## Abstract

Cells in the neurogenic region of an insect ectoderm have two alternative fates, making neurons or epidermis. The fates seem to be determined through a laterally inhibitory interaction among cells. That is, initially homogeneous cells are all competent to differentiate into neuroblasts. Once a cell has differentiated as a neuroblast, it inhibits its immediate neighbors from following this pathway. The differentiation process is simulated by a digital computer in a planar array of polygonal domains similar to a cell pattern. We find that the number of cells differentiating as neuronal precursors in insect neurogenesis is that expected under the hypothesis of lateral inhibition of cell differentiation between immediate neighbors.

## Introduction

It has often been proposed that lateral inhibition of cell differentiation mediates the development of a pattern of heterogeneous cells from a set of homogeneous cells (e.g., Wigglesworth, 1940; Mitchison and Wilcox, 1972; Korn, 1981). Recent work on early neurogenesis of insects has demonstrated that the membrane-bound proteins at cell surfaces mediate an inhibitory interaction among cells, with interactions limited to immediate neighbors (Hartenstein and Campos-Ortega, 1985; Doe and Goodman, 1985a,*b*; Technau and Campos-Ortega, 1987; Campos-Ortega, 1988). Once a cell in the neurogenic region has differentiated as a neuroblast, its immediate neighbors are inhibited from taking on the same fate and then enter epidermogenesis. The number ratio of epidermoblasts to neuroblasts in insect neurogenesis is reported to be between 3 and 4 (Hartenstein and Campos-Ortega, 1984; Doe and Goodman, 1985b; Campos-Ortega, 1985). The rationale underlying this ratio has not been examined.

The ratio could be estimated theoretically by computer simulation, with the assumption of lateral inhibition between immediate neighbors, if the cell arrangement and sequence of differentiation are defined. A computer simulation has been performed (Tanemura *et al*. 1991) and, as a result, the number ratio of inhibited cells to inhibiting ceils was found to be 3.2±0.11 (mean±standard deviation, averaged over simulations for 10 polygonal patterns with 30 repetitions per pattern, using a different series of random numbers in each repetition.) This simulation was undertaken in a square including 500 cells, with a periodic boundary condition. That is, the square including 500 cells is periodically repeated side by side in tessellation and the area simulated was considered to extend indefinitely without boundaries, unlike the actual neurogenic region. However, in the present simulation a more realistic differentiation area and boundary condition were used.

## Methods

### Polygonal patterns

Polygonal patterns are made by the method of Voronoi or Dirichlet geometry (Tanemura and Hasegawa, 1980; Honda and Eguchi, 1980; Honda, 1983). Disks of radius *r* are sequentially distributed at random in a rectangle (about 4:7) until there are 150 disks, where disk radius *r* is defined so that the total area of 150 disks is just half of the rectangle area (i.e., packing density=0.5). Series of random numbers are used to make x- and y-coordinates for random distribution of the disks. If a new disk overlaps with any disks already situated during the distributing process, the trial of positioning of the disk is discarded and another disk is tried with new random coordinates. The repetition of the above-mentioned procedure distributes the disks. Centers of disks are called Voronoi centers. The radii of disks (circles) are increased to make Voronoi domains, for which the domain boundaries are defined as intersecting lines between neighboring pairs of circles. The procedure to make Voronoi domains was performed with a computer program (Tanemura and Hasegawa, 1980; Tanemura *et al*. 1983).

### Boundary conditions

### Cell picking

Cells to be tested were picked in a polygonal pattern. If the chosen cell had not differentiated as a neuroblast and had no differentiated cells among its immediate neighbors, its differentiation was switched on. This procedure was iterated until all cells that were allowed to differentiate, according to the lateral inhibition rule, had done so.

In the simulation of Fig. 1A, a cell was picked at random using a series of random numbers. On the other hand, the procedure used in the simulation shown in Fig. IB and 1C involved picking cells closest to a boundary, as follows. A cell position is represented by a corresponding Voronoi center (*x,y*) where x-axis (*y*=0) and y-axis (*x*=0) are the lower and the left boundaries in Fig. 1, respectively. A distance of a cell from the left boundary is defined as *x*, and that from the right boundary is as (*x*right ^{—}*) where *x*=*x*_{right} is the right boundary. Cells are sequentially picked that are the smallest distance from the left boundary in Fig. IB, and the smallest distance from the left or right boundaries alternately in Fig. 1C regardless of their y-coordinate. The sequence of picking cells along the y-coordinate becomes random in order in Fig. IB and C, because there is little correlation between the *x*- and *y*-coordinates for the positions of randomly distributed cells beyond the distance of disk diameter.

## Results and discussion

A neurogenic region in a hemisegment of a grasshopper embryo consists of about 150 equivalent ectodermal cells and is rectangular, with widthilength about 4:7 (Doe and Goodman, 1985a,*b*). The anterior and posterior margins contact the neurogenic regions in neighboring anterior and posterior hemisegments, respectively. The lateral and medial margins contact the non-neurogenic ectodermal regions of the same hemisegment, the lateral ectoderm and the ventral midline ectoderm (or the mesectoderm in *Drosophila)*, respectively. In the computer simulation, a periodic boundary condition was used for the anterior and the posterior margins as in previous simulations. On the other hand, cells adjacent to the lateral and the medial margins were assumed not to differentiate as neuroblasts. That is, the area simulated was considered to extend indefinitely along the anteroposterior axis, but have boundaries at the lateral and the medial margins. Patterns of 150 polygons were generated as described in Methods and the simulation was performed with random cell-picking. Fig. 1A shows one of the results. The number ratio of epidermoblasts to neuroblasts is 3.9 on average (Table 1).

Observations of neurogenesis in grasshopper embryos showed that 30 neuroblasts form among about 150 equivalent ectoderm cells in a neurogenic region of each segment (Doe and Goodman, 1985b), i.e. the number ratio of epidermoblasts to neuroblasts is about 4 (= [150–30]/30). The value corresponds to that generated by the present computer simulation.

The sequence of neuroblast formation in grasshopper embryos is not in an anterior-to-posterior or medial-to-lateral gradient (Doe and Goodman, 1985*a*). It seems more likely that the neuroblast formation is spaced equally. Thus we picked cells at random in the computer simulation. On the contrary, in *Drosophila* neurogenesis, the segregating neuroblasts first form two longitudinal rows near lateral and medial margins, respectively, following the formation of the intermediate row with a few cells irregularly scattered (Harten -stein and Campos-Ortega, 1984). Therefore, in a subsequent computer simulation cells were not selected at random. Rather, the state of differentiation was tested first for a cell closest to the left boundary, second for a cell second-closest, and so forth. That is, the decisions of differentiation took place from left to right in Fig. IB. (Along the anteroposterior axis, the decision took place automatically at random as described in Methods.) The number ratio of epidermoblasts to neuroblasts was 3.3 (Table 1). Thus the number of neuroblasts was greater with sequential differentiation than with random selection. Other computer simulations gave almost the same number ratio if the decisions of differentiation took place alternately from both (left and right) sides toward the center (Table 1).

Observations of neurogenesis in *Drosophila* embryos showed that the neurogenic region extending over segments contains about 1800 cells and 440 of these segregate as neuroblasts (Campos-Ortega, 1985), i.e. the cell number ratio is about 3.1 (= [1800 – 440]/440). It is not possible to compare this value in detail with that of grasshopper neurogenesis, because these values were not measured for this purpose. However, we can conclude that our computer simulations with random and non-random tests show that observed cell number ratios are those expected under the hypothesis of lateral inhibition of ceil differentiation between immediate neighbors.

The sequence of segregation of neuroblasts in *Drosophila* seems to be correlated with location in the neurogenic region (Hartenstein and Campos-Ortega, 1985). The fate determination from both sides is observed in *Drosophila* neurogenesis. The simulation with determination from both sides (Fig. 1C) produced fairly regular longitudinal rows of cells, as shown.in Fig. 2, which resemble the pattern of segregating neuroblasts observed in the neurogenesis of *Drosophila*. Fig. 2 shows that the lateral inhibition hypothesis determines not only the number of neuroblasts, but also the cell pattern simulates partially during neuroblast formation in *Drosophila*.

## ACKNOWLEDGEMENTS

We thank Professor Masatoshi Takeichi (Kyoto University, Kyoto, Japan) for information on the insect neurogenesis, and Dr Jay E. Mittenthal (University of Illinois, Urbana, USA) for critical reading of the manuscript.

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