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In order to describe the process of bond formation in a rigorous
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

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