fashion a mathematical formalism is required that simultaneously

describes chemical reactions and diffusion. We postpone a full pre-

sentation of reaction--diffusion systems until Chapter 7 and here, fol-

lowing Bell, consider a simpli¬ed two-step model for bond formation,

schematically shown by the following relation

d+ r+

A + B ’’ A B ’’ C . (4.1)

←’ ←’

r’

d’

Here A and B are the two CAMs, AB is an ˜˜encounter complex” (in

which, by de¬nition, the two molecules are close enough to allow the

chemical reaction between them to take place), and C is the bound

state of A and B. The encounter complex arises purely by diffusion.

Thus the rate constants d+ and d’ are related to D(A) and D(B), the

diffusion coef¬cients of A and B within the cell membrane (Dembo

et al., 1979), by

d+ = 2π[D (A) + D (B )] (4.2a)

d’ = 2[D (A) + D (B )]R ’2 . (4.2b)

AB

It is assumed here that the reactants form an encounter complex

whenever they are separated by R A B , the encounter distance. Note

that the units of d+ and d’ correspond to the concentrations of A,

B, and C measured in numbers of molecules per unit area. Thus the

equilibrium constant K e for the encounter step is K e = d+ /d’ = πR 2 B ,

A

which is the area of a disc of radius R A B .

Equation 4.1 represents two simple processes, the second of which

corresponds to the purely reactive part of bond formation:

dN A B

= d+ N A N B ’ d’ N A B ’ r+ N A B + r’ N C , (4.3a)

dt

dN C

= r+ N A B ’ r’ N C , (4.3b)

dt

N A , N B , N A B , and N C being the concentrations of A, B, AB, and C,

respectively. Following Bell™s simpli¬ed picture, we assume that the

86 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

reaction complex is in equilibrium with the reactants and the product

(i.e., dN A B /dt = 0). Then the rate constants in the overall reaction

k+

A + B ’’ C (4.4)

←’

k’

can be expressed as

d+ r+

k+ = (4.5a)

d’ + r+

d’ r’

k’ = . (4.5b)

d’ + r+

The above expressions show that if the encounter complex is much

d’ ) then k+ ≈ d+ ,

more likely to react than to dissociate (i.e. r+

that is, the forward reaction is diffusion limited (i.e., its progres-

sion in time is controlled by the rate of diffusion). In this case

k’ ≈ d’ r’ /r+ ; from Eq. 4.2 we can see that it also depends on the

diffusion coef¬cients. Thus if the diffusion coef¬cients are small, as

for receptors in the viscous membrane, both forward and reverse rate

constants are small. (Note, however, that the equilibrium constant

K e = k+ /k’ = R 2 B r+ /r’ , which is independent of the diffusion coef-

A

¬cients.)

We now apply the above model to describe the formation of ho-

mophilic bonds between identical CAMs (such as cadherins) on jux-

taposed cells. Let the total number of cadherins per unit area of the

membrane be N1 and N2 and the corresponding numbers for unbound

(free) molecules be N 1f and N 2f . The number of bound molecules N b

is then determined by

N i = N if + N b (i = 1, 2). (4.6)

The kinetic equation that describes the bond formation reaction (4.4)

where A ’ N 1f , B ’ N 2f , C ’ N b has the form

dN b

= k+ N 1f N 2f ’ k’ N b , (4.7)

dt

k+ and k’ being given by Eqs. 4.5a, b. Using Eq. 4.6 this can be rewrit-

ten as

dN b

= k+ (N 1 ’ N b )(N 2 ’ N b ) ’ k’ N b (4.8)

dt

and solved for the time evolution of N b (t). Equation 4.8 is a nonlin-

ear ¬rst-order differential equation whose explicit solution is compli-

cated (Bell, 1978) and is not essential for understanding the relevant

features of bond formation. It is clear from Eq. 4.8 that the rate of

bond formation is maximal when the cells are ¬rst brought together

(t = 0), so that N b (t = 0) = 0 and

dN b

= k+ N 1 N 2 . (4.9)

dt max

4 CELL ADHESION, COMPARTMENTALIZATION, AND LUMEN FORMATION 87

At long times equilibrium will be approached and the right-hand side

of Eq. 4.8 will approach zero, resulting in

1/2

2

1 1 1 1

eq

= N1 + N2 + ’ N1 + N2 + ’ 4N 1 N 2 ,

Nb (4.10)

2 K 2 K

K being the equilibrium constant.

While the mathematical manipulations leading to Eq. 4.10 may

seem involved, the following simple application demonstrates its use-

fulness and shows the value of such analysis. If K is known, Eq. 4.10

can be used to determine the number of homophilic bonds per unit

area of the cell surface connecting it to adjacent cells. For a rough

estimate we consider cells of the same type and therefore take N1 =

N2 = N. Since the cases 1/K N or 1/K N lead to the unrealis-

eq

tic results of either no adhesive bonds (N b ≈ 0) or all CAMs being

eq

in bound pairs (N b = N ), we assume that 1/K ≈ N . Using Eq. 4.10,

eq

we then obtain N b ≈ 0.38N , indicating that at any given time about

40% of the CAMs form adhesive bonds.

Strength of adhesive bonds

The stability of adhesive bonds is determined by the free-energy

changes associated with the electrostatic, van der Waals, or hydrogen-

bond interactions between CAMs. Such bonds are reversible and each

will be broken and reformed given suf¬cient time. However, since,

as we have seen above, up to 40% of CAMs might be bound at any

given instant of time, the probability of all the receptors being simul-

taneously unbound is very small. Thus the separation of two cells

from one another requires a force that is capable of fairly rapidly (see

below) rupturing every bond.

The formation of an adhesive bond is accompanied by the lowering

of the system™s free energy. Moreover, at equilibrium a bond has a well-

de¬ned length. Thus in order to disrupt a bond, work must be done to

increase the separation between the molecules. The situation is repre-

sented schematically in Fig. 4.4. The minimum in the free energy at rb

corresponds to the equilibrium bound state of two neighboring cells.

Performing work on the system by ˜˜pulling” on the bond allows the

free-energy barrier to be overcome, after which the two cells separate

and eventually settle into new energy minima. Clearly a force applied

over a range r0 from the minimum will rapidly rupture the bond. This

force can thus be written as f0 = E 0 /r0 (remember, the energy change

equals the work done and is given by force times distance). Taking the

binding energy between cadherins lacking cytoskeletal attachments

(which such CAMs have in vivo) as about 1 kcal/mole (as measured by

Sivasankar et al., 1999) and r0 as about 1 nm (the typical linear di-

mension of the binding cleft on an antibody, Pecht and Lancet, 1976),

as a rough estimate we obtain f0 ≈ 50 pN, which is comparable with

the results of studies on single molecules by Baumgartner et al. (2000)

using atomic force microscopy. The latter authors found that about

40 pN was required to separate a single adhesion dimer of vascular

88 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

FREE

ENERGY

rb

r0

Separation

E0

Fig. 4.4 Typical variation of the adhesive free energy of two neighboring cells with the

separation between opposing cell surfaces. For simplicity we assume the connection is

established via a single homophilic pair of CAMs. For small distances the cells repel each

other owing to electrostatic or mechanical forces. Repulsion corresponds to large

positive values of the free energy. The minimum at rb can be considered as the

equilibrium binding energy between the two CAMs. Bond rupture between the two

CAMs and thus the separation between the cells takes place when the adhesive free

energy changes sign, turning from negative to positive. (Compare with the curve shown

in Fig. 2.12).

endothelial (VE) cadherins. Since the cadherins in this experimental

system also lacked cytoskeletal attachments, the measured force of

separation is probably an underestimate of the biological value.

In the absence of external forces two CAMs will dissociate in a

time t > „0 , where „0 is the average lifetime of the adhesive bond.

To describe the situation in which a constant force f stresses the

bond, Bell, applying results from the theory of solids, suggested the

following equation for the modi¬ed lifetime „ , the Bell equation:

„ = „0 exp(’γ f /kB T ). (4.11)

Here γ is a parameter (with units of length) whose value has to be

determined empirically; in the case of a solid γ accounts for its struc-

ture and imperfections.

We now are in a position to make the phrase ˜˜fairly rapid ruptur-

ing” of the bond more quantitative. Consider an adhesive cadherin

dimer between two cells, one of which is attached to a substratum

and the other to a spring. By stretching the spring to a given length

quickly (in the ideal case, instantaneously), one can measure the time

it takes for the cadherin--cadherin bond to dissociate under a constant

4 CELL ADHESION, COMPARTMENTALIZATION, AND LUMEN FORMATION 89

force. Since bond lifetime is a statistical quantity, performing such a

measurement a number of times yields a probability distribution for

„ ( f ). Fitting the average of this distribution to Eq. 4.11 provides values

for „0 and γ (Marshall et al., 2003).

Another application of Eq. 4.11 is based on pulling the spring with

varying velocity. Baumgartner et al. (2000) created cadherin adhesion

dimers in the following way. Cadherins derived from the vascular en-

dothelial cells lining the inner surface of blood vessels (VE cadherins)

were adsorbed onto mica and also covalently attached to the tip of

an atomic force cantilever, which acted as the spring. The cantilever

was brought close to the mica and adhesion between the cadherins

was established. Subsequently the cantilever was retracted with velo-

city v, and the rupture force fc (v) was measured. It was found that fc

depended logarithmically on v. This is consistent with the Bell equa-

tion if it is assumed that „ (v) = z/v (z is the distance over which the

spring has to be pulled before the bond ruptures), since in this case

Eq. 4.11 yields fc = kT /γ ln(v„0 /z). From the ¬t to the experimental

data the authors deduced „0 = 0.55 s and γ = 0.59 nm. This result

implies that for VE cadherins (lacking cytoskeletal attachments) the

reverse rate constant in Eq. 4.8 is k’ = 1/„0 = 1.8 s’1 .

The preceding discussion pertains to the force needed to rupture a

single adhesive bond on the cell surface. An extension of this analysis

to estimate the force required to separate two cells attached to each

other by N b complementary CAMs is presented in Box 4.1.

The Bell model has a number of limitations. For example, it ig-

nores the possible interaction between receptors on the same cell

surface (cis interactions) or the role of the cytoskeleton in adhesion,

factors that have been shown important for cadherin function (Whee-

lock and Johnson, 2003). It treats bond formation as a simple chemical

reaction (between the members of the encounter complex), whereas

in reality it is an intricate multistep process (Adams et al., 1996, 1998;

Gumbiner, 2000). Despite these de¬ciencies the Bell model remains

the basis of our physical understanding of cell--cell adhesion.

Box 4.1 Separating two adhering cells

We consider two cells adhering to each other through Nb complementary CAMs.

If the force needed to separate the two cells is F then, assuming that each bond is

equally stretched, the force per bond is F /Nb . Applying Bell™s equation, Eq. 4.11,

to this situation, the reverse rate constant in Eq. 4.8 (with N1 = N2 = N) should

be replaced by k ’ exp(γ F /k B TNb ) to yield

dNb

= k + (N ’ Nb )2 ’ k ’ Nb exp(γ F /k B T Nb ). (B4.1a)

dt

Since both terms in Eq. B4.1a are positive, as long as the second term is smaller than

the ¬rst, Nb will increase in time. If the force is large and the second term exceeds

the ¬rst, Nb will rapidly go to zero; there will be no ¬nite solution to dNb /dt = 0,

90 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

which now yields an equation similar to Eq. 4.10 in which K has been multiplied by

exp(’γ F /k B T Nb ). Thus, at a critical value of the force F c the equilibrium number