J. Phys. Chem. 1996, 100, 11471-11479
11471
When Bivalent Proteins Might Walk Across Cell Surfaces Willem Vanden Broek and Nancy L. Thompson* Department of Chemistry, UniVersity of North Carolina, Chapel Hill, North Carolina 27599-3290 ReceiVed: December 11, 1995; In Final Form: March 19, 1996X
A large variety of signal transduction processes involve bivalent attachment of protein ligands to molecular determinants on cell surfaces. In these processes, there may be significant dynamic conversion between dissociated, monovalently bound, and bivalently bound ligands. The possibility that these dynamics might occur to a substantial degree suggests that bivalent proteins might move laterally across cell surfaces by sequential detachment and attachment of single binding sites. It is the purpose of this work to theoretically define when this putative mode of translational mobility, called “walking”, might occur. Two aspects of the physiological significance of this question are addressed: the average number of steps walked before dissociation, and the average translational diffusion coefficient arising from the walking mechanism. Both symmetric and asymmetric bivalent ligands are considered.
Introduction Molecular events at cell surfaces are central to signal transduction and subsequent cellular response. Although it has long been known that antibodies are bivalent for cell surface antigens, recent work has demonstrated that many other protein ligands (e.g., fibrinogen [Farrell et al., 1992; Bennett et al., 1988], growth factors [Fretto et al., 1993; Cunningham et al., 1992], insulin [De Meyts et al., 1994], cell adhesion molecules [Osborn et al., 1994], and opioids [Lutz and Pfister, 1992]) are also bivalent in their capacity to bind to molecular determinants on cell surfaces. In such cases, there may be significant dynamic conversion between dissociated, monovalently bound, and bivalently bound ligands. The possibility that these dynamics might occur to a substantial degree suggests that bivalent or multivalent proteins might move laterally across cell surfaces by sequential detachment and attachment of single binding sites [Subramaniam et al., 1986; Smith et al., 1979]. It is the purpose of this work to theoretically define when this putative mode of translational mobility, called “walking”, might occur in a biologically significant manner.
Figure 1. Mechanisms for bivalent surface binding. Two mechanisms which might give rise to the walking of bivalent ligands across cell surfaces are considered. Left: a symmetric ligand walks by sequential detachment and attachment of identical binding sites. Right: an asymmetric ligand walks by sequential detachment and attachment of nonidentical binding sites. When the sites are identical, this mechanism is equivalent to that shown on the left.
The rate laws for the mechanism shown in eq 1 are
d B ) -(k1 + k-1)B + 2k-1C dt d C ) k1B - 2k-1C dt
Symmetric Bivalent Ligands Introduction. The simplest mechanism which incorporates the key features that determine when a bivalent, symmetric ligand might exhibit significant walking across a cell surface is shown in Figure 1. This mechanism may be written as k-1
k1
A 79 B {\ }C k -1
The factors of 2 in the terms corresponding to conversion from bivalently to monovalently bound states arise from the molecular symmetry. The solution for the density of surface-bound ligands during surface dissociation, F(t), is
F(t) ) B(t) + C(t) ) N(f1e-λ1t + f2e-λ2t)
(1)
where A denotes ligands that are dissociated from the surface, B denotes ligands bound to the surface by one site, and C denotes ligands bound to the surface by two sites. In this mechanism, the conversion of monovalently to bivalently bound molecules is modeled as an isomerization. Thus, we have assumed that different bivalent molecules do not compete for the same binding sites [Pisarchick and Thompson, 1990; Lauffenburger and Linderman, 1993], either for steric reasons or because the cell surface binding sites are far from completely occupied. In addition, we have assumed that the site release constants are equivalent, i.e., the ligand is not cooperative. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(95)03662-8 CCC: $12.00
(2)
(3)
where N is the initial density of surface-bound molecules, λ1 and λ2 are the two characteristic rates, and f1 and f2 are the fractional amplitudes associated with the rates. Characteristic Rates. The characteristic rates associated with eqs 2 and 3, normalized by k-1, are
λ1,2 1 ) /2[(K1 + 3) ( xK12 + 6K1 + 1] k-1
(4)
K1 ) k1/k-1
(5)
where
is the equilibrium constant that describes conversion from monovalent to bivalent surface binding. The rates λ1 and λ2, © 1996 American Chemical Society
11472 J. Phys. Chem., Vol. 100, No. 27, 1996
Vanden Broek and Thompson
Figure 2. Characteristic rates for symmetric molecules. The normalized rates λ1/k-1 and λ2/k-1 for symmetric molecules were calculated by using eq 4 and are shown as a function of the equilibrium constant K1.
Figure 3. Fractional amplitudes for symmetric molecules. The amplitudes f1 and f2 for symmetric molecules were calculated by using eq 7 and are shown as a function of the equilibrium constant K1.
as normalized by k-1, are shown in Figure 2 as a function of K1. When bivalent attachment is not thermodynamically favored (K1 f 0), λ1 ≈ 2k-1, λ2 ≈ k-1, and all molecules dissociate with a rate (λ1 or λ2) which has a magnitude on the order of the monovalent dissociation rate. When bivalent attachment is favored (K1 f ∞), λ1 ≈ k1 and λ2 ≈ 2k-12/k1. Some molecules dissociate with a rate (λ1) faster than the monovalent dissociation rate. This result is a consequence of the feature that monovalently bound molecules experience a parallel-type reaction (as in fluorescence quenching) in which they may either dissociate from the surface or become bivalently bound. Other molecules dissociate with a slow rate (λ2). Fractional Amplitudes. To solve for the fractional amplitudes, we assume that a density N of surface-bound ligands exists at time zero in preequilibrium on the surface, i.e.
B(0) )
2N K1 + 2
C(0) )
K1N K1 + 2
These initial conditions imply that
f1,2 )
[
1 12
K12 + 5K1 + 2
(6)
]
(K1 + 2)xK12 + 6K1 + 1
(7)
The fractional amplitudes f1 and f2 are shown in Figure 3 as a function of K1. When bivalent attachment is not favored (K1 f 0), all molecules dissociate with the monovalent dissociation rate (f1 ≈ 0, f2 ≈ 1, and λ2 ≈ k-1). The result that f1 ≈ 0 and f2 ≈ 1 also holds when bivalent attachment is favored (K1 f ∞). In this case, the dissociation rate is λ2 ≈ 2k-12/k1. The maximum value of f1 (≈0.041) and the minimum value of f2 (≈0.959) occur when K1 ) 1/3. Therefore, for all values of K1, the dominant dissociation rate is the slower rate λ2. Number of Steps Walked, Diffusion Coefficient, and Distance Traveled. The average surface residency time, T, and the average time between walking steps, ∆T, are
T)
[
][ ]
f2 (K1 + 4)(K1 + 1) 1 f1 + ) λ1 λ2 k-1 2(K1 + 2) ∆T )
1 1 + k1 k-1
(8)
(9)
Equation 8 follows from the expressions for λ1,2 and f1,2 given in eqs 4 and 7, and eq 9 is illustrated in Figure 4a. The average
Figure 4. Time between walking steps for symmetric and asymmetric molecules. (a) For symmetric molecules, the average lifetime of the monovalently bound state is 1/k1 and the average time of the bivalently bound state is 1/k-1, so that the average time between walking steps, ∆T, is given by eq 9. (b) For asymmetric molecules, both sites must detach and attach for the molecule to move, on the average, from a given position. The average lifetimes of the monovalently bound states are 1/k1 and 1/k2, and the average lifetimes of the bivalently bound states are 1/k-1 and 1/k-2, so that the average time between walking steps, ∆T, is given by eq 25.
number of steps walked is the ratio of these two times, or
n ) T/∆T ) [K1(K1 + 4)]/[2(K1 + 2)]
(10)
The values of n calculated from this equation are shown in Figure 5a. For low values of K1, n ≈ K1 f 0. For high values of K1, n ≈ K1/2 f ∞. The average diffusion coefficient D for symmetric molecules undergoing translational motion via the walking mechanism is
D)
( )
K1 w2k-1 w2 ) 4∆T K1 + 1 4
(11)
where w is the average distance between walking steps, and ∆T has been expressed according to eq 9. The diffusion coefficient D, as normalized by (w2k-1)/4, is shown in Figure 5b as a function of K1. D ranges from zero (K1 f 0) to (w2k-1)/ 4 (K1 f ∞). The average distance traveled by the walking mechanism is
s ) x4DT ) wxn
(12)
When Bivalent Proteins Might Walk Across Cell Surfaces
J. Phys. Chem., Vol. 100, No. 27, 1996 11473 Asymmetric Bivalent Ligands Introduction. The mechanism for bivalent, asymmetric molecules is shown in Figure 1. This mechanism may be written as k-2
k1
k2
-1
-2
k-1
A 79 B1 {\ } C [\ ] B2 98 A k k
(13)
where A denotes ligands that are dissociated from the surface, B1 and B2 denote ligands bound to the surface by one or the other site, and C denotes ligands bound to the surface by both sites. As for symmetric molecules, we have modeled the conversion of monovalently to bivalently bound molecules as an isomerization and assumed that the site release constants are equivalent, i.e., the ligand is not cooperative. The rate laws for the mechanism shown in eq 13 are
d B ) -(k1 + k-2)B1 + k-1C dt 1 d B ) -(k2 + k-1)B2 + k-2C dt 2 d C ) k1B1 + k2B2 - (k-1 + k-2)C dt
(14)
When k2 ) k1 and k-2 ) k-1, eqs 14 reduce to eqs 2 with B(t) ) B1(t) + B2(t). The solution for the density of surface-bound molecules, F(t), is
F(t) ) B1(t) + B2(t) + C(t) ) N(f1e-λ1t + f2e-λ2t + f3e-λ3t) (15) where N is the initial density of surface-bound molecules, λ1, λ2, and λ3 are characteristic rates, and f1, f2, and f3 are the fractional amplitudes for these rates. Characteristic Rates. The characteristic rates associated with eqs 14 and 15 are given by the solutions to the cubic equation
( ) ( ) ( )
λi 3 λi 2 λi +p +q +r)0 k-1 k-1 k-1
(16)
where
p ) -[(K1 + 2) + (K2 + 2)x] q ) (K1 + 1) + (K1K2 + K1 + K2 + 3)x + (K2 + 1)x2 r ) -[(K1 + 1)x + (K2 + 1)x2] K1 )
Figure 5. Number of walking steps, diffusion coefficient, and distance traveled for symmetric molecules. The following quantities were calculated for symmetric molecules and are shown as a function of the equilibrium constant K1: (a, top) the average number of steps walked n (eq 10); (b, middle) the average walking diffusion coefficient D, normalized by (w2k-1)/4, where w is the average step distance and k-1 is the dissociation rate constant (eq 11); and (c, bottom) the average distance walked s, normalized by w (eq 12).
The value of s, normalized by w, is shown in Figure 5c. For low values of K1, s ≈ wK11/2 f 0; for high values of K1, s ≈ w(K1/2)1/2 f ∞.
k1 k-1
K2 )
k2 k-2
x)
k-2 k-1
(17) (18)
In eqs 17 and 18, K1 and K2 are the equilibrium association constants that describe the conversions between monovalently and bivalently bound ligands. The dimensionless parameter x is a measure of the asymmetry in the off rates. By choosing k-2 e k-1, this parameter is restricted to range from zero to one. The normalized rates calculated from eqs 16 and 17 and the general analytical solution to a cubic equation are shown in Figure 6. As shown, some molecules dissociate with a rate (λ1 or λ2) faster than the higher monovalent dissociation rate (k-1), while others dissociate with a rate (λ3) slower than this constant. Figure 6a-c shows the rates for the case when the equilibrium association constants are equivalent (K1 ) K2). The rates λ1 and λ2 increase, and the rate λ3 decreases, with increasing K1
11474 J. Phys. Chem., Vol. 100, No. 27, 1996
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Figure 6. Characteristic rates for asymmetric molecules. The normalized rates (a,d,g) λ1/k-1, (b, e, h) λ2/k-1, and (c, f, i) λ3/k-1 for asymmetric molecules were calculated by using eqs 16 and 17 and are shown as a function of the equilibrium association constants K1 and K2, and the dimensionless parameter x. In a-c, the normalized rates are shown for K1 ) K2 ) 100 (line); K1 ) K2 ) 10 (dash); K1 ) K2 ) 1 (dash-dot); K1 ) K2 ) 0.1 (dash-dot-dot); and K1 ) K2 ) 0.01 (dot). In (d-f), the normalized rates are shown for K1 ) 100, with K2 ) 100 (line); K2 ) 50 (dash); K2 ) 20 (dash-dot); K2 ) 10 (dash-dot-dot); and K2 ) 1 (dot). In (g-i), the normalized rates are shown for K2 ) 100, with K1 ) 100 (line); K1 ) 50 (dash); K1 ) 20 (dash-dot); K1 ) 10 (dash-dot-dot); and K1 ) 1 (dot).
When Bivalent Proteins Might Walk Across Cell Surfaces ) K2. Figure 6d-f shows the rates for the case when K1 ) 100 and K2 e K1. The rate λ1 is high and approximately constant, the rate λ2 decreases with K2, and the rate λ3 increases with K2. Figure 6g-i shows the rates for the case when K2 ) 100 and K1 e K2. The rates λ1 and λ2 increase with K1, and the rate λ3 decreases with K1. For all values of K1 and K2 (Figure 6a-i), all three rates increase with the degree of offrate symmetry (x). When x f 1 and K1 ) K2, λ1 is equal to the rate given by the positive root in eq 4, λ2 ) k1 + k-1, and λ3 is equal to the rate given by the negative root in eq 4. When x f 0 (k-2 , k-1), the system is highly asymmetric in the offrates. To first order in x, eqs 16 and 17 yield
λ1 ≈ (K1 + 1) + x k-1
λ2 ≈ 1 + K2x k-1
λ3 ≈x k-1 (19)
Therefore, in this limit, λ1 ≈ k1 + k2 + k-1, λ2 ≈ k2 + k-1, and λ3 ≈ k-2. Fractional Amplitudes. To solve for the values of f1, f2, and f3, we have assumed that, at time zero, a density N of molecules is in a preexistent equilibrium on the surface, i.e.
K2,1N B1,2(0) ) K1K2 + K1 + K2
K1K2N C(0) ) K1K2 + K1 + K2
J. Phys. Chem., Vol. 100, No. 27, 1996 11475 to Figure 6, one sees that, for these cases, λ3 , k-1. Furthermore, when K1 ) K2 . 1, λ3 ranges from k-2 when x f 0 (eqs 18 and 19) to 2k-12/k1 when x f 1 (eq 4). Number of Steps Walked, Diffusion Coefficient, and Distance Traveled. The average surface residency time, T, and the average time between walking steps, ∆T, are
T) ∆T )
fi ) K2(d2j - d2k) + K1(d1k - d1j) + K1K2(d1jd2k - d1kd2j)
[
]
× d1i(d2j - d2k) + d2i(d1k - d1j) + (d1jd2k - d1kd2j) 1 + d1i + d2j (21) K1K2 + K1 + K2
[
]
where j ) 2 and k ) 3 when i ) 1, j ) 3 and k ) 1 when i ) 2, j ) 1 and k ) 2 when i ) 3, and, for i ) 1 to 3
d1i )
1 K1 + x - λi/k-1
d2j )
x 1 + xK2 - λi/k-1
(22)
The amplitudes calculated from eqs 21 and 22 are shown in Figure 7. Figure 7a-c shows the amplitudes for the case when the equilibrium association constants are equivalent (K1 ) K2); Figure 7d-f shows the amplitudes for the case when K1 ) 100 and K2 e K1; and Figure 7g-i shows the amplitudes for the case when K2 ) 100 and K1 e K2. When x f 1 and K1 ) K2, f1 reduces to the amplitude given by the negative root in eq 7, f2 ) 0, and f3 reduces to the amplitude given by the positive root in eq 7. When x f 0 (high asymmetry in the off rates), eqs 19, 21, and 22 reduce to
f1 ≈ 0 f2 ≈
n)
f3 ≈
K2(K1 + 1) K1 + 1 f K1K2 + K1 + K2 K1 + 2
(
1 1 1 1 1 + + + 2 k1 k-1 k2 k-2
)
(25)
[
]
f2 f3 f1 T + + × ) ∆T (λ1/k-1) (λ2/k-1) (λ3/k-1) 2xK1K2 (26) xK1K2 + K1K2 + xK2 + K1
[
]
The values of n calculated from this equation are shown in Figure 8a,d,g. When x f 1 and K1 ) K2, n equals the value previously calculated for symmetric molecules (eq 10 and Figure 5a). When x f 0, eqs 19, 23, and 26 imply that
n≈
2K22(K1 + 1) (K2 + 1)(K1K2 + K1 + K2)
f
2K1 K1 + 2
(27)
where the limit is for the case when K1 ) K2. By using eq 25, one finds that the diffusion coefficient D for asymmetric molecules undergoing translational motion via the walking mechanism is
D)
(
)
2xK1K2 w2k-1 w2 ) (28) 4∆T xK1K2 + K1K2 + xK2 + K1 4
where w is the average distance between walking steps. The diffusion coefficient D, as normalized by (w2k-1)/4, is shown in Figure 8b,e,h as a function of K1, K2, and x. When x f 1 and K1 ) K2, the expression in eq 28 for D reduces to that shown in eq 11 (Figure 5b). When x f 0 (high asymmetry in the off rates), D f 0. When both K1 and K2 are . 1 2 2x w k-1 D≈ x+1 4
(
)
(29)
The average distance traveled by the walking mechanism is given by eq 12 with n from eq 26. The values of s, normalized by w, are shown in Figure 8c,f,i. When x f 1 and K1 ) K2, the values of s shown in this figure reduce to those previously calculated for symmetric molecules (eq 12 and Figure 5c). When xf0
s ≈ wK2
K1 1 f K1K2 + K1 + K2 K1 + 2
(24)
Equation 25 is illustrated in Figure 4b. The average number of steps walked is the ratio of these two times, or
(20) These initial conditions imply that the fractional amplitudes are given by
f1 f2 f3 + + λ1 λ2 λ3
x
2(K1 + 1)
(K2 + 1)(K1K2 + K1 + K2)
x
fw
2K1 K1 + 2 (30)
where the limit is for the case when K1 ) K2.
(23)
where the limits are for K1 ) K2. As shown, for all but small values of K1 and/or K2, f3 is the dominant amplitude and most molecules dissociate with the slowest rate (λ3). By referring
Discussion The purpose of this work is to theoretically predict when bivalent protein ligands bound to cell surfaces might undergo significant translational motion via sequential detachment and attachment of individual binding sites. There are two somewhat
11476 J. Phys. Chem., Vol. 100, No. 27, 1996
Vanden Broek and Thompson
Figure 7. Fractional amplitudes for asymmetric molecules. The amplitudes (a, d, g) f1, (b, e, h) f2, and (c, f, i) f3 for asymmetric molecules were calculated by using eqs 21 and 22 and are shown as a function of the equilibrium association constants K1 and K2, and the dimensionless parameter x. The line notations are as specified in Figure 6.
When Bivalent Proteins Might Walk Across Cell Surfaces
J. Phys. Chem., Vol. 100, No. 27, 1996 11477
Figure 8. Number of walking steps, diffusion coefficient, and distance traveled for asymmetric molecules. The following quantities were calculated for asymmetric molecules and are shown as a function of the asymmetry parameter x and the equilibrium constants K1 and K2: (a, d, g) the average number of steps walked n (eq 26); (b, e, h) the average walking diffusion coefficient D, normalized by (w2k-1)/4, where w is the average step distance and k-1 is the dissociation rate constant (eq 28); and (c, f, i) the average distance walked s, normalized by w (eqs 12 and 26). The line notations are as specified in Figure 6.
11478 J. Phys. Chem., Vol. 100, No. 27, 1996 independent aspects to the physiological significance of this question. First, the molecules must experience a high enough number of walking steps before dissociating, so that they travel a significant distance along the cell surface. Second, the molecules must do so on a time scale that is rapid enough to be of biological significance. For symmetric molecules, the average number of steps walked before surface dissociation n (eq 10 and Figure 5a) depends only on, and increases with, the equilibrium constant that describes the conversion between monovalently and bivalently bound ligands, K1. The parameter n is significant only for higher values of K1; i.e., n ) 10 when K1 ) 18, n ) 100 when K1 ) 198, and n ) 1000 when K1 ) 1998. For these higher values of K1, D ≈ (w2k-1)/4 (eq 11 and Figure 5b). Thus, for w ) 100 Å, D ) 2.5 × 10-13 cm2/s for k-1 ) 1 s-1, D ) 2.5 × 10-10 cm2/s for k-1 ) 103 s-1, and D ) 2.5 × 10-7 cm2/s for k-1 ) 106 s-1. For asymmetric molecules, the average number of steps walked before surface dissociation (Figure 8) increases with both of the monovalent-to-bivalent equilibrium constants, K1 and K2. The parameter n is significant only for high values of K1 and K2. If either of these constants is low, n is significantly reduced because the molecules dissociate from the surface via the monovalent state associated with the low equilibrium constant. In addition, the parameter n increases with greater symmetry in the off-rates (x f 1). For high values of K1 and K2, and x > 0, D is on the order of (w2k-1)/4 (Figure 8). It should be noted that if the sites to which the bivalent molecules bind are themselves translationally mobile, the effective diffusion coefficient will be approximately given by the sum of that shown in Figures 5 or 8 and that for the binding sites. Thus, the physiological effect of walking will be important only if the walking diffusion coefficient is equal to or greater than the diffusion coefficient of the binding sites. There are several assumptions inherent in the theory presented in this paper. First, we have modeled the conversion of monovalently to bivalently bound molecules in the simplest possible manner, i.e., as an isomerization. Thus, we have assumed that different bivalent molecules do not compete for the same binding sites [Pisarchick and Thompson, 1990; Lauffenburger and Linderman, 1993], either for steric reasons or because the cell surface binding sites are far from completely occupied. Second, we have assumed that the bivalent ligands are not cooperative. This assumption is mathematically realized by setting the site release constants to be equivalent in the two mechanisms. Third, we have assumed that surface-bound ligands do not experience significant intermolecular interactions; i.e., that the bound ligands do not attract or repel each other [Saxton, 1994; Saxton, 1993]. Fourth, we have assumed that the molecules, when bound monovalently to the surface, experience enough rotational motion so that when they become bivalently bound, the new bound site will have travelled a finite distance from its previously bound position. Because the average lifetimes of the monovalently bound states are 1/k1 and 1/k2, these times must be greater than the rotational correlation time of the cell surface binding sites. Finally, we have not considered the possible mode of translational mobility in which molecules dissociate from the surface and then quickly rebind to a nearby site. Addressing this mode of translational mobility would require examining coupled reaction and diffusion equations [Axelrod and Wang, 1994; Hsieh and Thompson, 1994; Thompson et al., 1981]. The theory outlined in this paper may be compared with experimental results for bivalent proteins interacting with model membrane surfaces. In previous work, we have experimentally measured the monovalent-to-bivalent equilibrium association
Vanden Broek and Thompson constants for two different anti-dinitrophenyl monoclonal antibodies at substrate-supported planar membranes containing dinitrophenyl-conjugated phospholipids. For one antibody, K1 ≈ 10 [Pisarchick and Thompson, 1990; Vanden Broek and Thompson, 1996]; for the other, K1 ≈ 5 [Vanden Broek and Thompson, 1996]. For these conditions, the average number of steps walked is predicted to be (eq 10) n ≈ 3-6, and the average distance travelled is predicted to be (given w ≈ 100 Å) (eq 12) s ≈ 200 Å. For these antibodies, k-1 ≈ 1 sec-1 [Pisarchick et al., 1992; Vanden Broek and Thompson, 1996]. Thus, D ≈ 10-13 cm2/s (see above). One would expect more physiologically relevant walking for antibodies with weaker Fab dissociation rates (k-1 g 105 s-1) and higher monovalent-tobivalent equilibrium constants (K1 g 100). There are several experimental methods which might be able to probe for the walking phenomenon predicted in this work. First, we have recently combined total internal reflection fluorescence microscopy with pattern photobleaching recovery to provide one of the first measures of the diffusion coefficient of protein that is weakly bound to a membrane surface [Abney et al., 1991; Huang et al., 1994]. Second, one might be able to observe the predicted, distinct walking steps with single-particle tracking methods [Ghosh and Webb, 1994; Fein et al., 1993]. The theoretical results described in this paper should serve as a guide toward identifying those systems in which one would expect to observe walking using these techniques. Acknowledgment. This work was supported by NIH Grant GM-37145 and by NSF Grant DMB-9024028. References and Notes Abney, J. R., B. A. Scalettar and N. L. Thompson. 1992. Evanescent interference patterns for fluorescence microscopy. Biophys. J. 61: 542552. Axelrod, D. and M. D. Wang. 1994. Reduction-of-dimensionality kinetics at reaction-limited cell surface receptors. Biophys. J. 66: 588-600. Bennett, J. S., S. J. Shattil, J. W. Power and T. K. Gartner. 1988. Interaction of fibrinogen with its platelet receptor. Differential effects of R and γ chain fibrinogen peptides on the glycoprotein IIb-IIIa complex. J. Biol. Chem. 263: 12948-12953. Cunningham, B. C., Ultsch, M., De Vos, A. M., Mulkerrin, M. G., Clauser, K. R. and Wells, J. A. 1991. Dimerization of the extracellular domain of the human growth hormone receptor by a single hormone molecule. Science 254: 821-825. De Meyts, P., B. Wallach, C. T. Christoffersen, B. Urso, K. Gronskov, L. J. Latus, F. Yakushiji, M. M. Ilondo, and R. M. Shymko. 1994. The insulin-like growth factor I-receptor. Structure, ligand-binding mechanism and signal transduction. Hormone Res. 42: 152-169. Farrell, D. H., P. Thiagarajan, D. W. Chung and E. W. Davie. 1992. Role of fibrinogen R and γ chain sites in platelet aggregation. Proc. Natl. Acad. Sci. U.S.A. 89: 10729-10732. Fein, M., J. Unkeless, F. Y. Chuang, M. Sassaroli, R. da Costa, H. Vaananen and J. Eisinger. 1993. Lateral mobility of lipid analogues and GPIanchored proteins in supported bilayers determined by fluorescence bead tracking. J. Membr. Biol. 135: 83-92. Fretto, L. J., A. J. Snape, J. E. Tomlinson, J. J. Seroogy, D. L. Wolf, W. J. LaRochelle and N. A. Giese. 1993. Mechanism of platelet-derived growth factor (PDGF) AA, AB and BB binding to alpha and beta PDGF receptor. J. Biol. Chem. 268: 3625-3631. Ghosh, R. N. and W. W. Webb. 1994. Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecules. Biophys. J. 66: 1301-1318. Hsieh, H. V. and N. L. Thompson. 1994. Theory for measuring bivalent surface binding kinetics using total internal reflection with fluorescence photobleaching recovery. Biophys. J. 66: 898-911. Huang, Z., K. H. Pearce and N. L. Thompson. 1994. Translational diffusion of fluorescently labeled bovine prothrombin fragment 1 reversibly bound to substrate-supported planar membranes: Measurement by fluorescence photobleaching recovery with evanescent interference patterns. Biophys. J. 67: 1754-1766. Lauffenburger, D. A. and J. J. Linderman. Receptors: Models for Binding, Trafficking and Signaling, Oxford University Press: New York, 1993; pp 164-167. Lutz, R. A. and H. P. Pfister. 1992. Opioid receptors and their pharmacological profiles. J. Receptor Res. 12: 267-286. Osborn, L., C. Vassallo, B. G. Browning, R. Tizard, D. O. Haskard, C. D. Benjamin, I. Douglas and T. Kirchhausen. 1994. Arrangements of domains, and amino acid residues required for binding of vascular cell
When Bivalent Proteins Might Walk Across Cell Surfaces adhesion molecule-1 to its counter receptor VLA-4 (R4β1). J. Cell Biol. 124: 601-608. Pisarchick, M. L., D. Gesty and N. L. Thompson. 1992. Binding kinetics of an anti-dinitrophenyl monoclonal Fab on supported phospholipid monolayers measured by total internal reflection with fluorescence photobleaching recovery. Biophys. J. 63: 215-223. Pisarchick, M. L. and N. L. Thompson. 1990. Binding of a monoclonal antibody and its Fab fragment to supported phospholipid monolayers measured by total internal reflection fluorescence microscopy. Biophys. J. 58: 1235-1249. Saxton, M. J. 1993. Lateral diffusion in an archipelago. Single-particle diffusion. Biophys. J. 64: 1766-1780. Saxton, M. J. 1994. Anomalous diffusion due to obstacles: A Monte Carlo study. Biophys. J. 66: 394-401. Smith, L. M., J. W. Parce, B. A. Smith and H. M. McConnell. 1979.
J. Phys. Chem., Vol. 100, No. 27, 1996 11479 Antibodies bound to lipid haptens in model membranes diffuse as rapidly as the lipids themselves. Proc. Natl. Acad. Sci. U.S.A. 76: 4177-4179. Subramaniam, S., M. Seul and H. M. McConnell. 1988. Lateral diffusion of specific antibodies bound to lipid monolayers on alkylated substrates. Proc. Natl. Acad. Sci. U.S.A. 83: 1169-1173. Thompson, N. L., T. P. Burghardt and D. Axelrod. 1981. Measuring surface dynamics of biomolecules by total internal reflection fluorescence with photobleaching recovery or correlation spectroscopy. Biophys. J. 33: 435-454. Vanden Broek, W. and N. L. Thompson. 1996. Equilibrium and kinetics of membrane binding for bivalent and monovalent monoclonal antibodies measured using total internal reflection fluorescence microscopy. Biophys. J. 68: A406.
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