Team:Evry/plasmid splitting

From 2012.igem.org

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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 :
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<h3>Elementary events</h3>
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Xenopus' embryogenesis is modelled as a Poisson stochastic process where two distinct but successive events can happen :
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<h3>Equations</h3>
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The values used to represent the mean time between mitosis and normal distribution parameters will be discussed in the results and calibration sections.
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Each compartment is modeled by a differential equation representing the evolution of the auxin quantity as a function of the time.
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<h3>Simulation</h3>
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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.<br/><br/>
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Realisations of this stochastic process where simulated using the convenient variable time-step Gillespie Algorithm implemented in Matlab by our team.
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<center>
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<img src="https://static.igem.org/mediawiki/2012/2/25/Ode-syst.png" alt="ODE system" />
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<h2>Calibration</h2>
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As this model has been made in order to better understand how our experimental choice of plasmid injection instead of more complex nucleus integration would affect the efficiency of our constructs, calibration is of much importance.
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The first step after having implemented the algorithm was to tune its parameters in order to match experimental data. As the growth rate (or mean time between divisions, one being the inverse of the other) is a key parameter in order to have simulations with representative time scales, we carefully calibrated it. Using different available data about Xenopus' development, we were able to retrieve its growth in time, and along development stages :
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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 <a href="#hyp2">hypothesis 2</a>. Their mathematical formulation is the following:<br/><br/>
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<img src="https://static.igem.org/mediawiki/2012/7/7e/Fluxes.png" alt="mathematical expression of fluxes" />
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Where:
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    <li>S <strong>in m^2</strong>, represents the area of the exchange surface between the two compartments.</li>
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    <li>P <strong>in m^2</strong>, represents the permeability of the membrane between the specified compartments.</li>
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    <li>C <strong>in [quantity] / m^3</strong>, represents the concentration of auxin in the specified compartment</li>
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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.
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<h2>Calibration</h2>
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<h2>Results</h2>
<h2>Results</h2>

Revision as of 23:40, 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?

Sketch View of the plasmid repartition model

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
    2. 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
    3. 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
    4. 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

Elementary events

Xenopus' embryogenesis is modelled as a Poisson stochastic process where two distinct but successive events 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
The values used to represent the mean time between mitosis and normal distribution parameters will be discussed in the results and calibration sections.

Simulation

Realisations of this stochastic process where simulated using the convenient variable time-step Gillespie Algorithm implemented in Matlab by our team.

Calibration

As this model has been made in order to better understand how our experimental choice of plasmid injection instead of more complex nucleus integration would affect the efficiency of our constructs, calibration is of much importance.
The first step after having implemented the algorithm was to tune its parameters in order to match experimental data. As the growth rate (or mean time between divisions, one being the inverse of the other) is a key parameter in order to have simulations with representative time scales, we carefully calibrated it. Using different available data about Xenopus' development, we were able to retrieve its growth in time, and along development stages :

Results

Conclusion

References

References:

Other possible topologies

With auxin in the external medium:
tadpole + external compartments

With a specific receptor organ:
tadpole + other compartments

Retrieved from "http://2012.igem.org/Team:Evry/plasmid_splitting"