Definitions:

A model is a symbolic representation of an object’s or phenomenon’s aspects in the real world.

“All models are false. Some are useful.” Georges Box

Modeling is the process that allows the development of a model. It’s taking into account:

• The phenomenon to represent
• A specific formal system (equation, diagram..)
• Objectives (what we want to do with the model)
• Data (for variables) and knowledge (relation between variables) available or accessible by experimentation or observation

The tasks to obtain the model depend on the biological situation and the formal system chosen. Nevertheless, it must:

• Have a formalization work, which is the model writing
• Manipulate the model in the formal system to make it "usable" and to study its properties
• Objectives (what we want to do with the model)
• Establish relationships with other representations (computer program, graph function)
• Interpret and compare different representations obtained in the formal world with the biological reality (often that reality is seen through experimental data)

Situation:

After the destruction of the biofilm by “Biofilm Killer” bacteria, we want to have the choice to create either a surfactant to prevent the recolonization of the surface, or a positive biofilm. The switch is done by environmental condition: two inducers can be added to select one behaviour or another.

Biological system to model:

For this, we have created the following construction, with a double regulation:


figure 1: The construction of the biological model, The elements of this model are: 2 promoters (Pxyl and Plac), 2 repressors (LacI and XylR proteins) and 2 inducers (IPTG and Xylose), and also sfp and abrB genes for Sfp and AbrB proteins.

This system is a gene-regulatory network, where two different states are possible:

• formation of a naturally toxic bio-surfactant through sfp gene, which has antimicrobial properties that prevents the recolonization of the surface. The surfactant used is surfactine, which is regulated by sfp gene. This is the COAT option.
• creation of a positive biofilm by the inhibition of the main biofilm repressor abrB gene. This is the STICK option.

Figure 2: The two possible states: surfactant formation for the top construction or biofilm formation for the second construction



Figure 3: LacI and XylR repressors inhibit Plac and Pxyl promoter, the following constructions are inhibited.
There are two inducers in the system, IPTG (isopropyl β-D-1-thiogalactopyranoside) and Xylose (monosaccharide of the aldopentose type). In the absence of inducers, both constructions are inhibited. If only one of them is present, the inhibition disappears and the corresponding construction is enabled.

For example, in the presence of Xylose, XylR proteins will form an enzymatic complex with their Xylose sugar. Thus, the inhibition of Pxyl caused by XylR will disappear and there will be a bigger production of Sfp, AbrB and LacI proteins. Sfp production induces surfactine production, and AbrB production involves the repression of the biofilm formation. Eventually, LacI production will inhibit XylR production, so there will be stabilisation of Pxyl activation. In opposition, in the presence of IPTG, LacI proteins will bind to their ligand, and Plac promoter will be free. So XylR proteins will be overproduced, limiting Sfp and AbrB productions. Thus, there will be no surfactine in the environment, biofilm formation can begin.

Aim of the model

With this model, we want to verify the biological system, to be sure that the switch is possible. We also want to predict the behaviour of this biological system depending on the quantity of inducers present in the environment.

However....

We are working in a Bacillus Subtilis strain and some parameters such as XylR values on binding/unbinding kinetic, binding/unbinding kinetic, association constants... cannot be found in the literature and most of the existing values come from a E. Coli strain. Furthermore, we are finishing the final construction and its characterization is underway. Parameters will be measured very soon.

Because of this lack of information, we will create a theoretical model in order to have the main system behaviour.

Furthermore, as we were mainly biologists in the team, we thought interesting to explain how we can easily obtain a mathematical model from a biological system.

Basic knowledge:

We want to transform the biologic system into mathematical equations in order to be able to quantify the quantity of inducers (input) needed to obtain a particular behaviour (output).

Black box model

Elements of the model

First list of variables:

We want to have as output the concentration of LacI and XylR depending of the input of inducers concentration. We know that the repressors can bind either to their promoter (Plac and Pxyl respectively) or to their inducer (IPTG and xylose). Thus, first of all, we have the following variables in the system:

The model variables at first glimpse

Binding and unbinding kinetics:

We decided to analyse the relation between repressors, promoters and inducers depending on the law of mass action: they will bind and unbind with a kinetic k and km

Binding and unbinding kinetic par iGEM_Lyon_2011 Binding and unbinding kinetics for the variables
The model variables
Equations of the model

Now, we can find the equations, based on the behaviour of each element. There will be 3 types of equations, each of them related to the nature of the variable, i.e. a promoter, an inducer or a repressor.

Promoters:
As describe in the elements on the model, there are Plac and Pxyl promoters and each of them can be free (with no repressor binds on it) or occupy (with the corresponding repressor associated).
Thanks to the law of mass action and binding and unbinding kinetics, we obtain the equations like this:

So we obtain the following equations:
Promoters equations

Inducers:
With the same method, we obtain inducers equations. Inducers equations

Repressors: Now, for the repressors, the method is quite similar
Assumption/Hypothesis

• we just need LacI concentration for surfactant production and not Sfp and AbrB concentration because there is a proportional link between them. If there are LacI proteins produced, there will be also Sfp and AbrB protein.