# Team:USTC-China/modeling

### From 2012.igem.org

MODELING

## I. Mathematical models

The function of the anticro is to repress the expression of cro. That means the anticro can strongly prevent the lambda phage from turning into lytic life cycle. It is the toggle switch in the genome of lambda phage that control the decision of lysogenic or lytic life cycle and it is the target of our anticro. So, we study the steady state property of two system, without and with our designed genetic circuit separately. Finally, we give the phase graphs to show the steady state property visibly and draw the stream diagram to simulate the fate of each point on the graph.

We study the steady state property of two system, without and with our designed genetic circuit separately.

## Toggle Switch without designed genetic circuit

The chemical reactions are:

According to the Law of mass action, we can easily write down the equation approximations to reduce these equations.

First of all, we do quasi-steady state approximation for dimerization reactions

where and are the total concentration of and .

Besides, we decide to take the degradation into account in the very end step. So, considering the conservation of binding site

Where R is the total binding site number, we get

The above equations give us the expression of .
The final step is to write down the evolution equations for and .

where

## Toggle Switch with designed genetic circuit

There is a difference if we add our designed genetic circuit:

where refers to .

Three additional equation should be taken into account

The above gives us the expression for , which is a function of . It is then straightforward to derive from the following

Once obtained , substitute it into the following to get the final equation

## Result

We plot the stream diagram for and :

We can see the apparent two steady states on this phase graph. The stream diagram shows the changing trend of each point.

After adding our designed circuit, the stream flow diagram is

This result demonstrates the efficiency of the anticro of repressing the lytic life cycle.

## Value Table

## II. Computer simulations

In order to show the visible process of our engineered bacteria defending against the bacteriophage, we write a JAVA program to simulate the evolution of the colony which is invaded by the bacteriophage. The program interface is showed below:

Here is the description of our JAVA program: (if you want to skip the description and operate the program directly, please click here.)

The blue box: presents the lysogenies

The red box: presents the phage in the host is in lytic life cycle

The grey box: presents the normal bacterium

The amount of this three kinds of bacteria have been set as three changeable parameters in our program. The other three changeable parameters are:

**P1**: The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

**P2**: The lysogen suicides at possibility of P2.

**P3**: At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.

Model:

Grid model of solid LB medium

1. The cell can only reproduce new cells when there are empty grids around it.

2. The newly assembled and emitted phages will only infect the cells in the grids around its host.

3. The step length of time for this JAVA applet to operate is the time of a cell cycle.

4. The basic level of expression of the lysis gene and the quorum sensing among cells are ignored.

5. We adopt the Monte-Carlo simulation to deal with the process of competence among cells

Description of the algorithm

The process of infection:

1. When the host is lysed by the newly reproduced phages, the cells in the grids around it will be infected by the emitted phages and the grid the host occupied will return to be empty.

2. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

The process of growth of the cell:

1. The normal cell can only reproduce new cells when there are empty grids around it.

2. The lysogen suicides at possibility of P2 and reproduce a new lysogen at empty grid around it.

3. If a empty grid abut both a normal cell and a lysogen and both the two cell will reproduce new cell, the offspring of the lysogen win the empty grid at possibility of P3.

Procedures of the algorithm

1. Arrange three kinds of cells(grey: the normal cell, red: the host in which the phage is at lytic life cycle, blue: the lysogen) at the screen according to the parameters at random.

2. The red cell is lysed and releases the grid it occupied. All the cells in the grids abut the lysed host will be infected.

3. Scan the screen again. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

4. The normal cell sets marks represent the reproduction of normal cells (called the normal marks) in the grids around it.

5. The lysogen suicides at possibility of P2. Otherwise, it sets marks represent the reproduction of lysogens (called the lysogenic marks) in the grids around it. If some of these grids around the lysogen have been marked as reproduction of normal cells, then the lysogen marks these grids as competence states.

6. At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.

7. Those grids with normal mark are occupied by the newly reproduced normal cells and those grids with lysogenic mark are occupied by the lysogenies.

Here is our JAVA program: click here.

## Reference

- Timothy S. Gardner*, Charles R. Cantor* & James J. Collins*: Construction of a genetic toggle switch in Escherichia coli. NATURE, VOL 403, 20 JANUARY 2000
- Moises Santillan*, Michael C. Mackey: Why the Lysogenic State of Phage λ Is So Stable: A Mathematical Modeling Approach. Biophysical Journal volume 86, issue 1, Jan 2004, page 75-84
- Paul J. Darling, Jo M. Holt and Gary K. Ackers: Coupled Energetics of l cro Repressor Self-assembly and Site-specific DNA Operator Binding II: Cooperative Interactions of cro Dimers. J. Mol. Biol. (2000) 302, 625-638
- Chunbo Lou, Xili Liu, Ming Ni: Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch. Molecular Systems Biology 6:350