Team:Evry/plasmid splitting

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With the version 3 of the algorithm, using a normal distribution with std=0.1 for repartition after early mitosis, it yields the following characteristics. Nevertheless, looking at these two first moments only can be deceiving as we will see with a finer representation of the actual distribution.
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A first, we tried to compute directly the distribution (density function) of plasmids. It is simply the density of a random variable Np=N1xN2x....xNp where Ni follows a normal distribution. For p=2, there exists a direct solution but for any higher number, only the first moments can be computed easily.<br/>
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With the version 3 of the algorithm, using a normal distribution with std=0.1 for repartition after early mitosis, it yields the following characteristics. We can see the distribution is even more heterogeneous than previously. Nevertheless, looking at these two first moments only can be deceiving as we will see with a finer representation of the actual distribution.
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<br/><div class="caption">Distribution characteristics, for the version 3 of the algorithm, normal distribution for repartition with std=0.1, at 14h</div>
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Revision as of 03:56, 27 October 2012

Plasmid splitting

(Check out improvements since regional Jamboree !)

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 one-cell 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?

Sketch View of the plasmid repartition model
Sketch view of plasmid repartition after several mitosis

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.


  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)

  3. Plasmids repartition occurring at mitosis can be represented by a normal distribution

  4. 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.

  5. On the considered stages of development, only cell division occurs

  6. 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 :

  1. 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
  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.


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 )
Number of plasmids against time
Data for growth rate

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)022.683.195.79.812.1514.74
Growth Rate (h^-1)1.333.62.211.0550.60.50.05
Fit of cell growth
Calibration of the growth rate. Green is Data, Red simulation

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 bi-modal 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
m=0.5 std=0.05
Plasmid distribution std=0.05

m=0.5 std=0.1
Plasmid distribution std=0.1

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.

Uniform distribution on 0.4 0.6
Plasmid distribution for a uniform plasmid splitting in the range : [40%,60%]

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.

Improvements with the Version 3 of the model

Since regional jamboree, we tried to develop further our models in order to make them even more close to reality.

Investigating early cell cycles dynamics

As stated, early divisions are very important for the whole morphogenesis. In the very detailed review of the cell cycle in Xenopus we found that the 12 first divisions occur within 8h and are synchronous. We therefore adapted our algorithm to mimic this behaviour. In this third version of our algorithm, early divisions are therefore synchronous, with time between mitosis following a Erlang distribution as before. In order to calibrate our model we performed multiple simulations and tuned the early division rate in order to exit this early division period after 8h in average. With an early rate of 1.45 mitosis/hour the average (on 243 simulations) exit time was 8.03 h, with a standard deviation (due to stochasticity) of 1.06 h. This can be seen on the following exit time distribution :

End of early division part of our simulation for early division rate of 1.45
End of early division part of our simulation for early division rate of 1.45

Investigating the early plasmid repartition

As showed by our experiences, plasmids usually end up in localized tissues of Xenopus. A good explanation of this behaviour could come from the fact that DNA in early stage cells is wrapped by vitellus and could not diffuse in the first mitosis events. In this section we describe our though process in testing this assumption. The first step was done by comparing our reporting protein expression in tadpoles and comparing it with known fatemaps.

Early cells are well characterised and it is possible to know precisely the actual offspring of every cell at least until the 32 cell blastula. These are represented by coordinates, here for example D2.2 mean the cell who is the "second" daughter of the second daughter of the "Dorsal" daughter of the first cell. Her sister is therefore D2.1 and her own offspring will be noted D2.2.1 and D2.2.2

Names Of Blastomere cells
Extracted from (Moody)


We then compared our experimental results (for example this one) with (reversed) fate map data from (Moody), conveniently gathered in this Xenbase.org page.

Fate maps from Xenbase
Screen shot of the interactive fatemaps from Xenbase.org



Tissues with expressionBlastula Cells involved
Neural FoldV121 V122
Intestine - Branchial BasketV121 V111 V122 V112
Skin (Head)D111 D121 V121
Optical Nerve - Tail muscles - Branchial BasketV121 V111 V122 V112

Table summarizing the expression we saw in our experiments and their most probable origin in the 32 cell blastula


Although it is sometimes difficult to precisely identify the origin of the tissues where our plasmids were expressed, these comparisons are totally coherent with our assumption that plasmids don't diffuse in the early mitosis. When diffusion exactly starts is still unclear as reverse fate maps show usually various possible origin. Anyway, inhomogeneity in the first division is almost certain from these comparison and could last until the 3rd mitosis. Specific experiments are needed in order to clearly confirm and quantify this behaviour.



Pursuing the early repartition

In order to better understand how our plasmids are diffused during early divisions we performed the following experiment :
We injected zygotes with plasmids having a fluorescent tag and took microscope pictures in order to see properly the diffusion of plasmids. This experiment is difficult as only a few fecundated eggs will develop (and therefore, if the zygote we have put under the microscope doesn't develop, we can't have very early stages recorded. The time to put another embryo under the microscope and it is already well advanced.)
In these two experiments we achieved to track plasmids repartition between (approximatively) 2h45 after fecundation until 6h50 after fecundation. Pictures for the first 6 time steps of both experiments are reported here (the focus was fuzzy after) :


Plasmid repartition experiment t1 to t6
6 first time steps of our repartition plasmid experiment
Plasmid repartition experiment n2 t1 to t6
6 first time steps of our second repartition plasmid experiment


New assumption on early plasmid repartition

In the second experiment, the embryo rotated, allowing us to see a larger part of the surface.
Based on what we can see in the pictures, it appears that roughly only one quarter (to 1/8, depending on what happens inside the embryo) of the blastula has plasmids. This mean there is no diffusion during the 2 (or 3) first mitosis. Actually, it is obvious that reality is not as straight forward but this simplification seems acceptable and coherent with all the data we gathered from 2h45 to 6h50 at least.
Accordingly, in the version 3 of the algorithm, there is no diffusion of plasmids during the two first mitosis.

Plasmid repartition


A first, we tried to compute directly the distribution (density function) of plasmids. It is simply the density of a random variable Np=N1xN2x....xNp where Ni follows a normal distribution. For p=2, there exists a direct solution but for any higher number, only the first moments can be computed easily.
With the version 3 of the algorithm, using a normal distribution with std=0.1 for repartition after early mitosis, it yields the following characteristics. We can see the distribution is even more heterogeneous than previously. Nevertheless, looking at these two first moments only can be deceiving as we will see with a finer representation of the actual distribution.





Distribution characteristics, for the version 3 of the algorithm, normal distribution for repartition with std=0.1, at 14h



Distribution of plasmids in cells, for the version 3 of the algorithm, normal distribution for repartition with std=0.1, at 14h


Analysis :
We can see in this distribution that most of the cells have a "small" amount of plasmids. Zero plasmid is the most common feature (75,1%), the mean was 134 plasmids but we could find a cell with 38436 plasmids in it. This very heterogeneous distribution can explain our experimental observations : even in the tissues where GFP was expressed, we saw many "spots" of GFP rather than a smooth mean light intensity. This could be explained by the fact there should be some threshold in terms of number of plasmids by cell under which expression is not measurable. Cells above the threshold are very irregularly distributed and explain the spot pattern.



Determining the number of cells


In order to be able to use these images for our model we need to calibrate them. Time is the most convenient way of doing so but embryo growth is subject to many variations, some known as temperature, others unknown and so considered as random. The best way to have our model matching these experiments is to use the number of cells. As we only can see a face of the embryo we had to make an estimation based on this sole face.
In order to do this we tagged a few cell in pictures (the most distinguishable) as represented in this picture :



Example of tagged cells
Example of tagging cells


We then extracted "masks" of the surface of tagged cells as well as the embryo shape :

Example of tagged cells masks
Example of mask


Importing these masks in Matlab © we computed the true surface of the tagged cells, as well as that of the whole embryo. The true surface is not exactly what we can see here because the embryo has a spherical shape (the surface of cells near the center are correct but those in peripheral area are underestimated). We therefore made some Matlab routines (available here) in order to correct the surface based on the assumption of a spherical shape.
In order to count how many cells are in the pictures, we computed how many spheres could fit in the global volume using the maximal packing factor (0.74). Results are reported in the previous picture of our experiments.
Note that as we have mostly shot the animal pole, whose cells are smaller, we used data from Atlas of Xenopus Development and Xenbase to compute the surface disparity between animal pole cells and vegetative pole cells. The ratio (for radii) was estimated around 1.5 and we assumed half cells are in both category.

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 pro-nucleus and chromatin dynamics.

Code

You can download the Matlab code used to perform these simulations here

References:

  1. Course material Drik Drasdo : Modelling of multi cellular tissues, Paris VI lectures 2012
  2. Atlas of Xenopus Development G.Bernardini, M.Prati, E.Bonetti, G.Scari (1999)
  3. Nieuwkoop & Faber (Xenbase.org) retrieved on 15 september 2012
  4. Nicolas Pollet's data
  5. Transgenesis procedures in Xenopus. A.Chesneau, L.M.Sachs, N.Chai et al. Biology of the cell (2008)
  6. Techniques and probes for the study of Xenopus tropicalis development. Khokha MK, Chung C, Bustamante EL et al. Dev Dyn. (2002)
  7. The Xenopus Cell Cycle: An Overview, Anna Philpott & P. Renee Yew, Molecular Biotechnology (2008)
  8. Fates of the Blastomeres of the 32-Cell-Stage Xenopus Embryo, S.A Moody, Developmental Biology 122,(1987)