A Design for DNA Computation of the OneMax Problem David Wood Junghuei Chen Eugene Antip ov Bertrand Lemieux Walter Cedeno ~ Computer Sci. Chem. & Bio chem. Chemical Eng. Plant & Soil Sci. Hewlett-Packard Univ. of Del. Univ. of Del. Univ. of Del. Univ. of Del. Centerville Rd Newark DE Newark DE Newark DE Newark DE Wilmington DE Abstract details of the lab oratory op erations, and some prelimi- nary lab oratory results can b e found elsewhere [5, 38]. Elements of evolutionary computation and 1.1 Conventional Evolution Computation molecular biology are combined to design a DNA evolutionary computation. The tradi- Evolution Computation usually maintains a p opula- tional test problem for evolutionary compu- tion of candidate solutions, often represented as strings tation, OneMax problem is addressed. The of binary bits. In each generation, some less promising key feature is the physical separation of DNA candidates are likely to b e eliminated. Their replace- strands consistent with OneMax \ tness." ments are generated from relatively promising candi- dates by cloning and/or mutation and/or recombina- tion. Evolutionary Computation comes in many, many 1 Intro duction di erentvarieties [3, 19, 20], but most are variations on this outline: Evolution is a concept of obtaining adaptation through the interplay of selection and diversity. Analogies from Begin with an initial p opulation, often chosen ran- evolution have b een used in b oth conventional comput- domly. ing and molecular biology. In computing, the general term is \evolutionary computation." The paradigm in Rep eat the following steps until convergence. molecular biology is known as \in vitro evolution." 1. Evaluate tness of candidates. In this pap er we identify elements of evolutionary com- 2. Select candidates to breed, favoring the more putation and molecular biology that we combine to t over the less t. address the traditional OneMax problem. Our design 3. Delete some of the less t. can b e mo di ed to address some other problems as 4. Breed replacements, inducing variation. well [5], as is brie y discussed in Section 5. Wecho ose evolutionary computations that manipulate bitstrings using op erations of p ointwise mutation and crossover. 2 DNA Suits Evolutionary These op erations can b e p erformed by mo di cations Computation of techniques from molecular biology. Section 4.1 forms the heart of this pap er. The cru- Several means of DNA computation have b een ad- cial lab oratory op eration is physically separating DNA dressed. The rst was, of course, by Adleman [1, 2]. strands by their \ tness" for solving the OneMax prob- Recentoverviews can b e found in [17] and [22]. See lem. In this section, we identify two-dimensional de- also the DNA computing bibliography of Dassen and naturing gradient gel electrophoresis as an appropriate Pierluigi [9]. lab oratory technique for this crucial separation. From the b eginning of DNA based computing to the We are careful to make the following p oint. In this pa- present there have b een calls [11,24,29, 6] to consider per we only aspire to provide the design for some evo- carrying out evolutionary computations using genetic lutionary computations using p opulation sizes larger materials in the lab oratory. So far, there have b een than is practical with conventional computers. More three such exp eriment designs prop osed, including two one must physically separate DNA strands according in a recent DIMACS Workshop [4,5]. The very rst to their \ tness" for solving the problem at hand. design was presented ab out twoyears ago [10], but has not yet b een carried out in the lab oratory. 2.2 DNA Evolutionary Computation With our design, computing time using DNA is pro- Compared to Sup ercomputers p ortional to the numb er of generations. This mo- tivates incorp orating b oth p ointwise mutation and The following oversimpli ed estimates indicate DNA crossover, attempting to minimize the numb er of gen- computing techniques could in the future compare erations required. favorably to sup ercomputers in some cases. Favor- For that matter, one mightwant to consider any able cases include executing evolutionary computa- additional evolutionary analogies that might reduce tions having very simple tness evaluations and ex- the numb er of required generations. These might in- tremely large p opulations of candidate solutions. clude transp osition, inversions, introns, etc. These Consider a p opulation represented by a total of p bits. have b een included in researchinevolutionary com- Executing a evolutionary computation by any means putation using conventional computers. Computing will require at least g Op tness evaluations, where paradigms inspired byevolution evolutionary compu- g is the numb er of generations used. Now, assume tations seem particularly suited to implementation us- the tness evaluation of a candidate solution pro cesses ing DNA. This is b ecause evolutionary computations all the bits of the candidate solution. A state-of-the- often use bitstrings, crossover, and p ointwise muta- 12 art tera op sup ercomputer p erforms ab out 10 op er- tion, all of which are done with DNA in nature. ations p er second. Thus, wehave a rough estimate for sup ercomputer time complexity, 2.1 DNA Advantages for Evolutionary Computation 12 17 T g p 10 seconds g p 10 days: 1 DNA computing techniques are desirable for evolution- To compare this to DNA computing, let us assume the ary computation for several reasons, some of which are tness evaluation of an entire p opulation can b e done listed b elow. in the lab oratory in one day [38]. Thus, the time com- plexity of a DNA approachtoevolutionary computa- These techniques might pro cess, in parallel, p op- tion is approximately T g days. Essentially no new ulations which are billions of times larger than is lab oratory techniques or equipmentwould b e needed usual for conventional computers. One exp ecta- to use gram quantities of DNA. This corresp onds to tion for larger p opulations is: they may generate 21 p opulations represented by ab out p =10 bits. For high- tness individuals in fewer generations. p opulations of this size, we see from Eq 1 that the time complexity of DNA implementation of evolution- Massive information storage is available using ary computation T g days compares favorably to DNA. Grams of DNA could b e used. A gram of sup ercomputer time complexityof 21 DNA contains ab out 10 bases. This information 21 content is approximately 2 10 bits, greatly ex- 4 T g 10 days: 2 ceeding the 200 p etabyte storage of all the digital magnetic tap e pro duced in 1995 [37]. In addition, a further factor of 1; 000 may b e consid- ered within reachby using kilograms of DNA. Biolab oratory op erations on DNA inherently in- volve errors. DNA lab oratory errors may b e re- Some caution is needed in interpreting this compar- garded as noise. Noise is more tolerable in execut- ison. The comparison is based on unprecedentedly ing evolutionary computation than in executing large p opulations. Still miniscule compared to the deterministic algorithms. Evolutionary computa- size of the search space, of course! As far as we know, tion has b een found to b e robust in the presence it is unclear exactly how b ene cial large p opulation of noise in some studies [16,15,26,23]. sizes mightbe. The classical \schema theorem" of Holland [20]says a p opulation of P distinct candidates Mo di cations to the current molecular biology 3 prob es O P p otential solutions. However the appli- technology suce to implement crossover and cability of this result, like many others in evolutionary p ointwise mutation [5, 38]. computation, is actively debated [12, 35]. This may b e an appropriate p oint to rep eat our pri- However, selecting DNA strands for \breeding" in evo- mary goal. In this pap er we only aspire to provide the lutionary computation is challenging. This is b ecause design for some evolutionary computations using p op- ulation sizes larger than is practical with conventional computers. - 3 The OneMax Problem This is a traditional test problem for evolutionary com- putation. It involves binary bitstrings of xed length. An initial p opulation usually randomly generated is given. The ob jectiveistoevolve some bitstrings to Electrophoresis match a presp eci ed bitstring generally taken to b e all 1s. It is easy to miss the p oint here: one is not trying to solve a signi cant problem | indeed, the solution is implicit in the problem sp eci cation. OneMax is a + Low High able to arti cial ly classical test to con rm that one is Denaturant Concentration evolve to a solution starting from a given initial popu- lation. Figure 1: DGGE using p erfect matched double stranded DNA. The strands movedownward from a The OneMax problem has b een extensively studied reservoir across the top of the gure. The sp eed of [13, 15, 14], but with assumptions ab out selection that vertical strand migration is retarded as strands come do not seem to hold in our situation. apart denature as shown schematically on the gure. From [5], with p ermission. 4 The OneMax Problem via DNA 4.1 The Heart of the Matter: Physical Separation of DNA by OneMax Fitness by their initial placement from left to right; that is, The crucial part of the DNA implementation of evolu- byhow strongly they are denatured pulled apart. tionary computation is to identify a lab oratory pro cess On the left, where no denaturant is encountered, the that will physically separate DNA strands according strands move relatively quickly downward.