Models for the orientation of chemotactic
cells and growth cones;
the formation of netlike structures
From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company
A chemotactically-sensitive cell has to detect minute
concentration differences to orient the extension of pseudopods
Activation
a s a 2
2a

 ra a  Da 2  ba
t
b
x
Inhibition
b
 2b
2
 s a  rb h  Db 2
t
x
External asymmetry => s
2% + max 1 % random fluctuation
Pattern-forming systems can provide this sensitivity. At the homogeneous
steady state, these systems are instable. Minute external influences determine
the position at which a maximum will form. However…
Problem: re-orientation of a polarized cell
… once the pattern is formed, even a much stronger external asymmetry is
unable to re-orient the internal polarity, as shown in the simulation above. Thus,
the problem in chemotactic orientation is not only to provide the initial sensitivity
but to maintain this sensitivity during the navigation.
A possible solution of the re-orientation problem:
the activation oscillates
Meinhardt and Gierer (1974)
J. Cell Sci. 15,321-346
If the antagonist – in the simulation above an inhibitor - has a longer half-life than
the activator, the activation oscillates. Such system enters periodically into a
sensitive phase, in which small asymmetries are decisive. The sensitivity is
maintained: after changing the external gradient, the next peak forms at the new
side.
Such a mechanism would predict that there are only short time windows in
which the system is sensitive. Experiments indicate, however, that the cells
maintain their sensitivity permanently. Moreover, more than one pseudopod can be
formed at a given time…
Observations show that several protrusions may exist
simultaneously in one cell. They appear and disappear not in
synchrony
Multiple peaks would be formed if the range of the antagonist does not cover
the whole cell. However, …
…then, a maximum also can appear at the side disfavored by the external
cue. The cell would be unable to detect an external gradient. Thus, for
chemotactically sensitive cells, multiple peaks cannot result from a limited
range of the antagonist.
The role of saturation of autocatalysis in the generation
of multiple peaks
With a range of the antagonist that covers the whole cell,
without spread of the self-enhancing reaction and without
a saturation of the autocatalysis, the peak would be very
narrow. Moreover, the location of the peak would be locally
fixed.
a
s a2

....
t h (1  sa a 2 )
With saturation (sa > 0):
the peak is broader …
…and random fluctuations
can lead to multiple peaks
(without signal: Multiple
peaks at random position)
Permanent sensitivity and multiple signals can be generated
by a system with a saturating self-enhancement, in which the
newly formed signals become subsequently quenched
J. Cell Sci. (1999)
112, 2867-2874
Since the total extent of the activated area is regulated, the disappearance of
some signals provides the opportunity that signals can be generated, possibly at
an updated position.
The equation used:
s ( a 2 / b  ba )
a
2a

 ra a  Da 2
2
t
( sc  c)(1  sa a )
x
n
b
 rb  a / n rb b
t
1
c
 bc a  rc c
t
The equation describes the interaction between an autocatalytic activator a, the
rapidly distributed inhibitor b and a local inhibitor c. The inhibitor b, responsible for the
cell-wide competition, is assumed to equilibrate so rapidly that the concentration can
be described by averaging (the sum-symbol in the second equation, n = number of
spatial elements; “cells”). The external gradient is assumed to modify the ability of
the regions at the cell cortex to perform the self-enhanced reaction. It is subsumed in
the space-dependency of s (blue in the preceding simulations). The simulations are
made on a circle resembling the spatial elements of the cell cortex
(J. Cell Sci 112, 2867 (1999)
Netlike structures
Filamentous branching structures are frequent in all higher organism. The figure above
shows three nerve cells in the brain of a fly (left), blood vessels on a chick embryo
(center) and tracheae of an insect. Filaments may consist of long extended single cells
or of filamentous arrangement of many cells.
Veins of leaves
Formation of a netlike structure:
a trace behind a shifting signal
Differentiation 6, 117-123 (1976)
Assumed is an activator
- inhibitor system. A high activator concentration
leads to the differentiation of the exposed cell . Differentiated cells remove a
substance
from the surrounding cells. Since the activator production
depends on this substrate, the activator maximum becomes quenched in newly
differentiated cells and shifts toward a region of higher substrate concentration.
Long filaments of differentiated cells are formed behind wandering activator
maxima.
If the moving tips of the filaments become sufficiently remote and enough
space is available, a baseline activator production in the differentiated cells can
trigger a new maximum – the initiation of a branch.
Formation of a netlike structure:
a trace behind a shifting signal
Differentiation 6, 117-123 (1976)
Assumed is an activator
- inhibitor system. A high activator concentration
leads to the differentiation of the exposed cell . Differentiated cells remove a
substance
from the surrounding cells. Since the activator production
depends on this substrate, the activator maximum becomes quenched in newly
differentiated cells and shifts toward a region of higher substrate concentration.
Long filaments of differentiated cells are formed behind wandering activator
maxima.
If the moving tips of the filaments become sufficiently remote and enough
space is available, a baseline activator production in the differentiated cells can
trigger a new maximum – the initiation of a branch.
Net-like structures – the same simulation in
another plot
Signalling for elongation: the activator
(Delta/Notch in the case
of blood vessels)
Signalling for elongation: the inhibitor
Substrate or trophic substance
(VEGF in blood vessels,
auxin in plants)
Differentiation
Regeneration of a net-like structure
Differentiation 6, 117-123 (1976)
After removal of some veins, e.g., by an injury, substrate accumulates in the
deprived region. This attracts new veins. The resulting pattern is similar but not
identical
Control of vessel density
more VEGF
higher baseline
inhibition
Differentiation
6, 117-123 (1976)
Regulation of the density of a net. Tumors attract new blood vessels and cause an
extensive sprouting. A piece of tumor tissue grafted into the cornea of a rabbit
cause a massive invasion of blood vessels (redrawn after Folkman, 1976). A piece
of cartilage - a tissue that repels blood vessels - largely suppresses such invasion
when grafted in front of the tumor. Left: in the model an increase of level of trophic
factor (green; in blood vessels it is the Vegetal Endodermal Growth Factor, VEGF)
in the upper half leads to higher rate of branching. The influence of the cartilage is
simulated by an increased of a baseline inhibitor production in the center of the field
(shaded). Veins preferentially circumvent this area.
Path finding towards a target region
A local source of the trophic substance leads to a directed
extension of a filament towards the source region. Branching
occurs preferentially in this target area;
An open problem: the formation of closed loops
Since in this mechanism filaments are elongated into regions not yet
sufficiently supplied by veins, there is no inherent tendency to make
connection with other filaments. Closed loops are, however, common in
leaf venation of higher plants.
The plant hormone auxin has many properties of the depleted
substrate - the trophic factor - in the model but the active transport of
auxin in vein patterning is not yet included.
Thus, according to the model, for net-like structures we need:
1. A signal that determines at which position and in which direction the filament
should be elongated or where to initiate a new branch (e.g., an
activator-inhibitor system; Delta/Notch in the case of blood vessels)
2. The elongation of the filaments depends on a trophic factor; it is removed by
the filaments (auxin, NGF, VEGF); elongation goes up-hill
3. An irreversible determination that makes the filament different from the
remaining cells
a
sca 2

t
b
ra a  Da  a ba d
b
 sca 2 rbb  Db b bb d
t
c
 bc  rc c  cc c d  Dc c
t
rd d 2
d

t
1  sd d
rd d
bd a
An alternative way to generated closed loops:
a patch-forming system inhibits a stripe-forming
system, and vice versa:
Stripes are formed at a distance to the patches
Koch and M., (1994) Rev. Modern Physics 66, 1481
Also a net-like structure with closed loops: the
vein pattern of insect wings…
…is still waiting for modeling
Conclusions:
Intracellular pattern-forming reactions allow the generation of signals for
pseudopod formation within a cell. The cells can obtain a high sensitivity
for minute concentration differences imposed by external cues.
A peaks can be quenched by a second short-ranging but long -lasting
antagonist, allowing the formation of new signals for pseudopods,
possibly at an updated position. In this way, a cell can maintain its
sensitivity.
According to the model, the dynamic signalling continues even in the
complete absence of external signals, as it is observed.
Local signals can lead to a local elongation. The result are filamentous
branching structures as they are common in all higher organisms. The
mechanism allows the regeneration of net-like structures after partial
removal. Examples of the postulated trophic factors have been identified
(VEGF, Auxin). In blood vessel formation the predicted involvement of a
lateral inhibition system to specify the tip cells is realized by the
Delta/Notch system. The formation of closed loops in plants is still an
open problem.