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doi: 10.1242/10.1242/dev.00118

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Influence of cell fate mechanisms upon retinal mosaic formation: a modelling study

Stephen J. Eglen^*, and David J. Willshaw

Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, Edinburgh EH1 2QL, UK
^* Present address: Department of Anatomy and Neurobiology, Washington University School of Medicine, 660 S. Euclid, St Louis, MO 63110, USA

View larger version (34K): [in a new window]   Fig. 1. Example of lateral inhibition among a small group of cells. (A) Sample of undifferentiated cells (grey squares) created with the exclusion zone model (dmin=15.96 µm, =0.2; A=1 mm2; N=1000). Six central cells are numbered for plots in (C,D). Scale bar: 20 µm. In this case, all-neighbours inhibition (ANI) was used, so cell 1 has five neighbouring cells (labelled 2-6). By contrast, cell 3 has six neighbours, because its Voronoi polygon has six edges. If nearest-neighbour inhibition (NNI) was used, cell 1 would have just one neighbour (cell 2). (B) The same cells once the levels of Delta and Notch have stabilised. Open circles denote primary fate cells, filled circles are secondary fate cells. Cell 1 acquires primary fate, cells 2-6 adopt secondary fate. (C,D) The levels of Delta and Notch during development for the six cells highlighted in A. Time is measured in arbitrary units (a.u.).

View larger version (50K): [in a new window]   Fig. 2. Example outcomes of the all-neighbours and nearest-neighbour inhibition methods. Scale bar: 100 µm. In each plot, only a central region of the simulated retina is shown. (A) An initial population of 1000 cells was created using the exclusion zone model (dmin=11.28 µm, =0.1; A=1 mm2; N=1000). All-neighbours inhibition (ANI) then transformed this population into primary (open circles) and secondary fate (filled circles) cells. Here, the cell count ratio is 2.64:1. (B) As in A, but using nearest-neighbour inhibition (NNI). Here, the cell count ratio is 1.06:1.

View larger version (15K): [in a new window]   Fig. 3. Effect of packing intensity upon the cell count ratio for different cell fate methods. For each of the six packing intensities tested, 20 simulations for each cell fate method were run from different initial populations created using the exclusion zone model (A=1 mm2; N=1000). The symbols for the random and left-right methods have been shifted horizontally so that symbols do not overlap.

View larger version (13K): [in a new window]   Fig. 6. Effect of different cell death methods upon the regularity index and packing factor of the mosaics. An initial mosaic of 400 cells was created using the exclusion zone model (dmin=10 µm, =0.008; A=4 mm2). Cells were removed one-by-one until 40% of cells had been deleted. Starting with the same initial mosaic, all three methods (selecting cells either by smallest Voronoi area, smallest nearest-neighbour distance or at random) were tested. Hence the results for 0% cell death (showing the initial regularity) are the same for each method. Ten different initial mosaics were created, from which the mean and s.d. of the regularity index were calculated for each method.

View larger version (18K): [in a new window]   Fig. 4. Regularity index of the mosaics of undifferentiated cells, and subsequent primary and secondary fate cells, as a function of packing intensity of the undifferentiated mosaic. This figure summarises the same simulations as described in Fig. 3. The symbols for undifferentiated and secondary fate cells have been horizontally shifted slightly so that symbols do not overlap. (A) Effect of all-neighbours inhibition upon the regularity index. (B) Effect of nearest-neighbour inhibition. (C) Effect of the largest one-step method. (D) Effect of the smallest one-step method. The effects of random and left-right methods were similar to C,D and so are not shown here.

View larger version (18K): [in a new window]   Fig. 5. Packing factor of the mosaics of undifferentiated cells, and subsequent primary and secondary fate cells, as a function of packing intensity of the undifferentiated mosaic. This figure summarises the same simulations as described in Fig. 3 with the same conventions as Fig. 4.

View larger version (20K): [in a new window]   Fig. 7. Evaluation of hypotheses for formation of on- and off-centre RGC mosaics. (A) Initial populations of 500 on-centre and 500 off-centre cells were created independently using the exclusion zone model (dmin=10 µm, =0.04; A=1 mm2). Up to 40% of cells were deleted (using the distance method) independently from each population. The mean and s.d. of the regularity index over ten simulations, with different initial conditions, are plotted for just the on- or off-centre cells (open or filled circles) and for the combined population of on- and off-centre cells (grey squares). Broken lines show the range of regularity indexes for the on- or off-centre RGCs, from Table 1. (B) Same as A, but dmin=15 µm (=0.09). (C) As in A, but an initial population of 1000 cells was created using the exclusion zone model (dmin=10 µm, =0.08) and then divided into on- and off-centre subpopulations by randomly assigning each cell to one of two types. (D) As in A, but an initial population of 1000 cells was created using the exclusion zone model (dmin=0 µm, =0) and divided into on- and off-centre subpopulations by nearest-neighbour inhibition (NNI). The dmin value in B was chosen so that the regularity of the on- and off-centre mosaics were similar to those in D before cell death.