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Model using Ordinary Differential Equations (ODE)

Overview

This global model represents the tadpole as a three compartment system:
1. The skin that produces (or receives) auxins, denoted S in the equations;
2. The blood that transport auxins to the organs, denoted B;
3. The organs (called receptors) that interacts with auxin molecules, denoted R.

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:
1. The auxin concentration inside a compartment is homogeneous
This condition is inherent to this kind of modeling.
2. All the auxins that enter the receptor come from blood.
Considering blood speed and the big density of capillaries, we assumed that the auxin coming from other tissues can be neglected. This assumption will remain while the receptor isn't close to skin.
3. 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.
4. Permeabilities are symetric.
We have considered that for 2 issues X and Y the permeability from X to Y is equal to the one from Y to X.

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 equation is composed of two kinds of terms: synthesis or arrival (called born) and degradation (called die). The creation term can represent either a creation of auxin in the compartment or an arriving quantity of auxin from another one. The following equations represent the case where auxin flows in from water. In the case where we consider auxin being synthesized directly in skin, this term would be the mean rate of synthesis. This rate being computed integrating data from the plasmid repartition model and the auxin production model, as described in our derivation from PDEs. 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:

• S in m², represents the area of the exchange surface between the two compartments.
• P in m², represents the permeability of the membrane between the specified compartments.
• C in [quantity] / m³, represents the concentration of auxin in the specified compartment



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	Experiment Proposal
Permeabilities	P	similar molecule in litterature	here	link
Contact surfaces	S	calculated	here	-
Volumes	V	calculated	here	-
Degradation rate	Ddie	unknown	unknown	link
Creation rate	Dborn	computed	our derivation from PDEs	-

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*10³. You can see the effect of these variations:

• For permeabilities:
• For die rate:
For the geometrical parameters, we observed variations going from ref*0.3 to ref*2.We've obtained:
• For Surfaces:
• For Volumes:

Results

By modifying the initial conditions and c_w we have observed globaly 3 different behaviours of the system:

• This first graph is obtained considering no auxin in the tadpole and c_w=5.10^-3mol.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.

  

• 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.

  

• Here , the concentration in water has been devided by ten c_w=0.5.10^-3mol.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.

  

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:

1. Jean Philibert, One and a Half Century of Diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals 2, 2005

Other possible topologies

With auxin in the external medium:

With a specific receptor organ:

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