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Model using Ordinary Differential Equations(ODE)

Overview

The first model we developed 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.

The parallel with (electrical) engineering is made easy: the skin represents a generator that will add a quantity to the system; The blood represents wires, that convey this quantity throughout the system; Finally the organs are the sinks that use the quantity to work.

This very idealized view of the tadpoles allows to make some interesting simplifications: 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.

This over-simplication of the problem causes the model to give very imprecise quantitative results but its strength is in allowing us to make some qualitative predictions about the success or failure of some 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. 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.
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.

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, the J terms represent the fluxes 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 they 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	Ddie	estimated	?
Creation rate	Dborn	computed	plasmid repartition model

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.

  

• In the second graph, the initial value of skin is equal to 10 mmol.L^-1 andc_w=0.5.10^-3mol.L^-1

  

Conclusion

References

References:

Other possible topologies

With auxin in the external medium:

With a specific receptor organ:

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