DNA Splicing Systems By William DeLorbe and Dr. Elizabeth Goode

What is DNA? Deoxyribonucleic acid, or DNA, is the genetic material of all living things. The double stranded form, dsDNA, is a double helical ladder-like structure.

What is DNA? The “rungs” are made of nucleotides held together in a very specific orientation by hydrogen bonding. The ladder has a sugar phosphate backbone.

What is DNA?

What is a splicing system? Mathematically speaking a splicing system is a model of dsDNA along with restriction enzymes and ligase. These enzymes cut and paste the dsDNA forming new strands of DNA. The system will eventually reach a final state. By modeling these reactions we hope to determine what dsDNA are present in that final state.

The model consists a set of initial strands, denoted by I, a set of cutting rules encoding the action of the restriction enzymes and a set of pasting rules encoding the action of the ligase enzyme. What is a splicing system?

A word w = psq containing the string s can be cut using the cutting rule as follows: Cutting Rules

A word w = psq containing the string s can be cut using the cutting rule as follows: Cutting Rules

The sticky ends of the newly produced cut pieces are represented by. Cutting Rules

Two words = and = containing the can paste using the pasting rule as follows: Pasting Rules

Two words = and = containing the can paste using the pasting rule as follows: Pasting Rules

Iterated Cutting & Pasting

Initial Set of dsDNA The type of splicing system we model here has an initial set of many strands of the same kind of dsDNA. Each initial strand is the cancatenation of n copies of a certain sequence of DNA called an F-piece. Each F-piece serves as a single unit of measure.

Sequence Design In order to control our experimental model, we engineer the F sequence to meet certain criteria. The sequence pattern of the initial molecules is such that the restriction enzyme involved can only cut between two F-pieces. No other cuts will occur.

Initial Set An F piece is a sequence vxu of dsDNA. If, for example n = 5, each initial strand would have the sequence below: vxuvxuvxuvxuvxu

Initial Set An F piece is a sequence vxu of dsDNA. If, for example n = 5, each initial strand would have the sequence below: vxuvxuvxuvxuvxu The F piece is engineered such that the cancatenation loci contain a sequence uv that is recognized by a restriction enzyme.

Initial Set We must consider the ends of the DNA. The 5’ ends of dsDNA have a phosphate group attached. This donates the energy for the covalent bonds formed during pasting. Our DNA has two dephosphorylated ends. No pasting (blunt-end ligation) can occur on the ends of our initial molecules.

Molecule Types The initial molecules of DNA have two dephosphorylated blunt ends. For our model we began with DNA that are 10 F-pieces long. These molecules are called N-type molecules. They are written as N^10.

Molecule Types When saying N^10 it is understood that the N-type molecule we are looking at is 10 F- pieces long. If we wanted to simply refer to any N-type molecule we would write it as N^k. Where k represents any whole number greater than 1. For the other molecules k can also be equal to 1.

N-Type Molecules N-type molecule N^3 molecule

A-Type Molecules When an N-type molecule is cut at one site the result is two molecules, each having one sticky end and one blunt end. These are called A-type molecules.

B-Type Molecules The third molecule to be produced is one in which both ends are sticky overhangs. This means that the molecule can paste on both sides, or even paste to itself. This molecule is the result of an A-type molecule being cut in such a way as to remove its dephosphorylated end. We call these molecules B-type molecules, or B^k type molecule.

B-Type Molecules

Types of Molecules cont. The last type of molecule we have in our system is the result of a B-type molecule pasting to itself. When this happens the end result is a circular strand of DNA. This form of molecule is called a C-type molecule or C^k type molecules.

C-Type Molecule C-type molecules

Molecule Types N-type molecule A-type molecule B-type molecule C-type molecule

Possible cuts and pastes to the Molecules The first type of molecules we look at are the N-type molecules. There is one main reason for this -- no other molecule can be formed until a cut occurs to these molecules. To derive our equations we need to determine what will affect the total number of these molecules.

Cut and paste For our system we let be the probability a paste will occur if two complementary sticky overhangs come in contact. We let be the probability that a cut will occur at a restriction enzyme site in any given strand of DNA.

Change in the number of N^k molecules The N^k type molecules have two ways in which their total number can be changed. One such way is when two shorter A-type molecules paste together to form an N^k-type molecule. (The length of the two A-type molecules together must equal k.) This would yield a gain in N-type molecules. The other way would be a cut, which would be a loss to the total number of molecules.

Change in the number of N^k molecules The equation for the total change in the number of N^k molecules in one time step is below:

Changes in the number of A^k molecules There are six ways for a net change in the number of A-type molecules to occur. Three of which are gains and three of which are losses.

Changes in the number of A^k molecules A net gain occurs when a larger A-type molecule is cut to form an A^k molecule. It also occurs when a large N^m molecule is cut. The final way it occurs is when smaller A- type molecule pastes with a smaller B-type molecule.

Changes in the number of A^k molecules A smaller A-type molecule pastes with a smaller B-type molecule. A larger N^m molecule is cut. A larger A-type molecule is cut to form an A^k molecule.

A^k molecules cont. A net loss occurs when an A^k-type molecule is cut. Another loss occurs when an A^k-type molecule pastes with another A-type molecule. The final loss occurs when an A^k-type molecule pastes with any B-type molecule.

A^k molecules cont. A^k-type pastes with another A-type molecule. A^k-type pastes with any B-type molecule. A^k-type molecule is cut.

Changes in the number A^k molecules The equation for the total change in the number of A^k molecules in one time step is below:

Changes in the number of B^k molecules There are eight different ways the total number of B-type molecules can change, four of which are gains and four of which are losses.

Changes in the number of B^k molecules The first gain occurs when a larger A-type molecule is cut. The next gain is when a larger B^m type molecule is cut. Another gain occurs when two smaller B-type molecules paste together. The last way a net gain occurs is when a C^k molecule is cut.

Changes in the number of B^k molecules Smaller B-type molecules paste together. C^k molecule is cut. A^m-type molecule is cut. B^m type molecule is cut.

B^k molecules cont. The first way a net loss can occur is when a B^k molecule is cut. The next loss is when a B^k molecule pastes with an A-type molecule. A third loss occurs when a B^k molecule pastes with another B-type molecule. The final loss occurs when a B^k molecule pastes to itself.

B^k molecules cont. B^k molecule pastes to itself. B^k pastes with an A-type molecule. B^k pastes with a B-type molecule. B^k molecule is cut.

Changes in the number B^k molecules The equation for the total change in the number of B^k molecules in one time step is below:

Changes in the number of C^k molecules There are only two ways a change to the total number of C^k molecules can occur. The first change, a gain, is when a B^k molecule pastes to itself. The last change, a loss, is when a C^k molecule is cut, forming a B^k molecule.

Changes in the number C^k molecules The equation for the total change in the number of C^k molecules in one time step is below:

Verifying the model is consistent To show that these equations represent a closed system we must verify that the change in the total number of F- pieces is zero throughout the experiment.

Counting F-pieces We take our original ODE’s and multiple through by and assign new variables to certain terms in order to facilitate cancellation.

Counting F-pieces The variables below represent the total number of F-pieces contributed by each of the 4 different types of molecules.

Counting F-pieces M represents the total number of F-pieces contributed by all of the molecules in our closed system.

Ex 1: Counting F-pieces For example, the expression below represents the total number of F-pieces in N-molecules that exist by virtue of a paste operation between A-type molecules.

Ex 2: Counting F-pieces This expression represents the total number of F-pieces in A-molecules that exist by virtue of a paste operation between a shorter A-molecule and B-molecule.

Ex 3: Counting F-pieces For example, the expression below represents the total number of F-pieces in N-molecules that exist by virtue of a paste operation between A-type molecules. Recall:

Counting F-pieces All terms in all of the differential equations were evaluated and simplified in a fashion similar to that seen above. Careful analysis revealed that the total change M’ in the number of F pieces is zero throughout the reaction, as required.

Methods for solving We used the Forward Euler technique in Excel as our first approach to solve the system. This technique, although it does work, is not very accurate. We then used the dSolve command in Mathematica. This method worked well for the problems and yielded graphical results consistent with our theoretical predictions.

Starting with N^10 molecules The results obtained from Mathematica were predictable. At a very early stage the program shows the appearance of A-type molecules smaller than length ten. This is predicted because the first cuts to the N^10 molecules can only produce A^k molecules for k < 10.

Reduction in N^10 molecules N^10 molecules decrease in number.

Appearance of A^k molecules A^k molecules for k < 10.

New A^ks yield multiple N^ks At the next stage we see the appearance of N-type molecules both shorter and longer than length ten. These arise as the new A^k molecules for k < 10 begin to paste together to form multiple N molecules. The cycle continues as longer N molecules are cut to generate longer A molecules, which in turn get cut and contribute again to long N molecules.

Multiple N^k molecules N^k molecules for k = 2, 5, 7 & 9

Multiple N^k molecules N^k molecules for k = 11, 13, 15, & 17

Longer A^k molecules Once longer N molecules are available they are cut to generate longer A molecules.

Multiple N^k molecules N^k molecules for k = 19 & 20

Other Observations During the execution of the model system we see very little change in the number of B and C-type molecules.

Multiple B^k molecules B^k molecules for k = 2, 4, 6, & 8

Conclusion The equations acted as we had predicted, and with further investigation we hope to reach more substantial conclusions.

Equations and number of F- pieces

N and A Equations for F-pieces

B equations for F-pieces

C equations for F-pieces

N and A equations for ODE

B equations for ODE

C equations for ODE

Final equations