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

2 2

2a a a

a a ar a

x

sD b

bt

22

2b b

b br h D

t xas

Activation

Inhibition

External asymmetry => s2% + 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.

With saturation (sa > 0):the peak is broader …

…and random fluctuations can lead to multiple peaks

(without signal: Multiple peaks at random position)

2

2....

(1 )a

a a

t s a

s

h

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:

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)

2 2

2 2

( / )

( )(1 )a

a ac a

a b ba ar a D

t s c s a

s

x

1

/n

b b

br a n r b

t

c c

cb a r c

t

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

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.

more VEGF

higher baseline inhibition

Differentiation 6, 117-123 (1976)

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:

2

a a a a

a scar a D b d

t b

2

b b b

bsca r b D b b d

t

c c c c

cb r c c c d D c

t

2

1d

d dd

r ddr d b a

t s d

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

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.