Team:Montreal/Modeling

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Revision as of 19:29, 2 August 2008

Modeling

Theorists: Vincent Quenneville-Bélair and Alexandra Ortan

Background

Using coupled differential equations, we are modeling the Repressilator, which is a network of three genes, whose product proteins are repressing each other's growth. This cycle is taking place in each of a colony of cells, who communicate amongst themselves by exchanging an autoinducer molecule (AI, or PX here). The model attempts to take into account a sparse, heterogeneous distribution of many cells (grouped in clusters) with depletion of the autoinducer molecule and leakage while assuming continuous levels of the different proteins, as well as a continuous spatial distribution of these proteins.

For this purpose, the model is currently being coded up in Mathematica. The simulations, based on that continuous model, are generated using xCellerator and NDelayDSolve for different cell configurations (fig. 1); the results obtained are graphs of the concentration of molecules in the system versus time (fig. 2). That is, our interest lies in understanding the behaviour of the clusters with respect to each other (for instance, the phase difference or the synchronization time as in fig. 4-5). As of now, a low number (~4) of cells in two clusters is being used for testing, but a higher one (>100) in more clusters will be reached later.

Journal

The Mathematica notebook used to model the repressillator incorporates xCellerator to create the differential equations representing the chemical reactions at play. The system was then solved using NDSolve, a standard numerical solver in Mathematica -- which is, however, unable to deal with delay equations. Hence, the equations were not evaluated at retarded times at first. Indeed, our first approach was to add null cells in the network. The nulls cells, or pseudo cells, are just grid point where no internal cell reaction occurs, but only diffusion to the next grid point. However, the results depended to heavily on the number of such null cells. As we did not have a mean of estimating a realistic distribution of such cells, they did not seem reliable enough. Furthermore, null cells were taking similar computing time as of interesting/standard cells -- eating away the computational resources available.

We are now replacing NDSolve with NDelayDSolve to solve the system and we are therefore able to evaluated our equations at retarded times. We are not using null cells anymore. The value of the delays corresponds to the time the molecules of AI diffuse the intercellular distance: the reasoning being that each cells effectively "sees" the concentration of another cell at an earlier time -- the time the molecules take to diffuse between them. We have one worry for now with using NDelayDSolve rather than NDSolve: Mathematica warns that it is not able to reach the prescribed accuracy/precision during the computation. When checking the compatibility of the solution found with the equations, we find "reasonable" errors most of the time, but periodic (roughly at the same time as the peaks as seen in fig. 2) sudden kicks to high discrepancies (fig. 6).

For a given distance between the two clusters of cells (fig. 1), we plot the concentration of AI in each cells versus time. From there, we extract two informations: synchronization time and phase difference (fig. 4-5). The general trend in fig. 4 seems to be that the longer the distance, the longer it takes to synchronize, but we also noticed that after a certain distance, the two clusters are no longer in synchrony. Unless they just take much longer to synchronize than the windows of the graph -- in which case we would need to run the simulation for a longer time to observe synchronization. As for the graph of the phase difference (fig. 5) made to different scales, we have yet to figure out what could be the pattern. The results are also presented in the form of parametric plots involving the concentration of AI (fig. 3) and some short movies can be generated to show the spatial distribution of the concentration of AI as time moves forward (fig. 2c).

Figure 1a: This is a Voronoi graph representing the cells in the model. However, he cells are represented internally by points as in fig. 1b.	Figure 1b: The cells are represented in the model as points.	Figure 2a: Simulation using Mathematica of the Repressilator Network with four cells shown in fig. 1. The plot shows the concentration of AI against time (in arbitrary units).	Figure 2b: Similar to fig. 2a, but with a different distance between the clusters.
Figure 2c: Movie showing the spatial distribution of the concentration of AI as time goes forward.	Figure 3a: The concentration of AI of two cells in different clusters against each other.	Figure 3b: The concentration of X and AI in a given cell.	Figure 4: Time taken to have the two clusters synchronized (error of 5) versus distance (in arbitrary units).
Figure 5a: Phase difference between the clusters versus distance between the clusters with step 1.	Figure 5b: Similar to the other fig. 5, but with step 2.	Figure 5c: Similar to the other fig. 5, but with step 10.	Figure 6: Checking the compatibility of the solution with the equations. Ideally, the graph would be identically zero -- meaning that the solution agrees everywhere with the equations.

Future Goals

Eventually, if everything goes smoothly, we will use more realistic rates for the various reactions: some might be found in the literature, others will be experimentally measured. If the processing power is not too high, we will aim at increasing the number of cells and clusters to something more spectacular. Furthermore, we will attempt to do a parameter analysis over the various rates, so that we will try to find the critical values for which peculiar behaviour happens in the system of clusters.

References

• xCellerator: http://xlr8r.info/

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