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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 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 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

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 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 )

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)	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

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.

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.
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

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 vitalin and could not diffuse in the first mitosis. 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. Here the coordinates, for example 1.2.2 mean the cell who is the "second" daughter of the second daughter of the first cell. Her sister is therefore 1.2.1 and her own offspting will be noted 1.2.2.1 and 1.2.2.2

Extracted from Fates of the Blastomeres of the 32-Cell-Stage Xenopus Embryo

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

Plasmid distribution std=0.05

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.

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.

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