A Design for DNA Computation of the OneMax Problem - CiteSeerX

A Design for DNA Computation of the OneMax Problem David Wood

Junghuei Chen

Computer Sci. Univ. of Del. Newark DE

Chem. & Biochem. Univ. of Del. Newark DE

Eugene Antipov Bertrand Lemieux Chemical Eng. Univ. of Del. Newark DE

Abstract Elements of evolutionary computation and molecular biology are combined to design a DNA evolutionary computation. The traditional test problem for evolutionary computation, OneMax problem is addressed. The key feature is the physical separation of DNA strands consistent with OneMax \ tness."

1 Introduction Evolution is a concept of obtaining adaptation through the interplay of selection and diversity. Analogies from evolution have been used in both conventional computing and molecular biology. In computing, the general term is \evolutionary computation." The paradigm in molecular biology is known as \in vitro evolution." In this paper we identify elements of evolutionary computation and molecular biology that we combine to address the traditional OneMax problem. Our design can be modi ed to address some other problems as well [5], as is brie y discussed in Section 5. We choose evolutionary computations that manipulate bitstrings using operations of pointwise mutation and crossover. These operations can be performed by modi cations of techniques from molecular biology. Section 4.1 forms the heart of this paper. The crucial laboratory operation is physically separating DNA strands by their \ tness" for solving the OneMax problem. In this section, we identify two-dimensional denaturing gradient gel electrophoresis as an appropriate laboratory technique for this crucial separation. We are careful to make the following point. In this paper we only aspire to provide the design for some evolutionary computations using population sizes larger than is practical with conventional computers. More

Plant & Soil Sci. Univ. of Del. Newark DE

Walter Cede~no Hewlett-Packard Centerville Rd Wilmington DE

details of the laboratory operations, and some preliminary laboratory results can be found elsewhere [5, 38].

1.1 Conventional Evolution Computation Evolution Computation usually maintains a population of candidate solutions, often represented as strings of binary bits. In each generation, some less promising candidates are likely to be eliminated. Their replacements are generated from relatively promising candidates by cloning and/or mutation and/or recombination. Evolutionary Computation comes in many, many dierent varieties [3, 19, 20], but most are variations on this outline: Begin with an initial population, often chosen randomly. Repeat the following steps until convergence. 1. Evaluate tness of candidates. 2. Select candidates to breed, favoring the more t over the less t. 3. Delete some of the less t. 4. Breed replacements, inducing variation.

2 DNA Suits Evolutionary Computation Several means of DNA computation have been addressed. The rst was, of course, by Adleman [1, 2]. Recent overviews can be found in [17] and [22]. See also the DNA computing bibliography of Dassen and Pierluigi [9]. From the beginning of DNA based computing to the present there have been calls [11, 24, 29, 6] to consider carrying out evolutionary computations using genetic materials in the laboratory. So far, there have been three such experiment designs proposed, including two

in a recent DIMACS Workshop [4, 5]. The very rst design was presented about two years ago [10], but has not yet been carried out in the laboratory. With our design, computing time using DNA is proportional to the number of generations. This motivates incorporating both pointwise mutation and crossover, attempting to minimize the number of generations required. For that matter, one might want to consider any additional evolutionary analogies that might reduce the number of required generations. These might include transposition, inversions, introns, etc. These have been included in research in evolutionary computation using conventional computers. Computing paradigms inspired by evolution evolutionary computations seem particularly suited to implementation using DNA. This is because evolutionary computations often use bitstrings, crossover, and pointwise mutation, all of which are done with DNA in nature.

2.1 DNA Advantages for Evolutionary Computation DNA computing techniques are desirable for evolutionary computation for several reasons, some of which are listed below.

 These techniques might process, in parallel, pop-

ulations which are billions of times larger than is usual for conventional computers. One expectation for larger populations is: they may generate high- tness individuals in fewer generations.  Massive information storage is available using DNA. Grams of DNA could be used. A gram of DNA contains about 1021 bases. This information content is approximately 2  1021 bits, greatly exceeding the 200 petabyte storage of all the digital magnetic tape produced in 1995 [37].  Biolaboratory operations on DNA inherently involve errors. DNA laboratory errors may be regarded as noise. Noise is more tolerable in executing evolutionary computation than in executing deterministic algorithms. Evolutionary computation has been found to be robust in the presence of noise in some studies [16, 15, 26, 23].  Modi cations to the current molecular biology technology suce to implement crossover and pointwise mutation [5, 38]. However, selecting DNA strands for \breeding" in evolutionary computation is challenging. This is because

one must physically separate DNA strands according to their \ tness" for solving the problem at hand.

2.2 DNA Evolutionary Computation Compared to Supercomputers The following oversimpli ed estimates indicate DNA computing techniques could in the future compare favorably to supercomputers in some cases. Favorable cases include executing evolutionary computations having very simple tness evaluations and extremely large populations of candidate solutions. Consider a population represented by a total of bits. Executing a evolutionary computation by any means will require at least O( ) tness evaluations, where is the number of generations used. Now, assume the tness evaluation of a candidate solution processes all the bits of the candidate solution. A state-of-theart tera op supercomputer performs about 1012 operations per second. Thus, we have a rough estimate for supercomputer time complexity, p

g

p

g

T

   10?12 seconds    10?17 days (1) g

p

g

p

:

To compare this to DNA computing, let us assume the tness evaluation of an entire population can be done in the laboratory in one day [38]. Thus, the time complexity of a DNA approach to evolutionary computation is approximately  days. Essentially no new laboratory techniques or equipment would be needed to use gram quantities of DNA. This corresponds to populations represented by about = 1021 bits. For populations of this size, we see from Eq 1 that the time complexity of DNA implementation of evolutionary computation (  days) compares favorably to supercomputer time complexity of T

g

p

T

T

g

  104days g

:

(2)

In addition, a further factor of 1 000 may be considered within reach by using kilograms of DNA. Some caution is needed in interpreting this comparison. The comparison is based on unprecedentedly large populations. (Still miniscule compared to the size of the search space, of course!) As far as we know, it is unclear exactly how bene cial large population sizes might be. The classical \schema theorem" of Holland [20] says a population of distinct candidates probes O( 3 ) potential solutions. However the applicability of this result, like many others in evolutionary computation, is actively debated [12, 35]. This may be an appropriate point to repeat our primary goal. In this paper we only aspire to provide the ;

P

P

design for some evolutionary computations using population sizes larger than is practical with conventional computers. This is a traditional test problem for evolutionary computation. It involves binary bitstrings of xed length. An initial population (usually randomly generated) is given. The objective is to evolve some bitstrings to match a prespeci ed bitstring (generally taken to be all 1s). It is easy to miss the point here: one is not trying to solve a signi cant problem | indeed, the solution is implicit in the problem speci cation. OneMax is a classical test to con rm that one is able to arti cially evolve to a solution starting from a given initial population.

The OneMax problem has been extensively studied [13, 15, 14], but with assumptions about selection that do not seem to hold in our situation.

4 The OneMax Problem via DNA 4.1 The Heart of the Matter: Physical Separation of DNA by OneMax Fitness The crucial part of the DNA implementation of evolutionary computation is to identify a laboratory process that will physically separate DNA strands according to their \ tness" as candidate solutions to the OneMax problem. For this task our design uses so-called two-dimensional denaturing gradient gel electrophoresis (2d DGGE).

 A rst important fact for DNA computing is that

2d DGGE can detect even a single base mismatch

in double stranded DNA. Indeed, this was and early application of 2d DGGE in molecular biology [21].  A second important fact is that our preliminary experiments with 2d DGGE [5, 38] demonstrate a surprisingly smooth transition through a large dynamic range of mismatching. Let us brie y review the nature of 2d DGGE. Figure 1 shows a 2d DGGE from our laboratory using double stranded DNA, with no mismatches anywhere. The mixture of double stranded DNA is placed uniformly along the top of the gel. The strands travel vertically downward in the gel as a result of an applied electric eld. However, their speed of migration is determined

Electrophoresis

3 The OneMax Problem

-

+

Low

High Denaturant Concentration

Figure 1: DGGE using perfect matched double stranded DNA. The strands move downward from a reservoir across the top of the gure. The speed of vertical strand migration is retarded as strands come apart (denature) as shown schematically on the gure. (From [5], with permission.)

by their initial placement from left to right; that is, by how strongly they are denatured (pulled apart). On the left, where no denaturant is encountered, the strands move relatively quickly downward. In the center, they move more slowly because they encounter intermediate denaturing. At the extreme right, the stands are able to move only very slowly because the strands are almost completely pulled apart. We heuristically reason that when we repeat the above experiment with a mixture of imperfectly matched DNA strands, we expect that in every vertical section the better matched strands will migrate downward relatively faster. This is the basis for our physical separation by tness for the OneMax problem, and forms the heart of our experimental design. To the best of our knowledge, no one had ever before demonstrated what happens when 2d DGGE is used with a population having a very great diversity of mismatchings. (The question has not arisen in molecular biology applications.) Populations of strands all having considerable mismatches might regrettably not be distinguishable at all. Fortunately, we found otherwise. A mixture with a distribution of imperfect matches exhibits surprisingly smooth vertical spreading in our preliminary experiments [5, 38].

Purify, ReadOut Selected Candidates

Separate by 2-D DGGE

Reserve Hybridize Candidates and Targets

Selection

PCR Amplify-Mutate

One-Point Crossover Purify Candidates

Candidate Pool

Target Strands

Initial Candidate Solutions

Variation

Breeding Figure 2: Outline of DNA implementation of evolutionary computation for OneMax problem. The candidate pool appears in the upper left. Selection using 2d DGGE appears at the lower left. Selection, puri cation and ampli cation of the more t candidate strands appears at the bottom. Breeding using crossover appears at the upper right. Various (programmable) pathways through the Reserve box are indicated.

Fitness Evaluation

4.2 Outline of DNA Implementation Design Our implementation is given by the following outline. The same information is shown in Figure 2.

DNA Evolutionary Computation for OneMax Begin with an initial population of candidates. Repeat the following steps until convergence. 1. Evaluate tness by hybridizing to target strands and physically separate on a 2d gel. 2. Select candidates to breed. 3. Amplify selected candidates incorporating pointwise mutation and reserve a portion. 4. Breed candidates, using crossover. 5. Combine reserved and bred candidates, obtaining a new generation.

4.3 A New Fitness Selection is Combined With Existing Breeding Techniques As mentioned earlier, modi cations to current technology suce to implement crossover and pointwise mutation. Thus, we combine modi ed forms of three robust biolaboratory techniques for implementing DNA evolutionary computation [38].

 A new application of 2d denaturing gradient gel

electrophoresis (2d DGGE): the more t candidate strands of DNA can be physically separated from much less t candidates according to how well they match xed complementary \target" DNA strands.  We have designed a single-point crossover [5, 38], as it is a common choice for evolutionary computation. This is a more closely controlled modi cation of an existing crossover technique due to Stemmer [27, 28, 30].  Existing \dirty" polymerase chain reaction [18, 25, 31, 32, 33] can incorporate pseudo-random mutation.

4.4 Design of Candidate Solutions and Target DNA Strands Figure 3 shows our DNA strand design. Recall that single stranded DNA has a direction, conventionally denoted by 50 ! 30. Any single strand of DNA seeks out and sticks (hybridizes) to an oppositely oriented \complementary" strand | Ts sticking to As and Gs

sticking to Cs. However, perfectly complementary matching is not required for hybridization, although such imperfect matches are less strongly bonded. In our design, target strands (complementary to \all ones") are xed. The role of the target strands is analogous to computer subroutines that count the number of ones in a candidate solution to the OneMax problem. A target strand and a perfect candidate strand are shown in the gure. Imperfect candidate strands are not shown, but have 40 Ts or Cs instead of 40 consecutive Ts. At the 30 end of all candidate strands there is a universal section complementary to the clamp section of the target strands. The clamp sequence is designed to (1) resist denaturing, (2) encourage correct alignment and (3) avoid secondary structure (strands hybridizing to themselves). The candidate strands are longer to facilitate separation, as needed, of target and candidate strands using denaturing gel electrophoresis [36]. All 50 ends of the candidate strands are extended by a universal tail sequence. Since candidate strands have known sequences at both ends, they can be ampli ed using the polymerase chain reaction [36].

4.5 Fitness Evaluation of DNA encoded Candidate Solutions As mentioned before, a mixture with a large variation of imperfect matches exhibits vertical spreading in our experiments [5, 38]. See Figure 4. In this particular experiment, each of the 40 variable positions of the imperfect candidates is chosen to be T with 8 probability or C with 2 probability. :

:

4.6 Fitness Selection of Candidate Solutions Selection is done by literally cutting out portions of the 2d gel and extracting the DNA strands from it. It is possible to \weight" these samples by choosing the shapes of the excised portions of the 2d gel and by adjusting the concentrations of the extracted DNA. This allows a wide latitude for selection criteria. There are many classical selection schemes arising from various considerations. Of course, we are motivated to use selection schemes natural to our computing medium and the available laboratory procedures. On any vertical line in a 2d DGGE the lowest candidates are presumably the most t. However, the nature of variation from left to right is not yet clear. It will be important to examine the possible variety of selection techniques that can be used after candidates

Clamp

Candidate

20

40

Tail 60

80

Target Tail-Primer

Perfect Candidate

5' - CCTATCCGTGGCGAGGCACCTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTCACCGTGGCTCGCACCGAG

Universal Target

3' - GGATAGGCACCGCTCCGTGGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Tail Primer

3' - AGTGGCACCGAGCGTGGCTC

Figure 3: Design of target and a perfect candidate. Imperfect candidates would have 40 Ts or Cs in place of the 40 Ts in the perfect candidate. (From [5], with permission.) have been physically separated by 2d DGGE. These selection schemes may or may not include any of the usual ones from classical evolution computation. For our proposed design, experiments and analysis will be needed to optimize a selection strategy. However, we recover candidate solutions that are expected to be more t, on average, than candidate solutions we discard. Naturally, it is desirable to maintain genetic diversity to prevent the loss of genetic materials which may be needed in later stages of evolution. These two conditions, exploitation and exploration, underlie the analogy with natural evolution.

Electrophoresis

-

5 Summary, Discussion and Prospects

+

Low

High Denaturant Concentration

Figure 4: DGGE vertically separates imperfect candidates. The speed of vertical strand migration is retarded as strands come apart (denature). This is due to two factors: (1) decreasing quality of targetcandidate matching (mishybridization). and (2) denaturant concentration, which increases from left to right. (From [5], with permission.)

We have provided a design for some evolutionary computations using population sizes larger than is practical with conventional computers. In two recent papers [5, 38] we further specify the laboratory operations and present some preliminary laboratory results. In evolutionary computation, no matter by what means, the heart of the matter is evaluating the tness of candidate solutions. By using 2d DGGE with DNA, rather than conventional computers, we gain enormous parallelism at the cost of a great loss of generality. However, the ability to separate strands by the degree of matching is sucient to address additional problems of signi cant interest. Two such classes of problems are cited in [5]. One of these classes still matches against xed target strands. The other class of problems matches co-evolving pairs of strands. In Royal Road problems, cited in [5] tness is determined by matching entire speci c groups of bits in candidate solutions. The target is again xed, but the course of the evolution exhibits many puzzling fea-

tures. The dynamics of Royal Road problems have been extensively studied. For only a few recent contributions see [34, 7, 8, 35]. Doing Royal Road problems using DNA would exploit huge population sizes. Such population sizes are infeasible with conventional computers. In a class of negotiation games, the degree of agreement between competitors is the key factor. Cold War is an example negotiation game cited in [5]. One could use 2d DGGE in the evolution of game strategies. Here, one would pairwise match co-evolving DNA strands. No longer would there be any xed target strands; no longer would one know solutions in advance. Thus, we see considerable scope for DNA computing to address some especially dicult problems using evolutionary computation, because conventional computers lack the massive parallelism and huge memory capacity inherent in molecular computation [6].

Acknowledgment We want to acknowledge partial support for Chen and Wood under NSF Grant No. 9980092 and DARPA/NSF Grant No. 9725021. Antipov was partially supported under an NSF Research Experience for Undergraduates supplement to the latter grant and additionally as a Howard Hughes Medical Institute Scholar.

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Variation

Initial Candidate Solutions

Candidate Pool

Purify Candidates

One-Point Crossover

Breeding

Fitness Evaluation

Target Strands

Hybridize Candidates and Targets Reserve

Separate by 2-D DGGE PCR Amplify-Mutate Purify, ReadOut Selected Candidates

Selection

Electrophoresis

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+

Low

High Denaturant Concentration

Electrophoresis

-

+

Low

High Denaturant Concentration

Clamp

Candidate

20

40

Tail 60

80

Target Tail-Primer

Perfect Candidate

5' - CCTATCCGTGGCGAGGCACCTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTCACCGTGGCTCGCACCGAG

Universal Target

3' - GGATAGGCACCGCTCCGTGGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Tail Primer

3' - AGTGGCACCGAGCGTGGCTC