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ssBioMed Cent
Journal of Biological Engineering
Open Acce
Research
Solving a Hamiltonian Path Problem with a bacterial computer
Jordan Baumgardner
1
, Karen Acker
2
, Oyinade Adefuye
2,3
,
Samuel Thomas Crowley
1
, Will DeLoache
2
, James O Dickson
4
, Lane Heard
1
,
Andrew T Martens
2
, Nickolaus Morton
1
, Michelle Ritter
5
, Amber Shoecraft
4,6
,
Jessica Treece
1
, Matthew Unzicker
1
, Amanda Valencia
1
, Mike Waters
2
, A
Malcolm Campbell
2
, Laurie J Heyer
4
, Jeffrey L Poet
5
and Todd T Eckdahl*
1
Address:
1
Department of Biology, Missouri Western State University, St Joseph, MO 64507, USA,
2
Department of Biology, Davidson College,
Davidson, NC 28036, USA,
3
Department of Biology, North Carolina Central University, Durham, NC 27707, USA,
4
Department of Mathematics,
Davidson College, Davidson, NC 28036, USA,
5
Department of Computer Science, Math and Physics, Missouri Western State University, St Joseph,
MO 64507, USA and
6
Natural Science and Math Department, Johnson C. Smith University, Charlotte, NC 28216, USA
Email: Jordan Baumgardner - jbaumgardner@missouriwestern.edu; Karen Acker - karen.acker@gmail.com;
Oyinade Adefuye - oyinadeadefuye@yahoo.com; Samuel Thomas Crowley - stc8033@missouriwestern.edu;
Will DeLoache - wideloache@davidson.edu; James O Dickson - jidickson@davidson.edu; Lane Heard - axenmoon@hotmail.com;
Andrew T Martens - a.t.martens@gmail.com; Nickolaus Morton - nmorton@missouriwestern.edu;
Michelle Ritter - mritter2@missouriwestern.edu; Amber Shoecraft - ashoecraft@jcsu.edu; Jessica Treece - jtreece@kcumb.edu;
Matthew Unzicker - mru8487@missouriwestern.edu; Amanda Valencia - avalencia@missouriwestern.edu;
Mike Waters - miwaters@davidson.edu; A Malcolm Campbell - macampbell@davidson.edu; Laurie J Heyer - laheyer@davidson.edu;
Jeffrey L Poet - poet@missouriwestern.edu; Todd T Eckdahl* - eckdahl@missouriwestern.edu
* Corresponding author
Abstract
Background: The Hamiltonian Path Problem asks whether there is a route in a directed graph
from a beginning node to an ending node, visiting each node exactly once. The Hamiltonian Path
Problem is NP complete, achieving surprising computational complexity with modest increases in
size. This challenge has inspired researchers to broaden the definition of a computer. DNA
computers have been developed that solve NP complete problems. Bacterial computers can be
programmed by constructing genetic circuits to execute an algorithm that is responsive to the
environment and whose result can be observed. Each bacterium can examine a solution to a
mathematical problem and billions of them can explore billions of possible solutions. Bacterial
computers can be automated, made responsive to selection, and reproduce themselves so that
more processing capacity is applied to problems over time.
Results: We programmed bacteria with a genetic circuit that enables them to evaluate all possible
paths in a directed graph in order to find a Hamiltonian path. We encoded a three node directed
graph as DNA segments that were autonomously shuffled randomly inside bacteria by a Hin/hixC
recombination system we previously adapted from Salmonella typhimurium for use in Escherichia coli.
We represented nodes in the graph as linked halves of two different genes encoding red or green
fluorescent proteins. Bacterial populations displayed phenotypes that reflected random ordering of
edges in the graph. Individual bacterial clones that found a Hamiltonian path reported their success
Published: 24 July 2009
Journal of Biological Engineering 2009, 3:11 doi:10.1186/1754-1611-3-11
Received: 30 March 2009
Accepted: 24 July 2009
This article is available from: http://www.jbioleng.org/content/3/1/11
' 2009 Baumgardner et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Page 1 of 11
(page number not for citation purposes)
by fluorescing both red and green, resulting in yellow colonies. We used DNA sequencing to verify

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
that the yellow phenotype resulted from genotypes that represented Hamiltonian path solutions,
demonstrating that our bacterial computer functioned as expected.
Conclusion: We successfully designed, constructed, and tested a bacterial computer capable of
finding a Hamiltonian path in a three node directed graph. This proof-of-concept experiment
demonstrates that bacterial computing is a new way to address NP-complete problems using the
inherent advantages of genetic systems. The results of our experiments also validate synthetic
biology as a valuable approach to biological engineering. We designed and constructed basic parts,
devices, and systems using synthetic biology principles of standardization and abstraction.
Background
Contemporary mathematical challenges to computation
Mathematicians and computer scientists alike are familiar
with the computational complexity associated with prob-
lems referred to as NP-complete [1]. Such problems are
included in a group of decision problems known as NP, or
nondeterministic polynomial, which have solutions that,
once found, can easily be shown to be correct. Although
many NP problems can be solved quickly, NP-complete
problems cannot, since their complexity grows combina-
torially with linear increases in the problem size. These
problems are significant because of their relationships to
each other: every NP-complete problem can be cast in the
form of any other using a polynomial-time algorithm,
meaning that an efficient algorithm for one NP-complete
problem can be used to solve all others. Expert computer
programmers learn to recognize patterns in their codes
that suggest a particular problem is NP-complete and, as a
result, either settle for an approximate solution or aban-
don their attempt to obtain an exact one. The first prob-
lem proved to be NP-complete was the Boolean
Satisfiability Problem (SAT), which is the problem of
determining whether or not the variables in a logical
expression can be assigned to make the expression true
[2]. Other NP-complete problems include the Knapsack
Problem, the Maximum Clique Problem, and the Pancake
Problem. A version of the Pancake Problem, the Burnt
Pancake Problem, was introduced in the only academic
publication by Bill Gates [3]. The NP-complete problem
addressed in this paper is the Hamiltonian Path Problem
(HPP), in which a path must be found in a directed graph
from a beginning node to an ending node, visiting each
node exactly once. Figure 1 shows a directed graph with a
unique Hamiltonian path from node 1 to node 5.
The serial approach that most silicon computer algo-
rithms use is not well suited for solving NP-complete
problems because the number of potential solutions that
must be evaluated grows combinatorially with the size of
the problem. For example, a Hamiltonian Path Problem
for a directed graph on ten nodes may require as many as
10! = 3,628,800 directed paths to be evaluated. A static
number of computer processors would require time pro-
portional to this number to solve the problem. Doubling
the number of nodes to 20 would increase the possible
number of directed paths to 20! = 2.43 × 10
18
, increasing
the computational time by 12 orders of magnitude.
Improvement in computational capability could come
from parallel processing and an increase in the number of
processors working on a problem. Significant break-
throughs in this regard may be possible with the develop-
ment of biological computing, because the number of
processors grows through cell division.
Biological computing
In a groundbreaking experiment, Leonard Adleman dem-
onstrated an alternative to the serial processing of silicon
computers by developing a DNA computer that could
carry out parallel processing in vitro to solve the HPP in
Figure 1[4]. The seminal work by Adleman inspired
others to develop DNA computers capable of solving
mathematical problems that are intractable to serial com-
A directed graph containing a unique Hamiltonian pathFigure 1
A directed graph containing a unique Hamiltonian
path. The seven nodes are connected with fourteen
directed edges. The Hamiltonian Path Problem is to start at
node 1, end at node 5, and visit each node exactly once while
following the available edges. Adleman programmed a DNA
computer to find the unique Hamiltonian path in this graph Page 2 of 11
(page number not for citation purposes)
puting [5-7].
(1 ο4 ο7 ο2 ο3 ο6 ο5).

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
We asked whether it would be possible to move DNA
computing inside bacteria that could function as a living
computer with billions of processors. Programming bac-
teria to compute solutions to difficult problems could
offer the same advantage of parallel processing that DNA
computing brings, with the following additional desirable
features: (1) bacterial systems are autonomous, eliminat-
ing the need for human intervention, (2) bacterial com-
puters can adapt to changing conditions, evolving to meet
the challenges of a problem, and (3) the exponential
growth of bacteria continuously increases the number of
processors working on a problem.
In a previous study, we reconstituted the S. typhimurium
Hin/hixC recombinase system for use in E. coli [8]. In
addition to its potential use in controlling the order and
orientation of transgenes and for modeling syntenic
genome relationships, the system has proved to be a use-
ful tool in the development of bacterial computers.
Recombination by Hin recombinase results in the inver-
sion of DNA fragments that are flanked by a pair of hixC
sites [9,10]. We demonstrated that Hin recombinase
could invert either a single DNA fragment or multiple
adjacent fragments in a single operation [8].
We used the Hin/hixC system to engineer living bacterial
cells to calculate a solution to a variation of the Burnt Pan-
cake Problem [8]. The problem involves sorting a set of
burnt pancakes so that they all have the same orientation
and are arranged in a particular order. Our biological rep-
resentation of a burnt pancake was a functional DNA unit
containing a promoter or a protein coding sequence, each
flanked by a pair of hixC sites. We used the selectable phe-
notype of antibiotic resistance to identify bacteria that
solved the BPP. Our results served as an important proof-
of-concept that bacteria can function as parallel proces-
sors in the computation of solutions to a mathematical
problem.
We sought to use our bacterial computing approach to
solve a Hamiltonian Path Problem, as Adleman did with
a DNA computer. With an appreciation for history, we
designed a DNA-encoded version of Figure 1 to encode
the HPP into DNA segments that could be inverted by Hin
recombinase. To test the feasibility of solving the HPP in
vivo, we designed, constructed, and tested bacteria that
announced their arrival at a solution to a proof-of-concept
three node HPP by producing colonies that fluoresced yel-
low.
Results
Genetic encoding of the Hamiltonian Path Problem
The design of our bacterial computer benefited from a
series of abstractions of DNA sequence into the edges and
nodes of a Hamiltonian path. The first abstraction treated
DNA segments as edges of a directed graph. DNA edges
flanked by hixC sites can be reshuffled by Hin recombi-
nase, creating random orderings and orientations of edges
of the graph. The second abstraction treated all nodes,
except the terminal one, as genes split into two halves
(Figure 2). The first (5') half of the gene for a given node
is found on any DNA edge that terminates at the node,
while the second (3') half of the gene is found on any
DNA edge that originates at the node. The final abstrac-
tion was an arrangement of DNA edges that represented a
HPP solution and exhibited a new phenotype. To place
Illustration of the use of split genes to encode a seven node Hamiltonian Path ProblemFigu e 2
Illustration of the use of split genes to encode a seven node Hamiltonian Path Problem. a. The manner in which
each of the directed edges in Figure 1 could be encoded in DNA is illustrated. The 5' half of each node gene is denoted by
and the 3' half is denoted by . DNA edges are depicted by gene halves connected by arrows and flanked by triangles that
represent hixC sites. Transcription in the direction of the solid arrow would terminate early and result in the expression of
only one marker gene. b. Hin-mediated recombination would randomly reshuffle the DNA edges into many configurations.
One possible example of an HPP solution configuration with its marker gene halves reunited is illustrated. Transcription in the Page 3 of 11
(page number not for citation purposes)
direction of the solid arrow would result in expression of the six marker gene phenotypes.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
our proposed improvement of DNA computing in the his-
torical context of the graph in Figure 1, we designed the
constructs shown in Figure 2. Each node in the graph is
represented by a gene that encodes an observable pheno-
type, such as antibiotic resistance or fluorescence. The
exception to this is node 5, which is represented by a tran-
scription terminator to ensure that it will be the last node
in the Hamiltonian path. Each 5' half of a gene is denoted
by the left half of a circle and each 3' half is denoted by the
right half of a circle. Gene halves connected by arrows and
flanked by triangular hixC sites are the flippable DNA
edges. The order and orientation of the DNA edges deter-
mines the starting configuration, an example of which is
illustrated in Figure 2a. Hin-mediated recombination of
the 14 DNA edges could produce 1.42 × 10
15
possible con-
figurations. Of these, a small fraction represent Hamilto-
nian paths with all of the node genes intact (see
mathematical modeling section below for details). An
example of one of these solution configurations is illus-
trated in Figure 2b. Bacterial colonies that contain an HPP
solution will express a unique combination of pheno-
types that can be detected directly or found by selection.
Splitting GFP and RFP genes
Once we were convinced that our proposed in vivo DNA
computer could solve a HPP, we chose a simpler three
node graph for our first biological implementation of the
problem. To execute our design, we needed to split two
marker genes by inserting hixC sites. For each gene to be
split, we had to find a site in the encoded protein where
13 specific amino acids could be inserted without destroy-
ing the function of the protein. We examined the three-
dimensional structure of each protein candidate, chose a
site for the insertion, built gene halves, and tested the reu-
nited halves with the 13 amino acid insertion for protein
function. We successfully inserted hixC sites into the cod-
ing sequences of both GFP and RFP without loss of fluo-
rescence [11]. We inserted the hixC site between amino
acids 157 and 158 in GFP, and between the structurally
equivalent amino acids 154 and 155 in RFP. Each of the
insertions extended a loop outside of the beta barrel struc-
ture of the fluorescent proteins. We also tested two hybrid
constructs to ensure that they would not fluoresce. We
assembled the 5' half of GFP with the 3' half of RFP and
the hybrid protein did not fluoresce red or green (data not
shown). Similarly, the 5' half of RFP placed upstream of
the 3' half of GFP did not cause fluorescence (data not
shown). In addition, none of the four half proteins fluo-
resced by themselves (data not shown). These results
demonstrated the suitability of the GFP and RFP gene
halves as parts for use in programming a bacterial compu-
ter to solve an HPP. Being able to split two genes enabled
us to design a bacterial computer to solve an HPP for a
Mathematical modeling of bacterial computational
capacity
We used mathematical modeling to examine several
important questions about the system. The first question
is whether the order and orientation of the DNA edges in
a starting construct affect the probability of detecting an
HPP solution. During an HPP experiment, billions of bac-
teria cells will attempt to find a solution by random flip-
ping of DNA edges catalyzed by Hin recombinase. We
developed a Markov Chain model in MATLAB using the
signed permutations of {1,2,...n} as the states of DNA
edges in the HPP. We assumed that each possible reversal
of adjacent DNA edges was equally likely. Using this tran-
sition matrix, we computed the probability that any start-
ing configuration would be in any of the solved states after
k flips. We conducted this analysis for a number of differ-
ent graphs. Figure 3 shows one example of the results, for
a graph with four nodes and three edges. The graph shows
a relatively quick convergence to equilibrium, as was the
case for all the graphs we analyzed. In this example, there
are 48 possible configurations of the edges, only one of
which is a solution. After about 20 flips, the probability
that the edges are in the solution state (or any other state)
is 1/48 ( | 0.02). Consideration of the reaction rate
reported for Hin recombinase [12] led us to conclude that
equilibrium could be reached in the 3-node, 3-edge exper-
iment that we intended to use as a proof-of-concept.
Assuming that E. coli divides every 20–30 minutes and
that we grow the cells for 16 hours, exceeding 20 flips
should occur even if Hin recombinase catalyzes only one
reaction per cell cycle.
Markov Chain model of solving a Hamiltonian Path ProblemFigure 3
Markov Chain model of solving a Hamiltonian Path
Problem. Each colored line represents a different starting
configuration of a graph with four nodes and three edges. As
the number of flips increases, the probability of finding a Page 4 of 11
(page number not for citation purposes)
three node directed graph. Hamiltonian path solution converges to 1/48, or about 0.02.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
We also used mathematical modeling to determine how
many bacteria would be needed to have high confidence
that, after Hin recombination, at least one cell would con-
tain a plasmid with a true HPP solution. For the example
of the graph in Figure 1, each HPP solution would have six
DNA edges in the proper order and orientation followed
by the remaining eight edges in any order and orientation.
Because there are 8! ways to order the eight remaining
edges, and two ways to orient each one, there are 8!·2
8
=
10,321,920 different configurations that are solutions,
one example of which is shown in Figure 2b. There is a
total of 14!·2
14
= 1.42 × 10
15
possible configurations of
the edges (14! ways to order the edges, and two ways to
order each one), many of which are not even valid con-
nected paths in the graph, much less Hamiltonian paths.
The probability of any one plasmid holding an HPP solu-
tion is p = (8!·2
8
)/(14!·2
14
). Assuming that the states of
different plasmids are independent and that a sufficient
number of flips has occurred to achieve a uniform distri-
bution of the 14!·2
14
possible configurations, the proba-
bility that at least one of m plasmids holds an HPP
solution is 1-(1-p)
m
. From this expression, we can solve
for m to find the number of plasmids needed to reach the
desired probability of finding at least one solution. For
example, if we wanted to be 99.9% sure of finding an HPP
solution, we would need at least one billion independent,
identically distributed plasmids. A billion E. coli can grow
overnight in a single culture. It should be noted, however,
that it may take longer than that for Hin recombination to
produce a uniform distribution of all possible plasmid
configurations. Since each bacterium would have at least
100 copies of the plasmid, the computational capacity of
a billion cells exceeds our needs by two orders of magni-
tude. Because the number of processors would be increas-
ing exponentially, the time required for a biological
computer to evaluate all 14!·2
14
configurations is a con-
stant multiple of log(14!·2
14
), or approximately
14·log(14), while the time required for a conventional
computer to evaluate the same number of paths would be
a constant multiple of 14!·2
14
.
A key feature of our experimental design is the simplicity
of detecting answers with phenotypes of red and green flu-
orescence resulting in yellow colonies. However, when
our design is applied to a more complex problem such as
the one presented in Figures 1 and 2, it is possible that a
colony with a correct phenotype might have an incorrect
genotype, resulting in a false positive. We considered the
question of whether there are too many false positives to
detect a true positive. Using MATLAB, we computed the
number of true positives for the 14-edge graph in Figure 1
to be 10,321,920 and the number of total positives to be
168,006,848. The ratio of true positives to total positives
starting node and the ending node than in the true solu-
tion states, putative solutions could be screened using
PCR. However, since the ratio of true to total positives gets
smaller with the size of the problem, this approach
becomes increasingly impractical. An alternative would be
to conduct high throughput DNA sequencing of pooled
putative solution plasmids.
Our mathematical modeling supported the conclusion
that our experimental design could solve Hamiltonian
Path Problems. As a proof-of-concept, we designed a sim-
ple directed graph with a unique Hamiltonian path and
programmed a bacterial computer to find that path.
Programming a bacterial computer
Figure 4a shows the directed graph with three nodes and
three edges that we chose to encode in our bacterial com-
puter. The graph contains a unique Hamiltonian path
starting at the RFP node, traveling via edge A to the GFP
node, and using edge B to reach the ending TT node. Edge
C, from RFP to TT, is a detractor. Figure 4b illustrates the
DNA constructs we used to encode a solved HPP as a pos-
itive control and two unsolved starting configurations.
Since the solution must originate at the RFP node and ter-
minate at the GFP node, DNA edge A contained the 3' half
of RFP followed by the 5' half of GFP. DNA edge B origi-
nated at GFP and terminated at TT, so its DNA segment
has 3' GFP followed by the double transcription termina-
tor. DNA edge C originated with the 3' half of RFP and ter-
minated at TT. Each of the 5' gene halves included a
ribosome binding site (RBS) upstream of its start codon in
order to support translation.
DNA constructs that encode a three node Hamiltonian Path ProblemFigure 4
DNA constructs that encode a three node Hamilto-
nian Path Problem. a. The three node directed graph con-
tains a Hamiltonian path starting at the RFP node, proceeding
to the GFP node, and finishing at the TT node. b. Construct
ABC represents a solution to the three node HPP. Its three
hixC-flanked DNA segments are in the proper order and ori-
entation for the GFP and RFP genes to be intact. ACB has
the RFP gene intact but not the GFP gene, while BAC has Page 5 of 11
(page number not for citation purposes)
is therefore approximately 0.06. Since all false positive
solutions must have at least one more edge between the
neither gene intact.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
As illustrated in Figure 4b, we designed an expression cas-
sette to contain the three DNA edges. To ensure the solu-
tion begins at the RFP node, the cassette starts with a
bacteriophage T7 RNA polymerase promoter, an RBS, and
5' RFP prior to the first hixC site. Construct ABC represents
one of two HPP solutions since it begins with the RFP
node, passes through GFP and ends with TT. Since both
the RFP and GFP genes are intact, downstream of the pro-
moter, in the correct orientation, and followed by the
transcriptional terminators, ABC colonies should express
both red and green fluorescence and appear yellow. A sec-
ond solution is ABC', in which forward DNA edges A and
B are followed by backwards DNA edge C. Bacteria con-
taining this configuration are expected to fluoresce yel-
low, since RFP and GFP are intact and in forward
orientation. Construct ACB has the RFP gene intact, in the
correct orientation, and uninterrupted by transcriptional
terminators, but its GFP gene halves are not united. As a
result, this construct is predicted to produce red colonies.
The BAC construct has neither RFP nor GFP intact and
should not fluoresce at all. The three plates on the left side
of Figure 5 show that all three constructs produced the
predicted phenotypes in the absence of Hin recombinase:
ABC colonies fluoresce yellow, ACB colonies fluoresce
red, and BAC colonies show no fluorescence.
Random orderings of edges in the directed graph were
produced by Hin-mediated recombination in a separate
experiment using each of the three starting constructs
ABC, BAC, or ACB. In a given experiment, bacteria were
cotransformed with 1) a plasmid conferring ampicillin
resistance and containing one of the three starting con-
structs and 2) a plasmid encoding tetracycline resistance
with a Hin recombinase expression cassette. The resulting
cotransformed colonies were grown overnight for isola-
tion of plasmids containing the Hin-exposed HPP con-
structs. The isolated plasmids were then used in a second
round of transformation into bacteria that expressed bac-
teriophage T7 RNA polymerase and plated on media con-
taining only ampicillin (Figure 5). Ampicillin-resistant
colonies were grown overnight to allow the T7 RNA
polymerase to transcribe each plasmid in its final flipped
state. Because each colony represented a single transfor-
mation event and Hin was no longer present, each colony
contained isogenic plasmids and thus only one configura-
tion of the three DNA edges. This experimental protocol
was followed for each of the three starting constructs.
Verifying bacterial computer solutions to a Hamiltonian
Path Problem
Once Hin recombinase reorders the DNA edges of each of
the constructs, a distribution of 48 possible configura-
tions is expected. The positive control ABC construct
should convert from its yellow fluorescent starting pheno-
type to the red and uncolored phenotypes of unsolved
arrangements. The ABC recombination plate pictured in
Figure 5 matched our prediction. We assumed that the
double transcriptional terminator would function in
reverse orientation, so that green colonies would not be
possible in the experiment. However, green colonies on
the ABC recombination plate indicate that TT did not
block further transcription. The ABC recombination plate
also shows a number of unusually colored colonies that
were not expected, which we discuss later.
The ACB starting construct was expected to undergo Hin-
mediated recombination to produce a variety of configu-
rations, including a solution that requires at least two
flips. Yellow fluorescent colonies representing putative
HPP solutions are visible on the ACB recombination
plate. The BAC starting configuration was three flips away
from the nearest solution. Several examples of yellow flu-
orescent colonies on the BAC recombination plate are
candidates for solutions to the HPP. As with the ABC
recombination plate, we found unexpected colony colors
on both the ACB and BAC recombination plates.
Detecting solutions to a Hamiltonian Path Problem with bac-terial computingFigure 5
Detecting solutions to a Hamiltonian Path Problem
with bacterial computing. Bacterial colonies containing
each of the three starting constructs ABC, ACB, and BAC
are shown on the left. Hin recombination resulted in the Page 6 of 11
(page number not for citation purposes)
Yellow fluorescent colonies on the ACB and BAC recom-
bination plates provided preliminary evidence that the
three plates of colonies on the right. The callouts include yel-
low colored colonies that contain solutions to the HPP.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
bacterial computer had solved both versions of the HPP.
We wanted to verify this result by sequencing plasmid
DNA to determine the genotypes of three yellow colonies
from each of the ABC, ACB, and BAC recombination
plates. All nine colonies had a genotype of ABC or ABC',
in which the third DNA edge is in reverse orientation (Fig-
ure 6). Both of these configurations represent a solution
to the HPP. These results verified that our bacterial com-
puter had found true solutions to a three node HPP con-
figured in two different starting orientations.
Discussion
Bacterial computer reveals novel phenotypes
We used the principles and practices of synthetic biology
to design and build a bacterial computer that solved a
Hamiltonian Path Problem. We successfully encoded a
directed graph with three nodes and three edges into DNA
and used Hin recombinase to rearrange the edges into a
Hamiltonian path configuration that yielded a yellow flu-
orescent phenotype. We verified genotype solutions to the
problem with DNA sequencing. Our engineered bacterial
computer system functioned according to our expecta-
tions and solved the HPP unassisted by human interven-
tion.
Synthetic biology often reveals unexpected behaviors in
engineered biological systems. We observed novel pheno-
types produced by our bacterial computer that we had not
predicted. We isolated bacteria with unexpected colors
such as green, orange, pink, yellowish-green, and pale yel-
low (Figure 7). One possible explanation for these results
is that some colonies may not be clonal. We replated col-
onies with unusual colors for colony isolation. Some col-
onies did exhibit more than one clone by producing
colonies of more than one color. For the colonies of novel
color that were truly clonal, promoterless transcription in
the reverse direction could have produced low level gene
expression [8,13]. For example, a construct that produced
red color because of an intact RFP gene expressed by the
T7 RNA polymerase promoter could have produced a low
level of green with expression from intact GFP gene in the
reverse orientation. Such a clone might appear to be
orange in color. Another explanation for novel colors is
mutation of the coding sequences for RFP and GFP,
although we consider this to be less likely. Our system is
behaving in unexpected ways in addition to its designed
purpose of finding a solution to the HPP, which opens up
new areas for investigation of Hin recombinase activity in
vivo.
An iterative approach to synthetic biology is to examine a
natural system, deconstruct it into component parts and
devices, design and build an engineered system that per-
forms new functions or tests hypotheses about the natural
system, and evaluate the behavior of the engineered sys-
tem. It should not be surprising that attempts to engineer
biology produce results that are not easily explained with-
DNA sequence verification of HPP solutionsFigure 6
DNA sequence verification of HPP solutions. Three
yellow fluorescent colonies from each of the three recombi-
nation plates were used for plasmid preparation and DNA
sequencing. The number of ABC and ABC' solution geno-
types found for each of the starting constructs is listed. The
order and orientations of GFP (green) and RFP (red) gene
halves for each of the starting constructs and solutions is
Clones isolated from HPP recombination platesFigure 7
Clones isolated from HPP recombination plates.
Selected colonies from ABC, ACB, and BAC recombination
plates were grown overnight and replated. The results
emphasize the diversity of colors produced by the bacterial
computer in the HPP experiment.Page 7 of 11
(page number not for citation purposes)
out further research. But ignoring unexpected behavior
would be a lost opportunity to advance our understand-
illustrated.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
ing of nature. Rather, the unpredictability of engineered
biological systems should return synthetic biologists to
another iteration of examination, deconstruction, design,
and testing. The unexplained behavior of our system was
a good example of the dual benefits of synthetic biology.
In addition to engineering a bacterial computer to solve
the HPP, our work provided unanticipated opportunities
for further investigation of the mechanism by which Hin
recombinase functions in vivo and the means by which a
complex population of plasmids is maintained in our bac-
terial computer.
Hin-mediated recombination non-equilibrium
A test of whether or not Hin recombinase has achieved
equilibrium in our experiments is to compare the pre-
dicted and observed frequencies of colony phenotypes.
For the three node directed graph, there are 3!·2
3
= 48
possible configurations of the three DNA edges. At equi-
librium, each of these is expected to occur at a frequency
of 1/48. As a result of observing green fluorescent colo-
nies, we will assume for the purpose of this analysis that
the double terminator did not function in reverse orienta-
tion in our experiments. With this assumption, only the
configuration C'AB results in green fluorescence, so colo-
nies with this phenotype are expected at a rate of 1/48, or
about 2%. However, green colonies appear at less than
this rate in all the experiments. Yellow fluorescence can be
produced only by the two configurations ABC and ABC',
yielding a rate of 2/48, or about 4%. However, yellow flu-
orescent colonies predominate in the ABC experiment
and are less common than 4% in ACB and BAC experi-
ments. Red fluorescence requires either a configuration
with C in the first position or one with A in the first posi-
tion but not forward B in the second position. There are
14 configurations that satisfy these criteria so the expected
frequency of red fluorescent colonies is 14/48, or about
29%. However, red fluorescence is the dominant color on
ACB plates and is rare on the ABC and BAC plates. Config-
urations with A', B, or B' in the first position or with C' fol-
lowed by any combination except AB will yield no
fluorescence. There are 31 configurations that meet these
criteria, so the expected frequency is 31/48, or about 65%.
However, uncolored colonies dominate the BAC plate
and do not approach this expected rate on ABC and ACB
plates. Overall, these results show that each experiment
retained a greater frequency of original colony color than
was predicted at equilibrium. This supports the conclu-
sion that Hin recombinase had not reached equilibrium.
These results are in agreement with the conclusion of our
previous study that Hin recombinase flipping had not
reached equilibrium after 11 hours, perhaps because we
chose to omit the Recombination Enhancer element [8].
Lim et al. reported that Hin recombinase requires negative
supercoiling in its substrate plasmid, and that recombina-
tion removes two negative supercoils during a reaction
[14]. The supercoiling density has been reported to be 8–
12 supercoils per plasmid [15], and if new supercoils were
not introduced until DNA replication, then perhaps Hin
recombinase can perform only 4–6 reactions with each
plasmid per generation. Although our mathematical
modeling revealed that equilibrium was achieved in 20
reactions, perhaps replication of plasmids early in the
experiment increased the frequency of starting configura-
tions to levels that could not be achieved by configura-
tions that require more recombination reactions. In other
words, the starting configurations might produce a type of
founder effect that was still visible on the final recombina-
tion plates and not accounted for in our mathematical
model.
Scaling Hamiltonian Path Problems
We considered the question of what would be required for
our bacterial computer to find the Hamiltonian path in
directed graphs of increasing size. In addition to listing
the GFP and RFP genes used to solve the three node
directed graph, Table 1 lists specific proposals for split
genes that could be used for directed graphs containing 4–
7 nodes. Each of the genes chosen produces a phenotype
that could be observed in the presence of the other pheno-
types. In addition to the GFP and RFP genes used in the
current study, Ε-galactosidase is proposed for its ability to
produce blue colonies and three antibiotic resistance
genes not used in the experimental protocol are proposed.
The graph in Figure 1 could be addressed if we were able
to insert a hixC site into the four additional genes without
disrupting the functions of the encoded proteins and in
such a way that a hybrid of halves of any two genes did not
replicate any of the six phenotypes. If we could split the
four additional genes, then we could program our bacte-
rial computer to solve the same HPP in vivo that Adelman
Table 1: Proposed split genes for solving increasingly larger
Hamiltonian Path Problems
Directed Graph Split Genes
3 nodes GFP, RFP
4 nodes GFP, RFP, Ε-Gal
5 nodes GFP, RFP, Ε-Gal, Chl
6 nodes GFP, RFP, Ε-Gal, Chl, Kan
7 nodes GFP, RFP, Ε-Gal, Chl, Kan, Eryth
N nodes N-1 Split Genes
Split genes that would be needed to program a bacterial computer to
find the Hamiltonian path in a directed graph with the number of
nodes indicated are listed. The 3 node proposed was successfully
implemented in the current study. GFP = green fluorescent protein
gene, RFP = red fluorescent protein gene, Ε-Gal = Ε-galactosidase Page 8 of 11
(page number not for citation purposes)
We have considered possible explanations for the
observed Hin-mediated recombination non-equilibrium.
gene, Chl = chloramphenicol resistance gene, Kan = kanamycin
resistance gene, Eryth = erythromycin resistance gene.

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
solved in vitro with a DNA computer. As indicated in Table
1, our approach could be used to find the Hamiltonian
path in a directed graph containing N nodes by using N-1
split genes. Notably, the effort required to split genes
increases linearly although the complexity of the problem
increases combinatorially.
As described earlier, one in every 138,378,240 of the pos-
sible configurations of the edges of the graph in Figure 1
is a Hamiltonian path. Since this is roughly the number of
plasmids in a typical experiment, finding a solution
would require a more efficient screening mechanism. We
could increase the probability of finding a true HPP solu-
tion if we enhanced Hin recombinase function by adding
the Recombination Enhancer [16] or if antibiotic selec-
tion were used at time points prior to the end of the exper-
iment. If even larger graphs were to be addressed,
selection for partial solutions would be necessary and the
problem might have to be divided into stages. For exam-
ple, bacteria that had successfully solved the first half of
the graph could be assigned a higher fitness than those
that had failed to reach this milestone. In this way,
directed evolution could be used to guide the population
of bacterial processors toward a final solution.
Conclusion
The manner in which the complexity of NP-complete
problems such as the HPP grows is combinatorial with
respect to linear increases in their size. This makes finding
solutions to such problems a formidable challenge to
computation. The success of our experiments to program
a bacterial computer to solve a three node HPP represents
an important step in the development of bacterial com-
puters that can address this challenge. We have estab-
lished that bacterial computers can function as a culture of
exponentially growing cells that can evaluate an exponen-
tially increasing number of solutions to an NP complete
mathematical problem and determine which of them is
correct.
The successful design and construction of a system that
enables bacterial computing also validates the experimen-
tal approach inherent in synthetic biology. We used new
and existing modular parts from the Registry of Standard
Biological Parts [17] and connected them using a standard
assembly method [18]. We used the principle of abstrac-
tion to manage the complexity of our designs and to sim-
plify our thinking about the parts, devices, and systems of
our project. The HPP bacterial computer builds upon our
previous work and upon the work of others in synthetic
biology [19-21]. Perhaps the most impressive aspect of
this work was that undergraduates conducted every aspect
of the design, modeling, construction, testing, and data
Methods
Construction of HPP parts and devices
Materials used in molecular cloning procedures were as
follows. Plasmid preparations were conducted using
either the Zippy Plasmid Miniprep Kit from Zymo
Research or the QIAprep Spin Miniprep Kit from Qiagen.
Gel fragment purifications were performed with either the
Zymo Research Zymoclean DNA Recovery Kit or the Qia-
gen QiaExII polyacrylamide gel purification kit. Compe-
tent E. coli JM109 or T7 Express I
q
competent cells were
purchased from New England Biolabs. Transformants
were plated on LB media or grown in LB broth containing
100 ug/ml amplicillin, or 50 ug/ml tetracycline, or both.
Polyacrylamide gel electrophoresis was conducted using
7% or 12% acrylamide in TBE buffer and agarose gel per-
centages ranged from 1% to 3% agarose in TAE buffer.
We designed and built all the basic parts used in our
experiments as BioBrick compatible parts and submitted
them to the Registry of Standard Biological Parts [17]. Key
basic parts and their Registry numbers are: 5' RFP
(BBa_I715022), 3' RFP (BBa_ I715023), 5' GFP
(BBa_I715019), and 3' GFP (BBa_I715020). All basic
parts were DNA sequence verified. The basic parts hixC
(BBa_J44000), Hin LVA (BBa_J31001) were used from
our previous experiments [8]. The parts were assembled
by the BioBrick standard assembly method [18] yielding
intermediates and devices that were also submitted to the
Registry. Important intermediate and devices constructed
are: Edge A (BBa_S03755), Edge B (BBa_S03783), Edge C
(BBa_S03784), ABC HPP construct (BBa_I715042), ACB
HPP construct (BBa_I715043), and BAC HPP construct
(BBa_I715044). We previously built the Hin-LVA expres-
sion cassette (BBa_S03536) [8].
After construction of the A, B, and C DNA edges, DNA
sequencing was performed to verify that they were correct.
These intermediates were combined to produce the three
HPP constructs ABC, ACB and BAC, which were also
sequence verified. The HPP constructs were then cloned
downstream of the bacteriophage T7 RNA polymerase
promoter, an RBS element, and the 5' half of RFP. The Hin
recombination expression cassette was used as previously
constructed [8]. It included the lactose promoter, RBS, the
coding sequence for Hin recombinase with a LVA degra-
dation tag, and a double transcription terminator. The
cassette was cloned into plasmid pSB3T5, which contains
a tetracycline resistance gene and an origin of replication
that allowed it to be maintained alongside the replication
origin of the pSB1A3 plasmids used for the HPP con-
structs.
Splitting genesPage 9 of 11
(page number not for citation purposes)
analysis. In order to split genes by insertion of hixC sites, we devel-
oped an online tool for primer design [22]. The software

Journal of Biological Engineering 2009, 3:11 http://www.jbioleng.org/content/3/1/11
requires input of the coding sequence for the gene to be
split and the point in the sequence where it is to occur.
Since the hixC site is 26 bp and the BioBrick scar is 6 bp
on each side of it, the insert needed to be 38 bp. This is not
a multiple of 3 and therefore disrupts the reading frame
after the insertion. Since choosing the 39th base will result
in either glutamate or aspartate and can slightly modify
the melting point of the primers, the software allows this
choice to be made. The output is a PCR primer pair for the
5' and 3' gene halves. We used this tool to generate prim-
ers for the GFP and RFP genes that we wished to split. The
resulting primers were used in PCR with cloned GFP and
RFP genes as templates. The resulting DNA was cloned
into the plasmid vector pSB1A3 and used for transforma-
tion. Putative clones were sequenced in order to choose
clones with no mutations.
Hin-mediated recombination of HPP constructs
ABC, ACB, and BAC starting constructs were used to trans-
form T7 Express I
q
competent cells. These cells express the
bacteriophage T7 RNA polymerase needed for expression
of the HPP node genes. The transformants were plated on
LB with ampicillin. After overnight incubation at 37°C,
the plates were allowed to incubate at room temperature
for an additional two days in order for fluorescence to
develop. Pictures of these control plates were then taken
for use in Figure 5.
Exposure of ABC, ACB, and BAC starting configurations to
Hin recombinase was accomplished by cotransformation
of JM109 cells with pSB1A3 plasmids containing the three
constructs and a pSB3T5 plasmid containing the Hin
expression cassette. The cotransformants were plated onto
LB agar with ampicillin and tetracycline. Colonies were
then pooled and grown in LB media overnight. Plasmid
DNA was purified from each of the three recombination
cultures and used to transform T7Express I
q
competent
cells. The transformants were plated on LB agar with amp-
icillin only so that the Hin expression plasmid would be
lost and no further recombination would occur. The
resulting plates were photodocumented for use in
Figure 5.
Verification of HPP solutions by DNA sequencing
Selected colonies from the ABC, ACB, and BAC recombi-
nation plates were used for plasmid preparations. The
plasmids were subjected to DNA sequencing using three
primers. Primer RFP1 has the sequence 5' CGGAAGGTT-
TCAAATGGGAACGTG 3' and binds to the 5' RFP gene
fragment that precedes each of the HPP constructs. Primer
GFP2 has the sequence 5' TACCTGTCCACACAATCT-
GCCCTT 3' and binds to the 3' GFP coding sequence,
which can occur in any of the three positions or in either
of the HPP constructs. All sequencing reactions were per-
formed by the Clemson University Genomics Institute.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JB, KA, OA, STC, WD, LH, ATM, NM, JT, MU, AV, MW,
AMC, and TTE designed, constructed, confirmed and sub-
mitted project parts to the Registry of Standard Biological
Parts, built and tested constructs to solve the HPP path-
way using a bacterial computer, and verified HPP solu-
tions. JOD, MR, AS, LJH, and JLP conducted mathematical
modeling of the HPP. JB, AMC, LJH, JLP, and TTE wrote
the manuscript. All authors read and approved the final
manuscript.
Acknowledgements
We wish to thank the iGEM founders, organizers, and community for pro-
viding a supportive environment for conducting synthetic biology research
with undergraduates. Thanks to Dr. K. Haynes for helpful manuscript com-
ments on the submitted manuscript and to two anonymous reviewers for
substantive remarks that improved the manuscript. Support is gratefully
acknowledged from NSF UBM grant DMS 0733955 to Missouri Western
State University and DMS 0733952 to Davidson College, HHMI grants
52005120 and 52006292 to Davidson College, and the James G. Martin
Genomics Program. JB, TC, LH, MR, JT, MU, and AV were supported by
the Missouri Western State University Summer Research Institute and Stu-
dent Excellence Fund. WD, AS, and OA were supported by the Davidson
Research Initiative. JOD and ATM were supported by HHMI. AMC, LJH,
and TTE are members of GCAT, the Genome Consortium for Active
Teaching [23].
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