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Figure 1: Comparison of crossover operators. The interface between different font or line thickness indicates the crossover point. A single crossover point is adequate to divide strings (genetic algorithms) and trees (genetic programming) into two fragments. Graphs (genetic graphs) containing cycles require more than one crossover point to divide the system into two fragments. Furthermore, when two graph fragments with different numbers of crossover points must be mated, it is possible to create new edges to satisfy the excess crossover points on one fragment. 
Genetic software techniques have been used for molecular design in the past. There is a patent covering genetic graphs for molecular design [Weininger 95]. The patent describes the straightforward and fairly obvious parts of mapping standard genetic algorithm techniques to molecular design and the nonobvious portions: the crossover algorithm and fitness functions. The crossover algorithm described in the patent uses two parameters: the digestion rate which breaks bonds, and the dominance rate which apparently controls how many parts of each parent appear in descendants. This algorithm may produce fragments rather than completely connected molecules. Our paper describes a crossover algorithm that always produces connected molecules and has no parameters. This crossover algorithm is the heart of our genetic method. Fitness functions are clearly nonobvious, but must usually be custom designed for each application. Our fitness function and those used in [Weininger 95] both use the Tanimoto index as a distance measure. [Weininger 95] describes a number of fitness functions. We have used allpairsshortestpath in most of our work. Daylight Chemical Information Systems, Inc., which holds the patent, reports using genetic techniques to discover lead compounds for pharmaceutical drug development and other commercial successes.
[Nachbar 98] used genetic programming to evolve molecules for drug design by sidestepping the crossover/cycles problem. Each tree node represented an atom with a bond to the parentnode atom and each childnode atom. Hydrogen atoms were explicitly represented and are always leaf nodes. Rings were represented by numbering certain atoms and allowing a reference to that number to be a leaf node. Crossover was constrained not to break or form rings. Ring evolution was enabled by specific ring opening and closing mutation operators.
In a personal communication, Astro Teller reported developing a graph crossover algorithm as part of his dissertation at Carnegie Mellon University. This technique was applied in Neural Programming, a system developed by Teller that combines neural nets and genetic programming. At the time this paper was written, the details of Teller's algorithm were not available in the literature.
Genetic graphs uses cyclic graphs to represent molecules. Vertices are typed by atomic elements. Edges can be single, double, or triple bonds. Valence is enforced. Heavy atoms are explicitly represented by vertices but hydrogen atoms are implicit; i.e., any heavy atom with unfilled valences is assumed to be bonded to hydrogen atoms but these are not represented in the data structure. Our genetic graph software evolves the population using crossover only; i.e.., mutation and reproduction are not implemented. These are trivial additions to the method, and we wanted to investigate the crossover operator first.
Each individual in the initial population is generated by choosing a random number of atoms between half and twice the number of atoms in the target molecule. Atoms are randomly chosen from the elements present in the target molecule. Bonds are then added at random to construct a spanning tree; i.e., at this point all atoms are connected into a single molecule. Then a random number of additional bonds are added to create rings. The type of each bond is selected at random from the set of bond types present in the target molecule. The number of rings is chosen to be between half and twice the number of rings in the target molecule. The number of rings, by our definition, is always = bonds  atoms + 1. For this definition, single, double, and triple bonds are counted as one bond. In addition, an unambiguous definition of the number of rings [Corey 69] is used. For this definition:
While this definition is precise and useful for coding, it comes to the interesting conclusion that cubane (a cubic molecule) has five rings, not six. Consider the six sides of cubane to be six rings in the set of rings. If any one of the six rings is removed from the set of rings, all bonds are still included in the set of rings.
For this work, tournament selection was used to choose parents in a steady state genetic system. Tournament selection means that each parent is chosen by comparing two randomly chosen individuals and taking the best. Steady state means that new individuals (children) replace poor individuals in the population rather than creating a new generation. The poor individuals are also chosen by tournament, but the worst individual is selected for replacement. By convention, after populationsize individuals have been replaced, we say that one generation is complete. The implementation follows this procedure:
The most difficult portion of implementing genetic graphs is the crossover operator described above as "ripping" molecules into two parts and combining parts from each parentmolecule. Crossover requires two procedures: one to rip molecules in half and a second to mate two "molecular halves." To rip a molecule in half we use the following procedure:
To mate fragments we use the following procedure
Figure 2 depicts crossover between butane and benzene to create a single child.
Figure 2: butane and benzene are ripped apart at random points. Then one fragment of butane and a fragment of benzene are mated. Note that benzene must be cut in two places. Also, during mating the benzene fragment has more than one cut bond. A random choice is made to connect this extra cut bond to a random atom in the butane fragment. Alternatively, the extra cut bond could have been discarded. 
A somewhat more complete but significantly more confusing explanation follows. The terms vertex and edge are used for atom and bond respectively to indicate that the algorithm may be applied to any graph structure.
This approach can open and close rings using crossover alone and can even generate cages and higher dimensional graph structures as long as there are rings in the population. Unfortunately, if there are no rings in the population none can be generated. Similarly, if the population consists entirely of rings, no chains can be generated. Also, once the population consists entirely of twoatomgraphs, no graphs with more than two atoms can be generated. Nonetheless, this approach is by far the most general of those we examined or found in the literature. In particular, unlike [Nachbar 98] no specialpurpose ring opening and closing operators are necessary. Unlike [Weininger 95] no parameters are necessary and multiple molecular fragments are never produced.
The computational resources required for genetic software to find a solution is a function of the size of the search space, among other factors. The space of all possible graphs is combinatorial and enormous. For molecular design this space can be radically reduced by enforcing valence limits for each atom. Thus, a carbon atom with one double and two single bonds will not be allowed to add another bond. Also, avoiding explicit representation of hydrogen atoms substantially reduces the size of the graph and therefore the search space.
The space of all molecules has very little in common with a multidimensional continuous Cartesian space. The space consists of a large number of discrete points with no derivatives. One may, however, think of each point as having neighbors so that the space may have local and global minima. We may define the neighbors of a molecule as all molecules that may be formed by certain mutations. These mutations are adding or deleting one bond, adding a bond and an atom, changing an atom type, or changing a bond type. This scheme is sufficient for a graph to describe the space of all molecules with a single additional operation of adding a single atom to the "null molecule," a molecule with no atoms or bonds.
The key to any genetic software solution is a good fitness function  for tournament selection a function that can determine if one molecule is better than another. This function must be very robust since the randomly generated initial molecules rarely make much chemical sense. Fitness functions must also make fine distinctions between any two molecules even if both are very good or very bad. These fine distinctions are necessary to avoid flat regions in the fitness space where no direction is given to evolution. Also, for our initial studies we wanted a fitness function that only required the graph of a molecule, not the xyz coordinates of each atom.This simplifies initial studies and avoids the necessity of equilibrating the structure of candidate molecules, a CPU intensive step. The allatomspairsshortestpath similarity test chosen [Carhart 85] is a robust graphonly fitness function. Each atom is given an extended type consisting of the atomic number and the number of single, double, and triple bonds the atom participates in. Then the shortest path between each pair of atoms is found. A bag is constructed with one element for each atom pair. Each element in the bag is represented by the sorted extended types of the two atoms and the length of the shortest path between them. The fitness of each candidate molecule is the distance between its bag and the similarly constructed bag of a target molecule. A bag is a set that allows duplicate elements. The distance measure used is the Tanimoto index. This is
a intersection b / a union b
where a is the candidate's bag and b is the target's bag. Two elements are considered identical for the purpose of the intersection and union operations if the atoms have the same extended types and the distance between them is identical. Each duplicate in the bag is considered a separate element for the purpose of intersection and union operators. This measure always returns a number between 0 and 1. We prefer fitness functions that return lower numbers for fitter individuals, so we subtract the Tanimoto index from one.
Search spaces with many local minima are difficult to search because many algorithms tend to converge to local minima and miss the global minimum. The allpairsshortestpath fitness function has many local minima over the space of all molecules whenever the target molecule contains rings. Consider benzene, a six membered ring, as the target molecule. Any ring other than a six membered ring will be at a local minimum because a bond must be removed, lowering the fitness in most cases, before bonds and atoms can be added to generate benzene. Thus, any target molecule containing rings will have associated search space containing many local minima. Interestingly, a small modification in the definition of the molecular search space eliminates the local minima. Consider mutations that replace an edge with a vertex and two edges and replace a vertex and two edges with one edge. These mutations can change the size of rings. If these mutations can create neighbors in the space of all molecules, then incorrectly sized rings are not local minima. While we think these additional neighbor definitions are unreasonable, they do illustrate the difficulty of understanding the space of all molecules.
The targets for our initial study were butane, benzene, cubane, purine, diazepam, morphine and cholesterol. All targets except butane contain rings and thus generate a search space with local minima. The fitness function can not only find similar molecules, which is useful in drug design, but can also lead evolution to the exact molecule used as a target. This proves that the algorithm can reach particular kinds of molecules and the number of generations to find the target provides a quantitative measure of performance. Unfortunately, our implementation of allpairsshortestpath is O(n^{3}) so finding larger molecules can be quite time consuming.
The genetic graphs program is implemented in Java. Java was chosen since its syntax is similar to C++, many useful libraries are available, garbage collection vastly simplifies memory management, and Java's array bounds testing and other bugreducing features substantially reduce debugging time and produce more robust code than C++, Fortran or C. A significant runtime penalty is paid for these advantages. With luck, future improvements in Java development and runtime environments will reduce the performance penalty. All production runs were executed on SGI workstations at the NAS Division of NASA Ames Research Center.
According to our hypothesis, genetic graphs algorithm can find any possible molecule. To partially test this hypothesis, we ran the program using several target molecules:
Stereochemistry and hydrogens are left out of the molecular diagrams since the software does not consider them.
Since the algorithm is stochastic, twenty runs were conducted for each target molecule. The number of generations and population size were varied in an attempt to have enough successful runs (at least 11) to calculate the median time to find the target. Once the target was found the run stopped. Runs also stopped after a fixed, maximum number of generations. A few of the best individuals were saved to see if the software produced molecules similar to the target. These may be useful for drug design.
In table 1 median, rather than mean, generations to find the target was chosen because the data varied widely and many runs did not complete [Claerbout 73]. Butane was usually found in the initial random population so data were not taken. With a population of 100, the benzene runs did not find the target often enough to calculate the median so the larger population size was used. At the time of writing, there was not sufficient data for diazepam, morphine, or cholesterol to calculate the median for twenty runs. However, each molecule has been found at least once:
Table 1.  
20 runs for each molecule 
Population size 
Median generations to find target 
Minimum generations to find target 
Number of runs that failed to find target 
Maximum generations 

Benzene  200  39.5  2  8  1000  
Cubane  100  46.5  13  0  1000  
Purine  100  245.0  19  4  1000  
In table 2 the modified cholesterol was used due to an error in the input file. Although cholesterol was not found, the molecule that was found is very similar.
Table 2.  
Molecule  Population size  Generation found  Fitness function  
Morphine  1000  208  allpairsshortestpath  
Diazepam  200  256  allpairsshortestpath  
Cholesterol minus the two methyl groups connected to the rings 
500  1765  allpairsshortestpath plus number of rings  
To see if the crossover operator was better than random search, we searched for purine under three conditions: crossover alone, generating random molecules using the same algorithm as for the initial population (random search), and a 5050 mix of crossover and random search as shown in table 3. Twentyone runs of 1000 generations on a population of 200 were conducted in each case.
Table 3.  
Case  Number of runs that found purine  Median generations to find purine  
Random search  0  N/A  
Crossover alone  21  37  
5050 mix of crossover and random search  21  48  
Clearly the crossover operator is better than random search.
The genetic graphs algorithm can clearly find small molecules given an appropriate fitness function, and can find more complex molecules although significant time is required. The variability of runs is remarkable. Note that eight runs failed to find benzene in 1000 generations, but one run found benzene in only two generations. A few of the 20 runs found benzene in only three generations. It's also interesting to note that cubane was always found although the median time to discovery was somewhat greater than for benzene; and the cubane run used a smaller population. Presumably, cubane's single bond type simplified the search. Finally, note the much larger median time to success for purine. Apparently the addition of nitrogen and the fused ring made finding the target significantly more difficult. Still, more purine runs were successful than benzene runs.
Finding moderate size molecules has proven difficult with the available computer resources. This is probably because our fitness function was O(n^{3}), but also due to problems with the cyclestealing batch system used to run this program on idle workstations. Most genetic software uses mutation as well as crossover. Mutation helps systems make small changes. While crossover seems to be capable of generating molecules, the performance is sufficiently poor that mutation operators might help a great deal. Other than the usual operators to add and remove atoms or bonds, it may be helpful to have a mutation operator that makes a random ring aromatic (alternating double and single bonds for certain sized rings). Generating aromatic rings with crossover alone is probably difficult (note the problems generating benzene) so a special mutation operator may be helpful.
Examining populations generated by diazepam, cholesterol, and morphine we noted that the best molecules all had the same ring structure. This did not appear to be true of the population as a whole. Thus, it is possible for the population to get stuck in local minima. Occasionally the best molecule in a population would be radically improved by a particularly fortuitous crossover, but this can take a long time. We found that a minor modification to the fitness function eliminated this difficulty in the cholesterol run. The fitness function was an equal combination of allpairsshortestpath and a modified Tanimoto index on the number of rings in the target molecule versus the candidate. This fitness function appears to do a better job of finding interesting analogues to the target molecule by keeping the ring structure diverse. With allpairsshortestpath alone, populations seem to converge on a single poor ring structure, at least for the best molecules in the population. Adding the distance between the number of rings generates more fit molecules and more diverse ring structures, at least in preliminary results.
Performance analysis demonstrates what might be expected — our O(n^{3}) fitness function took most of the CPU time. A faster fitness function would substantially speed calculations. Some runs with faster fitness functions have been made, but these simpler fitness functions do not drive evolution to find the target exactly so comparison with the above data is difficult. Genetic software lore suggests that the fitness function is exceptionally important [Kinnear 94]. Our results bear this out.
Fortunately, the algorithm is embarrassingly parallel since many runs are required. Also, there is significant potential for parallelism within runs since fitness function execution on each individual is completely independent. Furthermore, the algorithm can be easily restarted and can afford to lose some runs. Thus, genetic graphs is a good candidate for cyclescavenging batch systems such as Condor [Litzkow 88]. Large genetic graph production runs can therefore use otherwise wasted workstation and PC cycles at facilities with large numbers of these machines. Although some of the results presented here were simply run on workstations, all current jobs are run under Condor.
Molecular design may be viewed as searching the space of all possible molecules for molecules with particular properties. One might expect genetic graphs to have strengths and weaknesses similar to genetic algorithms applied to other problems. Most research to determine which problems are difficult for genetic algorithms has focused on "deception," where partial solutions to a problem point away from the full solution [Goldberg 89]. [Forrest 93] points out, however, that
"... deception is not the only factor determining how hard a problem will be for a GA (genetic algorithm), it is not even clear what the relation is between the degree of deception in a problem and the problem's difficulty for the GA. There are a number of factors that can make a problem easy or hard, including those we described in this article: the presence of multiple conflicting solutions or partial solutions, the degree of overlap, and the amount of information from lowerorder building blocks (as opposed to whether or not this information is misleading). Other factors include sample error [Grefenstette 89], the number of local optima in the landscape [Schaeffer 89], and the relative differences in fitness between disjointed desirable schemas [Mitchell 92]. Most efforts at characterizing the ease or difficulty of problems for the GA have concentrated on deception, but these other factors have to be taken into account as well.
"At present, the GA community lacks a coherent theory of the effects on the GA of the various sources of facilitation or difficulty in a given problem; we also lack a complete list of such sources, as well as an understanding of how to define and measure them. Until such a theory is developed, the terms 'GAhard' and 'GAeasy' should be defined in terms of the GA's performance on the given problem rather than in terms of the a priori structure of the problem." (formatting modified to fit the conventions of this paper)
The primary contribution of this paper is the creation of a crossover operator for graphs. This allows the use of genetic algorithm techniques using graph representations. As [De Jong 90] states:
"The key point in deciding whether or not to use genetic algorithms for a particular problem centers around the question: what is the space to be searched? If that space is wellunderstood and contains structure that can be exploited by specialpurpose search techniques, the use of genetic algorithms is generally computationally less efficient. If the space to be searched is not so well understood, and relatively unstructured, and if an effective GA representation of that space can be developed, then GAs provide a surprisingly powerful search technique for large, complex spaces." (italics added by [Forrest 93])
Chemists have known for over a century that graphs are the most natural representation of molecules. Furthermore, the space of molecules is not well understood or characterized. Therefore, it is reasonable to presume that searching the space of molecules using genetic graphs will be profitable in a number of domains. We hope that our crossover operator will make a contribution.
Although finding target molecules is a useful measure of the algorithm, we already know the target molecule. The real purpose of the similarity fitness function is to find molecules similar but not identical to the target. In particular, the ideal result is a wide variety of molecules dissimilar to each other but relatively similar to the target molecule. This provides a diverse set of candidate molecules for drug development, a process that takes millions of dollars and many years to complete. In the ideal case, one or more candidates will have the beneficial properties of the target without negative side effects. Preliminary analysis of collections of the best individuals from each generation suggests that these collections are quite diverse and share some of the properties of the target molecule. Most of the analog molecules found tend to be chemically unstable in physiological conditions. We are developing a fitness function that will penalize molecules that are unstable in the body.
Many fitness functions of interest require molecular conformation; i.e., xyz coordinates for each atom. For example, designing a molecule to fit in a protein receptor to inhibit the activity of a disease organism, as the new and fairly effective AIDS drugs do, require molecular conformation. To design a fitness function evolving molecular fit, the very bizarre molecules often created by crossover must have their energy minimized quickly. Most minimizers available today will not work well with without a "reasonable" start point. We are searching the literature for algorithms to minimize very "bad" molecules. Searching for molecules when conformation is important may be expected to be more difficult than the search is described in this paper. Energy minimization takes considerable CPU time, multiple conformations for molecules increase the size of the search space, and a different crossover operator that takes physical space into consideration may be necessary.
It may be possible to search for molecules using hill climbing algorithms. However, the space of all molecules has no derivatives and no orthogonal dimensions. Therefore, from each molecule one must examine all the neighbors to find the steepest hill to climb. Since even moderately sized molecules will have a huge number of neighbors in the molecular space defined above, hill climbing algorithms are expected to have great difficulties. It may be possible, however, to use hill climbing algorithms that climb any reasonable hill rather than the steepest one. This amounts to evolution using mutation where only profitable mutations are allowed to survive. We hope to test this technique in the near future.
Circuit design is another field for which genetic graphs should, in principle, be well suited. Genetic algorithms (using variable length strings) [Lohn 98] and genetic programming [Koza 97] have been used to design analog circuits. In the genetic programming case, a tree language to generate analog circuits compatible with the SPICE (Simulation Program with Integrated Circuit Emphasis) [Quarles 94] simulator was constructed and a 64 node (80MHz per node) parallel supercomputer was used to design the circuits. The system designed a lowpass filter, a crossover filter, a fourway source identification circuit, a cube root circuit, a timeoptimal controller circuit, a 100 dB amplifier, a temperaturesensing circuit, and a voltage reference source circuit. Thus, genetic programming can design graphstructured systems. However, we have found it extremely difficult to create a tree language that can generate any possible graph and support crossover cleanly. Therefore, it may be advantageous to directly evolve graphs rather than trees that represent cyclic graphs.
Finally, it might be interesting to explore the relationship between genetic graphs, applied to molecules, and combinatorial chemistry. Combinatorial chemistry is a new field undergoing rapid development where hundreds or thousands of molecules and simultaneously manipulated and examined. The recombination rules in genetic graphs are not constrained by physical reality, whereas reactions in combinatorial chemistry experiments clearly are. Combinatorial chemistry is also much faster than genetic graphs, even on massively parallel computers. Crossover and fitness function evaluation can take seconds or even minutes on a workstation, but thousands of simultaneous reactions can occur on a much smaller time scale.
Algorithms and software to evolve graphs using genetic techniques were developed and applied to drug design using a molecular similarity based fitness function. Early data suggest that the software can indeed discover a variety of small molecules. Significant additional work will be required to demonstrate that representing molecules as graphs and using genetic software techniques is of major benefit in molecular design.
Thanks to Rich McClellan for providing the mol file reading and atomic element code. Thanks to Creon Levit, Subash Saini, and Meyya Meyyapan of NASA Ames for their support. Thanks to Creon Levit, Jason Lohn and Bonnie Klein for reviewing this paper. Special thanks to the anonymous referee who pointed us to many important results in the search literature. This work was funded by NASA Ames contract NAS 214303.
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