NY

LA

NY

LA

NY

LA

Fig. 6.6 Schematic representation of the percolation transition. Top panel: Depiction

of the intact land-line telephone network between Los Angeles (LA) and New York

(NY). The straight segments may denote cables or optical ¬bers. The heavy brown lines

indicate an interconnected percolation network that extends from one end (i.e. LA) of

the system to the other (i.e., NY). Middle panels: If the connections are gradually

destroyed, contact between LA and NY will be maintained as long as a single connected

path exists between the two cities, that is, the number of connections is above the

percolation threshold. Bottom panel: Below the percolation threshold no connecting

path exists and no communication between the two cities is possible, although local calls

(depending only on small clusters) may still go through.

(˜˜elements”) of varying ¬nite lengths, is interconnected, and has the

appearance of being random. It also has redundancy: the signal from

LA can arrive in NY in a number of ways. If this network is subject to

a series of disasters, more and more of the links being randomly de-

stroyed, connection between the two cities would still be maintained

6 MESENCHYMAL MORPHOGENESIS 145

up to a critical number of ¬nite elements (although the time for the

signal to arrive will typically increase). The telephone service is def-

initely disrupted below a threshold where an interconnected cluster

of connections extending from LA to NY no longer exists, although

many ¬nite clusters of cables may still be present.

As the above example illustrates, the percolation transition is ac-

companied by important changes in the connectivity of the system,

a well-de¬ned topological characteristic. This de¬nes the percolation

transition as a phase transition, in the course of which the system

transforms from the non-interconnected phase onto the intercon-

nected one. The transition takes place at a critical point or ˜˜perco-

lation threshold.” In other systems that exhibit percolation transi-

tions, changes in topology may lead to drastic alterations in physical

properties. For example, above a critical concentration, elementary

conducting metallic islands, randomly distributed in an insulating

matrix, interconnect and the macroscopic conductance of the system

becomes ¬nite: it is capable of transmitting electric signals (Clerk

et al., 1990). In another example, the elastic subunits in an otherwise

inelastic amorphous medium interconnect above the threshold con-

centration. As a consequence, the system develops macroscopic elas-

tic properties and responds to mechanical signals (Nakayama et al.,

1994).

During ¬brillogenesis, type I collagen, in connective tissues and

in the MDT experiment, also undergoes such a transition, referred to

as the ˜˜gelation transition,” which is the basis of the phase behavior

discussed above. We will now describe the physics of such transitions

in terms of the percolation model (de Gennes, 1976a, b).

This model stipulates that gelation (the process in which an origi-

nally liquid system -- a sol -- with ¬nite viscosity and no elastic modu-

lus transforms into a different type of material -- a gel -- with in¬nite

viscosity and ¬nite elasticity) in a ¬lamentous macromolecular sys-

tem such as a collagen matrix is due to the gradual interconnection

of growing ¬bers. The state of the network can thus be characterized

by p, the number of connections formed between ¬bers, normalized

in such a way that when all the ¬bers are connected p = 1. For suf¬-

ciently small p only isolated small clusters of connected ¬bers exist.

However, at a threshold value p = pc , which de¬nes the sol--gel tran-

sition, the interconnections between isolated clusters lead to a con-

tinuous (˜˜spanning”) network. Percolation theory predicts power-law

behavior for macroscopic physical properties such as the elastic mod-

ulus or viscosity in the vicinity of the gelation point. Such behavior

is characteristic of scale-free systems (see Box 2).

In particular, the power law for the static elastic modulus or

Young™s modulus, E , (de¬ned in Eq. 1.7) is

E ∝ ( p ’ pc ) f , (6.1)

which is valid for p > pc . The corresponding power law for the zero-

shear viscosity, · (de¬ned in Eq. 1.2), is

· ∝ ( pc ’ p)’k , (6.2)

146 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

Box 6.2 Power-law behavior and scale-free networks

The power-law behavior of a physical quantity signals that the system it charac-

terizes is scale-free. Scale-free networks, of which percolation networks represent

one example, are ubiquitous across many disciplines (Barab´ si, 2002). The charac-

a

teristic power-law behaviors discovered and observed in these networks re¬‚ect

the fundamental self-organizing property of the underlying systems.

Why do power laws imply scale-free character? To explore this, let ξ represent

the size of a system in terms of length, and let a physical quantity f characterizing

the system be a power law function of ξ with characteristic exponent s:

f = C ξs. (B6.2a)

Here C is a constant (i.e., it is length- or scale-independent). Length, and thus ξ ,

can be measured in various units: microns, feet, meters, etc. Let us change the unit

of length. For example we can initially use a measuring tape in inches and then

replace it with a tape in centimeters. If ξ = A 1 l 1 = A 2 l 2 , where l 1 and l 2 are the

two different units of length and A 1 and A 2 express the magnitude of ξ in terms

of these units (ξ = 2.54 cm = 1 inch) then the change of units leads to

f = C A s l 1 = C 1l 1 = C A s l 2 = C 2l 2 .

s s s s

(B6.2b)

1 2

Here the C i = C A is (i = 1, 2) are still constants and thus f has the same power-law

dependence on length no matter what tape is used to make the length measure-

ments or, in other words, at what scale the system is studied. Note that if, for

example, f depends exponentially on ξ then changing measuring tapes leads to

f = C (el 1 ) A 1 = C (el 2 ) A 2 and the functional dependence on length is modi¬ed

(i.e., the power of the exponential would change). This example demonstrates an

important implication of power-law behavior: the system “looks” similar under any

magni¬cation, i.e., it is scale-free. This is literally the case for fractals (see Chapter 8);

these are systems (the convoluted shoreline of Norway is a celebrated example)

whose local geometry is identical no matter what strength of magnifying glass is

used to look at them (Mandelbrot, 1983).

which holds for p < pc . The values of f and k (which are positive

quantities and are called ˜˜critical exponents”) are either measured or

determined theoretically; in the latter case they will depend on the

speci¬c model used to describe gelation (Brinker and Scherer, 1990;

Sahimi, 1994).

In the course of gelation the system exhibits complex viscoelastic

behavior (see Chapter 1). Its viscosity and elasticity are both evident

when it is probed at various time scales. This is typically accomplished

using a viscometer with a cone or plate immersed in the system and

oscillating with frequency ω. The frequency-dependent response of

the evolving gel is measured in terms of viscoelastic moduli, the stor-

age modulus G (ω), and the loss modulus G (ω) (Fung, 1993). These

quantities are related to the Young™s modulus and the viscosity in the

limit of vanishing frequencies via E = lim G (ω) and · = lim [G (ω)/ω].

ω’0 ω’0

Thus E and · can be determined by measuring G (ω) and G (ω)

at progressively smaller frequencies and extrapolating to ω ’ 0.

6 MESENCHYMAL MORPHOGENESIS 147

Gelation in collagen solutions, as discussed here, is a time-dependent

phenomenon driven by the gradual interconnection and entangle-

ment of growing ¬bers. (Sol--gel transitions can also be driven solely by

changes in temperature, using existing macromolecules, as in the de-

natured counterpart of collagen, gelatin (Djabourov et al., 1988, 1993).

The analysis of gelation in terms of percolation theory as described

above is based implicitly on the assumption that the extent of bond

formation between ¬bers, characterized by the parameter p, is linear

in time, i.e., p ∼ t; thus pc ∼ tg , tg being the gelation time. This as-

sumption does not appear to be well justi¬ed in the case of rapidly

gelling collagen. Instead of relying on Eqs. 6.1 and 6.2, we will follow

the more general method of Durand et al. (1987), which analyzes gela-

tion by considering directly the time evolution of G (t; ω) and G (t; ω).

According to Durand et al. (1987), who were following de Gennes™

(1976a, b) original proposal based on the concept of percolation, at

the gelation transition the frequency-dependent viscoelastic moduli

exhibit power-law behavior:

G (ω) = Aω cos(π /2), (6.3a)

G (ω) = Aω sin(π /2). (6.3b)

Equations 6.3 thus predict identical scaling laws for G (ω) and G (ω) at

the gel point: G (ω) ∝ G (ω) ∝ ω (A and in Eqs. 6.3a, b are constants).

Furthermore, it also follows that at the gel point the critical loss angle

δc (δ is de¬ned as tan δ = G /G ) is related to the exponent by

π

δc = . (6.4)

2

The theory of de Gennes relates to the critical exponents f and k

introduced in Eqs. 6.1 and 6.2 via = f /( f + k). Thus the value of δc

does not depend on the frequency, a result that can also be obtained

independently of the percolation model (Chambon and Winter, 1987;

Martin et al., 1989; Rubinstein et al., 1989).

Equations 6.3 and 6.4 provide two independent methods of loca-

ting the sol--gel transition in time. In the ¬rst method the frequency

dependence of the viscoelastic moduli at various times is measured.

According to Eqs. 6.3, at the gel point, t = tg , the plots of G (tg ; ω)

and G (tg ; ω) versus log ω should yield straight lines with identical

slopes for the two functions. In the second method the experimental

results obtained for the viscoelastic moduli are used to calculate the

loss angle. According to Eq. 6.4, at tg the plot of this quantity as

a function of frequency should yield a straight line parallel to the

horizontal axis. Moreover, a consistency check on the two methods

relates the slope obtained in the ¬rst method to the value of the

critical loss angle obtained in the second method.

Detailed measurements performed on the solutions of assembling

collagen used in the MDT experiments have con¬rmed that at a well-

de¬ned point in time the storage and the loss modulus indeed exhibit

power law behavior with the same exponent, and the loss angle is in-

dependent of frequency (Forgacs et al., 2003; Newman et al., 2004).

148 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

From this behavior and from Eqs. 6.3, it can be inferred that collage-

nous ECMs are capable of undergoing a percolation phase transition.

This transition still occurs, moreover, in the presence of cell-sized par-

ticles in suf¬cient number to cause the viscoelastic properties of the

assembling matrices to differ sharply from the particle-free matrices

(Newman et al., 2004).

Network concepts have important implications for mesenchymal

morphogenesis. As we have seen above, translocating mesenchymal

cells typically move during both gastrulation and neural crest dis-

persal as if they were mechanically coherent media (Nakatsuji et al.,

1986; Newgreen, 1989). This kind of behavior will be facilitated if the

cells are not simply borne along in a ¬‚owing liquid but are phys-

ically linked to one another (however transiently) across the extra-

cellular medium, allowing them to move like a ¬‚ock of birds or a

swarm of bees. The formation of a percolating cluster of ECM ¬bers

is a way of ensuring that the mechanical activities of individual cells

are conveyed to their nearest neighbors and also to those farther

away.

Like epithelioid tissues, discussed in Chapter 5, mesenchymal tis-

sues can undergo what appears to be phase separation. This may occur

under physiological conditions, as in the formation of the limb bud

primordia along the ¬‚ank of vertebrate embryos (Heintzelman et al.,

1978; Tanaka and Tickle, 2004), during limb regeneration (Crawford

and Stocum, 1988), or in arti¬cial situations in which, for example,

cell masses derived from developing fore and hind limbs are placed

in contact with one another (Downie and Newman, 1994) (Fig. 6.7;

see also Chapter 8). The network concept helps us to understand how

mesenchymal tissues, the cells of which have no direct contact with

one another, may behave as distinct phases.

Finally, networks, even if randomly constructed, can serve as com-

munication media (Barab´si, 2002). In tissues, these communication

a

networks can operate within (Forgacs, 1995; Shafrir et al., 2000; Shafrir

and Forgacs, 2002) and between cells. Because cells can attach to

(Gullberg and Lundgren-Akerlund, 2002; Miranti and Brugge, 2002)

and exert force on (Roy et al., 1999; Zahalak et al., 2000; Freyman

et al., 2001) their microenvironments, if the microenvironment is me-

chanically linked, like collagen above the percolation transition, the

forces can be transmitted over macroscopic distances. Similar con-

siderations apply to the non-mechanical signaling among mesenchy-

mal cells mediated by the cytonemes (Ramirez-Weber and Kornberg,

1999; Bryant, 1999) and tethered nanotubes (Rustom et al., 2004), de-

scribed at the beginning of this chapter. The networks formed by

these connections from cell to cell across the ECM, like the ¬ber-based

mechanical ones described above, can be spanning or non-spanning.

As such, the nature of information transfer (i.e., whether it is global

or local) in cytoneme- or nanotube-based mesenchymal networks

may be analyzed by percolation models analogous to those described

above.