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	<h2>Model description</h2>		<h2>Model description</h2>
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		+	Xenopus' embryogenesis is modelled as a classical Poisson stochastic process where two distinct event can happen :
		+
		+	<ol>
		+	<li>A given cell divides, giving birth to 2 daughter cells. These new cell will divide themselves after a lapse of time represented by an Erlang distribution of variable mean and factor k=12</li>
		+	<li>The amount of plasmids initially present in the mother cell is split between daughters following a normal distribution</li>
		+	</ol>
	<h3>Equations</h3>		<h3>Equations</h3>

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Revision as of 23:11, 17 September 2012

Plasmid splitting

Overview

The idea of this model is to better understand the consequences of our experimental protocol
Our protocol consists in injecting a large amount of plasmid at the 1-cell stage, directly into the cytoplasm. When cells divide, the initial quantity of plasmid is split between daughter cells. Only a very infinitesimal proportion of plasmid will be integrated in the nucleus so most of the "effective" plasmids containing our constructs comes directly from this first injection.
This model has been created in order to answer critical questions about our experimental protocol :

• What is the average amount of plasmid we can expect to find in a cell at a given time?
• How uniform is the plasmid repartition among cells?
• Which known mechanisms in morphogenesis could play a role in the plasmid repartition?

Hypothesis

Various hypothesis are needed in order to model the plasmid repartition in time. Some of them are related to biological knowledge and will allow to get insight into the underlying mechanisms while others are more related to modelling choices and computational tractability.

1. Time between successive mitosis can be modelled using an Erlang distribution
The Erlang distribution with factor k is the sum of k exponential distributions with same mean. The use of this distribution is motivated by considering that biologically, a cell has to finish several elementary biological processes (such as replicating all its chromosomes) before being able to divide. Assuming (with over-simplification) that each of these processes has the same mean duration and follows an exponential law, as commonly assumed for Poisson processes, the overall time between two mitosis events will follow an Erlang distribution. (Ref : Drasdo 2012)
2. Plasmids repartition occurring at mitosis can be represented by a normal distribution
This seemed the more straightforward and natural choice of repartition. This hypothesis being closely related to the fundamental dynamics of mitosis during early cell divisions and to cytoplasm's physical properties, it will be further discussed in this page.
3. On the considered stages of development, only cell division occurs
This hypothesis is more for sake of simplicity than based on biological ground. The team obviously acknowledge the central role of cell death processes, and mainly apoptosis in morphogenesis, but this process is much more important for cell differentiation than it is for the overall growth rate (in terms of number of cells). Being mainly interested by the later, we will only consider cell growth.

Model description

Xenopus' embryogenesis is modelled as a classical Poisson stochastic process where two distinct event can happen :

1. A given cell divides, giving birth to 2 daughter cells. These new cell will divide themselves after a lapse of time represented by an Erlang distribution of variable mean and factor k=12
2. The amount of plasmids initially present in the mother cell is split between daughters following a normal distribution

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 hypothesis 2. Their mathematical formulation is the following:


Where:

• 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

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.

Calibration

Results

Conclusion

References

Other possible topologies

With auxin in the external medium:

With a specific receptor organ: