Team:Evry/ODE model
From 2012.igem.org
Model using Ordinary Differential Equations (ODE)
Overview

This global model represents the tadpole as a three compartment system:
 The skin that produces (or receives) auxins, denoted S in the equations;
 The blood that transport auxins to the organs, denoted B;
 The organs (called receptors) that interacts with auxin molecules, denoted R.
 The auxin concentration inside a compartment is homogeneous This condition is inherent to this kind of modeling.
 No auxins can go from the skin directly to the organs. We have chosen to neglect the exchanges between the skin and the other organs.
 The auxin flow follows the concentration gradient between compartments. This assumption is based on the fact that a small capillary is nothing more than a hole in a cell layer. So the exchanges between the surrounding cells and the capillary are mutual.
 S in m^{2}, represents the area of the exchange surface between the two compartments.
 P in m^{2}, represents the permeability of the membrane between the specified compartments.
 C in [quantity] / m^{3}, represents the concentration of auxin in the specified compartment
 For permeabilities:
 For die rate:
 For Surfaces:
 For Volumes:

This first graph is obtained considering no auxin in the tadpole and c_{w}=5.10^{3}mol.L^{1} which is the concentration we wanted to use for experiences. The result shows that the quantity of auxin in skin and in the receptor grow until a limit value.It seems that we can make an electrical analogy; both skin and receptor behave as condensators that are charging.
If there has been injected auxin in the receptor and in skin: n_{s}(t=0)=10 mmol n_{r}(t=0)=10 mmol, we can observe a decrease of auxin in receptor before an establishement of the equilibrium. To continue the electrical analogy, we see that the receptor behaves as a condenser that is decharging , whereas skin is a charging condenser.
Here , the concentration in water has been devided by ten c_{w}=0.5.10^{3}mol.L^{1}, while the initial value of skin is still n_{s}(t=0)=10 mmol. We observe first a big increase in the receptor due to the diffusion of the auxin intially injected in skin then a decrease because of the small water concentration in auxin.
The parallel with an electric circuit is straightforward: the skin represents a generator that will propagate intensity to the other systems compartments;
The blood represents wires, that convey this quantity throughout the system and interfaces are like resistors;
Finally the organs are the sinks that use this intensity to work.
Using this very idealized view of the tadpoles makes it possible to model it simply : The processes happening in the system can be approximated using Ordinary Differential Equations (ODE), one of the simplest form of differential equations; Plus, the organs repartition and shape are not taken into account. As showed in this page, these equations can be seen as approximations of more realistic PDEs where space considerations have been removed.
Although our model is very general and simple, it allows to have a quantitative view. As many parameters would require much experiment to be estimated correctly. At this stage this quantitative model only allows us to make some prediction about the global behaviour of the system and gives us hints about the reasons for the success or failure of experiments.
Assumptions
There are the different assumptions we were constrained to make in order to model the system:Model description
Equations
Each compartment is modeled by a differential equation representing the evolution of the auxin quantity as a function of the time.
Each equations are composed of two kinds of terms: creation and degradation. The creation term can represent either a creation of auxin in the compartment or an arriving quantity of auxin from another one. In the same way, the degradation term can either represent a natural degradation of molecules or a quantity leaving the compartment.
In this system,D_{die} is a degradation rate, the other D terms represent the flows between the different compartments. We made them depend on the concentrations of both the in and out compartments as explained in assumption 2. Their mathematical formulation is the following:
Where:
These flow equations are based on Newton's law of cooling where the difference between the concentrations of the two compartments gives the direction and magnitude of the flow. This allows us to model in a single equations the two opposite flows between the compartments.
Parameters
The parameters of the model are hard to estimate because direct experiences concerning auxin's behaviour in the xenopus where never made before, as it is a plant hormone.Moreover,the parameters are an aggregation of different physical values. This is due to the high level view provided by the model. The different parameters are:
Description  Symbol  Type  Values 

Permeabilities  P  calculated  here 
Contact surfaces  S  calculated  here 
Volumes  V  calculated  here 
Degradation rate  D_{die}  estimated  unknown 
Creation rate  D_{born}  computed  plasmid repartition model 
Sensitivity Analysis
There are 2 kinds of parameters in this model; those concerning the geometry of the tadpole and those concerning the behaviour of auxin in the tadpole's tissues. As auxin is a plant hormone, we had to evaluate these parameters by taking those of similar molecules. That's why the uncertainties on these measures are very big. For this category, we observed variations from reference value ref*10^{3} to ref*10^{3}. You can see the effect of these variations:
Results
By modifying the initial conditions and c_{w} we have observed globaly 3 different behaviours of the system:Conclusion
This model being simplified to the maximum, has let us, thanks to an estimation of very few parameters, to see the global behaviour of the system. It helps us to determine expected behaviours and helps us to establish possible reasons of dysfunctions. Nevertheless, quantitatively, this model is too imprecise because it doesn't take tadpole's geomoetry into account. This is the aim of our most elaborated model which will use partial equations.
References
References:
Other possible topologies
With auxin in the external medium:With a specific receptor organ: