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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

References

1.

Miguel Arenas. Computer programs and methodologies for the simula-

tion of dna sequence data with recombination. Frontiers in genetics, 4:9,

2013.

2. Robert Scharein. Knotplot, 2018.

3.

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