Team:Evry/plasmid splitting
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Revision as of 03:36, 27 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 onecell stage, directly into the cytoplasm. When cells divide, the initial quantity of plasmid DNA molecules is split between daughter cells. As there is no functional origin of replication in our plasmids, unless very rare event, our plasmids are not replicated throughout development.
This model was 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?
Assumptions
Various assumptions 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.

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 oversimplification) 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)

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.

On the considered stages of development, only cell division occurs
This assumption is more for the sake of simplicity than based on biological ground. The team obviously acknowledge the central role of 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 proliferation.
Model description
Elementary events
Xenopus' embryogenesis is modelled as a Poisson stochastic process where two distinct but successive events can happen :
 A given cell divides, giving birth to two daughter cells. These new cells will divide themselves after a lapse of time represented by an Erlang distribution of variable mean and factor k=4 for the four phases of cell cycle
 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 timestep 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.
Growth rate
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 (data at 25°C for X. tropicalis) :
(ref : Atlas of Xenopus Development, Xenbase, N. Pollet's data, Khokha et al.,2002 )Using this growth curve as target, we adjusted a piecewise linear growth rate function of time to match our simulation with data. Given that early divisions and most of morphogenesis is a very complex phenomenon, using a single growth rate was far too unrealistic. Moreover, differentiation is a key factor in explaining why the overall growth rate is to vary in time. In the end a well enough fitting growth rate function is given as an interpolation of :
Time (h)  0  2  2.68  3.19  5.7  9.8  12.15  14.74  
Growth Rate (h^1)  1.3  3  3.6  2.21  1.055  0.6  0.5  0.05 
Note : As our simulations are stochastic, they are very sensitive to early division time. Fitting data is therefore a difficult and long task as many trials are needed to fit data 'in average'. To have significant results, our simulations showed in the "Result" section are always corresponding to a sufficiently correct growth profile.
Plasmid repartition characteristics
Another important parameter of our model is the plasmid repartition between daughter cells. We first considered a simple normal distribution centred on 50% plasmids in each cell with a variable standard deviation representing inhomogeneity in both plasmids' spatial repartition in the cytoplasm (in early stages, the nucleus is tiny in comparison to the cytoplasm and the fate of injected DNA is unclear as discussed in [5]) and unequal volumes of daughter cells. Focusing on the later phenomena (the former being very hard to capture and assuming the volumetric effect was preponderant) we measured roughly the differences in cells radii from microscopical data at different stages to retrieve volume disparities.
A standard deviation of 0.1 seemed coherent with the radii disparities for early time steps (<8h, after what, no data was easily available) although we couldn't fit all points. This suggest that using the same standard deviation for the whole development is too simplistic.
Moreover, experimental results where we had injected GFP carrying plasmids seemed to show the distribution was much more unequal and a bimodal distribution could be more realistic to take into account the large disparity between cells in the animal and vegetal poles. Anyway, even with our simple normal distribution simulations shows that quickly, the standard deviation in the average number of plasmid by cell becomes larger than the average amount of plasmid itself. This shows a strong inhomogeneity and could be sufficient to explain our observations.
Therefore, a precise quantification of plasmids, specifically in the very first stages would be necessary to go further. We now believe that cytoplasm is much more dense that we thought and that plasmids nearly don't diffuse at all in the early stages. This belief comes from observing that GFP tagged plasmid seemed to only be expressed in some randomly selected tissues or organs. By coupling this information with fatemaps, it could be possible to quantify precisely how many divisions occur before plasmids get split.
This would radically change the repartition model as half or 3/4 (or more) of the organism could be totally plasmid free if the two first mitosis occur without diffusion of plasmids.
At last, this could help to improve this otherwise convenient injecting technique, by performing multiple small injections rather than a single big one.
Initial plasmid quantity
In order to estimate how many plasmids are injected in the egg, we performed a back of the envelope calculation taking into account :
 The mass of plasmid injected
 The weight per base of a double stranded plasmid
 The average length of our plasmids in base
The final figure is : 3.10^7 plasmids
Results
Normal distribution
In this section we provide the results of our simulation using a normal distribution for the repartition of plasmids among daughter cells.In this simulation the repartition follows a normal distribution of mean=0.5 with variable standard deviation
Uniform distribution
When considering that the repartition of plasmids between daughter cells follows a uniform (bounded) distribution, the distribution becomes even more inhomogeneous than with a normal distribution. In the subsequent example, the distribution in plasmids is uniform between 40% and 60%, having therefore a smaller variance than a normal distribution with std=0.1. It appears that the distribution of plasmids is far more irregular. This shows that the more irregular is the average split, the more heterogeneous will be the final distribution.
Experiment proposal
As stated before, the plasmid repartitions occurring with the early mitosis are the most important. In order to measure this early repartition many experiments can be performed. Picking one cell at different early stages and counting the amount of plasmids in it (using for instance plasmids tagged with a radioactive element) could allow us to gain very precise knowledge.
To asses later stage distributions, we could simply compare GFP level with reference cells (in which we injected a known, relatively low amount of plasmids) in order to compute the mean amount of plasmids and its standard deviation and compare it to simulations integrating a threshold quantifying the minimal number of plasmids required to observe GFP. As different distributions give rise to different profiles for mean and standard deviation, this measure would be very informative.
Conclusion
This model takes into account our very experimental technique, which is not so common in modelling for biology. But it is an important step to really link models to observations. Moreover, this question is closely related to still badly understood behaviour of early cells. Better parametrizing such a model could therefore give important insight into deep mechanisms such as the pronucleus and chromatin dynamics.
Code
You can download the Matlab code used to perform these simulations here
References
References:
 Course material Drik Drasdo : Modelling of multi cellular tissues, Paris VI lectures 2012
 Atlas of Xenopus Development G.Bernardini, M.Prati, E.Bonetti, G.Scari (1999)
 Nieuwkoop & Faber (Xenbase.org) retrieved on 15 september 2012
 Nicolas Pollet's data
 Transgenesis procedures in Xenopus. A.Chesneau, L.M.Sachs, N.Chai et al. Biology of the cell (2008)
 Techniques and probes for the study of Xenopus tropicalis development. Khokha MK, Chung C, Bustamante EL et al. Dev Dyn. (2002)