 ____.
~~~~
106.6
INDO
106.0 106.4 108.2 110.7 107.3 106.6 112.2
CNDO
EHT
107.1 108.6 106.7
150
W.G. RICHARDS AND MOIRA E. BLACK
77
(acetylcholine, histamine or dipeptides), conformation is largely specified by rotations about two bonds then ideally a conformational energy map should be produced, with the two variable angles as axes and the energies indicated by contours. To produce a complete map may involve a very large number of separate calculations so that until very recently this labour was avoided and results were presented in the less satisfactory manner of series of graphs each showing the variation in one angle while the other is held constant. Minima indicate stable conformational structures, the depths being indications of relative stability. The advent of complete conformational energy surfaces has highlighted an important point of thermodynamics ignored in most published work. If there are several minima then the relative populations of each region of the energy surface depend not on internal energy differences but on free energy differences. Entropy effects have to be considered. Put very crudely, the relative populations of two valleys in an energy surface will depend firstly on how difficult it is for a molecule to get out of the valley (the enthalpy) and on the ease with which molecules can get into the valleys due to their varying widths (or entropy). It is possible, although it is not yet generally done [17] t o compute free energy differences from the energy diagrams. Figure 2.2 shows energy surface situations in which there are more than one minimum. Figure 2.2(a) is a case where there is only one variable angle; a deep stable minimum x and a shallower one y are marked. Figure 2.2(b) concerns a hypothetical case where there are
(a 1
*
W
a w
L w
Yr X
1
ANGLE 8
Oi
9
3 16
Figure 2.2. Energy diagrams (a) With a single variable angle of rotation. (b) With two independently variable rotation angles.
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QUANTUM CHEMISTRY IN DRUG RESEARCH
two variable angles 01 and 0 2 , one deep maximum x and two equivalent shallower depressions y : these could represent a trans and equivalent gauche conformations of a disubstituted ethane-like molecule. In order to decide upon the relative populations of molecules in the conformations x and y , we need to be able to estimate the difference in free energy of these conformations. In an approximate way this may be done by using a classical Boltzmann partition function for each region. For the single angle case this would be defined as
2=
c 0
exp[-~(O)/kT]
and in the two variable angle case
where E ( 0 ) or ~ ( 0 1 0 2 )is the calculated internal energy for a conformation defined by 0 or (0 1 , 02), k is the Boltzmann constant and T is the temperature taken as say 37°C. The summations would be taken over a regular grid of points fine enough t o reflect the shape of the energy surface and bounded by an energy contour which is arbitrary but could be set at k T or 2kT. If we assume that all contributions t o the entropy are constant, other than those derived from the surface, and that any volume change on altering conformation is negligible, then
AGO = -kT ln'(Z,/Z,) per molecule. In the work on histamine it has been shown [ 171 that the mole fraction of trans conformer is 0.62 computed using free energy differences but 0.56 if calculated internal energy differences are used. This difference is significant and yet the actual energy surface is by no means an extreme case. T h ~ simportant thermodynamic point has been ignored in most of the published work on calculations of conformational equilibria and should be borne in mind when assessing such work. A further very important aspect of such calculations is also under-stressed. This is that the calculations are performed on idealised 'lone gas phase molecules'. Thus the possible effects of solvation are neglected completely. These effects may be unimportant in considering different conformations of a molecule if each conformation is solvated in the same way. However, occasionally calculations indicate specific intramolecular interactions which would probably be genuine in the gas phase but improbable in solution. Notably calculations
W.C. RICHARDS AND MOlRA E. BLACK
19
often suggest the presence of intramolecular hydrogen bonds, conformations with such bonds seeming exceptionally stable. In aqueous solution, however, any capacity for hydrogen bonding will surely be satisfied by the solvent even if infra-red spectra in carbon tetrachloride support the idea of intramolecular effects in that medium. It is fair t o say that theoretical calculations can give accurate predictions of the relative energies of conformations of molecules in the gas phase. If enough care is taken to compute free energies then relative populations can be calculated but the effect of solvent remains a problem. For this reason it is not safe to trust predictions made by theoretical calculations on one single molecular species. It is far safer to look at a series of similar molecules and consider differences between them when solvation effects may be constant and further as always some experimental test of the value of the calculations is really necessary to make them convincing.
CALCULATED PARAMETERS Theoretical calculations produce not only wave functions and energies but also the possibility of calculating a number of other molecular parameters. These may in turn be correlated with biological activity. We now give a brief resumC of the parameters which are most commonly encountered in applications to medicinal chemistry. CHARGE DENSITY
The square of a wave function gives a measure of charge density. The charge distribution within a molecule is usually defined in terms of the gross charges on the various nuclei and again this may be derived from the wave function. However, it must again be stressed that any idea of ‘the charge at a particular nucleus’ is a rather woolly concept. The only charge on the nucleus is its formal charge. What we are interested in is how the influence of this is modified by having electrons close to it, but ‘close to’ is also rather imprecise. When we have a bond between two atoms how can we assign the electrons to one nucleus or the other? This vexed question has been the subject of much debate but in pharmacological work the simplest method is normally adopted. This is the so-called ‘Mulliken population analysis’ [ 181 . If our molecular orbitals are LCAO functions, i.e.
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QUANTUM CHEMISTRY IN DRUG RESEARCH
then the net atomic population of a given ortital xi is defined as
The overlap population defined as
-
the population between atoms i and j is further
Sii being the integral Jxq d7 which we have met earlier. Since p i = Oii, we can write the gross population of orbital
as
pi =CO, i
Finally the total population at any nucleus n is given by adding all the values of
Pi for orbitals j which are centred on atom n. Most molecular orbital computer programmes will do this simple calculation automatically and print out a list of atomic populations. Thus it is important to know how meaningful are these quantities. From the atomic populations it is possible t o compute dipole moments which can be compared to experimental values. The qualitative nature of the agreement with experiment is easily summarised ab initio methods
CNDO, MIND0 IEHT EHT
good charge densities good charge densities good charge densities very exaggerated charge densities.
However all the methods do seem capable of reproducing trends in charge density variation within a set of similar molecules. ORBITAL ENERGIES
Corresponding to each molecular orbital & there is an orbital energy q.Of most interest to workers interested in correlations of properties and activities are the energy of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). These would be the orbitals involved in any donation or acceptance of charge.
W.G. RICHARDS AND MOIRA E. BLACK
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FRONTIER ELECTRON DENSITY [ 191
The frontier electron theory [19] was originally developed to explain the difference in reactivity at each position in an aromatic hydrocarbon. It is based on the intuitive idea that the reaction should occur at the position of the largest density of the electrons in the frontier orbitals, which are defined according t o the type of reaction: in an electrophilic reaction, the highest occupied molecular orbital (HOMO); in a nucleophilic reaction, the lowest empty molecular orbital (LEMO); in a radical reaction, both of these. This theory was later given a sound theoretical basis by Fukui, Yonezawa and Shingu [20] who then introduced the concept of superdelocalisability. Denoting the occupied molecular orbitals by 1, 2, ... m , and the unoccupied levels by m + 1, m + 2, ... N , the superdelocalisability, S,, is given for the three types of reaction by: (a) for an electrophilic reaction
(b) for a nucleophilic reaction
(c) for a radical reaction
where Crj is the coefficient of the rth atomic orbital in the jth molecular orbital, and Xi is the coefficient in the orbital energy, which is given as ~j = c( + Xjp. The orbital which mainly determines the value of S, in each type of reaction is the same as the frontier orbital previously considered. There are problems in the use of both frontier electron density and superdelocalisability. The latter concept was originally put forward considering the n-electron part of the molecule only, with the energies of the orbitals being given in units of the resonance integral of a C-C bond in benzene. This means that in a series of molecules there would be a common zero of energy. Using all-valence molecular methods, the energies are obtained in absolute terms, so that the zero
QUANTUM CHEMISTRY IN DRUG RESEARCH
82
of energy is in the unoccupied orbitals. This is obviously not consistent in a series of molecules. The frontier electron density strictly permits only a comparison of reactivities at different positions within the same molecule. In order t o extend this concept for use over a series of molecules, a further quantity, F , may be considered:
where f, is the frontier electron density, e is the energy of the appropriate frontier orbital. F may be thought of as a weighted frontier electron density, in the sense that ease of removal of the particular electron is also considered.
THEORIES OF DRUG ACTIVITY The receptor theory of drug action implies that the pharmacological properties of a compound are dependent not only on the nature and properties of the constituent groups within the molecule, but also on the way in which these groups are distributed in space. This follows from the idea that the receptor is a discreet, spatially organised structure, and that maximum activation of the receptor only occurs when there is a close fit between the drug and the receptor. However, it is conceivable that interaction of a drug molecule with a particular feature of the receptor may induce a change in the receptor that brings additional drug and receptor features into close proximity. Production of an observable biological effect, generally associated with the interaction of a drug with a receptor, involves a long chain of events, of which the action at the receptor is probably only the first step. The methods of investigation of the drug-receptor binding all follow from a relationship derived by Cammarata [21]. The interactions of a drug with a receptor is considered as analogous to the combination of substrate and an enzyme, so that: S+
a
R
e SR*
b
where S is the drug molecule, R is the receptor, SR* is the drug-receptor complex. From this relation, two main theories of drug action are derived. The kinetic theory of Paton [22] considers that the rate-determining step is the rate of combination of the drug with the receptors, i.e. a is the rate-determining step.
W.G. RICHARDS AND MOIRA E. BLACK
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The occupation theory of Ariens [ 2 3 ] considers that it is the number of occupied receptors that determines the magnitude of the response, i.e. b is the rate-determining step. Experimentally, it is difficult to distinguish between the two theories, but, fortunately, they lead to the same relationship between drug concentration and biological response. The 'activated' drug-receptor complex implicit in the Paton theory, or the relatively more stable drugreceptor complex consideration in-Ariens' theory, is assumed t o be the response-determining factor, so that drug activity, A , will be proportional to the free energy of formation of a drug-receptor complex AGRs. Considering only the first-order interactions:
AaAG,, =
tA
G t~ AGS t AGP t A G ~ ( A G+ A ~ G ~+ A@) tAG~(AG tA ~ G ~+ AGP)
t AGS(AG~ t A G t~AGP) t AG*(AG~t A
G t~A G ~ )
where Ace, AGS and A@ are independent contributions t o the free energy due to electronic and steric interactions between drug and receptor and t o conformational changes in the receptor, respectively. AGd is introduced to represent the free energy change due to desolvation which would acc'ompany the union of drug and receptor. If we consider the case where interaction terms are constant or negligible, this leads t o the following expression:
AaAG,,=
AGe t AGd t ACs t AGP t k .
The method used is an examination of the variation of the individual terms, particularly AGe and A@, for a series of related compounds, using molecular orbital calculations. Conformational analysis of the drug molecules may be carried out as described above, to determine which of the conformations could be the one which the drug adopts t o fit the receptor. Electronic charge distributions may also be calculated - the variation of charge on corresponding atoms in a series of molecules may be found t o match changes in their biological activities, which could be good evidence for the identity of the pharmacodynamic groups. At this point yet another caveat must be introduced about the use of calculations to estimate AGe for a series of compounds and ?hen seeking correlations between these parameters and observed biological activities. The calculations are performed on the unperturbed ground state of a molecule,
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QUANTUM CHEMISTRY IN DRUG RESEARCH
1 I
TRANSITION STATE
REACTION COORDINATE
Figure 2.3. 7%e transition state view of reactivity.
whereas differences in reactivities are more properly ascribed to differences in transition state (see Figure 2.3). The calculated parameters such as charge density differences in a series of compounds only tell us about the initial slopes of the reaction curves (position R in the figure). It is assumed that since for drug-receptor interactions the binding energy is low, AE is small enough for estimates of the initial slope of the curve to reflect the different activation energies. Although this may be true for the type of reaction which typifies drug-receptor combination it is not generally true in organic chemistry.
ACTIVITY AND BINDING TO RECEPTORS Most of the theoretical methods applied to series of similar compounds with subsequent searches for correlations between computed properties and measured biological activity have been based on the following logic; Calculated Parameter a Binding Free Energy a Biological Activity (1)
(11)
UII)
Thus stage I is correlated with stage 111. This will only be valid if I is proportional t o I1 and I1 to 111. The theories of drug activity depend on I1 correlating with 111. Recently [24] an attempt has been made to list the correlation of I with 11, within the framework and area of applicability of the more ambitious correlations. As data on binding in a biological environment, the binding between haptens and antibodies were taken. It seems not unreasonable to consider the binding site of an antibody to be a model for a receptor and what is needed is a set of data covering the binding of a series of rather similar molecules to the antibody. An appropriate system is the antibody obtained against the hapten consisting
W.G. RICHARDS AND MOIRA E. BLACK
85
of the phenyltrimethylammonium ion attached to protein by a para-azo linkage. Inhibition experiments can be used to measure the extent to which a variety of unattached ammonium ioqs prevent precipitation and hence binding free energies can be measured. It is likely that the binding between this particular hapten series and the antibody receptor site is electrostatic and chemical investigation suggests the presence of a carboxyl group in the receptor which could attract the positive ammonium ion. Since this ion is quaternary substituted, hydrogen-bonding and charge-transfer mechanisms seem unlikely. In fact, statistically significant correlations between the experimental binding and calculated charge on the nitrogen atom were found even when the simple Extended Hiickel molecule orbital approximation and Mulliken population analysis were used. A further encouragement t o the use of correlations of computed parameters with activity comes from some as yet unpublished work from this laboratory. We have found that once again there are significant correlations between computed properties and octanol-water partition coefficients. In this case the experimental measurement can be thought of as being analogous t o binding to a lipid phase. The relative attraction of a molecule to water or octanol seems to be dependent on charge and on orbital energies. Both these series of calculations lend support to the notion that it is possible to produce meaningful correlations between computed properties and biological activity. Once proved and satisfactory, the advantage of these correlations over empirical structure-activity relationships is that they may be applied to molecules before they are synthesised and hence provide a guide t o synthesis.
APPLICATIONS TO SERIES OF COMPOUNDS Rather than quote many examples we will restrict ourselves to a single example which illustrates the general principles and difficulties. A particularly impressive-looking correlation between computed properties and measured activities has been found by Peradejordi 1251 who studied the tetracyclines. The tetracyclines are believed t o operate by the means of forming a complex with the ribosome. The equilibrium constant for the binding reaction is related to the binding energy between the drug and receptor. If the receptor is unique, then the variation in binding for a family of tetracyclines will be dependent solely on variations in the drug molecule. Thus, in principle it should be possible to find a correlation between activity and calculated parameters related to binding such as electronic charges, Q , and indices of nucleophilic
86
QUANTUM CHEMISTRY IN DRUG RESEARCH
reactivity, N , or electrophilic reactivity, E. For a series of compounds labelled by the suffm j , the relative binding energies will then be given by
Aei =
c
[aiQi + biNi + c i E i ] .
i
Calculation can provide estimates of the quantities Qi,Ni and Ei for each atom i in the molecule and the coefficients ai, bj, Ci are obtained by fitting known data by multiple regression analysis. Peradejordi produced the following formula for the activity A ; l o g A j = 18.4 + 56 Qolo+ 1 7 b ' 0 , ~+ 48 Qolo
-Eolo
+
71 QoIz+ 18Eo12+ 3 Qc,
.
The fit is extremely good (Figure 2.4) and suggests that the variation in antibiotic power depends largely on the effect of substituents on the atoms 0 1 0 , 011,012 and C6.
log Ai (experimental) Figure 2.4. Typical correlation between calculated and experimental activity. (From Actua1iti.s de Chimie Thkrapeutique (R.Daudel (1971) p . 8).)
W.G. RICHARDS AND MOIRA E. BLACK
81
This is only one of a number of such correlations. The statistical success is striking but a few words of caution must be emphasised. Above all, it is notable that rather a large number of disposable parameters are employed. In the above case fourteen pieces of experimental data were fitted t o an expression with seven coefficients. It is not surprising that such a fit looks good. If more data are t o be used this frequently involves trying to include compounds with very different structures where the intricacies of binding may be substantially different. For these reasons the value of correlations, remembering the difficulties stressed earlier in this article, is not for the quantitative predictions which they offer but more for the insight into important features of molecules required for particular types of activity. If too many adjustable parameters are included, then this leads to highly complicated equations which are difficult t o interpret physically. It would seem t o be more useful and meaningful to consider only a small number of parameters, perhaps as few as one or two, and to obtain results which have some physical significance even if they are less statistically impressive. Correlations, even if they are of high statistical significance, are of little interest unless they lead to verifiable hypotheses or alternatively unless they can be used predictively. APPLICATIONS TO SINGLE COMPOUNDS Calculations on single compounds have most frequently been aimed at understanding the conformation of the molecule. Details of many such calculations have been given by Kier [ l ] and by Pullman [26] but here we again will limit comment t o a few illustrative references. Calculations have been performed by Kier in a number of species including acetyl choline [17], muscarine [27], muscarone [27], nicotine [28] and histamine [29]. In these calculations he uses the Extended Huckel Method and varies angles independently, presenting the results as sections through potential surfaces rather than conformational energy maps. In general the proposed conformations are in agreement with the limited structural data available from X-ray crystallography, despite the fact that the effects of the solvent are not Considered. In the case of histamine [29] Kier found that there are two relatively stable conformers which he hypothesised could be responsible for the two distinguishable physiological activities of the molecule [30]. More extensive work [31] which studied the conformer population ratios of a series of methyl substituted histamines by similar methods but also supported by n.m.r. measurements does not support the contention. This more recent work is satisfactory in the sense
88
QUANTUM CHEMISTRY IN DRUG RESEARCH
that it does produce agreement between calculated and observed conformer and tautomer ratios, but daunting because it suggests that it is necessary t o perform a vast number of calculations on an extensive set of similar compounds before being able to use the results with confidence. Conformational studies using the more trustworthy PCILO method have beell made on dipeptides [26] and on barbiturates [32]. In these instances conformational energy maps have been published and again the predicted stable conformations do seem to be in good accord with crystallographic data. Even the conformational angles of peptide linkages in high resolution X-ray crystal structures of proteins seem to be compatible with the calculations on the small units, leading to the hope that the energy maps may be of assistance in the prediction and explanation of protein structures. Since once again solvation has to be ignored, the differences between compounds are probably of more significance than any absolute numerical result. Calculations on single molecules can be made with great ease using standard computer programmes and only a few minutes of computer time. They may be of great assistance in helping a medicinal chemist to have a feeling for the structure, conformation and charge distribution in his molecule. As long as they are treated as aids to thinking, rather like molecular models, then they are probably useful. On the other hand because of all the approximations and dangers mentioned above, any calculations on a single gas phase species should not be taken as the basis of a theory.
DISCUSSION This article has taken a deliberately critical view of the application of quantum chemistry t o drug research since it is clear from the rapidly expanding number of publications in the field that many of the assumptions inherent in the work are not widely realised. It is very simple t o do a quantum mechanical calculation using a standard programme on a large computer: far simpler than for example running a spectrum. No one would publish the spectrum of a single compound and yet many theoretical papers contain far less information than a single spectrum. This facile work does quantum chemistry a disservice. All molecular behaviour is ultimately based upon quantum mechanics. In chemistry, after a shaky start, quantum mechanical methods, particularly molecular orbital theory has been abundantly fruitful, but it has been necessary to adopt a high standard of criticism and to be aware of the limitations of particular methods. The best quantum mechanical work on drug molecules is both interesting and useful. It may not answer all the questions but it can be very illuminating.
W.G. RICHARDS AND MOIRA E. BLACK
89
For example, it is possible that active compounds have to distort their conformation slightly when they encounter a receptor. Thus the structure of the molecule in the crystal or its conformation in free solution may not be the appropriate information which the medicinal chemist needs. Provided that the calculations can reproduce these known facts, however, they can also indicate in what manner the molecule could distort and just how much energy would be involved in any particular deformation. Molecular orbital calculations have been of some assistance already and providing enough care is taken they will certainly become more valuable. We prefaced this article with a cautionary quotation and it is appropriate t o finish with another of a more encouraging nature but which at the same time gives a true picture of the application of quantum mechanics t o drug research. It comes from Bertrand Russell. ‘Unless we can know something without knowing everything, it is obvious we can never know something.’ Molecular orbital calculations on pharmacologically interesting molecules do tell us something.
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11 12. 13 14. 15. 16.
L.B. Kier, Molecular Orbital Theory in Drug Research (Academic Press, New York, 1971). L.B. Kier (ed.), Molecular Orbital Studies in Chemical Pharmacology (Springer Verlag, Berlin, 1970) p. 31. R. Hoffmann, J. Chem. Phys., 39 (1963) 1397. R. Rein, N. Fukuda, H. Win, G.A. Clarke and F.E. Harris, J. Chem. Phys., 45 (1966) 4743. J.A. Pople and G.A. Segal, J. Chem. Phys., 44 (1966) 3289. R.N. Dixon, Mol. F’hys., 1 2 (1967) 83. J.A. Pople, D.L. Beveridge and P.A. Dobosh, J. Chem. Phys., 4 7 (1967) 2026. N.C. Baird and M.J.S. Dewar, J. Chem. Phys., 50 (1969) 1262. N.C. Baird, M.J.S. Dewar and R. Sustmann, J. Chem. Phys., 5 0 (1969) 1275. G. Diner, J.P. Malrieu, F. Jordan and M. Gilbert, Theoret. Chim. Acta, 15 (1969) 100. W.G. Richards and J.A. Horsley, Ab Initio Molecular Orbital Calculations for Chemists (Clarendon Press, Oxford, 1970). W.G. Richards, T.E.H. Walker and R.K. Hinkley, Bibliography of Molecular Orbital Calculations (Clarendon Press, Oxford, 1971). L.E. Sutton (ed.), Tables of Interatomic Distances and Configurations in Molecules and Ions (Special Publication of the Chemical Society, London, 1958) Vol. 11 (Main Volume) 1965; Vol. 1 8 (Supplement). L.C. Allen and J.D. Russell, J. Chem. Phys., 46 (1967) 1029. J.A. Pople, D.P. Santry and G.A. Segal, J. Chem. Phys., 4 3 (1965) S.129. M.S. Gordon and J.A. Pople, J. Chem. Phys., 49 (1 968) 4643.
90 17. 18. 19. 20. 21. 22. 23. 24. 25. 26 27. 28. 29. 30. 31. 32.
QUANTUM CHEMISTRY IN DRUG RESEARCH L. Famell and W.G. Richards, J. Theoret. Biol., 43 (1974) 389. R.S. Mulliken, J. Chim. Phys., 46 (1949) 497. K. Fukui, T. Yonezawa, and C. Nagata, Bull. Chem. SOC.Japan, 27 (1954) 423. K. Fukui, T. Yonezawa, and H. Shingu, J. Chem. Phys., 20 (1952) 722. A. Cammarata, J. Med. Chem., 11 (1968) 111. W.D.M. Paton, Proc. Roy. SOC.B., 154 (1961) 21. E.J. Ariens, Arch. Int. Pharmacodyn., 99 (1954) 32. R.RC, New and W.G. Richards, Nature 237 (1972) 214. F. Peradejordi, Aspects de la Chimie Quantique Contemporaine (CNRS, Paris, 1971) p. 261. B. Pullman, Aspects de la Chimie Quantique Contemporaine (CNRS, Paris, 1971) p. 261. L.B. Kier, Mol. Pharmacol., 3 (1967) 487. L.B. Kier, Mol. Pharmacol., 4 (1968) 70. L.B. Kier, J. Med. Chem., 11 (1968) 441. A.S.F. Ash and H.O. Schild, Brit. J. Pharmacol. Chemother., 27 (1966) 427. C.R. Ganellin, E.S. Popper, G.N.J. Port and W.G. Richards, J. Med. Chem., 16 (1973) 610,616. B. Pullman, J.L. Coubeils and P. Courrikre, J. Theoret. Biol., 35 (1972) 375.
2 Quantum Chemistry in Drug Research W.G. RICHARDS, M.A., D.Phi1. and MOIRA E. BLACK, B.A. Physical Chemistry Laboratory, University of Oxford INTRODUCTION
68
QUANTUM CHEMICAL METHODS General considerations Extended Huckel theory Iterative extended Huckel theory Complete neglect of differential overlap Intermediate neglect of differential overlap Perturbative configuration interaction using localised orbitals A b initio molecular orbital methods
69 72 72 73 73 74 74 75
THE CALCULATION O F GEOMETRIES
75
THE CALCULATION O F CONFORMATION
76
CALCULATED PARAMETERS Charge density Orbital energies Frontier electron density
79 79 80 81
THEORIES OF DRUG ACTIVITY
82
ACTIVITY AND BINDING TO RECEFTORS
84
APPLICATIONS T O SERIES O F COMPOUNDS
85
APPLICATIONS T O SINGLE COMPOUNDS
a7
DISCUSSION
88
REFERENCES
89
61
68
QUANTUM CHEMISTRY IN DRUG RESEARCH
INTRODUCTION A good guideline for medicinal chemists who are confronted with the increasing volume of literature on applications of quantum chemistry t o drug research is a quotation due to A.N. Whitehead: ‘There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain’. Drug research involves complicated biological problems where the relatively pure techniques of quantum chemistry might seem totally out of place. However, provided that a sufficiently critical attitude is maintained, much useful work can be done. The wealth of recent literature on the subject, which includes two books [1,2] provides examples both of the utility, and the dangers of naivety in the approach. Using the largest of electronic computers, good electronic wavefunctions can be computed for molecules, with up to about twenty atoms. In the realm of isolated gaseous molecules many properties can be calculated as accurately, or even on occasion more accurately than they can be measured. Examples include geometries and barriers to internal rotations. Physico-chemical experiments such as e.s.r. spectra or X-ray photoelectron spectra can be rationalised by means of theoretical calculation. Chemical reactivity remains a serious problem, but drug activity is in a sense a problem intrinsically more tractable. The activities of drugs fall into two main groups. There are those whose activities appear t o be related to their macroscopic properties, such as lipid solubility, and which are non-specific. Secondly, there are specific drugs where activity is closely allied to fine structural properties of the molecule. In such cases, minor alterations in structure can alter the activity of the drug by several orders of magnitude. We will only consider the latter type. The active molecule is envisaged as interacting with a receptor molecule in order to produce the response. This interaction is frequently considered as analogous to a ‘lock and key’ mechanism, although the search for receptors has not yet resulted in any very detailed understanding of their activity in terms of structure. For drugs which are highly specific, the classical methodology of pharmacology is to find an active compound; similar molecules are then synthesised by the organic chemist and tested for activity similar t o that of the lead compound (agonist activity), or for possible blocking (antagonist activity). The choice of molecules t o be prepared and screened may perhaps be made on the basis of the ease of synthesis. More logically, some relationship between structure and activity is preferred as a basis for testing. Measured physical properties of the
69
W.G. RICHARDS A N D MOIRA E. BLACK
molecule may be correlated with activity. In particular, Hansch has used a type of linear free energy relationship with some notable successes in such correlations. Theoretical calculations have two potential advantages over empirical correlations. Firstly, submolecular properties such as local charge densities or polarisabilities are easily calculated and may be more directly related to binding than overall molecular or bulk properties like partition. Secondly, and even more attractive, is the fact that the calculations may be performed without synthesising the molecule first. Hence, provided properties computed from wave functions correlate with activity, then predictions may be made about the activity of similar molecules and used as a guide to synthesis and screening. The organic chemistry and pharmacology will be as important as ever, but a little guidance could reduce the number of molecules which have to be tested. If care is taken in the calculation of the wavefunction, and series of related compounds are compared, then meaningful correlations can and have been obtained; geometries of active molecules may be predicted; tautomer and conformer populations can be determined and detailed charge distributions in the active molecules can be given. Perhaps most tempting of all, from the detailed charge distribution of the active molecule, something of the complementary nature of the receptor may be inferred.
QUANTUM CHEMICAL METHODS It is well known that all the problems of chemistry are in principlt solved in the Schrodinger equation, H+ = E $ . Were it possible to solve this equation for a system containing many electrons, then the resulting wave function could be used to calculate any property of the system. It is equally well known that beyond the example of the hydrogen molecule ion, H i , the equation cannot be solved exactly. However approximate solutions are possible and if computers are used, then these approximate wave functions do provide a basis for calculating observable properties to an almost arbitrary level of accuracy. The only problem is that to obtain very accurate results a considerable amount of expensive computer time must be employed. All the common approximate methods for obtaining wave functions express the wave function of the whole molecule, $, as a product of orbitals, @,which individually represent functions which deal with one single electron, i.e.
+
70
QUANTUM CHEMISTRY IN DRUG RESEARCH
Each @ is further expressed as a sum of atomic orbitals, appropriate coefficients, i.e. $1 = C,Xl -t c2x2 -t
x, multiplied
by
... .
This expression is often referred to as the linear combination of atomic orbital (LC AO) expansion. As a simple example we can consider the molecule LiH which has four electrons in two doubly occupied sigma-type molecular orbitals
This product, which has to be antisymmetrised to take account of the Pauli Principle, indicates the occupied molecular orbitals @ j . Each 4 can be expressed as a linear combination, for example, as
~ ~ , = ~ ~ x ~ ~ ~ , ~ ~ ) + c ~ X ( l s , N ) The two atomic orbitals, x, are known from work on atoms and the whole problem for the quantum chemist is to determine the coefficients Ci.Once the coefficients are known, the $ j and hence the overall wave function, $, are specified. One of the irritations of quantum chemistry is that it is bedevilled with a plethora of acronyms. Each refers to one of the methods of determining the coefficients. The acronyms and names of the methods most commonly applied to pharmacological problems are indicated in Figure2.1. In this figure the methods produce increasingly accurate answers as the table is read from the bottom to the top and from left to right. Generally the more accurate the method the more costly in terms of computer time, which also mounts rapidly as the number of electrons in the molecule under consideration increases. Roughly this increase in time is proportional to the fourth power of the number of electrons in the molecule. For molecules of interest to the medicinal chemist this means that only the relatively crude methods are employed. To assess the literature, a medicinal chemist need only know of the strengths and weaknesses of the particular techniques. An excellent review of some of the methods has been published by Hoyland [2] and below we summarise some of the salient points of the various molecular orbital methods. Before doing so one further important point must be stressed. The starting data for a particular calculation are merely the Cartesian coordinates of the various atoms in the molecule. These may be calculated directly
W.G. RICHARDS AND MOIRA E. BLACK
%
HARTREE MCK UMlT
t
I
INITIQ, EXTENW) BAYS
t
AB INITIO, MlNlMUM W S
I
INDO, MIND0
I -
AMXlRATE RWLTS
CI
CI
CI
CNDO
IEHT
71
pc'Lo
-
t
EHT
Figure 2. I . A classification of Molecular Orbital Methods. Accuracy improves from bottom to top of the dingram and left to right.
from the bond lengths and bond angles. The resulting wave function and energy of the molecule for this particular geometry is thus strictly appropriate only to the lone gas phase molecule. There is no simple way of including any solvent effects. Thus, applying the results to a biological situation where the molecule will be in a solution and possibly in an environment of uncertain pH requires an act of faith. It is an act of faith which is scarcely justified if one is only considering a single molecule. There may be more justification if one is concerned with a series of similar molecules, where solvation may be a constant factor, or different geometrical arrangements of the atoms in a single species where again differences in energies may be realistically computed. It is not too harsh a judgement to say that quantum mechanical calculations on drug molecules should only be given much credence if they are performed on a series of similar compounds or geometries and preferably if also there are some physico-chemical measurements which lend support to the values of the wave functions. We now give a brief resume of the molecular orbital methods commonly applied to biologically active molecules.
12
QUANTUM CHEMISTRY IN DRUG RESEARCH GENERAL CONSIDERATIONS
If we use our orbital expansion in the Schrodinger equation, it becomes, for each one electron orbital @,
Here H is a hamiltonian operator and expansion then we have
E
the orbital energy. In the LCAO
These equations can conveniently be transformed into the Roothaan equations
where
A set of such equations is at the heart of all molecular orbital methods. In principle a knowledge of all integrals of the types H l k and S Ik enable both the coefficient Clk (and hence the orbitals @)and the orbital energy E to be calculated. The various approximations differ in the methods used to deduce appropriate values for the integrals H l k and S l k which are frequently referred to as matrix elements. EXTENDED HUCKEL THEORY [3] (EHT)
This method ignores core Is electrons on all atoms except hydrogen but is capable of dealing with molecular orbitals of both u- and n-types. Integrals of the type Hll are replaced by experimental ionisation potentials and those of type Hlk deduced from ionisation potentials using the prescription
K is an empirical constant usually set equal to 1.75. All the overlap integrals S l k are actually calculated using so-called Slater atomic orbital functions for the X I .
W.G. RICHARDS AND MOIRA E. BLACK
73
The energy of the molecule is assumed to be a sum of the energies, E , of all the occupied molecular orbitals. This implies a neglect both of electron-electron and nuclear-nuclear repulsions. The theoretical basis of this method is very shaky indeed. However SO much experimental data is put into the calculation that little actual calculation remains and EHT can almost be considered as an interpolation formula connecting data. Despite its simplicity, it does appear to give sensible indications of the preferred geometries and conformations of molecules, particularly aliphatic hydrocarbons, although charge distributions are exaggerated and less reliable. Its greatest utility is the prediction of conformation but not barriers t o conformational change. ITERATIVE EXTENDED HUCKEL THEORY [ 4 ] (IEHT)
Simple EHT uses ionisation potentials for H 11 and also shows that two carbon atoms in a molecule may have different charges. Thus it would be more realistic to have slightly different ionisation potentials for these two carbon atoms since the ease of removing an electron will depend on the electronic charge surrounding the atom as well as its nuclear charge. IEHT achieves this by doing an EHT calculation, then using the charges produced t o modify the values of H 11 and repeating this cycle until the charges are self-consistent. The charge distiibutions are improved and predictions of dipole moment are reasonable. The method is as good as EHT for conformational prediction. COMPLETE NEGLECT OF DIFFERENTIAL OVERLAP [ 5 ] (CNDO)
This method is less empirical than Huckel approximations. Integrals of the type Hlk are no longer treated simply as empirical parameters. The actual nature of the operator H is considered and each matrix element H l k is expanded as an
appropriate set of integrals over atomic orbitals x, multiplied by products of the coefficients C. Since determining the coefficients C is part of the exercise, the method is again an iterative self-consistent field technique. Values of C in the expansion
are guessed, the integrals over functions x deduced and ultimately the Roothaan equations are solved giving new values of C and this is repeated until convergence is achieved.
I4
QUANTUM CHEMISTRY IN DRUG RESEARCH
Only valence electrons are considered and the integrals over the x functions are calculated in some simple cases, ignored, taken from experiment or treated as empirical parameters. In particular, many integrals involving x on two separate atoms are put equal to zero which is the origin of the name of the method. Computer programmes of this type can deal with molecules with up to about twenty atoms before the computing time becomes a serious problem. Generally this type of method gives good geometries but is unreliable for conformational studies. Charge distributions on the other hand do agree with experimental observation from dipole moment or e.s.r. studies. INTERMEDIATE NEGLECT OF DIFFERENTIAL OVERLAP [ 6 - 9 ] (INDO AND MINDO)
INDO differs from CNDO by the inclusion of some electron repulsion integrals which are neglected in the simpler method. One advantage is that the method does distinguish between singlet and triplet electronic states and further, e.s.r. hyperfine coupling constants can be predicted. MINDO differs from INDO in the procedures used to estimate the various integrals. It provides very good agreement with heats of formation of many compounds, but fails t o predict the relative stabilities of rotational isomers and gives poor dipole moments. The electron distribution in molecules containing hetero atoms is exaggerated. PERTURBATIVE CONFIGURATION INTERACTIONS USING LOCALISED ORBITALS LOCALISED ORBITALS [ 101 (PCILO)
This is a method based on the expansion of molecular orbitals in terms of bond orbitals rather than atomic orbitals. The main hypotheses of the CNDO approximation are maintained and the status of this method is roughly parallel to the CNDO calculations. Some improvement is obtained by including configuration interaction which is a technique of improving a wave function and hence an energy calculation by allowing the wave function to take some account of excited states of the molecule. There are an infinite number of excited states which could be included but the PCILO method restricts attention to the few closest in energy to the ground state of the molecule. In terms of predictions of geometry or conformation the results of the PCILO method are broadly similar to those of the CNDO method but they are obtained much more quickly. This is an important consideration since, if calculations can be performed using only small amounts of computer time, then it becomes possible to do many calculations on a molecule rather than just a few. PCILO
W.G. RICHARDS A N D MOIRA E. BLACK
15
has in this way been responsible for a significant advance in the application of quantum chemical methods to drug research: using PCILO methods it is now possible t o produce complete energy maps with contours when considering conformational energy. These give considerably more information than the earlier results which were normally presented as graphs of energy against some conformational twist. AB INITIO MOLECULAR ORBITAL METHODS [ 11,121
In ab initio methods all electrons including the core electrons are considered. No integrals are ignored or replaced by empirical parameters. This means that many millions of integrals have to be computed for a molecule containing more than about twenty atoms and the computer time needed which is roughly proportional to the fourth power of the number of electrons, mounts prohibitively. These rigorous techniques are improving steadily in both speed and accuracy but drug molecules are still beyond their practical capabilities especially when it is borne in mind that not one but perhaps several hundred calculations need to be performed on each molecular species if a thorough study is to be made. The only serious impact which ab initio methods have had on calculations on pharmacologically interesting molecules is that they provide bench-mark calculations on small molecules against which any approximation should be compared before applying i't to complicated species. This obvious test of reliability has not always been performed. THE CALCULATION OF GEOMETRIES The method which has to be followed to determine geometries by theoretical calculation is logical. We perform a calculation of the energy of the molecule for a given geometrical arrangement of the constituent atoms. One by one all bond lengths and bond angles are varied by small increments and the predicted geometry will be that structure which is calculated t o be the most stable. This logical procedure would involve many thousands of separate molecular orbital calculations and even in principle is limited t o calculations of the ab initio type since the empirical parameters used in the approximate work are only suitable for standard bond lengths. Thus an approximation to the idealised scheme is favoured. This involves starting with known bond distances and angles (perhaps from published crystal data) or using standard values. There is an invaluable set of data published by Sutton [ 131. Bond angles may be found by studying the variation of calculated
16
QUANTUM CHEMISTRY IN DRUG RESEARCH
energy with bond angle and the results are very satisfactory for most of the molecular orbital methods discussed above. Table 2.1 gives some typical results for the calculation of bond angles and lends confidence to the view that rigid geometries can be calculated theoretically.
THE CALCULATION OF CONFORMATION The theoretical calculation of geometry gives some indication of the reliability of the various molecular orbital methods but is of no real value to the medicinal chemist. Satisfactory physical methods may be used to determine geometries and even the use of molecular models is normally sufficient. The same cannot be said of conformation. Molecular models of most drug molecules show considerable flexibility due to the rotation of groups about single bonds and it is of vital importance to have some idea of the conformation of an active species as this may lead to possible inferences about the nature of the receptor. In some simple cases nuclear magnetic resonance techniques may give some information about conformation and X-ray crystallographic studies can be made on crystalline phases, but in general the problem of determining conformation and relative conformational preferences remains a difficult one. Quantum chemical calculations should be able t o treat conformational studies in precisely the same manner as investigations of geometries. A calculation is performed for a series of positions of one part of the molecule with respect to another and the energies are then compared. If there is only one bond about which rotation can occur the results can be presented simply as a graph of energy against angle. If, as is frequently the case Table 2.1. Predicted Bond Angles 114, 15, 161 Molecule
Experiment
Ab initio
H20 CHI NH3 MeOH
104.5 103.2 106.6 (IICH) 109.5 (HCO) 109.5 (COH) 105.9 (HCH) 108.8 (HCC) 110.1
110
C2H6 ____.
~~~~
106.6
INDO
106.0 106.4 108.2 110.7 107.3 106.6 112.2
CNDO
EHT
107.1 108.6 106.7
150
W.G. RICHARDS AND MOIRA E. BLACK
77
(acetylcholine, histamine or dipeptides), conformation is largely specified by rotations about two bonds then ideally a conformational energy map should be produced, with the two variable angles as axes and the energies indicated by contours. To produce a complete map may involve a very large number of separate calculations so that until very recently this labour was avoided and results were presented in the less satisfactory manner of series of graphs each showing the variation in one angle while the other is held constant. Minima indicate stable conformational structures, the depths being indications of relative stability. The advent of complete conformational energy surfaces has highlighted an important point of thermodynamics ignored in most published work. If there are several minima then the relative populations of each region of the energy surface depend not on internal energy differences but on free energy differences. Entropy effects have to be considered. Put very crudely, the relative populations of two valleys in an energy surface will depend firstly on how difficult it is for a molecule to get out of the valley (the enthalpy) and on the ease with which molecules can get into the valleys due to their varying widths (or entropy). It is possible, although it is not yet generally done [17] t o compute free energy differences from the energy diagrams. Figure 2.2 shows energy surface situations in which there are more than one minimum. Figure 2.2(a) is a case where there is only one variable angle; a deep stable minimum x and a shallower one y are marked. Figure 2.2(b) concerns a hypothetical case where there are
(a 1
*
W
a w
L w
Yr X
1
ANGLE 8
Oi
9
3 16
Figure 2.2. Energy diagrams (a) With a single variable angle of rotation. (b) With two independently variable rotation angles.
78
QUANTUM CHEMISTRY IN DRUG RESEARCH
two variable angles 01 and 0 2 , one deep maximum x and two equivalent shallower depressions y : these could represent a trans and equivalent gauche conformations of a disubstituted ethane-like molecule. In order to decide upon the relative populations of molecules in the conformations x and y , we need to be able to estimate the difference in free energy of these conformations. In an approximate way this may be done by using a classical Boltzmann partition function for each region. For the single angle case this would be defined as
2=
c 0
exp[-~(O)/kT]
and in the two variable angle case
where E ( 0 ) or ~ ( 0 1 0 2 )is the calculated internal energy for a conformation defined by 0 or (0 1 , 02), k is the Boltzmann constant and T is the temperature taken as say 37°C. The summations would be taken over a regular grid of points fine enough t o reflect the shape of the energy surface and bounded by an energy contour which is arbitrary but could be set at k T or 2kT. If we assume that all contributions t o the entropy are constant, other than those derived from the surface, and that any volume change on altering conformation is negligible, then
AGO = -kT ln'(Z,/Z,) per molecule. In the work on histamine it has been shown [ 171 that the mole fraction of trans conformer is 0.62 computed using free energy differences but 0.56 if calculated internal energy differences are used. This difference is significant and yet the actual energy surface is by no means an extreme case. T h ~ simportant thermodynamic point has been ignored in most of the published work on calculations of conformational equilibria and should be borne in mind when assessing such work. A further very important aspect of such calculations is also under-stressed. This is that the calculations are performed on idealised 'lone gas phase molecules'. Thus the possible effects of solvation are neglected completely. These effects may be unimportant in considering different conformations of a molecule if each conformation is solvated in the same way. However, occasionally calculations indicate specific intramolecular interactions which would probably be genuine in the gas phase but improbable in solution. Notably calculations
W.C. RICHARDS AND MOlRA E. BLACK
19
often suggest the presence of intramolecular hydrogen bonds, conformations with such bonds seeming exceptionally stable. In aqueous solution, however, any capacity for hydrogen bonding will surely be satisfied by the solvent even if infra-red spectra in carbon tetrachloride support the idea of intramolecular effects in that medium. It is fair t o say that theoretical calculations can give accurate predictions of the relative energies of conformations of molecules in the gas phase. If enough care is taken to compute free energies then relative populations can be calculated but the effect of solvent remains a problem. For this reason it is not safe to trust predictions made by theoretical calculations on one single molecular species. It is far safer to look at a series of similar molecules and consider differences between them when solvation effects may be constant and further as always some experimental test of the value of the calculations is really necessary to make them convincing.
CALCULATED PARAMETERS Theoretical calculations produce not only wave functions and energies but also the possibility of calculating a number of other molecular parameters. These may in turn be correlated with biological activity. We now give a brief resumC of the parameters which are most commonly encountered in applications to medicinal chemistry. CHARGE DENSITY
The square of a wave function gives a measure of charge density. The charge distribution within a molecule is usually defined in terms of the gross charges on the various nuclei and again this may be derived from the wave function. However, it must again be stressed that any idea of ‘the charge at a particular nucleus’ is a rather woolly concept. The only charge on the nucleus is its formal charge. What we are interested in is how the influence of this is modified by having electrons close to it, but ‘close to’ is also rather imprecise. When we have a bond between two atoms how can we assign the electrons to one nucleus or the other? This vexed question has been the subject of much debate but in pharmacological work the simplest method is normally adopted. This is the so-called ‘Mulliken population analysis’ [ 181 . If our molecular orbitals are LCAO functions, i.e.
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QUANTUM CHEMISTRY IN DRUG RESEARCH
then the net atomic population of a given ortital xi is defined as
The overlap population defined as
-
the population between atoms i and j is further
Sii being the integral Jxq d7 which we have met earlier. Since p i = Oii, we can write the gross population of orbital
as
pi =CO, i
Finally the total population at any nucleus n is given by adding all the values of
Pi for orbitals j which are centred on atom n. Most molecular orbital computer programmes will do this simple calculation automatically and print out a list of atomic populations. Thus it is important to know how meaningful are these quantities. From the atomic populations it is possible t o compute dipole moments which can be compared to experimental values. The qualitative nature of the agreement with experiment is easily summarised ab initio methods
CNDO, MIND0 IEHT EHT
good charge densities good charge densities good charge densities very exaggerated charge densities.
However all the methods do seem capable of reproducing trends in charge density variation within a set of similar molecules. ORBITAL ENERGIES
Corresponding to each molecular orbital & there is an orbital energy q.Of most interest to workers interested in correlations of properties and activities are the energy of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). These would be the orbitals involved in any donation or acceptance of charge.
W.G. RICHARDS AND MOIRA E. BLACK
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FRONTIER ELECTRON DENSITY [ 191
The frontier electron theory [19] was originally developed to explain the difference in reactivity at each position in an aromatic hydrocarbon. It is based on the intuitive idea that the reaction should occur at the position of the largest density of the electrons in the frontier orbitals, which are defined according t o the type of reaction: in an electrophilic reaction, the highest occupied molecular orbital (HOMO); in a nucleophilic reaction, the lowest empty molecular orbital (LEMO); in a radical reaction, both of these. This theory was later given a sound theoretical basis by Fukui, Yonezawa and Shingu [20] who then introduced the concept of superdelocalisability. Denoting the occupied molecular orbitals by 1, 2, ... m , and the unoccupied levels by m + 1, m + 2, ... N , the superdelocalisability, S,, is given for the three types of reaction by: (a) for an electrophilic reaction
(b) for a nucleophilic reaction
(c) for a radical reaction
where Crj is the coefficient of the rth atomic orbital in the jth molecular orbital, and Xi is the coefficient in the orbital energy, which is given as ~j = c( + Xjp. The orbital which mainly determines the value of S, in each type of reaction is the same as the frontier orbital previously considered. There are problems in the use of both frontier electron density and superdelocalisability. The latter concept was originally put forward considering the n-electron part of the molecule only, with the energies of the orbitals being given in units of the resonance integral of a C-C bond in benzene. This means that in a series of molecules there would be a common zero of energy. Using all-valence molecular methods, the energies are obtained in absolute terms, so that the zero
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82
of energy is in the unoccupied orbitals. This is obviously not consistent in a series of molecules. The frontier electron density strictly permits only a comparison of reactivities at different positions within the same molecule. In order t o extend this concept for use over a series of molecules, a further quantity, F , may be considered:
where f, is the frontier electron density, e is the energy of the appropriate frontier orbital. F may be thought of as a weighted frontier electron density, in the sense that ease of removal of the particular electron is also considered.
THEORIES OF DRUG ACTIVITY The receptor theory of drug action implies that the pharmacological properties of a compound are dependent not only on the nature and properties of the constituent groups within the molecule, but also on the way in which these groups are distributed in space. This follows from the idea that the receptor is a discreet, spatially organised structure, and that maximum activation of the receptor only occurs when there is a close fit between the drug and the receptor. However, it is conceivable that interaction of a drug molecule with a particular feature of the receptor may induce a change in the receptor that brings additional drug and receptor features into close proximity. Production of an observable biological effect, generally associated with the interaction of a drug with a receptor, involves a long chain of events, of which the action at the receptor is probably only the first step. The methods of investigation of the drug-receptor binding all follow from a relationship derived by Cammarata [21]. The interactions of a drug with a receptor is considered as analogous to the combination of substrate and an enzyme, so that: S+
a
R
e SR*
b
where S is the drug molecule, R is the receptor, SR* is the drug-receptor complex. From this relation, two main theories of drug action are derived. The kinetic theory of Paton [22] considers that the rate-determining step is the rate of combination of the drug with the receptors, i.e. a is the rate-determining step.
W.G. RICHARDS AND MOIRA E. BLACK
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The occupation theory of Ariens [ 2 3 ] considers that it is the number of occupied receptors that determines the magnitude of the response, i.e. b is the rate-determining step. Experimentally, it is difficult to distinguish between the two theories, but, fortunately, they lead to the same relationship between drug concentration and biological response. The 'activated' drug-receptor complex implicit in the Paton theory, or the relatively more stable drugreceptor complex consideration in-Ariens' theory, is assumed t o be the response-determining factor, so that drug activity, A , will be proportional to the free energy of formation of a drug-receptor complex AGRs. Considering only the first-order interactions:
AaAG,, =
tA
G t~ AGS t AGP t A G ~ ( A G+ A ~ G ~+ A@) tAG~(AG tA ~ G ~+ AGP)
t AGS(AG~ t A G t~AGP) t AG*(AG~t A
G t~A G ~ )
where Ace, AGS and A@ are independent contributions t o the free energy due to electronic and steric interactions between drug and receptor and t o conformational changes in the receptor, respectively. AGd is introduced to represent the free energy change due to desolvation which would acc'ompany the union of drug and receptor. If we consider the case where interaction terms are constant or negligible, this leads t o the following expression:
AaAG,,=
AGe t AGd t ACs t AGP t k .
The method used is an examination of the variation of the individual terms, particularly AGe and A@, for a series of related compounds, using molecular orbital calculations. Conformational analysis of the drug molecules may be carried out as described above, to determine which of the conformations could be the one which the drug adopts t o fit the receptor. Electronic charge distributions may also be calculated - the variation of charge on corresponding atoms in a series of molecules may be found t o match changes in their biological activities, which could be good evidence for the identity of the pharmacodynamic groups. At this point yet another caveat must be introduced about the use of calculations to estimate AGe for a series of compounds and ?hen seeking correlations between these parameters and observed biological activities. The calculations are performed on the unperturbed ground state of a molecule,
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QUANTUM CHEMISTRY IN DRUG RESEARCH
1 I
TRANSITION STATE
REACTION COORDINATE
Figure 2.3. 7%e transition state view of reactivity.
whereas differences in reactivities are more properly ascribed to differences in transition state (see Figure 2.3). The calculated parameters such as charge density differences in a series of compounds only tell us about the initial slopes of the reaction curves (position R in the figure). It is assumed that since for drug-receptor interactions the binding energy is low, AE is small enough for estimates of the initial slope of the curve to reflect the different activation energies. Although this may be true for the type of reaction which typifies drug-receptor combination it is not generally true in organic chemistry.
ACTIVITY AND BINDING TO RECEPTORS Most of the theoretical methods applied to series of similar compounds with subsequent searches for correlations between computed properties and measured biological activity have been based on the following logic; Calculated Parameter a Binding Free Energy a Biological Activity (1)
(11)
UII)
Thus stage I is correlated with stage 111. This will only be valid if I is proportional t o I1 and I1 to 111. The theories of drug activity depend on I1 correlating with 111. Recently [24] an attempt has been made to list the correlation of I with 11, within the framework and area of applicability of the more ambitious correlations. As data on binding in a biological environment, the binding between haptens and antibodies were taken. It seems not unreasonable to consider the binding site of an antibody to be a model for a receptor and what is needed is a set of data covering the binding of a series of rather similar molecules to the antibody. An appropriate system is the antibody obtained against the hapten consisting
W.G. RICHARDS AND MOIRA E. BLACK
85
of the phenyltrimethylammonium ion attached to protein by a para-azo linkage. Inhibition experiments can be used to measure the extent to which a variety of unattached ammonium ioqs prevent precipitation and hence binding free energies can be measured. It is likely that the binding between this particular hapten series and the antibody receptor site is electrostatic and chemical investigation suggests the presence of a carboxyl group in the receptor which could attract the positive ammonium ion. Since this ion is quaternary substituted, hydrogen-bonding and charge-transfer mechanisms seem unlikely. In fact, statistically significant correlations between the experimental binding and calculated charge on the nitrogen atom were found even when the simple Extended Hiickel molecule orbital approximation and Mulliken population analysis were used. A further encouragement t o the use of correlations of computed parameters with activity comes from some as yet unpublished work from this laboratory. We have found that once again there are significant correlations between computed properties and octanol-water partition coefficients. In this case the experimental measurement can be thought of as being analogous t o binding to a lipid phase. The relative attraction of a molecule to water or octanol seems to be dependent on charge and on orbital energies. Both these series of calculations lend support to the notion that it is possible to produce meaningful correlations between computed properties and biological activity. Once proved and satisfactory, the advantage of these correlations over empirical structure-activity relationships is that they may be applied to molecules before they are synthesised and hence provide a guide t o synthesis.
APPLICATIONS TO SERIES OF COMPOUNDS Rather than quote many examples we will restrict ourselves to a single example which illustrates the general principles and difficulties. A particularly impressive-looking correlation between computed properties and measured activities has been found by Peradejordi 1251 who studied the tetracyclines. The tetracyclines are believed t o operate by the means of forming a complex with the ribosome. The equilibrium constant for the binding reaction is related to the binding energy between the drug and receptor. If the receptor is unique, then the variation in binding for a family of tetracyclines will be dependent solely on variations in the drug molecule. Thus, in principle it should be possible to find a correlation between activity and calculated parameters related to binding such as electronic charges, Q , and indices of nucleophilic
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QUANTUM CHEMISTRY IN DRUG RESEARCH
reactivity, N , or electrophilic reactivity, E. For a series of compounds labelled by the suffm j , the relative binding energies will then be given by
Aei =
c
[aiQi + biNi + c i E i ] .
i
Calculation can provide estimates of the quantities Qi,Ni and Ei for each atom i in the molecule and the coefficients ai, bj, Ci are obtained by fitting known data by multiple regression analysis. Peradejordi produced the following formula for the activity A ; l o g A j = 18.4 + 56 Qolo+ 1 7 b ' 0 , ~+ 48 Qolo
-Eolo
+
71 QoIz+ 18Eo12+ 3 Qc,
.
The fit is extremely good (Figure 2.4) and suggests that the variation in antibiotic power depends largely on the effect of substituents on the atoms 0 1 0 , 011,012 and C6.
log Ai (experimental) Figure 2.4. Typical correlation between calculated and experimental activity. (From Actua1iti.s de Chimie Thkrapeutique (R.Daudel (1971) p . 8).)
W.G. RICHARDS AND MOIRA E. BLACK
81
This is only one of a number of such correlations. The statistical success is striking but a few words of caution must be emphasised. Above all, it is notable that rather a large number of disposable parameters are employed. In the above case fourteen pieces of experimental data were fitted t o an expression with seven coefficients. It is not surprising that such a fit looks good. If more data are t o be used this frequently involves trying to include compounds with very different structures where the intricacies of binding may be substantially different. For these reasons the value of correlations, remembering the difficulties stressed earlier in this article, is not for the quantitative predictions which they offer but more for the insight into important features of molecules required for particular types of activity. If too many adjustable parameters are included, then this leads to highly complicated equations which are difficult t o interpret physically. It would seem t o be more useful and meaningful to consider only a small number of parameters, perhaps as few as one or two, and to obtain results which have some physical significance even if they are less statistically impressive. Correlations, even if they are of high statistical significance, are of little interest unless they lead to verifiable hypotheses or alternatively unless they can be used predictively. APPLICATIONS TO SINGLE COMPOUNDS Calculations on single compounds have most frequently been aimed at understanding the conformation of the molecule. Details of many such calculations have been given by Kier [ l ] and by Pullman [26] but here we again will limit comment t o a few illustrative references. Calculations have been performed by Kier in a number of species including acetyl choline [17], muscarine [27], muscarone [27], nicotine [28] and histamine [29]. In these calculations he uses the Extended Huckel Method and varies angles independently, presenting the results as sections through potential surfaces rather than conformational energy maps. In general the proposed conformations are in agreement with the limited structural data available from X-ray crystallography, despite the fact that the effects of the solvent are not Considered. In the case of histamine [29] Kier found that there are two relatively stable conformers which he hypothesised could be responsible for the two distinguishable physiological activities of the molecule [30]. More extensive work [31] which studied the conformer population ratios of a series of methyl substituted histamines by similar methods but also supported by n.m.r. measurements does not support the contention. This more recent work is satisfactory in the sense
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QUANTUM CHEMISTRY IN DRUG RESEARCH
that it does produce agreement between calculated and observed conformer and tautomer ratios, but daunting because it suggests that it is necessary t o perform a vast number of calculations on an extensive set of similar compounds before being able to use the results with confidence. Conformational studies using the more trustworthy PCILO method have beell made on dipeptides [26] and on barbiturates [32]. In these instances conformational energy maps have been published and again the predicted stable conformations do seem to be in good accord with crystallographic data. Even the conformational angles of peptide linkages in high resolution X-ray crystal structures of proteins seem to be compatible with the calculations on the small units, leading to the hope that the energy maps may be of assistance in the prediction and explanation of protein structures. Since once again solvation has to be ignored, the differences between compounds are probably of more significance than any absolute numerical result. Calculations on single molecules can be made with great ease using standard computer programmes and only a few minutes of computer time. They may be of great assistance in helping a medicinal chemist to have a feeling for the structure, conformation and charge distribution in his molecule. As long as they are treated as aids to thinking, rather like molecular models, then they are probably useful. On the other hand because of all the approximations and dangers mentioned above, any calculations on a single gas phase species should not be taken as the basis of a theory.
DISCUSSION This article has taken a deliberately critical view of the application of quantum chemistry t o drug research since it is clear from the rapidly expanding number of publications in the field that many of the assumptions inherent in the work are not widely realised. It is very simple t o do a quantum mechanical calculation using a standard programme on a large computer: far simpler than for example running a spectrum. No one would publish the spectrum of a single compound and yet many theoretical papers contain far less information than a single spectrum. This facile work does quantum chemistry a disservice. All molecular behaviour is ultimately based upon quantum mechanics. In chemistry, after a shaky start, quantum mechanical methods, particularly molecular orbital theory has been abundantly fruitful, but it has been necessary to adopt a high standard of criticism and to be aware of the limitations of particular methods. The best quantum mechanical work on drug molecules is both interesting and useful. It may not answer all the questions but it can be very illuminating.
W.G. RICHARDS AND MOIRA E. BLACK
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For example, it is possible that active compounds have to distort their conformation slightly when they encounter a receptor. Thus the structure of the molecule in the crystal or its conformation in free solution may not be the appropriate information which the medicinal chemist needs. Provided that the calculations can reproduce these known facts, however, they can also indicate in what manner the molecule could distort and just how much energy would be involved in any particular deformation. Molecular orbital calculations have been of some assistance already and providing enough care is taken they will certainly become more valuable. We prefaced this article with a cautionary quotation and it is appropriate t o finish with another of a more encouraging nature but which at the same time gives a true picture of the application of quantum mechanics t o drug research. It comes from Bertrand Russell. ‘Unless we can know something without knowing everything, it is obvious we can never know something.’ Molecular orbital calculations on pharmacologically interesting molecules do tell us something.
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L.B. Kier, Molecular Orbital Theory in Drug Research (Academic Press, New York, 1971). L.B. Kier (ed.), Molecular Orbital Studies in Chemical Pharmacology (Springer Verlag, Berlin, 1970) p. 31. R. Hoffmann, J. Chem. Phys., 39 (1963) 1397. R. Rein, N. Fukuda, H. Win, G.A. Clarke and F.E. Harris, J. Chem. Phys., 45 (1966) 4743. J.A. Pople and G.A. Segal, J. Chem. Phys., 44 (1966) 3289. R.N. Dixon, Mol. F’hys., 1 2 (1967) 83. J.A. Pople, D.L. Beveridge and P.A. Dobosh, J. Chem. Phys., 4 7 (1967) 2026. N.C. Baird and M.J.S. Dewar, J. Chem. Phys., 50 (1969) 1262. N.C. Baird, M.J.S. Dewar and R. Sustmann, J. Chem. Phys., 5 0 (1969) 1275. G. Diner, J.P. Malrieu, F. Jordan and M. Gilbert, Theoret. Chim. Acta, 15 (1969) 100. W.G. Richards and J.A. Horsley, Ab Initio Molecular Orbital Calculations for Chemists (Clarendon Press, Oxford, 1970). W.G. Richards, T.E.H. Walker and R.K. Hinkley, Bibliography of Molecular Orbital Calculations (Clarendon Press, Oxford, 1971). L.E. Sutton (ed.), Tables of Interatomic Distances and Configurations in Molecules and Ions (Special Publication of the Chemical Society, London, 1958) Vol. 11 (Main Volume) 1965; Vol. 1 8 (Supplement). L.C. Allen and J.D. Russell, J. Chem. Phys., 46 (1967) 1029. J.A. Pople, D.P. Santry and G.A. Segal, J. Chem. Phys., 4 3 (1965) S.129. M.S. Gordon and J.A. Pople, J. Chem. Phys., 49 (1 968) 4643.
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QUANTUM CHEMISTRY IN DRUG RESEARCH L. Famell and W.G. Richards, J. Theoret. Biol., 43 (1974) 389. R.S. Mulliken, J. Chim. Phys., 46 (1949) 497. K. Fukui, T. Yonezawa, and C. Nagata, Bull. Chem. SOC.Japan, 27 (1954) 423. K. Fukui, T. Yonezawa, and H. Shingu, J. Chem. Phys., 20 (1952) 722. A. Cammarata, J. Med. Chem., 11 (1968) 111. W.D.M. Paton, Proc. Roy. SOC.B., 154 (1961) 21. E.J. Ariens, Arch. Int. Pharmacodyn., 99 (1954) 32. R.RC, New and W.G. Richards, Nature 237 (1972) 214. F. Peradejordi, Aspects de la Chimie Quantique Contemporaine (CNRS, Paris, 1971) p. 261. B. Pullman, Aspects de la Chimie Quantique Contemporaine (CNRS, Paris, 1971) p. 261. L.B. Kier, Mol. Pharmacol., 3 (1967) 487. L.B. Kier, Mol. Pharmacol., 4 (1968) 70. L.B. Kier, J. Med. Chem., 11 (1968) 441. A.S.F. Ash and H.O. Schild, Brit. J. Pharmacol. Chemother., 27 (1966) 427. C.R. Ganellin, E.S. Popper, G.N.J. Port and W.G. Richards, J. Med. Chem., 16 (1973) 610,616. B. Pullman, J.L. Coubeils and P. Courrikre, J. Theoret. Biol., 35 (1972) 375.
