Interesting question when you are more a biologist than a mathematician (too many complicated equations!!!) And as most of your team members were biologists/biochemists, we tried to explain the model easily for everyone.

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 protector 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 2: The two possible states: surfactant formation for the green construction or biofilm formation for the red 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.

We have done 2 models, one depending on the law of mass action and the second one based on the Toggle Switch.

How do we transform a biological system to a mathematical 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 (left) and Switch model (Right)

Elements of the models

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