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Introduction

In many areas in biology information about individual behavior can only be acquired by observing the average output of many individuals. For example, to study the temporal sequence of events in the cell cycle (e.g. Creanor & Mitchison 1994, Stuart & Wittenberg, 1995), measurements are often made on cultures of synchronized cells. Since the degree of synchrony may vary from one experiment to another, drawing conclusions from these average values of different cultures might lead to contradicting pictures of the cell cycle. It would be more desirable if the variability in a sample could be reduced. One approach is to develop even better synchronization methods. Alternatively one can try to synchronize the experimental output using a mathematical filter.

Creanor & Mitchison (1982, 1994) have put forward two such mathematical filters. Both their methods require a time series of experimental data for the cell cycle trait to be studied and information on the distribution of the cells over the cell cycle. Assumptions made for both methods are that the variability measured at a particular stage in the cell can be carried over to the starting point of the event studied. In other words this assumption states that progress through the cell cycle is deterministic. Here we put forward two other filtering methods which have the same requirements and the same assumptions. In contrast to the methods put forward by Creanor & Mitchison we do not assume any form of the kinetic function for the individual trait. In the methods presented here the distribution function for the cells over the cell cycle is used to transform uniquely the observed average kinetic data to the kinetics of an individual cell.

To show how these methods are used we apply them to data on CLN2 transcription in budding yeast cells from Stuart and Wittenberg (1995). In budding yeast, progress through the cell cycle is controlled at the G1/S boundary by processes involving G1-cyclins CLN1, CLN2 and CLN3. If all those cyclins are absent, cells cannot leave G1 and proliferation stops. If any one of those cyclins is available, cells are able to divide, but in all cases there is a noticeable change in cell cycle dynamics. CLN3 is a powerful inducer of CLN2 expression while apparently CLN1 and CLN2 are not, since CLN3-deleted cells need to grow twice as big before they leave G1. In the experiment (fig. 1-A) a strain of budding yeast cells with only CLN2 present is compared with a strain lacking the function of all CLN's. No noticeable difference in CLN2 expression is observed. However, as seen from the time series of the fraction of budded cells (fig. 1-B) the CLN2 strain undergoes division, while the strain with non-functional CLN's does not show any mitotic activity. It is precisely this time series of budded cells which shows the lack of synchrony, thereby supplying the information about the actual distribution of cells at time of budding. We fed approximations of this distribution to our two synchronization methods to estimate the time-course of CLN2 expression in this strain (fig. 1-C). Both methods and all approximate distributions clearly show an enhancement of CLN2 transcription over that in the strain without functional CLN's, a conclusion that cannot be drawn from the unprocessed data. Stuart and Wittenberg repeated the experiment with cells that grew much slower and observed a clear difference in CLN2 transcription between both strains (fig. 2), just as we conclude from our analysis of their original experiments.

In the analysis performed here, there is however one problem (indicated by Wittenberg, personal communication) we can not do away with. We do not know whether we can leave the data of the strain with non-functional CLN's untransformed. We did so while we have no information on the distribution of the cells and assume that the output measured for these cells is the same for all cells. However, the cells do chance in their constitution over time, and synchrony might not be perfect either. If we assume that the distributions for both strains are similar, transformation does not change the original outcome of the experiment.

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Figure 1:

   figure29
Figure 2:

The two methods we use differ in the way cell cycle variability is explained. In the first method, we follow a cohort of cells whose variability arises from incomplete synchronization at the start of the experiment. Each individual cell is assumed to follow a deterministic trajectory through the cell cycle, so we can represent its current state with its age. Since all cells move deterministically the initial age distribution of the cells will retain it shape until division. The measured output is always the average of the output of all cells.

In the second method initial synchronization is assumed to be perfect but there is an exponentially (or uniformly) distributed stochastic process for entering a specific stage of the cell. In this case the measured output is from cells having entered this specific stage.

In the following we will first introduce both methods. Thereafter we will present more detail how we transformed the Stuart & Wittenberg experiments, and will discuss application of the methods on data from Creanor & Mitchison (1994). Finally, in appendix B we show, using theoretical examples, how simmilar and how different the results for each transformation method approach can be.


next up previous
Next: The methods Up: Estimating the behavior of Previous: Estimating the behavior of

John Val
Mon Oct 14 15:36:06 EDT 1996