Definitions

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

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

Modelling is the process that enables the development of a model. It takes into account [1]:

• 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 and use 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 describe its properties (theoretical main behavior regardless of the values of the parameters)
• 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 either create a surfactant or establish a positive biofilm, both to prevent the recolonization of the surface by deleterious organisms. The toggle switch is done by environmental conditions : two inducers can be added to select one behavior or another.
For this, we have created the following construction, with a double regulation:


Figure 1: The construction of the biological model, with the following elements: 2 promoters (P_xyl and P_lac), 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 surfactin, whose production is activated by the sfp gene. This is the COAT option.
• Establishment of a positive biofilm by the inhibition of the main biofilm repressor gene abrB. This is the STICK option.

Figure 2: The two possible states, surfactant formation for the top operon or biofilm formation for the bottom construction.
LacI and XylR proteins are called repressors. They bind to their respective promoters (P_lac and P_xyl), thus preventing RNA polymerase binding. So proteins under the inactivated promoter are not produced.
There are also two inducers in the system: IPTG (isopropyl β-D-1-thiogalactopyranoside) and Xylose (monosaccharide of the aldopentose type). In the absence of these inducers, both constructions are inhibited. If only one of them is present, the corresponding inhibition disappears and the associated construction is expressed.



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


With this model, we pursue two main objectives :

• 1) Verify the design of the biological system, to be sure that the toggle switch is functional;
• 2) Predict the behaviour of this biological system depending on the presence of inducers to give usage guidelines for its industrialization.

However...

We are working in a Bacillus subtilis strain and some parameters such as XylR values on binding/unbinding kinetics to both inducer and promoter or production rate from P_xyl promoter cannot be found in the literature and most of the existing values come from an E. coli strain rather than B. subtilis. Furthermore, we are finishing the biological system construction and its characterization is on the way. Parameters will be measured very soon.

Because of this lack of information, we will create a theoretical model in order to characterize the global behavior of the system.

Moreover, as we are mainly biologists in the team, we thought it could be interesting to explain how we can easily obtain a mathematical model from a biological system.

Basic knowledge

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


Figure 3: The black box model:
there will be the STICK or COAT option depending on the inducers concentration

• mathematical equation;
• format:
• explanation: used in biology and physics to represent the growth or evolution of a quantity dx (i.e. population or concentration) proportional to the population size/effective concentration x during a period of time t;
• x is called a variable.

Elements of the model

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


Figure 4: The model variables at first glimpse

We decided to analyze the relation between repressors, promoters and inducers depending on the law of mass action. It is a branch of chemical kinetics, which states that the speed of a chemical reaction is proportional to the quantity of the reacting substances. These substances will bind with an association kinetic k and unbind with a dissociation kinetic k_m.



It is working either for LacI binding to its P_lac promoter than for XylR to P_xyl and also the inducers and LacI and XylR. This binding creates a new complex.

Figure 6: The model variables

Equations of the model

Now, we can find the equations, based on the behavior 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.
As described above, there are two promoters, P_lac and P_xyl, and each of them can be free (with no repressor bound on it) or occupied (with the corresponding associated repressor).
Thanks to the law of mass action and binding and unbinding kinetics, we obtain the equations like this:


With the same method based on law of mass action and binding/unbinding kinetic, we obtain the inducers' equations.


Now, for the repressors, the method is quite similar as before. However, we have to take into account that the proteins have a degradation rate (δ) depending on their nature and the environment. The quantity of protein produced at each time depends on the promoter under control (α).


LacI Equation

We obtain XylR's equation exactly as LacI's.

Figure 9: Repressors' equations

Parameters of the model

For this model, we need at least 12 parameters that characterize the variables and their relationship between each other. The available values have been measure mainly in an E. coli strain.

Name	Description	Unit	Value	Reference
Prod_Plac	Production rate from Plac promoter	mol.s-1	1.66E23	1
Prod_Pxyl	Production rate from Pxyl promoter	mol.s-1	***	**
k1	binding kinetic of LacI and IPTG	mol-1.s-1	1.2E5	2
km1	unbinding kinetic of LacI_IPTG	s-1	2.1E-1	2
k2	binding kinetic of XylR and Xylose	mol-1.s-1	***	**
km2	unbinding kinetic of XylR_Xylose	s-1	***	**
k3	binding kinetic of LacI and Plac	mol-1.s-1	5.1E6	2
km3	unbinding kinetic of PlacO	s-1	3.7E-2	2
k4	binding kinetic of XylR and Plac	mol-1.s-1	***	**
km4	unbinding kinetic of XylR and PlacO	s-1	***	**
δ_LacI	degradation rate of LacI	s-1		3
δ_XylR	degradation rate of XylR	s-1	***	**

Model parameters. *** for no value and ** for no reference

Hypotheses

The following hypotheses have been made for this model.

• 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 proteins.
• we assumed that there will be no degradation of IPTG due to its high stability[4] and no metabolism of Xylose in our condition[5].
• we are aware of LacI[6] and XylR[7] dimerisation as fundamental functional unit but they are not taken into account in this model.

Finally, we translated the biological system into mathematical equations. As we can see, there is a lot of paramaters for the binding and unbinding kinetics, degradation rates and productions from promoters. However, values for only a few of them are available. This is why we needed to simulate the behaviour of our system.
We want to have an overproduction of XylR in the presence of IPTG, for the STICK option. And an overproduction of LacI when there is xylose in the environment, for the COAT option.
According to the above theoretical model, we should generate the expected toggle switch (figure below), using the appropriate values for all parameters.


Figure 10: Expected results of the model
So far, we can consider the values for the P_lac promoter, LacI and IPTG that have been measured in E. coli to apply to our B. subtilis model. We have performed experiments with the P_xyl, xylose and XylR to evaluate whether this promoter can be modelled using similar values. When this step is performed, our model will allow us to determine two important concentration limits:

• 1) the lowest IPTG concentration for the induction of the COAT option;
• 2) the lowest xylose concentration for the induction of the STICK option.

We also thought of a way to obtain binding and unbinding kinetics of the proteins. Some methods such as Isothermal titration calorimetry (ITC) permit to determine the thermodynamic parameters of interactions in solution