Team:Lyon-INSA/modelling
From 2012.igem.org
Modelling
Interesting question when you are more a biologist than a mathematician (too many complicated equations!!!) And as most of our team members were biologists/biochemists, we tried to explain the model easily for everyone.
This is our Biological Modelling for Dummies !
Click on the title to show/hide the text.
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).
The black box model:
there will be the STICK or COAT option depending on the inducers concentration
Ordinary differential equation (ODE):
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. 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 km.
Binding and unbinding kinetic par iGEM_Lyon_2011
It is working either for LacI binding to its Plac promoter than for XylR to Pxyl and also the inducers and LacI and XylR. This binding creates a new complex.
Binding and unbinding kinetics in the model
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:
Plac Equation par iGEM_Lyon_2011
So we obtain the following equations:
Promoters equations
Inducers:
With the same method, we obtain inducers equations.
Inducer Equation par iGEM_Lyon_2011
Inducers equations
Repressors:
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 depending on their nature and the environment. The quantity of protein produiced at each time depends from the promoter under control because of different expression level.
Assumption/Hypothesis
Some hypothesis have been done for this model.
This is our Biological Modelling for Dummies !
What is modelling ?
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 tasks to obtain the model depend on the biological situation and the formal system chosen. Nevertheless, it must:
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)
Biological System description
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:
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 these inducers, both constructions are inhibited. If only one of them is present, the inhibition disappears and the corresponding construction is enabled.
Xylose Induction
IPTG Induction par iGEM_Lyon_2011
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, 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.
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:
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.
There are 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 inhibition disappears and the corresponding construction is enabled.
Xylose Induction
IPTG Induction par iGEM_Lyon_2011
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, 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.
Biological modelling for dummies !
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).
there will be the STICK or COAT option depending on the inducers concentration
Ordinary differential equation (ODE):
- mathematical equation
- format: $dx/dt$ (joli dessin)
- 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
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:
Binding and unbinding kinetics:
We decided to analyse 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 km.
Binding and unbinding kinetic par iGEM_Lyon_2011
It is working either for LacI binding to its Plac promoter than for XylR to Pxyl and also the inducers and LacI and XylR. This binding creates a new complex.
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:
Plac Equation par iGEM_Lyon_2011
So we obtain the following equations:
Inducers:
With the same method, we obtain inducers equations.
Inducer Equation par iGEM_Lyon_2011
Repressors:
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 depending on their nature and the environment. The quantity of protein produiced at each time depends from the promoter under control because of different expression level.
Assumption/Hypothesis
Some hypothesis have been done 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 protein.
- we assumed that there will be no degradation of IPTG and xylose BECAUSE OF ?????