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One of the most rewarding aspects of chemistry is the ability to create new forms of matter that were previously unknown and perhaps even unsuspected. Chemists have now made, or at least identified, more than 10 million different molecules. Around 400 000 new molecules are described in chemical research journals every year. Some of them occur naturally, whereas others are synthesised in the laboratory for the first time. Sometimes, the novel creations are compounds with highly interesting and useful properties: plastics that can conduct electricity, fuels that burn cleanly, pesticides that work selectively and, of course, new pharmaceutical drugs.

Developing such compounds can be expensive and time-consuming, however. Traditionally, it has involved making and testing thousands of chemically similar materials on a somewhat hit-or-miss basis in order to find the compound with the optimum properties. This is particularly true in the pharmaceuticals industry. Clearly, a way of knowing the properties of a chemical in advance would greatly help the chemicals industry in identifying and designing useful new materials. Chemists would like to find a short cut-a more quantitative strategy that would predict accurately the properties of a new chemical before they attempted to make it.

How can this be done? We might start from a compound’s molecular structure. Modern spectroscopic techniques-infrared spectroscopy and nuclear magnetic resonance in particular-have allowed us to determine precisely the structure of most known molecules. The structural information we can glean includes the number of atoms in the molecule, the kinds of atoms present, the number and types of bonds there are between the atoms, the lengths of the bonds, the arrangement of the atoms in space (the angles between bonds), and the way in which the atoms are linked by the bonds.

The next step is to find some reliable way of determining a substance’s physical and chemical properties-melting point, reactivity or biodegradability, and so on-from the structure. The earliest way of predicting chemical properties was to assume that each atom in a molecule would contribute a fixed amount to the property of interest. This could then be estimated by simply adding up the contributions from all the atoms in the molecule.

This approach works well for some properties: molecular weight, for example, derives from adding up the weights of all the atoms in a molecule. The molecular weight of a material can sometimes, in turn, offer a good guideline for other properties. It often correlates with the melting and boiling points of a substance, for example. Hydrocarbons (compounds consisting only of carbon and hydrogen atoms) with a low molecular weight such as methane (CH₄) and ethane (C₂H₆) are gases. Slightly heavier hydrocarbons such as octane (C₈H₁₈) and decane (C₁₀H₂₂) are liquids. Long-chain hydrocarbons such as polyethylene are, of course, solids.

It is in organic chemistry-the chemistry of carbon compounds-where it is perhaps easiest and most likely to be useful to be able to predict properties. Certain atoms or groups of atoms are known to give the molecules containing them a particular kind of chemical reactivity. For instance, carboxyl groups (made up of carbon, oxygen and hydrogen in the form COOH) in a compound make it acidic; plastics containing large numbers of fluorine atoms are usually inert, polytetrafluorethylene (PTFE) being a case in point. There are, however, many complicating factors that can make predictions inaccurate. If, for example, the bonds in a molecule are strained, the properties can be completely different from those expected. For this reason, organic molecules containing small rings of carbon atoms are notorious for their exotic and unpredictable chemistry.

In principle, chemists can calculate molecular properties from the fundamental theory that underpins all chemistry, known as quantum theory. The Schrodinger wave equation for the molecule provides a mathematical description of its electronic structure. Solving the equation yields reliable values for the energy levels of the electrons and the distribution of the electronic density in comparatively small molecules containing up to about 100 atoms. This approach is valuable for calculating some electronic properties: whether the material might be coloured or electrically conducting, for example. But it cannot predict the more general physical and chemical properties of materials in bulk, which is what the chemicals industry is interested in. These properties depend not only on the size and shape of the molecules but also on the ways in which molecules interact in large numbers.

To predict how a chemical will behave in bulk, chemists-particularly those in industry-are starting to use an exciting new method that can estimate these properties satisfactorily. The method is ‘topological’, which means that it considers the way atoms in a molecule are linked. Different linkages may result in a straight chain, a branched chain, a ring or some combination of these structures. Based on the pattern of linkages, there are various ways we can derive a number that characterises the molecular structure-the ‘topological index’. The aim is to assign to each structure a unique number that describes it mathematically. Using this approach, molecular structure can be related to molecular behaviour in a quantitative way. Once we have a relationship for a test set of molecules, we can predict how new molecules will behave.

The topological approach is based on a branch of mathematics known as graph theory. Mathematicians have studied graphs for more than 250 years. They were first used by Leonhard Euler in 1736 to solve a celebrated problem of his day known as the Konigsberg bridges problem . Graphs are convenient devices for depicting how things are connected. (Incidentally, they bear no relation to data plots that are also commonly referred to as graphs.) A map of the London Underground System is a perfect example of a graph. No matter how you draw the map the connections between the stations remain the same. Graphs can be described in terms of certain fixed topological characteristics that do not change no matter how the graph is drawn. For molecules, these are called topological indices.

Researchers have developed many different ways of deriving topological indices. In fact, these methods are so varied that there are now more than 120 different ways of obtaining them. The indices incorporate the particular features of a structure to differing extents, depending on what properties they are designed to predict. Some reflect mainly molecular size (the volume of space occupied by a molecule), others characterise mainly the shape (distribution of the volume in space), and yet others focus on the amount of branching in a molecular structure.

The graph represents a molecule stripped down to its bare essentials. Its atoms are depicted as points and the chemical bonds as lines. This type of representation takes no account of the lengths or angles of the bonds. Figure 1 shows some examples of graphs for three different molecules: methane (CH₄), which has one central carbon atom joined to four hydrogen atoms; ethene (C₂H₄), which has two carbon atoms joined by a double bond and two hydrogen atoms joined to each carbon atom; and benzene (C₆H₆), which has six carbon atoms joined in a ring with a hydrogen atom attached to each carbon atom. In benzene, the bonds between the carbon atoms are traditionally shown as three single and three double bonds but, in fact they are all equivalent. We can regard these bonds as 1 1/2, bonds produced by ‘smearing out’ the electron density associated with the extra bonding.

Molecular graphs do not reflect the different types of bonding. To allow for such differences we can weight its contribution to the molecular index appropriately. In the case of the benzene molecule, for instance, the carbon-carbon bonds can be represented by lines weighted by the factor 1 1/2. A double bond, as found in ethene, can be weighted by the factor 2. The points representing atoms can also be weighted to reflect different kinds of atoms. For example, we can use a weighting that depends on the size of the atom. Hydrogen atoms are usually given a zero weighting because they are very small and have little influence on other atoms in the structure.

How can we calculate a topological index? A simple way would be merely to count up the number of carbon atoms in a molecule. This is equivalent to counting the points in the topological graph. This number will, of course, stay the same no matter how the molecular graph is drawn. Chemists have used this very simple topological index to characterise hydrocarbon molecules for some 150 years. Today, it is called the carbon number index. Although this index works well in predicting certain properties of hydrocarbons, it reveals nothing about their structure. For instance, it does not differentiate between branched and unbranched hydrocarbons with the same number of carbon atoms.

The search for topological indices that would reflect molecular structure more subtly began in 1947, when Harry Wiener at Brooklyn College in New York suggested an index that would better characterise simple hydrocarbons. He defined his topological index as the sum of the number of bonds separating all the pairs of carbon atoms in a molecule. To illustrate how this index distinguishes between similar structures, take the two different structures that a molecule of butane (C₄H₁₀) can assume. One is a straight chain, the other a branched chain, as shown in Figure 2. If we calculate Wiener’s index for the straight chain, the sum of the bonds comes out to be 10 whereas for the branched chain the sum is only 9. For large molecules, these numbers can be computed by setting up a matrix (a mathematical array) that gives the number of links between any pair of atoms in the structure. Adding up all the links gives the Wiener index.

Chemists have used the Wiener index to correlate structure with a host of physical properties such as boiling point, viscosity and surface tension. It has also proved possible to correlate structure with more complicated properties, including biological characteristics such as the toxicity of industrial solvents, the antibacterial activity of organic molecules, and the narcotic effects of vapours and fluids released as effluents.

Once we have a correlation, we can plot a curve from which we can read off the properties of new molecules. Such curves can be used to predict the properties of surprisingly large molecules, and it is feasible, therefore, to obtain good estimates of many of the physical properties of polymers. Examples include predicting the melting point and refractive index for plastics, including polyethylene and polyesters.

The Wiener index has also been used in the study of solid materials to model how crystals grow, and where ‘alien’ atoms position themselves in crystal lattices to form so-called defects. The index has also helped researchers investigate how gas molecules attach themselves to crystal surfaces: hydrogen atoms on a platinum surface, for example. Understanding such processes is important in the chemicals industry, which relies heavily on precious-metal catalysts to speed up reactions. Elsewhere, the index has proved useful for investigating novel electronic and semiconducting materials such as polyacetylene and polydiacetylene. The index indicates how changes in molecular structure affect properties such as the absorption of light or electrical conductivity.

The most popular topological index to be applied so far is that proposed by Milan Randic, of Drake University in Des Moines, Iowa. In 1975, he introduced what is now known as the molecular connectivity index. This index weights each bond in a molecule according to the number of other bonds adjacent to it and then adds up all the weighted numbers.

For example, taking the molecular graph of butane in Figure 2a, we start by determining how many lines radiate from each of the points. For the points at the end of the chain this number is one, whereas for the two middle points the number is two. These numbers are then used to weight each bond by use of the formula: weighting = (n₁ x n₂) ^-1/2, where n is the number of radiating lines. The weighting for the first bond on the left is, therefore, (1 x 2) ^-1/2. The bond on the right will clearly also have the same weighting. The bond in the middle will have the weighting (2 x 2) ^-1/2. Summing these weighted contributions from each line in the graph gives the molecular connectivity index. In this case, the sum amounts to: 2(1 x 2) ^-1/2 + (2 x 2) ^-1/2 = 1.914.

Randic’s index is represented by the symbol ₁. The superscript 1 distinguishes it from later elaborations of this index introduced by Monty Kier at Virginia Commonwealth University in Richmond and Lowell Hall at Eastern Nazarene College in Quincy, Massachusetts. Their indices involved adding up molecular fragments other than bonds. They suggested adding up bond pairs (two adjacent lines in the graph) and bond triples (three adjacent lines either in a line or radiating from a single point), among others. They also proposed weighting the graph ‘vertices’ of the atoms of carbon or hydrogen in the molecule. Because their weighting is based on valence (the number of bonds that an atom can make), the superscript v is added to the symbol (¹)

These molecular connectivity indices have been applied successfully to an exceptionally wide range of physical, biological and pharmacological properties. Figure 3 shows a plot of the ¹ index for 138 different organic molecules against their partition coefficient between water and a fatty alcohol called octanol-a measure of how each substance distributes itself. The plot reveals an excellent correlation. (This coefficient provides a good measure for studying how effectively drugs can pass from the watery environments around body cells through the fatty layer of cell membranes.) Chemists have obtained similar plots for many other physical properties, including boiling point, vapour pressure and heat of formation (a measure of the energy needed to make a molecule).

Predicting taste and smell

On the biological side, chemical structure has been correlated with odour, taste, carcinogenicity and enzyme inhibition. Colman’s Mustard has exploited molecular connectivity indices to predict the taste and smell of ingredients in food. And Harp Lager has even used these indices to improve the taste of its beer. Topological indices are also becoming important in environmental studies. In the US, the Environmental Protection Agency (EPA) uses molecular connectivity indices to predict properties of compounds, such as toxicity, biodegradability and the amount of the material absorbed by various soils.

Although we can correlate molecular structure with such an astonishing variety of different properties using the Wiener and molecular connectivity indices, chemists are constantly looking for new ones. Particular kinds of topological index are needed to describe some properties. The so-called octane number of a fuel is a good example of such a property. The octane number reflects the ‘anti-knocking’ characteristics of fuel in petrol engines, and depends on the extent of the branching in its hydrocarbon molecules. Alexandru Balaban of the Polytechnic Institute in Bucharest, Romania, introduced a set of topological indices known as centric indices designed to reflect sensitively the branching in organic molecules. These indices gave excellent correlations with octane numbers.

Another property of hydrocarbons that is difficult to predict is their tendency to produce soot when burnt. Soot, which consists primarily of carbon, represents the final stage in a long and complex combustion process. The ‘sooting’ tendency of fuels measures how efficiently they burn in air and, therefore, how much pollution they are likely to cause.

Predicting sooting is particularly useful for ships, as some of the fuels they burn produce a lot of soot. My research group investigated this problem. Eventually, we discovered indices that correlated the sooting tendency with structure very well for about 100 hydrocarbon molecules of varying types. Although one index is often enough to obtain a good correlation, in this case we found that we needed two: the Balaban average distance sum connectivity index (a development of Randic’s index); and our own hydrogen deficiency index, which gives a measure of the multiple bonding present in a molecule. We combined these two indices and obtained the plot shown in Figure 4.

In the early days, researchers applied topological indices to sets of molecules that were closely related; for example, hydrocarbons forming a ‘homologous series’ that differ only in the number of links in the main carbon chain. Our own study on sooting tendency, however, included many different types of molecules, among which were some containing atoms other than carbon and hydrogen. This is why we needed to use a more complicated weighting technique. In fact, even in the case of molecules where the method might be expected to fail, we can often devise weighting procedures to predict properties accurately.

There are three major instances where the method might not work. First, sets of molecules of differing types can present problems. Secondly, behaviour that involves transforming the original molecule into some new structure, such as the oxidation of a molecule, may invalidate the method. Thirdly, some biological properties, such as toxicity, depend on molecules ‘docking’ at appropriate biological receptors in living organisms; their effect may be strongly dependent on subtle changes in the shape of the molecules.

Our study on sooting has shown that it is possible to overcome the first two constraints. The third can also be overcome in many cases, though predicting biological properties may occasionally require as many as five different indices. William Herndon and Laszlo von Szentpaly at the University of Texas at El Paso were able to predict very accurately whether polycyclic aromatic hydrocarbons would cause cancer in experimental animals. They ingeniously combined five different indices: the carbon number index, the square of this index, and three other indices that characterise molecular shape.

Many chemical companies and government agencies are using topological indices to predict the properties of new chemicals. On the medical front, Cornell University’s Medical Center in New York Hospital is interested in predicting the properties of ‘perfluorinated’ molecules in which all the hydrogen atoms have been replaced by fluorine atoms. Such molecules often display remarkable properties. They are extremely stable and can sometimes act as carriers of gases, including oxygen. This makes them suitable as substitutes for body fluids such as blood and the vitreous humour in the eye. We have shown that indices can predict the behaviour of perfluorinated molecules very accurately.

The Du Pont Chemical Company in Waynesboro, Virginia, is applying topological indices to predict the mechanical properties and stability of polymers such as polyurethanes. The idea is to develop new polymers with properties rivalling or even improving on those of natural fibres. And both the United States Air Force and Allied Signal Corporation are presently exploiting indices to predict the behaviour and design of advanced fuels for aeroengines and rockets.

One of the most promising and challenging areas where topological indices are now being used is in designing completely new drugs. Pharmaceuticals companies, including Pfizer, Glaxo and Upjohn, employ indices to search for molecular structures that are likely to be biologically active. They do this by looking for structures that are similar to some target structure that is already known to be active. Finding such structures-known as leads-is one of the most pressing problems for pharmaceuticals companies. Figure 5 shows the result of one such search performed on a database belonging to Upjohn. Using topological indices, the company can produce lists of the 100 most similar structures to any given target structure. The first three of these are shown in the figure.

The possibilities for applying topological indices are far from exhausted. New applications seem to be found almost daily. At present, topological indices are still used mainly in research to predict how chemical substances behave. It is one of the approaches that is helping to formulate strategies for research in organic chemistry. But, today, any material being sold commercially not only has to do its job well, but it also has to be environmentally safe. The EPA has developed a huge computer program and database which can calculate indices and predict the behaviour of chemicals that might have an impact on the environment. The time is not far off when this quantitative approach will be sufficiently developed to help in the routine design of new molecules for specific applications. The chemicals industry will then have a powerful new tool for designing not only more efficient drugs, safer pesticides, cleaner fuels, but also completely new materials with unusual properties.

Dennis Rouvray is a research professor in the chemistry department at the University of Georgia at Athens, Georgia.