(at time infinity). The time-dependent diffusion coefficient depends only on the intrinsic dissociation rate and a ârebinding parameterâ, which in...

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Lateral Diffusion from Ligand Dissociation and Rebinding at Surfaces† Alena M. Lieto,‡,§ B. Christoffer Lagerholm,§,| and Nancy L. Thompson*,§ Department of Physics & Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3255, and Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290 Received June 29, 2002. In Final Form: September 12, 2002 The lateral diffusion coefficient of a ligand which repeatedly dissociates and rebinds to sites on a planar surface is described. An analytical expression showing the manner in which the diffusion coefficient depends on the propensity for rebinding after dissociation, as the molecule moves through the solution and across the surface, is derived. The diffusion coefficient is time dependent and ranges from the surface diffusion coefficient (at time zero when a tagged molecule is placed on the surface) to the solution diffusion coefficient (at time infinity). The time-dependent diffusion coefficient depends only on the intrinsic dissociation rate and a “rebinding parameter”, which in turn depends on a group of constants including the intrinsic association and dissociation rates, the density of unoccupied surface binding sites, and the diffusion coefficient in solution.

Introduction A wide variety of biological processes are mediated by interactions between soluble ligands and cell surface receptors. Examples include immune processes which rely on interactions between soluble antibodies specific for pathogens and antibody receptors on immune cell surfaces,1 neurological processes in which soluble transmitters such as serotonin stimulate cells by binding to specific receptors,2 regulation of cellular growth and proliferation by interactions between specific growth factors and their cell-surface receptors,3 and blood hemostasis, which is mediated in large part by soluble proteins such as fibrinogen which associate with specific receptors on platelet surfaces.4 A long series of theoretical investigations has addressed the thermodynamic, kinetic, and transport characteristics of interactions between soluble ligands in three-dimensional solution and specific receptor sites on planar or spherical surfaces.5-35 A key feature of these processes is * To whom correspondence should be addressed: (919) 962-0328 (telephone); (919) 966-3675 (fax); [email protected] (electronic mail). † Part of the Langmuir special issue entitled The Biomolecular Interface. ‡ Department of Physics & Astronomy. § Department of Chemistry. | Present address: Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. (1) Daeron, M. Annu. Rev. Immunol. 1997, 15, 203. (2) Kim, J. H.; Huganir, R. L. Curr. Opin. Cell Biol. 1999, 11, 248. (3) Wells, J. A.; deVos, A. M. Annu. Rev. Biochem. 1996, 65, 609. (4) Zwaal, R. F. A.; Comfurius, P.; Bevers, E. M. Biochim. Biophys. Acta Biomembr. 1998, 1376, 433. (5) Adam, G.; Delbruck, M. Reduction of dimensionality in biological diffusion processes. In Structural Chemistry and Molecular Biology; Rich, A., Davison, N., Eds.; Freeman: San Francisco, CA, 1968; pp 198-215. (6) Berg, H. C.; Purcell, E. M. Biophys. J. 1977, 20, 193. (7) DeLisi, C. Q. Rev. Biophys. 1980, 13, 201. (8) Shoup, D.; Lipari, G.; Szabo, A. Biophys. J. 1981, 36, 697. (9) DeLisi, C.; Wiegel, F. W. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 5569. (10) Thompson, N. L.; Burghardt, T. P.; Axelrod, D. Biophys. J. 1981, 33, 435. (11) Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33. (12) Agmon, N. J. Chem. Phys. 1984, 81, 2811.

that they occur at the interface of a three-dimensional solution and a two-dimensional surface. This physical feature implies that the mechanisms by which ligandreceptor interactions proceed may be complicated by the interplay between the intrinsic chemistry of the ligandreceptor interaction and transport in solution. One phenomenon that is thought to be of particular importance is the process in which reversibly bound ligands dissociate from receptors, diffuse for a time in the nearby solution, and then rebind to the same or a nearby receptor on the cell membrane. Understanding the propensity for rebinding is therefore of central importance in under(13) Berg, O.; von Hippel, P. H. Annu. Rev. Biophys. Biophys. Chem. 1985, 14, 131. (14) Northrup, S. H.; Curvin, M. S.; Allison, S. A.; McCammon, J. A. J. Chem. Phys. 1986, 84, 2196. (15) Northrup, S. H. J. Phys. Chem. 1988, 92, 5847. (16) Zwanzig, R. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 5856. (17) Baldo, M.; Grassi, A.; Raudino, A. J. Phys. Chem. 1991, 95, 6734. (18) Zwanzig, R.; Szabo, A. Biophys. J. 1991, 60, 671. (19) Wang, D.; Gou, S.; Axelrod, D. Biophys. Chem. 1992, 43, 117. (20) Lauffenburger, D. A.; Linderman, J. J. Receptors: Models for Binding, Trafficking, and Signaling; Oxford University Press: New York, 1993. (21) Forsten, K. E.; Lauffenburger, D. A. J. Comput. Biol. 1994, 1, 15. (22) Axelrod, D.; Wang, M. D. Biophys. J. 1994, 66, 588. (23) Balgi, G.; Leckband, D. E.; Nitsche, J. M. Biophys. J. 1995, 68, 2251. (24) Goldstein, B.; Dembo, M.Biophys. J. 1995, 68, 1222. (25) Lauffenburger, D. A.; Forsten, K. E.; Will, B.; Wiley, H. S. Ann. Biomed. Eng. 1995, 23, 208. (26) Model, M. A.; Omann, G. Biophys. J. 1995, 69, 1712. (27) Agmon, N.; Edelstein, A. L. Biophys. J. 1997, 72, 1582. (28) Edelstein, A. L.; Agmon, N. J. Comput. Phys. 1997, 132, 260. (29) Zhao, X. L.; Yen, A.; Kopelman, R. J. Phys. Chem. B 1999, 103, 1930. (30) Ahn, J.; Kopelman, R.; Argyrakis, P. J. Chem. Phys. 1999, 110, 2116. (31) Lansky, P.; Krivan, V.; Rospars, J.-P. Eur. Biophys. J. 2001, 30, 110. (32) Shvartsman, S. Y.; Wiley, H. S.; Deen, W. M.; Lauffenburger, D. A. Biophys. J. 2001, 81, 1854. (33) Wofsy, C.; Coombs, D.; Goldstein, B. Biophys. J. 2001, 80, 606. (34) Levin, M. D.; Shimizu, T. S.; Bray, D. Biophys. J. 2002, 82, 1809. (35) Wofsy, C.; Goldstein, B. Biophys. J. 2002, 82, 1743.

10.1021/la0261601 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/05/2002

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Figure 1. Schematic of rebinding phenomenon. Molecules in solution, A, are in equilibrium with free surface binding sites, B, forming occupied surface binding sites, C. The association and dissociation rate constants are ka and kd, respectively. A single tagged molecule is placed at the origin at time zero. As time proceeds, the tagged molecule dissociates from the surface, explores the solution with diffusion coefficient DA, and rebinds to the surface at the same or a different position and later time. The equilibrium dissociation constant is Kd ) kd/ka, and the diffusion coefficient of the complexes on the surface is DC.

standing the mechanisms by which a large variety of biological processes occur. Rebinding is also important in the measurement of receptor-ligand interactions by using the surface plasmon resonance technique35-38 or total internal reflection fluorescence microscopy10,39-42 and can play a major role in the performance of surface-based biosensors43,44 and chromatographic materials.45,46 In a previous work, we presented a rigorous theoretical treatment of the rebinding process for the case in which monovalent ligands reversibly bind to monovalent sites on planar membrane surfaces.47 The coupled, partial differential equations that describe the reversible interaction of ligands in three-dimensional solution with sites on a planar surface were solved to find analytical solutions for the probabilities of finding a ligand on the surface or in solution, given initial placement on the surface. The general analytical solutions were used to find a simple expression for the probability that a molecule rebinds to the surface at a given position and time, after initial release at the origin. The previously derived probability expressions47 provide fundamental equations which form the basis for subsequent modeling of the effects of rebinding on ligand-receptor interactions. In a second work, the formalism was extended to find expressions for the average number and rate of rebinding events that have occurred, on the average, at a given time after placing a ligand on the surface.48 In the work described herein, we build on the previous theory to find an analytical expression for the manner in which the lateral diffusion coefficient is affected by repeated dissociation and rebinding at the surface. (36) Edwards, P. R.; Maule, C. H.; Leatherbarrow, R. J.; Winzor, D. J. Anal. Biochem. 1998, 263, 1. (37) Schuck, P.; Millar, D. B.; Kortt, A. A. Anal. Biochem. 1998, 265, 79. (38) Myszka, D. G.; He, X.; Dembo, M.; Morton, T. A.; Goldstein, B. Biophys. J. 1998, 75, 583. (39) Hsieh, H. V.; Thompson, N. L. Biochemistry 1995, 34, 12481. (40) Hsieh, H. V.; Thompson, N. L. Biophys. J. 1994, 66, 898. (41) Lagerholm, B. C.; Starr, T. E.; Volovyk, Z. N.; Thompson, N. L. Biochemistry 2000, 39, 2042. (42) Starr, T. E.; Thompson, N. L. Biophys. J. 2001, 80, 1575. (43) Sadana, A. Biosens. Bioelectron. 1998, 13, 1127. (44) Bourdillon, C.; Demaille, C.; Moiroux, J.; Saveant, J. M. J. Am. Chem. Soc. 1999, 121, 2401. (45) Ludes, M. D.; Wirth, M. Anal. Chem. 2002, 74, 386. (46) Hansen, R. L.; Harris, J. M. Anal. Chem. 1998, 70, 4247. (47) Lagerholm, B. C.; Thompson, N. L. Biophys. J. 1998, 74, 1215. (48) Lagerholm, B. C.; Thompson, N. L. J. Phys. Chem. B 2000, 104, 863.

Figure 2. Rebinding parameter b. The rebinding parameter b is a measure of the propensity for rebinding. This parameter was calculated from eq 3 and is shown as a function of (a) the dissociation rate constant kd, (b) the density of unoccupied sites B, and (c) the diffusion coefficient in solution DA. The equilibrium dissociation constant Kd equals (line) 1 nM, (dot) 1 µM, or (dash) 1 mM. In (a), B ) 100 molecules µm-2 and DA ) 100 µm2 s-1. In (b), kd ) 1 s-1 and DA ) 100 µm2 s-1. In (c), kd ) 1 s-1 and B ) 100 molecules µm-2.

Results Definitions. A reversible bimolecular reaction at a surface (the xy plane) is coupled with diffusion in solution (Figure 1). Molecules in solution, of concentration A, interact with unoccupied surface binding sites, of density B, at thermodynamic equilibrium. The density of occupied surface binding sites (complexes) is C. The kinetic association and dissociation rate constants are ka and kd, respectively, and the equilibrium dissociation constant is

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Kd ) kd/ka. The total surface site density is N, B ) NKd/(A + Kd), and C ) NA/(A + Kd). We imagine the case in which a given molecule is tagged and placed on the surface at the origin at time zero. The system remains in chemical equilibrium while the tagged molecule explores the surface and solution with time. Of interest is the lateral diffusion coefficient, D(t), which depends on the degree to which repeated dissociation from and rebinding to the surface occurs. D(t) is related to the diffusion coefficient of ligands in solution, denoted by the constant DA, and the diffusion coefficient of the complexes on the surface, denoted by the constant DC. Temporal Probability Densities. As shown previously,47,48 the stochastic behavior of the system is described by two basic probability functions, U(t) dt and Y(t) dt. U(t) dt is the probability that the tagged molecule dissociates from the surface between times t and t + dt, given that it bound at time zero. Y(t) dt is the probability that the tagged molecule rebinds to the surface between times t and t + dt, given that it dissociated at time zero. These functions are related to V(t), the probability that the tagged molecule has not dissociated from the surface by time t given that it bound at time zero, and Z(t), the probability that the tagged molecule has not rebound by time t given that it dissociated at time zero. These four functions are47,48

U(t) ) V(t) ) 1 -

Y(t) ) -

d -kdt e ) kd e-kdt dt

∫0 U(t′) dt′ ) e-k t t

d

()

kd d w(ib(kdt)1/2) ) b dt πt

Z(t) ) 1 -

(1)

1/2

- kdb2w(ib(kdt)1/2)

∫0t Y(t′)dt′ ) w(ib(kdt)1/2)

In eqs 1, w(iη) ) exp(η2) erfc(η) and the “rebinding parameter” b (which depends on kd, Kd, DA, and B) is described below. The integral of Y(t) over all time equals 1; i.e., dissociated molecules always rebind to a planar surface of infinite extent, and Z(∞) ) 0. Similarly, the integral of U(t) over all time is 1 and V(∞) ) 0; bound molecules always eventually dissociate. The difference between the dissociation and rebinding processes is found in the functional forms of U(t) and Y(t). Also of interest in this mathematical problem is the series of probability functions, Pn(t), describing the possible histories of dissociation and rebinding at a given time t after placing a tagged molecule at the origin on the surface at time zero. Each successive function Pn(t) represents one additional dissociation or rebinding event. P0(t) is the probability that the tagged molecule has not dissociated by time t; P1(t) is the probability that the tagged molecule dissociated and has not rebound by time t; P2(t) is the probability that the tagged molecule dissociated, rebound, and has not dissociated by time t; P3(t) is the probability that the tagged molecule dissociated, rebound, dissociated, and has not rebound by time t; P4(t) is the probability that the tagged molecule dissociated, rebound, dissociated, rebound, and has not dissociated by time t; and so forth.

As previously described48

P0(t) ) V(t) P1(t) ) P2(t) ) P3(t) )

P4(t) )

∫0t dt1 U(t1)Z(t - t1)

∫0t dt2 ∫0t

2

∫0t dt3 ∫0t

dt2

∫0t dt4 ∫0t

dt3

3

4

(2)

dt1 U(t1)Y(t2 - t1)V(t - t2)

∫0t

dt1

∫0t

dt2

2

U(t1)Y(t2 - t1)U(t3 - t2)Z(t - t3) 3

∫0t

2

dt1

U(t1)Y(t2 - t1)U(t3 - t2)Y(t4 - t3)V(t - t4)

In these expressions, t1 is the time of the first dissociation, t2 is the time of the first rebinding event, t3 is the time of the second dissociation, and t4 is the time of the second rebinding event. Hence, 0 e t1 e t2 e t3 e t4 e t. Functions Pn(t) with values of n > 4 may be found by including additional integrals to the functions shown in eqs 2. The sum over all n of Pn(t) equals 1; i.e., the functions Pn(t) describe all possible molecular histories at a given time t.48 The Rebinding Parameter b. The probability functions in eqs 1 and 2 depend only on the time, t, the intrinsic dissociation constant, kd, and the parameter b, which is given by47,48

b)

( )

B kd Kd DA

1/2

(3)

This parameter indicates the propensity for rebinding to the surface after dissociation. When b f 0, rebinding does not occur. In this case, Y(t) ) 0, Z(t) ) 1, P0(t) ) exp(-kdt), P1(t) ) 1 - exp(-kdt), and the remaining functions Pn(t) are zero. When b f ∞, molecules immediately rebind to the sites from which they dissociate. In this case, Y(t) ) δ(t), Z(t) ) 0, Pn(t) ) (kdt)n/2 exp(-kdt)/(n/2)! for even n, and Pn(t) ) 0 for odd n. Figure 2 shows the theoretical values of the parameter b as a function of Kd, kd, B, and DA. The parameters Kd, kd, and DA are assumed to be constants, for a given experimental situation, within the context of the model. The density of unoccupied binding sites, B, depends not only on the total binding site surface density, N, but also on Kd and the concentration of ligands in solution, A, which compete with the ligand of interest for the binding sites (see above). The values of Kd, kd, DA, N, and A allow one to calculate the value of b and therefore predict the degree to which rebinding is important. A nuance is that if the sites cluster, rebinding is expected to be more significant in regions of clustering, where B will be higher than the average over the whole surface, and less significant in the remaining regions, where B will be lower than the average. Average Times Spent Free and Bound. At time zero, when the tagged molecule of interest is placed on the surface, the lateral diffusion coefficient is DC. As time proceeds, the diffusion coefficient ranges from DC to DA, and the manner in which this process occurs depends on the fractions of the elapsed time that the molecule has spent on the surface and in the solution at a given time t. The average time spent free in solution may be calculated as

tf(t) ) tf1(t) + tf2(t) + tf3(t) + tf4(t) + ...

(4)

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where (see eqs 2)

∫0t dt1 (t - t1)U(t1)Z(t - t1)

tf1(t) ) tf2(t) ) tf3(t) )

∫0t dt2 ∫0t

2

dt1 (t2 - t1)U(t1)Y(t2 - t1)V(t - t2)

∫0t dt3 ∫0t

dt2

∫0t dt4 ∫0t

dt3

3

(5)

∫0t

dt1 (t - t3 +

∫0t

dt2

2

t2 - t1)U(t1)Y(t2 - t1)U(t3 - t2)Z(t - t3)

tf4(t) )

4

3

∫0t

2

dt1 (t4 - t3 +

t2 - t1)U(t1)Y(t2 - t1)U(t3 - t2)Y(t4 - t3)V(t - t4) and so forth. Similarly, the average time spent bound to the surface is

tb(t) ) tb0(t) + tb1(t) + tb2(t) + tb3(t) + tb4(t) + ...

(6)

where

tb0(t) ) tV(t) tb1(t) )

(7)

∫0t dt1 t1U(t1)Z(t - t1)

tb2(t) )

∫0t dt2 ∫0t

tb3(t) )

tb4(t) )

2

dt1 (t - t2 + t1)U(t1)Y(t2 - t1)V(t - t2)

∫0t dt3 ∫0t ∫0t dt4 ∫0t

3

dt2

∫0t

dt1 (t3 - t2 + t1)

∫0t

dt2

2

U(t1)Y(t2 - t1)U(t3 - t2)Z(t - t3)

4

dt3

3

∫0t

2

dt1 (t - t4 +

t3 - t2 + t1)U(t1)Y(t2 - t1)U(t3 - t2)Y(t4 - t3)V(t - t4) By combining eqs 4-7, one sees that ∞

tf(t) + tb(t) ) t

∑ Pn(t) ) t

(8)

n)0

As shown in the Appendix, analytical expressions for tf(t) and tb(t) may be found by Laplace transforming eqs 5 and 7, summing the series in Laplace transform space, and then inverse Laplace transforming. The results are tf(t) ) t - 2b

( ) t πkd

1/2

+

b2 - 1 + kd

[(b - 1)b2 + b]w(-ib1(kdt)1/2) - [(b2 - 1)b1 + b]w(-ib2(kdt)1/2)

Figure 3. Average times spent free in solution, tf(t), and bound to the surface, tb(t). Panel a shows the values of kdtf(t) calculated from eq 9, and panel b shows the values of kdtb(t) calculated from eq 10. The rebinding parameter b equals (line) 0, (dot) 1, (dash) 10, and (dot-dot-dash) 100. At all times and for all b values, tf(t) + tb(t) ) t.

The manner in which tf and tb depend on the time t and the rebinding parameter b are shown in Figure 3. For low values of b, tf quickly approaches t - 2b(t/πkd)1/2, while tb approaches 2b(t/πkd)1/2. For high values of b, this process occurs more slowly. Lateral Diffusion Coefficient. The motion of the tagged molecule in the x - y plane is described by the diffusion coefficient

2

(b1 - b2)kd

D(t) )

(9)

tf(t)DA + tb(t)DC t

(12)

By using eqs 9 and 10 in eq 12, one finds that

{

and

( )

t tb(t) ) 2b πkd

1/2

D(t) ) DC + {DA - DC} 1 -

1 - b2 + kd

2b b2 - 1 + + 1/2 k dt (πkdt)

}

[(b - 1)b2 + b]w(-ib1(kdt)1/2) - [(b2 - 1)b1 + b]w(-ib2(kdt)1/2)

[(b2 - 1)b2 + b]w(-ib1(kdt)1/2) - [(b2 - 1)b1 + b]w(-ib2(kdt)1/2)

(b1 - b2)kd

(b1 - b2)kdt

2

(10) where

b1,2 )

1 [-b ( (b2 - 4)1/2] 2

(11)

(13) The diffusion coefficient D(t) as given in eq 13 ranges from DC at time zero (when the tagged molecule is on the surface) to DA at time infinity (when the majority of the elapsed time has been spent free in solution). This dependence of

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D(t) on the elapsed time is shown in Figure 4a. The rate of the transition from DC to DA depends on the rebinding parameter b. When b ) 0 (no rebinding), the transition is rapid (for a given kd value) and reflects the first surface dissociation event. Here, eq 13 reduces to

[

]

1 (1 - e-kdt) [DA - DC] kdt

[D(t)]b)0 ) DC + 1 -

(14)

When b approaches infinity (high propensity for rebinding), the transition is very slow, and D(t) ≈ DC. In this case, the tagged molecule never escapes from the surface. In general, lower propensities for rebinding result in rapid translocation to the solution where the particle diffuses with constant DA, whereas higher propensities for rebinding repeatedly return the molecule to the surface where it diffuses with constant DC. The rate of the transition can be characterized by a half-time, t1/2, defined by the equation D(t1/2) ) (DC + DA)/2. As shown in Figure 4b, for low b values, kdt1/2 is small. In this limit (b , 1), kdt1/2 ≈1.59 + 1.48b. For high b values, kdt1/2 is large and ≈1.59b2. Also of interest are the short and long time limits of D(t). These limits are calculated from eqs 11 and 13 as

[

[

]

D(t) - DC DA - DC

]

D(t) - DC DA - DC

)

tf0

)1-

tf∞

kd t 2

(15)

2b (πkdt)1/2

Discussion The theoretical formalism describing the dissociation and rebinding behavior of a ligand reversibly bound to a planar surface developed in two previous works47,48 has been extended here. In the earlier works, the coupled differential equations describing the reversible interaction of ligands in three-dimensional solution with sites on a planar surface were solved to obtain analytical solutions for the probabilities of finding a ligand on the surface or in solution, given initial placement on the surface. These functions were then used to find an expression for the probability that a ligand rebinds to the surface at a time t after its dissociation, given by Y(t)47 (eqs 1). A series of probability functions, Pn(t) (eqs 2), was defined to describe the possible histories of dissociation and rebinding at a time t after placing a tagged molecule on the surface.48 These functions were used to obtain analytical expressions for the average number and rate of rebinding events at a given time after placing a ligand on the surface. In the work presented here, the theory has been extended to calculate an analytical expression describing the manner in which the lateral diffusion coefficient of the ligand depends on repeated surface dissociation and rebinding events as the molecule moves through solution and across the surface. Expressions were derived representing the average times a molecule spends free in solution and bound to the surface (eqs 9-11, Figure 3) by summing over average times associated with all possible dissociation/rebinding histories Pn(t). These expressions were combined with the constant surface and solution diffusion coefficients to obtain the predicted time-dependent diffusion coefficient, D(t) (eqs 12 and 13, Figure 4a). The diffusion coefficient ranges from the surface diffusion coefficient, DC, at time zero (when the tagged molecule is placed on the surface) to the solution diffusion coefficient, DA, at time infinity. The time dependence of the transition

Figure 4. Lateral diffusion coefficient D(t). (a) D(t) was calculated directly from eqs 14, 13, and 11. The rebinding parameter b equals (line) 0, (dot) 1, (dash) 10, and (dot-dotdash) 100. (b) The half-time t1/2 was calculated numerically from eqs 13 and 11. The lines show the approximate forms for kdt1/2, at low and high values of b, as described in the text.

between DC and DA depends only on the intrinsic dissociation rate and a “rebinding parameter”, b (eq 3), which in turn depends on the dissociation rate, kd, the density of unoccupied surface binding sites, B, the diffusion coefficient in solution, DA, and the equilibrium dissociation constant, Kd (Figure 2). The half-time for this transition was calculated numerically and is shown in Figure 4b as a function of the rebinding parameter b. Limiting expressions for D(t) are given for the cases of no rebinding (b ) 0) and infinite rebinding (b ) ∞), and for both short and long times (eqs 14 and 15). In general, both the previously presented theory47,48 and the new extension described in this work predict that rebinding effects will be significant if the value of the rebinding propensity, b, is on the order of 1 or larger (e.g., Figures 3 and 4). In the absence of competing molecules where rebinding is more prominent, A ) 0 and b ) 1/2 kaN(kdDA)-1/2 (eq 3). Thus, when N g k-1 a (kdDA) , b g 1, and significant rebinding occurs. Typical association rate constants at membrane surfaces and typical ligand diffusion coefficients in solution are on the order of ka ≈ 106 M-1 s-1 and DA ≈ 10-6 cm2/s, respectively. Therefore, for kd ) 1 s-1, rebinding will be significant when N g 6000 molecules/µm2. This value is about equal to the average cell surface density of a highly abundant receptor. Higher values of N are expected in regions where cell surface receptors cluster. For weaker binding (kd ) 100 s-1), very

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high receptor densities are required for significant rebinding to occur (N g 60 000 molecules/µm2) whereas for tighter binding (kd ) 0.01 s-1), the requirement on the receptor density is less stringent (N g 600 molecules/ µm2). As shown in Figure 4, for the typical case where DC < DA, rebinding with b g 1 leads to significantly slower exploration of the cell surface. In a previous work, total internal reflection with fluorescence photobleaching recovery (TIR-FPR) was used to verify, to the extent that it was possible, the theory for rebinding on which the results of this work are based.41 A related method, total internal reflection with fluorescence correlation spectroscopy (TIR-FCS),42 might also be used to verify the theory. To correlate the value of b measured by these methods with the apparent lateral diffusion coefficient as predicted by eq 13, it might be useful to systematically vary the characteristic distance of the observed surface area. In TIR-FCS, the surface area is defined by the size of an aperture at an intermediate image plane. In TIR-FPR, the surface area is defined either by the extent to which the incident beam is focused41 or by using evanescent interference patterns.49 A more direct method of measuring the degree to which the lateral diffusion coefficient is reduced by rebinding after dissociation would be to use single molecule tracking methods to follow the paths of individual ligands as they move near model or natural membranes containing receptors.

By defining

ξ)

tf(R) )

{

kd -W′(R) [1 - ξ + ξ2 - ξ3 + ...] + kd + R

[

]

tfn(R) )

(-1)(n+2)/2nkdn/2 2(kd + R)(n+2)/2

(A1)

tfn(R) )

(-1)

{

(A4)

Equation A4 sums exactly to

tf(R) )

kdW(R) 2

R [kdW(R) + 1]

(A5)

Inverse Laplace transforming equation A5 yields eq 9. Laplace transforming t to R in eqs 7 and subsequent terms yields

(-1)n/2(n + 2)kdn/2

tbn(R) )

2(kd + R)(n+4)/2

Xn/2(R)

(A6)

n ) 0, 2, 4, 6, ... tbn(R) )

(-1)(n-1)/2(n + 1)kd(n+1)/2 W(R)X(n-1)/2(R) (n+3)/2 2(kd + R) n ) 1, 3, 5, ...

(n-2)/2

X′(R)X

(R) Then

n ) 2, 4, 6, ... (n+1)/2

}

X′(R){1 + kdW(R)} [1 - 2ξ + 3ξ2 - ...] kd + R

Laplace transforming t to R in eqs 5 and subsequent terms yields

kd W′(R) kd + R

(A3)

one finds that

Appendix: Average Times Spent Free and Bound

tf1(R) ) -

kd X(R) kd + R

tb(R) )

(n+1)/2

kd

(kd + R)(n+1)/2 n-1 W(R)X′(R) X(n-3)/2(R) + W′(R)X(n-1)/2(R) 2 n ) 3, 5, 7, ...

}

1 + kdW(R)

W′(R) )

d W(R) dR

X(R) ) RW(R) - 1

X′(R) )

d X(R) dR

(A2)

(49) Huang Z.; Pearce, K. H.; Thompson, N. L. Biophys. J. 1994, 67, 1754.

(kd + R)

{1 - 2ξ + 3ξ2 - 4ξ3 + ...}

(A7)

Equation A7 sums exactly to

tb(R) )

where

W(R) ) LtfRw(ib(kdt)1/2)

2

1 R [kdW(R) + 1] 2

(A8)

Inverse Laplace transforming equation A8 gives eq 10. Acknowledgment. This work was supported by NSF Grant MCB-0130589, ACS-PRF Grant 35376AC5-7, and North Carolina Biotechnology Center Grant 2000-ARG-0026. LA0261601