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(The FASEB Journal. 2000;14:823-827.)
© 2000 FASEB

Modeling of mosaic patterns in chimeric liver and adrenal cortex: algorithmic organogenesis?

GABRIEL LANDINI*1 and PHILIP M. IANNACCONE{dagger}

* Oral Pathology Unit, School of Dentistry, The University of Birmingham, Birmingham, B4 6NN, England, U.K.; and
{dagger} Department of Pediatrics and the Children’s Memorial Institute for Education and Research, Northwestern University Medical School and Children’s Memorial Hospital, Chicago, Illinois 60614, USA

1Correspondence: Oral Pathology Unit, School of Dentistry, The University of Birmingham, St. Chad’s Queensway, Birmingham, B4 6NN, England, U.K. E-mail: G.Landini{at}bham.ac.uk


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
If organogenesis were a completely deterministic process, then the amount of information required to store the spatial position and fate of every cell in vertebrate organisms would be larger than the total information that could be contained in their genomes. This suggests that the instructions of developmental mechanisms involved in organogenesis, coded in DNA, must be at least in part procedural or algorithmically based. Chimeric mosaic patterns in rat livers have been shown to be isotropic and to have fractal profiles (D~1.3) whereas adrenal gland mosaics show a less irregular radial pattern, with lower fractal dimension (D~1.2) than in the liver. These findings suggested a possible model of parenchyma generation. We propose that during organogenesis in both liver and adrenal cortex, the same basic mechanism is directed to organ mass enlargement, whereas the differences observed in mosaic patterns between the organs could be due to the control of a single parameter, namely, a form of contact inhibition. Computer simulations in two dimensions returned comparable results in both the fractal dimension value of mosaic patches and appearance of the mosaic ‘tissues’, as observed histologically in chimeras. This suggests that position information and locomotion of cells would not be required to produce the mosaic pattern observed in chimeras.—Landini, G., Iannaccone, P. M. Modeling of mosaic patterns in chimeric liver and adrenal cortex: algorithmic organogenesis?


Key Words: chimeras • cell division • fractals • computer model


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
ORGANOGENESIS IN VERTEBRATES can be studied in the mosaic tissues of experimental chimeras. These are animals produced by either amalgamation of embryonic tissues of genetically distinguishable strains or transgenic methodology, so that later in development cells can be individualized according to their original parental lineages (1 , 2) . The micro/mesoscopic patterns in chimeras may indicate how the parenchyma was generated. The mosaic patterns in the same organs are similar across animals, which suggests that the procedures that govern organogenesis are conserved; however, they are different in different organs, implying that mechanisms may be organ dependent. The mosaic pattern in rat liver characteristically shows ‘islands’ or ‘patches’ of one cell type embedded in the other, with neither characteristic patch size nor anisotropy (Fig. 1A ). It was previously shown that the complexity of the outlines of the ‘patches’ in the liver are fractal (3) and the value of their fractal dimension is consistently ~ 1.3 across locations within the organ and across animals (4) . ‘Fractals’ are objects that exhibit scale invariance, the amount of complexity of some measured physical property (perimeter, density, area) independent of scale (5 , 6) . A measure of such complexity is given by the so-called fractal dimensions. This means that fractal geometry provides a mathematical medium to quantify irregular morphologies.



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  		Figure 1. Binarized images of the autoradiograms of adrenal cortex radial pattern (A) and the fragmented liver pattern (B) (field widths 1.05 and 2.20 mm, respectively). Note the similarities between these images and simulations in Fig. 2 . The chimeras were produced by morula aggregation from PVG rats of two types that are congenic for the expression of MHC class I (PVG-RT1a and PVG). I125-labeled monoclonal antibody R2/15S, which recognizes the MHC class I molecules, was applied to 6 µm thick cyrostat sections. Slides were coated with NTB2 nuclear tract emulsion and exposed at -20°C for up to 7 days. After exposure, the slides were developed in photographic developer to visualize the silver grain accumulation over sites of antibody reaction (1 2 3) .

In the adrenal cortex, however, the mosaic pattern is characterized by radial cords of cells in rats and mice (7 , 8) (Fig. 1B ). The fractal dimension of the outlines of the adrenal patches is lower than those in the liver (D ~ 1.2) (9) . Many methods of constructing fractals rely on iteration and recursion of a small number of rules. ‘Iteration’ means repetition, and ‘recursion’ consists of taking as input the result of the previous procedure; they obviously take place in cellular systems. The fact that mosaic patches are fractal has suggested that perhaps organogenesis may involve iterative mechanisms with simple rules (3 , 4) , which introduces the advantage of minimizing the amount of information storage necessary for correct distribution of parenchymal cells during development. Here we investigated the 2-dimensional version of an algorithmic model of clonal growth to test whether liver and adrenal mosaic patterns may be due to iteration and recursion of a simple set of rules governed by contact inhibition.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Pushing cell model
The pushing cell model is the generalization of the early model of clone growth described by Ransom (10) . Several incarnations of the original model have been used to simulate cell patterning (11 , 12) , chimera mosaics (13) , and pattern fragmentation (14) , but no one investigated alternative patterns (such as those in adrenal cortex) or the inclusion of contact inhibition. In the model, growth takes place in a nonperiodic square lattice, where a small solid cluster of cells or primordium (each cell occupying a lattice site) is introduced. The dividing cells are selected randomly, after a random choice for the division direction. To allocate the daughter cell, a subsequent ‘pushing’ of the neighboring cells is necessary. We introduced a new parameter, f, which represents the maximal force a dividing cell could exert on the tissue to allocate the daughter cell. This parameter is an integer value chosen at the start of the simulation. For each candidate cell to divide, if the number of cells to be pushed in the direction of division (to allocate space for the daughter cell) is larger than f, then no division takes place and another cell is selected. Basically, the iteration of pushings shuffles the tissue, producing a mixing effect, but is limited to an outer cortical zone of width f. The force f may have a number of counterparts in nature, for example, cell adhesion, which would have to be overcome by a certain force. In a real situation this could be determined by atomic force microscopy (15) or micromanipulation (16) .

Although we call this parameter ‘contact inhibition’, similar results could be obtained by different principles such as modulation of nutrient concentration across the tissue, cell signaling, or other mechanisms that are a function of the distance between cells and a source outside the organ mass.

The original model described in ref 10 corresponds to the special case of our model when f = {infty}. A further enhancement to the model was introduced by labeling cells as ‘not movable’ at locations where they could not push their neighbors in any direction. This avoided calculations on those labeled cells, as they would not replicate, and therefore dramatically speeded up the computation in cases with low f. The algorithm of our pushing model in a square lattice can be summarized as follows:

1) User selects the pushing force f of the cells and the proportion (p) of marked cells.

2) Create a primordium and default all cells as ‘movable’.

3) Select a candidate cell for division.

4) If cell is ‘movable’, then go to 5); otherwise go to 3).

5) Select the direction of division. The possible cardinal directions in which division could occur in a square lattice, considering eight neighbors: N-S (or S-N), E-W (or W-E), NE-SW (or SW-NE), and NW-SE (or SE-NW),

6) Determine the direction of least effort. The whole cluster needs to be shuffled to create space for the daughter cell by pushing all cells in the selected direction until the nearest empty space available is found. For example, in the N-S direction, there are two possible positions in which the growth can take place: north or south. Whichever is the nearest to the first vacant position is selected (or randomly chosen if they are the same).

7) Check the pushing force required in the direction of least effort. If the number of neighbors to be pushed is smaller than f, then the cell is allowed to divide and the tissue shuffled; otherwise, it does not divide.

8) If cell does not divide, then check to see whether it could move in any other directions, if it cannot, then label it as ‘nonmovable’.

9) Repeat from 3).

All simulations were started with a primordium of 100 cells (compacted in a square of 10x10). Twenty-five percent of the cells (seeded randomly) were of one parental line and marked in a different color from the rest (P=0.25); other than their color, their behavior was the same as unmarked cells of the other parental line. This percentage was chosen because it produces the largest number of marked clusters when f = {infty} (14) , and so would maximize the data for the fractal analysis. It was shown that for f = {infty} the complexity of the fragmentation is not affected by the relative proportions (parameter p) of the two cell types in either simulation (14) . The fractal dimension of the mosaic patches in the liver was found to be constant, independent of the proportion of marked cells (4) .

Two hundred and eight simulations were performed using f values of 2, 4, 8,16, 32, 64, 128, and 256 (26 simulations of each) and grown to up to 4 x 105 cells.

Fractal analysis
The degree of irregularity of the patches was determined with the so-called ‘area-perimeter relation’ or ‘slit island analysis’ (6) . This method is useful for determination of the fractal dimension of outlines of various sized objects that share the same complexity, using a single resolution or scale rather than investigating each patch separately with a length-resolution method (17) . The areas (A) and perimeter lengths (L) of the patches were measured, plotted in logarithmic scales, and the fractal dimension (D) of the islands’ coastlines was estimated from the expression:

where n is the slope of the linear regression of log(A) on log(L) and

The areas and perimeters were calculated using Freeman’s algorithm (18 , 19) on the marked patches and considering 8 neighborhood connectivity for deciding pixel membership to patches.

Because of a finite size effect of the pixels (Euclidean shapes) on the very small patches, only data included in the largest two and a half orders of magnitude of the perimeter lengths were used (these were called ‘filtered patches’). In addition, data from the individual simulations were integrated in eight groups (called here integrated data) corresponding to the different values f. The fractal dimension for these integrated data groups was also estimated.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
A sample of simulations with 104 cells using the pushing model is shown in Fig. 2 , using various proportions of marked cells (P) and various pushing forces (f) while keeping the same random number generator seed and initial conditions for each simulation.



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  		Figure 2. Simulations with 104 cells with varying values of P (proportion of marked, black, cells) and f (the pushing force). The same random number generator seed and initial conditions have been used in all simulations of this figure. Note the similarities between the patterns generated at low f and that observed in the adrenal (low irregularity, radial sliced cake pattern), and also between patterns generated at high f and that observed in the liver (both in Fig. 1 ).

For small f there was little mixing of the tissue and consequently small numbers of patches. In addition, the outlines of the patches are less irregular than for simulations with larger f, giving a radial ‘sliced cake’ appearance. This feature is completely absent in the simulations with f = {infty}.

The increase in the total number of patches and filtered patches with f is shown in Fig. 3 and the size distribution of patch sizes for all the groups is shown in Fig. 4 . The data approaches a hyperbolic distribution, which is an inverse power law relationship between the number of patches N(a) and patch size a:

in which K is a constant (theoretical number of single cell patches in the simulation) and b is an exponent that describes the decay in number of patches with increasing patch size (b ~ 2). Table 1 shows the distribution parameters derived empirically by regression analysis of the first two orders of magnitude of the integrated data (log transformed and subsequent fitting using the least squares method). Since the regression was estimated for a data set that integrated 26 data sets per group, K' (K/26), which corresponds to the number of single-cell patches per simulation, was also calculated. There are about three orders of magnitude more fragmentation (in number of patches) in the simulations with large f than in the simulations with f = 2.



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  		Figure 3. Number of patches in relation to the pushing force (f). Note the small numbers of patches for low f. The points used to determine D are within the two and a half largest orders of magnitude of the perimeter length (this avoids the finite size effect of small patches). Both axes are in logarithmic scale.



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  		Figure 4. Patch size distribution and the pushing force (f). Note the all sets approach a power law with exponent ~ -2. The dotted reference line has a slope of -2.0.


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  		Table 1. Parameters of the patch number-size distribution (first two orders of magnitude of integrated data)a

The patterns produced not only resemble qualitatively those observed in chimeric liver and adrenal, but the degree of irregularity of the simulated patches also shows the same variability as those observed histologically. The change in fractal dimension of the patches estimated by the area-perimeter relation as a function of f is shown in Fig. 5 . Patches produced in simulations with small f have smaller dimension than those with higher f, with an asymptotic approach to D~1.3.



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  		Figure 5. The fractal dimension of the outlines of marked patches as a function of the pushing force (f). For small f, the dimension is lower (less irregular objects), whereas for f >64, D remains ~1.3 (note that f is in logarithmic scale).


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
Regarding development, a central issue is understanding the different levels at which organogenesis is controlled. Is the process deterministic or procedural? This has interesting consequences, for example, in terms of information storage in DNA coding. In the extreme (and unlikely) case of organogenesis being a fully deterministic process, the information specifying cell positioning would be expected to be stored or predetermined. There are at least two facts that do not seem to agree with this possibility. First, this would make the whole process very prone to error, since a failure of positioning of a single cell at an early stage would carry and amplify the error throughout development. Second, the genome may not be large enough to contain all the positional information for all cells. One possibility is that in liver and adrenal, a very accurate cell positioning may not be crucial, but this would require another type of control—perhaps an iterative algorithm in which some basic strategy is repeated until a critical event is reached.

Chimeras provide an interesting model for studying organogenesis. As the two cell lineages in the organ are ‘not aware’ of their parental line differences, the mosaic pattern geometry can be considered a fingerprint of the mechanisms acting during development.

The mosaic pattern observed in adrenal glands has been shown to correspond to a different class than that in the liver. The adrenal mosaics show little or no fragmentation, with a radial pattern that extends through most of the cortex (7) . Centripetal migration of adrenal cortex cells was first suggested by Graham in 1916 (20) . More recently, Morley et al. (8) , on the basis of mosaic patterns in adrenals of transgenic mice, also suggested that centripetal migration is responsible for such radial patterns. In this study, however, we suggest that the same results can be obtained not by active migration, but by iterative cell division and pushing by daughter cell placement biased by contact inhibition. This invites a new interpretation of the mosaic pattern: a developmental mechanism that requires no guidance, positioning information, or active cell migration during the stages of rapid parenchyma growth and still results in the same type of mosaic pattern.

Perhaps past interpretation of cell migration in the adrenal cortex has been addressed in terms of an implicit renormalization of the organ size. When size renormalization of the growing organ is applied (consider a shrinkage of the organ proportional to the number of cells, so the organ is always the same size), the cells would appear to migrate toward the inner layers of the cortex, although in absolute terms they do not. If no renormalization is considered, then the only cells that seem to migrate relative to the center of the organ are the daughter cells at the outer layers of the cortex.

In the computer model, cell division took place while the force needed to shuffle the tissue and allocate the daughter cell was less than a certain threshold (f). Contact inhibition constrained the tissue to grow only at the ‘edges’, where the number of cells to push was small.

The width of the growth zone is therefore dependent on f, and after the tissue diameter reaches 2 f, the growth rate becomes linear with the diameter of the ‘organ’.

The radial patterns appeared to be due to the inactivity of the central areas (lack of mixing) and division of the more superficial ones, all without a requirement for active migration, guided cell movement, or storage of cell positioning. In the liver simulation, where all cells are able to push their neighbors (despite their number), the model produces constant mixing of the tissue. In this case, the tissue growth is exponential. This realistically reflects the period of parenchymal growth during organogenesis or nonneoplastic compensatory growth (regeneration) after partial hepatectomy (21) . The dynamic nature of the growth model in the adrenal cortex similarly would extend through the parenchymal growth phase and is also a realistic reflection of the embryological development of the organ (8) . Obviously, sustaining exponential growth is only possible for limited periods; solid malignant tumors are known to behave in this manner, after which central inactivity or necrosis of the tissue occurs and growth also enters a linear phase.

In conclusion, the proposed model suggested that the two dissimilar mosaic patterns that develop in chimeric liver and adrenal gland may be closely related in algorithmic terms and that the differences rely solely on a cell division restriction parameter (f). Despite the simplicity of such modeling, the principles considered here appear to have a real counterpart in some cellular systems; there is emerging evidence for genetic control of the probability of cell division (22) and of the position of the daughter cell after division (23 , 24) .

Further mathematical interpretation of the mosaic patterns and model simulations will undoubtedly contribute to a better understanding of the possible mechanisms of regulation and rapid expansion of parenchyma compartments during organogenesis.;1>


   FOOTNOTES
 
Received for publication May 21, 1999. Revised for publication November 30, 1999.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 

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