1. Topological approach to cell division in tissues
In 1855 Rudolph Virchow postulated that every cell in the organism comes
from another cell through the process of cell division ("Omnis
cellula e cellula"). There is a multitude of observations showing
highly ordered patterns of cell divisions in tissues. However, the question:
How do the cells divide in the organism? remains to be answered.
We know that the laws of Nature appear to us when we find some restrictions,
limitations inherent in the very nature of physical bodies and of the processes
that we are studying. In other words, we can say that laws of Nature actually
represent the predictable outcomes that must occur under these intrinsic
limitations. Do such limitations exist for dividing cells? If so, is it
possible to theoretically construct a tissue with these limitations applied
and to make a computer simulation of cell division? The following limitations,
indeed, exist: 1) the cell divides into two cells, 2) the cells in a tissue
are connected with each other and can not move freely within the tissue.
If we accept that the tissue is not just a sac with cells that can move
around, but an integral structure, we can regard the tissue as a crystal
with the cells as the elements of a crystal. There is, however, a great
difference here. In crystals, new elements are added from the surrounding
solution into the specific sites on the surface of the lattice, but the
new elements in the tissue, the cells, are added by the division of the
existing cells inside the tissue. When cells divide in the embryo, it grows
and becomes a different structure. But, in the adult tissues, their size
and structure are kept unchanged despite of the fact that some of them
renew their cells, i.e. replace the old dying cells with new ones, sometimes
at a considerable rate.
How do these tissues avoid suffering a great number of "defects"
as a result of repeated cell division? It has been shown previously on
a model of the intestinal crypt (Ref. 1) that maintaining permanent structure
and size is possible when the division, although going on inside the tissue,
proceeds from cell to cell by a wave following certain directions in the
"lattice" of a tissue-crystal and obeying some topological rules.
Most importantly, no breakdown of the cell contacts ever needs to occur.
This model, built from purely topological considerations, resulted in periods
between successive cell generations coinciding with the periods found in
the previous model deduced from the principle of the limited number of
cell generations in the organism (Ref. 2).
2. Properties of the crypt of intestinal epithelium
|Photo 1. Intestinal epithelium separated from the underlying tissue. The villi are seen as "caps".
|Photo 2. View from the crypts' side. "Caps" are seen from the inside.||
|Photo 3. A row of the villi connected by crypts is stretched by two needles.||
The animal intestine is lined inside with a one-cell thick layer of tissue
- the epithelial layer. This lining covers a curved surface with hill-shaped
villi, and it also goes, uninterrupted, into the finger-shaped depressions,
forming there the crypts of Lieberkuhn situated around the villi. In the
crypts, we find the fastest dividing cells in the organism. The cells multiply
only in crypts and they move on to the villi and are eventually shed from
the tips of the villi into the intestine. The entire cell layer (which
can be regarded as a two-dimensional lattice) has no gaps; the cells are
tightly packed and even connected with each other by cell-cell junctions.
The photographs (made at different magnifications) show the epithelial
layer separated from the underlying tissue.
One villus is "fed" by cells coming from several crypts surrounding it. One
crypt usually "feeds" two villi to which it is attached on two
sides. The shape of the crypt approximates to a cylinder closed at the
|| || ||
|Photo 4. Two villi with some crypts
|| || ||Photos 5, 6.
Left - a crypt with its nuclei stained.
Right - the same crypt, focus on the dividing nuclei (darker bodies) near the crypt axis.
These are the parameters of the crypt essential to our program:
|Photo 7. Surface of the epithelium of the villis.
1) The size of the crypts and villi is different in various parts of the intestine.
In the small intestine of a rat, the crypt length is about 30 cells and
its circumference is about 22 cells.
2) The period between successive cell
generations is about 11 hours, although it is so only for the cells situated
above the bottom.
3) In the bottom region, there are long-living, rarely
dividing cells, some of which are considered to be the predecessors (stem
cells) of the cells in the crypt cylinder. It is believed that around the
crypt circumference, near the bottom, there are 4 to 8 of these stem cells
present at a given moment.
4) In the upper part of the cylinder and on
the villus the cells do not divide. It is believed that their division
capability is exhausted because they have achieved the terminal stage as
a specialized cell line.
5) Another phenomenon, relevant to our program,
is the cell death observed in the region of the crypt bottom. Although
the data here are not so definite, the death of these cells is believed
to occur quite regularly and is even called a "programmed cell death".
The above parameters remain essentially constant throughout the life of
|A schematic view of the section made along the crypt axis. Continuous row of cells going to the villi is shown.
Our program does not consider the cells on the villi. We also do not consider
in this version the cessation of divisions in the upper part of the cylinder.
Earlier, one mechanism that might account for the cessation of divisions
was proposed (Ref. 1). The program does not take into consideration two
other phenomena: 1) In addition to the main cell type, crypts contain other
cell types differing biologically and having different behavior with respect
to cell division. 2) Before a crypt cell divides into two, it changes its
shape: its surface on the outside of the cylinder becomes quite small and
the cell mass concentrates nearer to the cylinder axis. After the division,
the shape is restored.
We would like to note here that a comparison can be made between the structure
of the crypt and the structure of other proliferating animal and even plant
tissues. The proliferating cells seem to originate near the bottom of a
cylinder or a cup, or near the center of a whorl; they subsequently move
away from the center.
3. Topological presumptions in the program
(Note: For a more detailed explanation, please, consult
Ref. 1 )
This program considers
topological events on the two-dimensional (although, curved) surface of
the cell layer forming a cylinder closed at the bottom. This is an idealized
model: it is presumed that the cells are packed in a hexagonal lattice
with all vertices being 3-rayed. However, to form the hemisphere of the
bottom, we need to introduce six pentagons ("structural" pentagons)
into the lattice. Crypts of any chosen size and shape are built in the
part of the program called Crypt Builder.
The division wave.
Cell divisions are performed in the main program. The
cells divide in a wave. On the shown here screenshot from the program,
the wave proceeds from left to right. The front of the wave is formed by
two cells, a heptagon (blue) and a pentagon (red). The dividing cell is
always a heptagon (its expected division plane is indicated by the manually
drawn dotted line). The wave proceeds along the row of hexagons in such
way that the division plane of the heptagon is directed, on one end, to
the hexagon in front of it (making it a heptagon) and, on the other end
- to the pentagon (making it a hexagon). The wave leaves behind it two
rows of hexagonal sister cells (light green) that gradually grow into regular
Initiation of the division wave.
This program differs from the
previously published model (Ref. 1) in the mode of initiation of the division
waves. In the old model, the initiation was accomplished by cell division
between "structural" pentagons in the bottom. It resulted in
the formation of two heptagon/pentagon pairs initiating two waves. In this
program, we used a different mechanism - the death (removal) of the "structural"
pentagon, which results in the formation of a new "structural"
pentagon from a neighboring hexagon and the formation of one heptagon/pentagon
The first figure (above, on the left) shows how, upon removal of the
pentagon, its borders collapse forming 3-rayed vertices and its space is
taken by the neighboring cells. The number of sides in the neighboring
cells changes: two pentagons and one heptagon are formed.
The second figure shows that when the heptagon starts dividing, it
directs the plane of division to one pentagon, leaving the other pentagon
as a "structural" pentagon replacing the one that was removed.
The fates of these pentagons, of course, could have been reversed, had
the heptagon directed the division plane toward the lower pentagon, and
the resulted division wave would have taken a different direction. Moreover,
when the pentagon was removed, the 3-rayed vertices could have been formed
in five different configurations, resulting in the appearance of the heptagon
in five different places. With these choices combined, ten different division
waves are possible.
The two figures below show this division wave after it passed one and
two divisions respectively:
Other situations where a pair heptagon/pentagon appears as a point
of initiation are conceivable, although, in the present version of the
program the initiation is restricted to the removal of a pentagon.
Therefore, the program operates with six potential "dying"
cells ("structural" pentagons) and six "stem" cells
(the appearing heptagons), all situated near the crypt bottom.
The program allows maintaining the crypt diameter and its structure
in the steady state indefinitely. It shows which division waves are needed
for the steady state proliferation and which waves will lead to the growth
of the crypt diameter or to various crypt distortions.
1. Pyshnov, M. B., J. theor. Biol. (1980), v. 87, 189-200.
2. Pyshnov, M. B. and Reggirt, S. A., J. theor. Biol. (1977), v. 68,
247-257. (Reggirt S. A. was, at the time, the nom de plume of S.
A. Trigger. S. A. Trigger is also actively participating in the organizational
part of the work on this program, as well as in the scientific discussions.)