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Analyzing the Topological Transformation Probability of DNA using Models of Cre Recombinase

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This study was undertaken to better understand the enzymatic activity of DNA recombinases, specifically CRE-lox recombinase reactions. To model this enzyme activity, a simulation through the topological modeling program KnotPlotTM and the BFACF algorithm was used to study recombination on all knots with up to seven crossings. The data collected from the computational simulations were analyzed to produce a transition probability matrix in order to predict the topological transformations of DNA knots treated with Cre Recombinase. The probability matrix was used to determine the steady state for the recombination reaction and the efficiency of each knot transforming into unknot. The accuracy of each simulation trial was analyzed and compared to experimental data performed on the transformation of PSC1.3i by Cre recombinase. This research has potential pharmaceutical applications that can be furthered to improve the efficiency of enzyme activity in transforming circular DNA chains into the unknotted form.
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Analyzing Topological Transformation
Probabilities of DNA Using Computational
Models of CRE Recombinase
STELLA LI1, JANANI SEKAR2, JEFFREY YANG3
1Robert Louis Stevenson School, 2James Logan High School, 3The Harker
School
COSMOS @ UC Davis
August 3, 2018
1 Abstract
This study was undertaken to better understand the enzymatic activity of
DNA recombinases, specifically CRE-lox recombinase reactions. To model
this enzyme activity, a simulation through the topological modeling program
KnotPlot
TM
and the BFACF algorithm were used to study recombination
on all knots with up to seven crossings[
4
]. The data collected from the
computational simulations was analyzed to produce a transition proba-
bility matrix in order to predict the topological transformations of DNA
knots treated with Cre Recombinase. The probability matrix was used to
determine the steady state for the recombination reaction and the efficiency
of each knot transforming into unknot. The accuracy of each simulation
trial was analyzed and compared to experimental data performed on the
transformation of PSC1.3i by Cre recombinase. This research has potential
pharmaceutical applications that can be furthered to improve the efficiency
of enzyme activity in transforming circular DNA chains into the unknotted
form.
2 Introduction
This topic was selected as our area of research because of our interest in
studying the applications of knot theory and topology in biological systems.
Furthermore, the idea that the probability of the different knots that appear
after recombination has yet to be modeled using computational methods
1
motivated us to try and understand this enzyme better. As an intermediate
step, recombination is a step in which enzymes may repair damaged DNA
or shift the topology of the DNA itself. For this project, we accept the general
assumption within the biological community that an unknot is the preferred
state for an organism’s DNA despite a few possible advantages to knotting
[
3
]. We also included a wet lab portion of our experiment where we looked
at the transitions on knot types after Cre reactions with plasmid. Essentially,
we reacted the plasmid with Cre twice: once to knot it and once to create a
different knot. We hypothesize that Cre activity is not inherently random
as the enzyme will attempt to unknot the substrate it binds to. Overall, our
goals for this proejct are to better understand the behavior of Cre enzyme
recombinase and compare our model’s accuracy to real laboratory data.
3 Methods
All simulations were done using KnotPlot
TM
, a program that displays pro-
jections and manipulates different mathematical knots. KnotPlot
TM
allows
the user to run different algorithms, analyze characteristics, and collect data
on various knot types. A KnotPlot
TM
script randomly transformed a knot
into another knot based on a chosen site for recombination. A Java program
was then used to generate a probability matrix, a table where both axis have
all the knots, and the corresponding box is the probability out of one that the
specific knot transition occurs, and MatLab
TM
was used to produce a visual
representation of the data. In the wet lab experiments, Cre recombinase that
we used was from New England Bio along with the plasmid DNA.
3.1 KnotPlotTM Simulation
The first step to begin the simulation was to modify a script in KnotPlot
TM
that recombined all knots between 0_1 and 7_7 which was written by
Michelle Flanner, a member of the Arsuaga - Vázquez lab [
3
]. This script
was run for two trials for two versions: the first KnotPlot
TM
script was
written for 15 knots between 0_1 and 7_7, and the second script was writ-
ten for 27 knots between 0_1 and 7_7*. The second version included the
mirrored image knots, making a distinction between the two data tables.
For the KnotPlot
TM
script, each knot was first loaded in as a smooth knot
as seen by Figure 1. The knot was then converted to a simple cubic lat-
tice model of a knot, as in Figure 2, so that it can expand and contract for
more possible recombination sites to be chosen from. BFACF is a valid
algorithm to model recombination because it is proven to explore the entire
chain uniformly instead of modeling recombination on a local area of DNA.
2
Furthermore, BFACF employs the Markov chain algorithm that deforms
cubic lattice links for stochastic Monte- Carolo sampling.This depiction with
the beads is necessary so that the BFACF algorithm can expand or contract
the knot based on the move that is randomly chosen, but it preserves knot
type. After BFACF is run for 100,000 times with a z-value that changes the
distribution of randomly simulated moves, a random location is chosen
where the recombination process will be simulated [
3
]. These programs,
created by memers of the Arusaga-Vázquez lab, run in the platofrm pro-
vided by KnotPlot
TM
. For the purpose of this simulation, the program only
models sites in inverted repeat, meaning the sites are in opposite directions,
ensuring that the result of recombining a knot will be a knot, instead of
a link. Then, the knot is contracted, so it can be more easily identified by
the built-in ID command in KnotPlot
TM
. Java program was used to count
the number of times each knot transformation happens, creating a spread-
sheet of the normalized data which does not take into account the unknown
knots and the knots that are over 7 crossings. In order to noramalize the
data, we have to keep uniform paramters for each knot, so any knots that
are created over seven crossings will not be considered along with knots
unknown by the ID method. We summed the total number of these two
types of knots and substracted it from the total number of trials for each
knot. Then, we divided the total number of a knot transition over that value,
scaling the probability to the right proportion. The Java Code is in this
github repository: https://github.com/Jyang2602/COSMOSFinalProject.
Figure 1: Example of 51from KnotPlotTM [2]
3
Figure 2: Knot 51in a Simple Cubic Lattice [2]
3.2 Finding Steady State and Verifying Simulation Ac-
curacy
To analyze the data produced from the simulation, a transitionprobability
matrix was created. This was done by organizing the data for every trial
in a spreadsheet using the Java program, then using MatLab
TM
to calculate
the eigenvectors and eigenvalues. Because the KnotPlot
TM
program recom-
bined every input knot, the total of all the decimal probabilities for every
knot transition for each input knot would ideally sum to one. To test that
the KnotPlot
TM
simulation worked correctly which was indeed the case, a
MatLab
TM
program was used to calculate the eigenvalues and correspond-
ing eigenvectors for the probability matrix. The first eigenvalue was one,
showing that our data was normalized without any outliers, and the total
transformation probability for every knot column was one. A steady state
for the corresponding eigenvector displayed the values that recombined
knots would converge to after multiple repetitions.
3.3 Data Validation Using Chi-Square Test
Four chi-square tests were performed on our data: The first test was done to
test the accuracy and reproducibility of the simulation without considering
chirality of the trial which consists of 15 data points, one for each knot from
0
1
to 7
7
. The second test was done to test the accuracy and reproducibility of
4
the simulation considering chirality with 27 data points, one for each knot.
The third test was done to evaluate the quality of the approximation of the
simulation without considering chirality: the idea that a knot is different
from its mirror image. 15 data points were taken from the average steady
state vectors of the first simulation; 15 other data points were taken by
combining the possibilities of a knot and its chiral counterpart in the second
simulation. A fourth test was conducted as a positive control to validate
the the test. 27 data points were taken from the average steady state vectors
of the first simulation; 27 other data points were obtained by a random
uniform distribution probability.
3.4 Improving Simulation by Setting More Parameters
To better simulate Cre activity, the KnotPlot
TM
script for recombination had
restrictions for choosing the location for the process: the site where the
knot is cut to be recombined is chosen between one-third the length of the
entire knot because that is the accurate tendency for Cre. The code was
changed in the parameter for the distance along the arc of the length where
the recombination site can be chosen from. Knot length was also set to a
certain length using the KnotPlot
TM
command, bfacf run to length, where x
is the length for the knot. Boundaries for recombination were set between
a length of 1 and 50 since a BFACF algorithm set every knot’s length to
150. The same data excel sheet was created, except with 2,000 trials for each
knot. The two data tables were compared to each other to find discrepancies.
Refer to figure[14] the appendix.
3.5 Laboratory Experiments on In Vitro Cre Reactions
In order to obtain experimental data for comparison to the KnotPlot
TM
sim-
ulations, in vitro Cre reactions were conducted in a laboratory environment.
3.6 Testing the Effect of GelRed on Cre Reaction
The first in vitro experiment was done to test the effect of GelRed on Cre-loxP
reaction. This test was necessary because the DNA would be extracted
from the stained gel and treated with Cre again. Before incubating the Cre
reaction, GelRed was added so that the concentration of GelRed in the tube
reached 1X, the same as the concentration in the agarose gel. A control was
established by a standard Cre reaction. The control and the Cre+Nicking
reaction with GelRed were loaded side-by-side into a gel and electrophoresis
was run to analyze the effect of GelRed on Cre recombination. Looking
at figure 3, the Cre reaction with GelRed in the tube is in lane 19, the Cre
5
reaction without GelRed in the tube is in lanes 7 and 20. In lanes 7 and 20,
there is band defined a little bit under “1” marker on the right, suggesting
the presence of visible knotted DNA there; however, in lane 19, there is no
band, suggesting that the knot is more supercoiled due to the GelRed. In the
interest of our experiment, we will deal with the assumption that GelRed
does not change the topology enough to effect the second Cre reaction.
Figure 3: Lab Graph for Cre Proof
3.7 Cre + Nicking reaction on plasmid DNA
3.7.1 Preparing Reaction tubes
In order to to nick the products of Cre recombination, we made a total
reaction volume of 50 microliters stored in an eppendorf consisting of 40.9
microliters of distilled water, 1 microgram of PCS1.3I plasmid DNA of den-
sity 325 micrograms per milliliter, 5 microliters of 10x Cre reaction buffer,
and 1 microliter of 15x CRE. Afterwards, we incubated the reaction at 37oC
for 60 minutes. Then, we removed 6 microliters from the solution and re-
placed it with 5 microliters of CutSMart nicking buffer and 1 microliter of
Nb.BbvCl enzyme: This step was taken to rid the extra torsion or super-
coiling in the DNA. We terminated the reaction by adding 10 microliters of
Loading Purple Dye.
6
3.7.2 Making a Separate Gel
To make sure that some DNA without any GelRed can be isolated, a gel that
has GelRed in one half of the gel and no GelRed in the other half was made
by placing a ruler in the middle of the gel and pouring gel solutions of the
same concentration but one without GelRed. The gel with the ruler in the
middle is displayed by figure 4, and the gel is split up so we can control the
effect of GelRed on Cre in the experiment. The product of the gel is in figure
5.
Figure 4: Setup for Gel
Figure 5: Finished Gel
3.8 Isolate DNA of a Certain Knot Type and Rerun Cre
Reaction
We loaded our Cre into this gel separately with the same items on both sides.
For the left side of the gel, the one with GelRed, lane 1 was the ladder, lane 2
was plasmid, lane 3 was nicked plasmid, lane 4 was linear plasmid, and lane
5 to lane 9 were nicked cre. The same lane loading was done for the other
side of the gel without GelRed. We ran the gel for 20 hours at 40 Volts. Our
second gel was run with slightly different procedures. Because not enough
DNA showed up, we allowed the Cre reaction to run for 2 hours instead
7
so the band will be brighter. Then, we identified each knot type based on
the distance that the DNA fragments migrate on the half of the gel with
GelRed loaded. To quantify the probability More complicated knot types
would migrate further from the anode. We cut along straight of both sides
of the gel for all bands that we were interested in, and we suspended the
DNA in a buffer solution to let it sink over two day. Once that happened, we
obtained the DNA of each knot type. Alternatively, a dialysis machine can
be used to remove GelRed from DNA solutions. Afterwards, we treated the
DNA with ligase to rebuild the phosphodiester bond broken by Nb.BbvCl.
In order to look at the probability of knot transitions, we performed another
Cre + Nicking reaction on the obtained plasmid DNA, following the same
procedure and volumes to prepare the reaction. Run a gel overnight again
and look at changes in probability based on light intensity.
3.8.1 Gel Electrophoresis Analysis and Comparison to Simula-
tion
We will quantitatively analyze the percentage of each resulting knot type by
examining the brightness of each band after the second Cre reaction. Then,
we will compare the lab data with the one found through the computational
simulation.
4 Data
To visualize the probability matrices generated by the Java program, proba-
bility maps were made for each type of trial using MatLab
TM
. The following
graphs show in order: the average recombination probability of knots in
the first two trials, the trials did not separate mirrored knots, the average
recombination probability found in the next 2 trials with the mirrored image,
and the recombination probability of knots in the final trial which further
limited the arc length to model Cre. Each section will have a graph of a heat
map of the probabilities with the raw data in the appendix.
4.1 First KnotPlotTM Simulation Combining Chirality
The first trial of the KnotPlot
TM
simulation did not distinguish between a
knot and its mirror image. The following data table was created to show
how many times each knot transformed into every other knot. The sum
of every column in the table is 500 because each knot was run 500 times.
The same process was repeated for one more trial. Refer to image[15] in
8
appendix for the raw data. This heat map shows the general trends for the
normalized probabilities.
Figure 6: Probability Map for Knot Types without its Mirrored Image
4.2 Second KnotPlotTM Simulation Considering Chi-
rality
The next two trials in KnotPlot
TM
did distinguish between a knot and its
mirror image: in the mirror image of a knot, the orientation of the crossings
is the opposite. Knots 0.1, 4.1, and 6.3 are achiral, or can be redrawn to look
identical to their mirror image. Thus, these knots were considered the same
as their mirror image in our simulation. We wanted to have both graphs
and data to test a theory of accuracy if we consider chirality or not. The
following heat map was created from the average of the trials with separate
mirrored knots. The raw data for the trials with mirrored images can be
found in figure [16].
9
Figure 7: Data for Trial with Mirrored Images
4.3 KnotPlotTM Simulation with More Restrictions
The final trial in this study involved a simulation that limited the arc length
between recombination sites to 1
/
3 the length of the knot to more accurately
model the distance between recombo sites for Cre. The data from this trial
and all the previous trials were then normalized using the Java program
turned into a square probability matrix. The following heat map shows the
data from the normalized probability matrix from the final trial. The raw
data for these trials can be found in figure [17].
10
Figure 8: Probability Map for Trial with Cre Boundaries
5 Results and Conclusion
5.1 Data Validation Using Chi-Square Test
Four chi-square tests were performed on our data: The first test was done to
test the accuracy and repeatability of the simulation without considering
chirality with 15 data points for each trial. The second test was done to test
the accuracy and repeatability of the simulation considering chirality with
27 data points, one for each knot. The third test was done to evaluate the
quality of the approximation of the simulation without considering chirality:
the idea that a knot is different from its mirror image. 15 data points were
taken from the average steady state vectors of the first simulation; 15 other
data points were taken by combining the possibilities of a knot and its chiral
counterpart in the second simulation. A fourth test was conducted as a
positive control to validate the the test. 27 data points were taken from the
average steady state vectors of the first simulation; 27 other data points were
obtained by a random uniform distribution probability.
To determine whether the null hypothesis was accepted or rejected, the
χ2
value for each test was compared to the chi square distribution table with
corresponding degrees of freedom. With a degree of freedom of 14, the
11
χ2
value of 3.3582 from the trials from the first simulation gave a p-value
less than 0.05. Therefore the null hypothesis was accepted, suggesting that
V1=V2
for this simulation, where
V1
is the steady state vector of trial 1 and
V2is the steady stae for trial 2 of the first set of data.
With a degree of freedom of 26, the
χ2
value of 1.9234 from the trials from
the second simulation which consider chirality to be different gave a p-value
less than 0.05. Therefore the null hypothesis was accepted, suggesting that
V1=V2
for this simulation, where
V1
is the steady state vector of trial 1 of
simulation 2 and V2is the steady state for trial 2 of the second set of data.
With a degree of freedom of 15, the
χ2
value of 10.4782 from third test
gave a p-value less than 0.05. Although this
χ2
is considerably higher than
the
χ2
obtained from testing two trials of the same simulation, the p-value is
still within 0.05. Therefore the null hypothesis was accepted, suggesting that
V1=V2
for this simulation. In this test,
V1
is the average steady state vector
for the simulation that does not consider chirality, and
V2
is the average
steady state vector for the simulation that does consider chirality. This
test showed that the simulation which did not distinguish chirality is a
good approximation of the simulation considering chirality, which further
confirmed our hypothesis that chirality does not influence recombination
probability.
With a degree of freedom of 26, the
χ2
value of 826.0035 from the positive
control gave a p-value greater than 0.05. Therefore the null hypothesis was
rejected, suggesting that
V16=V2
for this simulation, and the steady state
vector does not conform to a uniform distribution. The results of the positive
control test suggested that the chi square test is effective in determining
whether two sets of data are precise to a certain degree.
5.2 General Trends in the Data
5.2.1 Common Transitions
Based on our data, efficiency calculations, and analysis of results, we are
able to model the general process of recombination. The probability maps
that were generated in MatLab
TM
show a high probability that a knot will
recombine to itself. Furthermore, our data suggests that knots are likely to
recombine into a unknot, which supports the belief that DNA prefers to be
in an unknotted state. Our data also shows that the knots most likely to
recombine into a unknot, besides the unknot itself, are knots 3.1, 5.1, and 7.1,
which can be found in figure [9]. Once again, the orientation of the knots is
important as inverted repeat sites as the knots will purely be transformed
into knots and not links.
12
Figure 9: Torus Knots 31, 51, 71[2]
These knots are all classified as torus knots because of their shape, which
may contribute to why they were able to recombine into an unknot in fewer
steps than other knot types. After performing a type 1 tangle (inverted
orientation) at any position of the torus knot, the knot transforms into a
twisted unknot. Then a Reidemeister I move can be performed on the
twisted knot to reach the unknotted state.
5.2.2 Consdering Pathway Efficiency
Using the probability matrix from the simulation differentiating chirality, a
probability table was computed to represent the probability that one knot
transforms into the unknot in a certain number of steps. The probability
of a knot transforming to an unknot in one step is given by the probability
matrix. To calculate the probability that a certain knot transforms into an
unknot in two steps, the row vector of each knot transforming to an unknot
in one step was multiplied by the column vector of the knot transforming
into each knot. This operation summed up the possibility of a knot trans-
forming into another knot then transforming into an unknot. For n+1 steps
of recombinations, the row vector of each knot transforming to an unknot in
n step was multiplied by the column vector of the knot transforming into
each knot. This is shown in figure 10, where all the knot’s probabilities of
being an unknot after 20 steps is shown.
For the second pathway probability graph, referring to figure 11, the
assumption that an unknot is the optimal state for DNA inside the cell was
accepted. Therefore, in the transformation pathway model, a knot stays
unchanged as an unknot once it reaches the unknotted state. Essentially, we
changed the table so that once a knot reaches an unknot, it will always stay
there. To match the probability matrix with this assumption, the probability
that an unknot converts into an unknot was set to be 1. The calculation to get
the matrix and the graph are the same as in the first simulation. This graph
13
represents the probability that a knot transforms into an unknot within
certain steps. Assuming our model stays consistent, at a larger number of
steps, all knots should converge to one value.
Figure 10: Pathway Probability at 20 steps
Figure 11: Pathway Probability within 20 steps
5.2.3 Looking at the Data Trial with Restrictions
The final trial in our experiment that limited the arc length between recom-
bination sites to one-third the knot length, which is proportional to the
distance between LoxP sites, altered our results. The first major difference
in this trial was that the probability of becoming an unknot increased for
most knot types, especially the ones with more intersections. More specifi-
14
cally, only six out of twenty-seven knots had a lower probability of turning
into an unknot in this trial: 3_1, 4_1, 5_1, 5_1*, 7_1 and 7_1*. These results
suggest that if a knot is more complicated, when the recombination sites are
restricted to a certain distance, Cre is more likely turn the knot into an un-
knot. The transformation probability of the simpler knots, however, stayed
largely the same. This may be because of the simpler structure of these knots
which allow them recombine into an unknot at a high probability already.
In addition, knots 3_1, 4_1, and 7_1 are all a similar shape, the twist knot
family, so this may contribute to why their probability of turning into an
unknot decreases during this simulation. To summarize, referencing the
probability map of the data, see figure 8, with the boundaries for Cre, the
shades on the map are all slightly darker for the knot transition to itself,
shifting the data towards less variance in knot types.
5.3 Wet Lab Results
Looking at the two gel images, there were clear bands at different lengths
considering the ladder. From figure 12 and 13, each well from four and on are
Cre reactions with DNA. Furthermore, each band represents different knot
complexities, though there are many limitations with this experiment. First,
we are unable to accurately differentiate which knot each band corresponds
to, and to make this model better would require more advanced procedures.
One idea would be to manually look at each knot in the microscope and
then label the eppendorfs based on manual classification. However, two
important conclusions from this experiment is that Cre reacts with DNA
topology affected by GelRed and that some complex knots may occur in
high probabilities. The bright bands that indicate relatively complex knots,
however, are all above the band for the plasmid. This means that the original
plasmid DNA is more dense than the resulting knots, indicating that the
original plasmid is of a more complex knot type. This agrees with our
hypothesis that Cre tends to convert DNA strands into less complex forms.
From the two images, each one has a few bands that do not follow the
general trend of decreasing probabilities. These bands should be further
studied to see which knot type it is, and if this trend continues, perhaps
there is a biological advantage to these knots.
15
Figure 12: Cre Lab Trial 1
Figure 13: Cre Lab Trial 2
5.4 Sources of Error for Computational Model
The first source of error arises in the number of trials: our first data tables
were run with 500 iterations for each knots, and our second data table was
run with 2000. Despite the number of iterations, more trials are needed
to find a more accurate average of the number of trials. For example, the
difference in probabilities between trials is not significant for all knots up to
6 crossings. However, knots above 7 crossings have a significantly greater
difference in terms of probability, suggesting a need for more trials. An-
other error occurs in choosing the z-value of the BFACF algorithm[
3
]. The
z-value represents the probability that the polygon chain knot grows larger
or smaller with each step. Each individual knot has its own z-value which
should allow the knot to stabilize at a length of 200. Due to the time con-
straint, we were unable to determine the z-value manually for each knot,
and thus, we made the assumption that its effect is minimal as we included
16
a command to always run the knot to a certain length before applying re-
combination. This assumption, however, may have skewed the data in a
certain direction. Round-off error is an issue as well because the eigenvector
for the column corresponding to eigenvalue one does not add up to 1. This
inconsistency is most likely attributed to double float point rounding issues
in the design of Java.
5.5 Sources of Error for Lab
The most pertinent error in our lab was the fact that DNA bands still ap-
peared on the gel without GelRed. Without the stain, the DNA should not
have appeared under the UV light, but this fact may also be attributed to
the purple loading dye added to stop the reaction. Our initial plan was
to use the stained gel as a marker for where to cut the unstained gel, but
since both gels had DNA bands under the UV light, we suspended both the
DNA in solution. This, however, means that our DNA will be affected by
GelRed for the second Cre reaction. Even though the Cre reaction worked
with the DNA affected by GelRed, figure 3 shows that the GelRed band is
still slightly lower. A second reaction may magnify this difference.
6 Future Research
Further research for this project entails first and foremost generating more
experimental data. By scaling up the lab experiments in the gel, more DNA
can be extracted so that more data points can be generated. Our experiment
only includes one gel, so there is not enough data to be compared to our
probability map. With more advanced programs, we could run more trials
where the script stops once the knot is transformed an unknot. This method
would be more accurate in determining the efficiency of enzymes on turning
knots into unknots. Changing the recombo range could also be interesting
to see how the data is affected by how size of the range of the recombo
site. If the arc length where the recombination site can be chosen from
increases, interesting further research could be to mathematically model the
steady state value of the knot becoming an unknot. Our hypothesis for this
sub-topic would be that as the arc length increases, the chance of an uknot
in the steady state decreases. Clearly, our data changed quite a bit when we
limited the range between 1 and 50. We would presume that limiting the
range even more would cause the knot to turn to itself or an unknot even
more often. Looking at the accuracy of our model, we could look into the
Dr. Arena’s paper on varying accuracy of DNA models: Trying out different
applications to model DNA with Cre activity will allow us to narrow down
17
our errors as well[
1
]. Furthermore, we dismissed the fact that CRE only
acts on negative writhe, and a next step in an accurate simulation would be
to include that. Lastly, this research only considers inverted repeat DNA
sequences for enzyme binding sites, ensuring that a recombinase will turn a
not into a knot. A more comprehensive model would also look into direct
repeat DNA sequences for enzymes that would create links between DNA
strands instead of only knots.
7 Acknowledgements
A huge thank you to all of the people who supported us throughout this
project. From our peers to our TA’s and all of our professors. However,
to those outside of the COSMOS circle who helped out, such as Michelle
Flanner, your help is greatly appreciated as well. To especially Ali Heydari,
Keith Fraga, Professor Vázquez, and Professor Arsuaga, your support day
in and day out has made this research paper and program in general such a
wonderful educational opportunity for us.
8 Appendix
Figure 14: KnotPlotTM Script [2]
18
Figure 15: Data for Trial with No Mirrored Images
Figure 16: Data for Trial with Mirrored Images
Figure 17: Data for Trial with Cre Boundaries
19
Figure 18: Script to Calculate Chi Value
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Miguel Arenas. Computer programs and methodologies for the simula-
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2. Robert Scharein. Knotplot, 2018.
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Robert Stolz, Masaaki Yoshida, Reuben Brasher, Michelle Flanner, Kai
Ishihara, David J Sherratt, Koya Shimokawa, and Mariel Vazquez. Path-
ways of dna unlinking: A story of stepwise simplification. Scientific
reports, 7(1):12420, 2017.
4.
EJ Janse Van Rensburg and SG Whittington. The bfacf algorithm and knot-
ted polygons. Journal of Physics A: Mathematical and General, 24(23):5553,
1991.
20
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