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 O p 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. In the cen-
to their \ tness" as candidate solutions to the One-
ter, they move more slowly b ecause they encounter
Max problem. For this task our design uses so-called
intermediate denaturing. At the extreme right, the
two-dimensional denaturing gradient gel electrophore-
stands are able to move only very slowly b ecause the
sis 2d DGGE .
strands are almost completely pulled apart.
A rst imp ortant fact for DNA computing is that
We heuristically reason that when we rep eat the ab ove
2d DGGE can detect even a single base mismatch
exp eriment with a mixture of imperfectly matched
in double stranded DNA. Indeed, this was and
DNA strands, we exp ect that in every vertical section
early application of 2d DGGE in molecular biol-
the b etter matched strands will migrate downward rel-
ogy [21].
atively faster. This is the basis for our physical sepa-
ration by tness for the OneMax problem, and forms
A second imp ortant fact is that our preliminary
the heart of our exp erimental design.
exp eriments with 2d DGGE [5,38] demonstrate
a surprisingly smooth transition through a large
To the b est of our knowledge, no one had ever b e-
dynamic range of mismatching.
fore demonstrated what happ ens when 2d DGGE is
used with a p opulation having a very great diversityof
mismatchings. The question has not arisen in molec- Let us brie y review the nature of 2d DGGE. Figure 1
ular biology applications. Populations of strands all shows a 2d DGGE from our lab oratory using double
having considerable mismatches might regrettably not stranded DNA, with no mismatches anywhere. The
b e distinguishable at all. Fortunately,we found oth- mixture of double stranded DNA is placed uniformly
erwise. A mixture with a distribution of imp erfect along the top of the gel. The strands travel vertically
matches exhibits surprisingly smo oth vertical spread- downward in the gel as a result of an applied electric
ing in our preliminary exp eriments [5, 38]. eld. However, their sp eed of migration is determined er app ears at the v er left. Selection, puri cation and w x are indicated. ebo olutionary computation for OneMax problem. The candidate ys through the Reserv a w nofev tatio e path arious programmabl t. V Figure 2: Outlinep o ol of app ears DNA in implemen the upp er left. Selection using 2d DGGE app ears at the lo ampli cation of the more t candidate strands app ears at the b ottom. Breeding using crosso Target Strands upp er righ Initial Variation Candidate Solutions One-Point Crossover Candidate Pool Purify Candidates
Hybridize Candidates and Targets
Reserve
Separate by 2-D DGGE Breeding PCR Amplify-Mutate
Purify, ReadOut Selected Candidates Fitness Evaluation Fitness Selection
sticking to Cs. However, p erfectly complementary 4.2 Outline of DNA Implementation Design
matching is not required for hybridization, although
Our implementation is given by the following outline.
such imp erfect matches are less strongly b onded.
The same information is shown in Figure 2.
In our design, target strands complementary to \all
ones" are xed. The role of the target strands is anal-
DNA Evolutionary Computation for One-
ogous to computer subroutines that count the number
Max
of ones in a candidate solution to the OneMax prob-
lem.
Begin with an initial p opulation of candidates.
A target strand and a p erfect candidate strand are
Rep eat the following steps until convergence.
shown in the gure. Imp erfect candidate strands are
1. Evaluate tness byhybridizing to target
not shown, but have 40 Ts or Cs instead of 40 consec-
strands and physically separate on a 2d gel.
utive Ts.
2. Select candidates to breed.
0
At the 3 end of all candidate strands there is a uni-
3. Amplify selected candidates incorp orating
versal section complementary to the clamp section of
p ointwise mutation and reserve a p ortion.
the target strands. The clamp sequence is designed to
1 resist denaturing, 2 encourage correct alignment
4. Breed candidates, using crossover.
and 3 avoid secondary structure strands hybridiz-
5. Combine reserved and bred candidates, ob-
ing to themselves . The candidate strands are longer
taining a new generation.
to facilitate separation, as needed, of target and candi-
date strands using denaturing gel electrophoresis [36].
4.3 A New Fitness Selection is Combined
0
All 5 ends of the candidate strands are extended bya
With Existing Breeding Techniques
universal tail sequence. Since candidate strands have
known sequences at b oth ends, they can b e ampli ed
As mentioned earlier, mo di cations to current technol-
using the p olymerase chain reaction [36].
ogy suce to implement crossover and p ointwise mu-
tation. Thus, we combine mo di ed forms of three ro-
bust biolab oratory techniques for implementing DNA
4.5 Fitness Evaluation of DNA enco ded
evolutionary computation [38].
Candidate Solutions
As mentioned b efore, a mixture with a large variation
A new application of 2d denaturing gradient gel
of imp erfect matches exhibits vertical spreading in our
electrophoresis 2d DGGE : the more t candi-
exp eriments [5,38]. See Figure 4. In this particular
date strands of DNA can b e physically separated
exp eriment, each of the 40 variable p ositions of the im-
from much less t candidates according to how
p erfect candidates is chosen to b e T with :8 probability
well they match xed complementary \target"
or C with :2 probability.
DNA strands.
Wehave designed a single-p oint crossover [5, 38],
4.6 Fitness Selection of Candidate Solutions
as it is a common choice for evolutionary compu-
tation. This is a more closely controlled mo di -
Selection is done by literally cutting out p ortions of
cation of an existing crossover technique due to
the 2d gel and extracting the DNA strands from it.
Stemmer [27,28,30].
It is p ossible to \weight" these samples bycho osing
the shap es of the excised p ortions of the 2d gel and by
Existing \dirty" p olymerase chain reaction [18,
adjusting the concentrations of the extracted DNA.
25, 31, 32, 33] can incorp orate pseudo-random
This allows a wide latitude for selection criteria.
mutation.
There are many classical selection schemes arising
from various considerations. Of course, we are moti-
4.4 Design of Candidate Solutions and
vated to use selection schemes natural to our comput-
Target DNA Strands
ing medium and the available lab oratory pro cedures.
Figure 3 shows our DNA strand design. Recall that On anyvertical line in a 2d DGGE the lowest can-
single stranded DNA has a direction, conventionally didates are presumably the most t. However, the
0 0
denoted by5 ! 3 .Any single strand of DNA seeks nature of variation from left to right is not yet clear.
out and sticks hybridizes to an opp ositely oriented It will b e imp ortant to examine the p ossible varietyof
\complementary" strand | Ts sticking to As and Gs selection techniques that can b e used after candidates Clamp Tail Candidate 20 40 60 80 Target Tail-Primer Perfect Candidate 5' - CCTATCCGTGGCGAGGCACCTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTCACCGTGGCTCGCACCGAG
Universal 3' - GGATAGGCACCGCTCCGTGGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Target
Tail Primer 3' - AGTGGCACCGAGCGTGGCTC
Figure 3: Design of target and a p erfect candidate. Imp erfect candidates would have 40 Ts or Cs in place of the
40 Ts in the p erfect candidate. From [5], with p ermission.
have b een physically separated by 2d DGGE. These
selection schemes mayormay not include anyofthe
usual ones from classical evolution computation.
For our prop osed design, exp eriments and analysis will
b e needed to optimize a selection strategy.However,
we recover candidate solutions that are exp ected to
verage, than candidate solutions we
- b e more t, on a
discard. Naturally, it is desirable to maintain genetic
diversity to prevent the loss of genetic materials which
may b e needed in later stages of evolution. These two
conditions, exploitation and exploration, underlie the
analogy with natural evolution.
5 Summary, Discussion and Prosp ects
Electrophoresis
Wehave provided a design for some evolutionary com-
putations using p opulation sizes larger than is practi-
cal with conventional computers. In two recent pap ers
[5,38]we further sp ecify the lab oratory op erations and
present some preliminary lab oratory results.
+ Low High
olutionary computation, no matter by what
Denaturant Concentration In ev
means, the heart of the matter is evaluating the tness
of candidate solutions. By using 2d DGGE with DNA,
Figure 4: DGGE vertically separates imp erfect can-
rather than conventional computers, we gain enormous
didates. The sp eed of vertical strand migration is
parallelism at the cost of a great loss of generality.
retarded as strands come apart denature . This is
due to two factors: 1 decreasing quality of target-
However, the ability to separate strands by the degree
candidate matching mishybridization . and 2 de-
of matching is sucient to address additional prob-
naturant concentration, which increases from left to
lems of signi cantinterest. Two such classes of prob-
right. From [5], with p ermission.
lems are cited in [5]. One of these classes still matches
against xed target strands. The other class of prob-
lems matches co-evolving pairs of strands.
In Royal Road problems, cited in [5] tness is deter-
mined by matching entirespeci c groups of bits in can-
didate solutions. The target is again xed, but the
course of the evolution exhibits many puzzling fea-
[5] Junghuei Chen, Eugene Antip ov, Bertrand tures. The dynamics of Royal Road problems have
Lemieux, Walter Cedeno, ~ and David Harlan b een extensively studied. For only a few recent contri-
Wo o d. DNA computing implementing genetic al- butions see [34, 7, 8, 35]. Doing Royal Road problems
gorithms. In Laura Landweb er, Erik Winfree, using DNA would exploit huge p opulation sizes. Such
Richard Lipton, and Stephan Freeland, editors, p opulation sizes are infeasible with conventional com-
Preliminary Proceedings DIMACS Workshop on puters.
Evolution as Computation, pages 39{49, DI-
In a class of negotiation games, the degree of agree-
MACS, Piscataway NJ, January 1999. Available
mentbetween comp etitors is the key factor. Cold
on request from DIMACS. Pap er found at URL:
War is an example negotiation game cited in [5]. One
http://www.cis.udel.edu/~wo o d/pap ers/DIMACS 99.ps.
could use 2d DGGE in the evolution of game strate-
gies. Here, one would pairwise match co-evolving DNA
[6] Junghuei Chen and David Harlan Wo o d. Com-
strands. No longer would there b e any xed target
putation with biomolecules. Proceedings of the
strands; no longer would one know solutions in ad-
National Academy of Sciences, USA, 97 4 :1328{
vance.
1330, 2000. Commentary.
Thus, we see considerable scop e for DNA computing to
[7] James P. Crutch eld and Erik van Nimwegen.
address some esp ecially dicult problems using evolu-
Opti-
tionary computation, b ecause conventional computers
mizing ep o chal evolutionary search: Population-
lack the massive parallelism and huge memory capac-
size dep endent theory. SFI Working Paper 98-
ity inherent in molecular computation [6].
10-090, 1998, 18 pages. Pap er found at URL:
http://www.santafe.edu/pro jects/evca/evabstracts.htmlo e e
Acknowledgment
[8] James P. Crutch eld and Erik van Nimwegen.
The evolutionary unfolding of complexity. In
Wewanttoacknowledge partial supp ort for Chen
Laura Landweb er, Erik Winfree, Richard Lipton,
and Wood under NSF Grant No. 9980092 and
and Stephan Freeland, editors, Proceedings of the
DARPA/NSF Grant No. 9725021. Antip ovwas par-
DIMACS Workshop on Evolution as Computa-
tially supp orted under an NSF Research Exp erience
tion, New York, 1999, to app ear. Springer-Verlag.
for Undergraduates supplement to the latter grant and
additionally as a Howard Hughes Medical Institute
[9] J. H. M. Dassen and F. Pierluigi. A bibliography
Scholar.
of molecular computation and splicing systems.
HTML source:
http://www.wi.LeidenUniv.nl/ pier/dna.html,
References
BibTeX source:
http://www.wi.LeidenUniv.nl/ pier/dna.bib.
[1] Leonard M. Adleman. Molecular computation
This bibliography is also ho oked into
of solutions to combinatorial problems. Science,
http://liinwww.ira.uka.de/bibliography/, The
266:1021{1024, Novemb er 1994.
Collection of Computer Science Bibliographies.
[2] Leonard M. Adleman. Computing with DNA.
[10] R. Deaton, R. C. Murphy, J. A. Rose,
Scienti c American, 279 2 :54{61, August 1998.
Max H. Garzon, Donald R. Franceschetti, and
[3] Thomas Back, David B. Fogel, and Zbigniew
S. E. Stevens Jr. A DNA based implementation
Michalewicz, editors. Handbook of Evolutionary
of an evolutionary search for go o d enco dings for
Computation. Institute of Physics Publishing,
DNA computation. In Proceedings of the 1997
Philadelphia, 1997.
IEEE International Conference on Evolutionary
Computation, pages 267{271, Indianap olis, Indi-
[4] Thomas Back, Jo ost N. Kok, and Grzegorz
ana, April 13{16, 1997. IEEE Press.
Rozenb erg. Evolutionary computation as a
paradigm for DNA-based computing. In Laura
[11] Alan Dove. From bits to bases: Computing
Landweb er, Erik Winfree, Richard Lipton, and
with DNA. Nature Biotechnology, 16 9 :830{832,
Stephan Freeland, editors, Preliminary Pro-
Septemb er 1998.
ceedings DIMACS Workshop on Evolution as
[12] David B. Fogel. Some recent imp ortant foun- Computation, pages 67{88, DIMACS, Piscat-
dational results in evolutionary computation. In away NJ, January 1999. Available on re-
K. Miettinen, P. Nittaanmaki, M. M. Makela, and quest from DIMACS. Pap er found at URL:
J. P eriaux, editors, Evolutionary Algorithms in http://www.wi.LeidenUniv.nl/~joost.
[23] Riccardo Poli, William B. Langdon, and Una-May Engineering and Computaer Science,chapter 4,
O'Reilly. Analysis of schema variance and short pages 55{71. John Wiley and Sons, New York,
term extinction likeliho o ds. In John R. Koza, 1999.
Wolfgang Banzhaf, Kumar Chellapilla, Kalyan-
[13] Philipp e Gigure and David E. Goldb erg. Popu-
moy Deb, Marco Dorigo, David B. Fogel, Max H.
lation sizing for optimum sampling with genetic
Garzon, David E. Goldb erg, Hitoshi Iba, and Rick
algorithms: A case study of the Onemax prob-
Riolo, editors, Genetic Programming 1998: Pro-
lem. In John R. Koza, Wolfgang Banzhaf, Ku-
ceedings of the ThirdAnnual Conference, pages
mar Chellapilla, Kalyanmoy Deb, Marco Dorigo,
284{292, University of Wisconsin, Madison, Wis-
David B. Fogel, Max H. Garzon, David E. Gold-
consin, USA, 22-25 July 1998. Morgan Kaufmann.
b erg, Hitoshi Iba, and Rick Riolo, editors, Genetic
Programming 1998: Proceedings of the ThirdAn-
[24] Rob ert Po ol. Forget silicon, try DNA. New Sci-
nual Conference, pages 496{503, Universityof
entist, 151 2038 :26{31, July 13, 1996.
Wisconsin, Madison, Wisconsin, USA, 22-25 July
1998. Morgan Kaufmann.
[25] D. L. Rob ertson and F. G. Joyce. Selection in
vitro of an RNA enzyme that sp eci cally cleaves
[14] David E. Goldb erg and Kalyanmoy Deb. A com-
single-stranded DNA. Nature, 344 6265 :467{468,
parative analysis of selection schemes used in ge-
March 29, 1990.
netic algorithms. In Foundations of Genetic Al-
gorithms, FOGA'91, proceedings, 1991.
[26] M. Rudnick and David E. Goldb erg. Signal, noise
and genetic algorithms. Technical Rep ort 91005,
[15] David E. Goldb erg, Kalyanmoy Deb, and
Illinois Genetic Algorithms Lab oratory IlliGAL ,
James H. Clark. Genetic algorithms, noise, and
june 1991.
the sizing of p opulations. Complex Systems,
6 4 :333{362, August 1992.
[27] Willem P. C. Stemmer. DNA shuing by random
fragmentation and reassembly: In vitro recom-
[16] David E. Goldb erg and B. L. Miller. Genetic algo-
bination for molecular evolution. Proceedings of
rithms, selection schemes, and the varying e ects
the National Academy of Science, U.S.A., 91:389{
of noise. Evolutionary Computation, 4 2 :113{
391, 1994.
131, 1996.
[28] Willem P. C. Stemmer. Rapid evolution of a pro- [17] Tino Gram, Stephan Bornholdt, Michael
tein by DNA shuing. Nature, 370:389{391, 1994. Gram, Melanie Mitchell, and Thomas Pellizzari.
Non-Standard Computation. Wiley-VCH, Wein-
[29] Willem P. C. Stemmer. The evolution of molecu-
heim, 1998.
lar computation. Science, 270:1510{1510, Decem-
[18] Rachel Green, Andrew D. Ellington, David P.
b er 1, 1995.
Bartel, and Jack W. Szostak. In vitro genetic
analysis: Selection and ampli cation of rare func-
[30] Willem P. C. Stemmer. Sexual PCR and As-
tional nucleic acids. Methods, 2:75{86, 1991.
sembly PCR. In Rob ert M. Meyers, editor, The
Encyclopedia of Molecular Biology and Molecular
[19] Jorg Heitkotter and David Beasley. The
Medicine,volume 5, pages 447{457. VCH, New
hitch-hiker's guide to evolutionary computa-
York, 1996.
tion FAQ for comp.ai.genetic . Web page
at http://alife.santafe.edu/ joke/encore/www/,
[31] Jack W. Szostak. In vitro genetics. Trends in
Septemb er 1999.
Biochemical Sciences, 172 3 :89{93, 1992.
[20] John H. Holland. Adaptation in natural and arti-
[32] H. J. Thiesen and C. Bach. Target detection as-
cial systems. MIT Press, Boston, 1992.
say TDA | A versatile pro cedure to determine
DNA-binding sites as demonstrated on SP1 pro-
[21] L. S. Lerman, K. Silverstein, and E. Grinfeld.
tein. Nucleic Acids Research, 18 11 :3203{3209,
Searching for gene defects by denaturing gradient
1990.
gel electrophoresis. Trends in Biochemical Sci-
ences, 172 3 :89{93, 1992.
[33] C. Tuerk and L. Gold. Systematic evolution of lig-
[22] Carlo C. Maley. DNA computation: Theory, prac- ands by exp onential enrichment | RNA ligands
tice, and prosp ects. Evolutionary Computation, to bacteriophage-T4 DNA-p olymerase. Science,
6 3 :201{229, 1998. 249 4968 :505{510, August 3, 1990.
[34] Erik van Nimwegen, James P. Crutch eld, and
Melanie Mitchell. Statistical dynamics of the
Royal Road genetic algorithm. Theoretical Com-
puter Science, to app ear.
[35] Michael D. Vose. The Simple Genetic Algorithm.
MIT Press, Cambridge, 1999.
[36] J. Watson, M. Gilman, J. Witkowski, and
M. Zoller. Recombinant DNA. Scienti c Amer-
ican Bo oks, W.H. Freeman and Co., New York,
second edition, 1992.
[37] Roy Williams. Data p owers of ten. Web page at
http://www.ccsf.caltech.edu/~roy/dataquan.
[38] David Harlan Wo o d, Junghuei Chen, Eugene An-
tip ov, Bertrand Lemieux, and Walter Cedeno. ~ In
vitro selection for a OneMax DNA evolutionary
computation. In David Gi ord and Erik Win-
free, editors, DNA Based Computers V: DIMACS
Workshop, June 14-15, 1999, DIMACS series in
discrete mathematics and theoretical computer
science, Providence, to app ear. American Mathe-
matical So ciety. Target Strands Initial Variation Candidate Solutions One-Point Crossover Candidate Pool Purify Candidates
Hybridize Candidates and Targets
Reserve
Separate by 2-D DGGE Breeding PCR Amplify-Mutate
Purify, ReadOut Selected Candidates Fitness Evaluation Fitness Selection - Electrophoresis
+ Low High Denaturant Concentration - Electrophoresis
+ Low High Denaturant Concentration Clamp Tail Candidate 20 40 60 80 Target Tail-Primer Perfect Candidate 5' - CCTATCCGTGGCGAGGCACCTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTCACCGTGGCTCGCACCGAG
Universal 3' - GGATAGGCACCGCTCCGTGGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Target
Tail Primer 3' - AGTGGCACCGAGCGTGGCTC