INTRODUCTION

Top Abstract Introduction Results Discussion Appendix A Appendix B References

Numerous biochemical processes are mediated by interactions between soluble ligands and cell-surface receptors. Considerable effort has been devoted to understanding the role of ligand-receptor interactions in the mechanisms of these processes. A series of theoretical investigations into the thermodynamic, kinetic, and transport characteristics of interactions between macromolecules in three-dimensional solution and sites on a planar or spherical surface has been presented (e.g., Adam and Delbrück, 1968; Berg and Purcell, 1977; DeLisi and Wiegel, 1981; Shoup and Szabo, 1982; Berg and von Hippel, 1985; Northrup et al., 1986; Northrup, 1988; Zwanzig, 1990; Wang et al., 1992; Axelrod and Wang, 1994; Forsten and Lauffenburger, 1994; Lauffenburger et al., 1995; Goldstein and Dembo, 1995; Balgi et al., 1995; Model and Omann, 1995). These investigations have suggested that a phenomenon 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 surface. Evidence for the rebinding process has been experimentally obtained for a variety of ligand-receptor systems, including haptens with IgE-coated mast cells (Erickson et al., 1987; Goldstein et al., 1989; Erickson et al., 1991); bovine prothrombin fragment 1 with negatively charged substrate-supported planar membranes (Pearce et al., 1992); antibodies with immobilized peptides (Duschl et al., 1996); lipoprotein lipase with immobilized heparin sulfate (Lookene et al., 1996); and neurotransmitters in synapses (Otis et al., 1996).

In this work, a rigorous theoretical treatment of the rebinding process is presented. An analytical solution for the spatial and temporal dependence of the probabilities of finding the molecule on the surface or in the solution, given initial placement at the origin, is derived. This general analytical solution is 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 (not placement) at the origin. The probability expressions provide fundamental equations forming the basis for subsequent modeling of ligand-receptor interactions in particular geometries.

	RESULTS

Top Abstract Introduction Results Discussion Appendix A Appendix B References

General considerations

Consider a reversible bimolecular reaction at a surface (the xy-plane) coupled with diffusion in solution (Fig. 1). A concentration of molecules in solution, A, is in equilibrium with a density of molecules on the surface, C, and a density of unoccupied, immobile surface binding sites, B. We imagine a case where a tagged molecule is 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. The reaction mechanism may be written as

(1)

where k_a and k_d are the kinetic association and dissociation rate constants, respectively. Of interest are the probabilities P_C(x, y, t) and P_A(x, y, z, t) for finding the tagged molecule on the surface or in solution, respectively, at time t > 0. These functions may be used to calculate parameters that describe rebinding of the tagged molecule at the surface.

View larger version (8K): [in this window] [in a new window]   FIGURE 1   Schematic of rebinding phenomenon. Molecules in solution (open circle) (A) are in equilibrium with free surface binding sites (B) and occupied surface binding sites (C). The association and dissociation rate constants are ka and kd, respectively. A single tagged molecule (closed circle) is placed at the origin at time zero. As time proceeds, the tagged molecule dissociates from the surface, explores the solution with diffusion coefficient D, and rebinds to the surface at a different position and later time.

Differential equations

We begin with the differential equations that govern the reaction:

(2)

(3)

where D is the diffusion coefficient of the tagged molecule in solution and ² is the three-dimensional Laplacian. Because the system is in chemical equilibrium, the density of free surface sites, B, the concentration of molecules in solution A, and the density of bound molecules, C, are constant and the differential equations are linear rather than nonlinear.

Initial and boundary conditions

At time zero, we define the probability of locating the molecule on the surface by a normal distribution around the origin with a small width, a, and the probability of finding the molecule in solution as zero:

(4)

As a  0, the initial surface probability is a Dirac delta function located at the origin. Nine of the ten required boundary conditions are

(5)

The final boundary condition describes the flux at the surface:

(6)

General solutions for P_C(r, t) and P_A(r, z, t)

To describe the rebinding process, we have analytically solved the set of equations given above (Eqs. 2-6) for the surface density probability and the solution concentration probability. The details are given in Appendix A. The expressions for P_C(r, t) and P_A(r, z, t), where r = (x, y), are

(7)

(8)

In these equations, q is an integration variable that results from a Fourier transform with r  q, J₀(qr) is the zero-order Bessel function, and the w function is defined as (Abramowitz and Stegun, 1974)

(9)

The rates _i, and the fractional amplitudes f_i, of the three terms containing w functions with arguments dependent on the _i, are given by

(10)

(11)

where i  j  k. The expressions shown in Eq. 7 and Eqs. 9-11 are similar to those derived previously in a different context (Thompson et al., 1981).

Characteristic rates and distances

It is instructive to write P_C(r, t) and P_A(r, z, t) in terms of physically significant characteristic rates and distances whose relative sizes determine the shapes of these functions. The resulting expressions are also useful for generating more concise plots of P_C(r, t) and P_A(r, z, t) (see below). One characteristic rate is the surface dissociation rate, k_d. Another characteristic rate is

(12)

In Eq. 12, N is the total density of surface binding sites (occupied and unoccupied) and K_d is the equilibrium dissociation constant. A characteristic length may be found by using the solution diffusion coefficient and the intrinsic dissociation rate. This length is defined as

(13)

Dimensionless forms of the probability functions

By writing the time and lengths in dimensionless forms as

(14)

the probability expressions may be written in dimensionless form as

(15)

(16)

The parameter c is a dimensionless integration variable, and

(17)

(18)

The parameter b

If the time t is cast as a product with k_d, and the lengths r, z, and a are scaled by the length , then the shape of the dimensionless probability densities, Q_C(, ) and Q_A(, , ) (Eqs. 15 and 16), written in terms of the dimensionless variables, is entirely determined by the parameters  and b, where (see Eqs. 12 and 18)

(19)

Typical values of the parameter b for various experimentally relevant conditions are shown in Fig. 2.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   Typical values of the rebinding parameter b. The values of the rebinding parameter b are shown as a function of (a) the solution concentration, A, (b) the association rate, ka, and (c) the dissociation rate, kd. For all plots, the solution diffusion coefficient D equals 106 cm2s1 and the total surface site density N = 1 molecule/µm2 (line), 102 molecules/µm2 (dash), or 104 molecules/µm2 (dot). In (a), ka = 106 M1s1 and kd = 1 s1. In (b), A = 106 M and kd = 1 s1. In (c), ka = 106 M1 s1 and A = 106 M. Values were calculated using Eq. 19.

The parameter b is interpreted as a measure of the likelihood of prompt rebinding. For example, if N approaches zero or D approaches infinity, the parameter b approaches zero. In these limits, rebinding does not occur either because there are no surface binding sites available for occupation by the tagged molecule, or because the tagged molecule quickly diffuses away from the surface before rebinding. On the other hand, if N approaches infinity or D approaches zero, rebinding is promoted and the parameter b approaches infinity.

The parameter b depends in a more complex manner on the solution concentration A than on D and N. As the concentration of untagged molecules in solution is increased, the surface density C is increased, rebinding is blocked and b decreases to zero. In this case, untagged molecules in solution compete with the tagged molecule for surface binding sites. At low solution concentrations A, b equals a limiting value given by

(20)

In this limit, even though very few of the surface binding sites are occupied, b does not become infinitely large. The parameter b' describes the extent of rebinding in the case where competition from solution phase molecules is negligible.

The dependence of the parameter b on the association rate constant k_a is also nonlinear. As k_a is decreased, b decreases to zero. This limit describes the case in which the surface acts as a reflecting wall. When the constant k_a is increased to infinity, the parameter b reaches a limiting value given by

(21)

The parameter b", which does approach infinity if the solution concentration is zero, can be understood as describing the extent of rebinding when the association rate constant is high but the surface binding sites are partially occupied.

The dependence of the parameter b on the dissociation rate constant k_d is somewhat complex. The limit of no rebinding, b  0, is reached if k_d approaches either zero or infinity. When k_d is very large, the surface acts as a reflecting wall. When k_d is very small, the equilibrium dissociation constant is small, binding of untagged ligands is promoted, and the surface binding sites to which the tagged molecule might rebind are blocked. As a function of k_d, the parameter b peaks when

(22)

This condition is the one in which the surface binding sites are half-occupied. In this case, the extent of rebinding is maximized by balancing the inhibitive effects of low and high kinetic dissociation constants.

It is instructive to write the parameter b in terms of the density of occupied surface binding sites, the solution concentration, and the length , i.e.,

(23)

As shown, b is the ratio of two lengths. The first length, C/A, is the thickness of a slab in solution that contains a density of untagged molecules equal to the density of untagged molecules on the surface. The second length,  (Eq. 13), is the average distance traveled by diffusion in solution during the average surface residency time, k_d¹. If   C/A, the molecule diffuses away from the surface and rebinding is not favored. On the other hand, if   C/A, diffusion in solution is comparatively slow, and rebinding is favored. Fig. 3 a shows the values of the characteristic length  as a function of the dissociation rate k_d, and Figs. 3 b and c show the values of the length C/A for typical values of N, K_d, and A.

View larger version (12K): [in this window] [in a new window]   FIGURE 3   Typical values for the characteristic lengths  and C/A. In (a), the values of the length  (Eq. 13) are shown as a function of the dissociation rate, kd, with the solution diffusion coefficient, D, equal to 106 cm2s1. Also shown are the values of the length C/A (Eq. 23), where C is the surface density, as a function of (b) the solution concentration, A, and (c) the equilibrium dissociation constant, Kd. In (b) and (c), the total surface site density, N, equals 1 molecule/µm2 (line), 102 molecules/µm2 (dash) or 104 molecules/µm2 (dot). In (b), Kd = 106 M. In (c), A = 106 M.

Limiting solutions for P_C(r, t) and P_A(r, z, t)

A closed form solution may be obtained for P_C(r, t) in the limit of no rebinding (b  0). In this case (Eqs. 17 and 18),

(24)

By using these values in Eqs. 7 and 15, one finds that (Abramowitz and Stegun, 1974)

(25)

In this limit, the tagged molecule dissociates from its initial binding site near the origin in an exponential fashion with rate k_d, and is never again found on the surface.

In the limit of infinite rebinding, b  , simple forms for both P_C(r, t) and P_A(r, z, t) can be found. In this case (Eqs. 17 and 18)

(26)

By using these values in Eqs. 7, 8, 15, and 16, one finds that

(27)

(28)

In this limit, the tagged molecule remains bound at the origin and is never found in the solution.

Plots of the general solutions for Q_C(, ) and Q_A(, , )

The surface probability density Q_C(, ) and the solution probability density Q_A(, , ) can be calculated by numerical integration of Eqs. 15 and 16. Figs. 4-6 show the results of these calculations for four values of the rebinding parameter b (0.01, 1, 100, 10⁴). Two values of the parameter  are considered, which correspond to a = 30 Å and D = 10⁶ cm²s¹. In the first set of plots, the intrinsic surface dissociation constant is large (k_d = 100 s¹), the characteristic length  is small ( = 1 µm), and  = 3 × 10³. In the second set, the dissociation constant is small (k_d = 0.01 s¹), the length  is large ( = 100 µm), and  = 3 × 10⁵.

View larger version (29K): [in this window] [in a new window]   FIGURE 4   Surface density probability, QC(, ), near the origin. PC(r, t) = (D/kd) QC(, ) is the probability density of finding a tagged molecule at the surface at position r and time t, given the distribution at time zero from Eq. 4, where kd is the surface dissociation rate,  is a dimensionless form of the surface position, and  is a dimensionless form of the time (Eqs. 13 and 14). QC(, ) was calculated by numerical integration of Eq. 15, for a particle radius, a, of 30 Å, and a solution diffusion coefficient, D, of 106 cm2s1. In (a-d), kd = 100 s1, implying  = 1 µm and  = 3 × 103, and in (e-h) kd = 0.01 s1, implying  = 100 µm and  = 3 × 105 (Eqs. 13 and 14). The rebinding parameter b (Eq. 19, Fig. 2) equals (a and e) 0.01, (b and f) 1, (c and g) 100, or (d and h) 104. The probability density was plotted for  equal to 0 (line), 0.1 (dash), 0.5 (dot), 1 (dot-dash), and 5 (dot-dot-dash). The surface density probability, QC(, ), was normalized by the surface density probability at the origin at time zero, [QC(, )],=0.

View larger version (31K): [in this window] [in a new window]   FIGURE 5   Surface density probability, QC(, ), far from the origin. PC(r, t) = (D/kd) QC(, ) is the probability density of finding a tagged molecule at the surface at position r and time t, given the distribution at time zero from Eq. 4, where kd is the surface dissociation rate,  is a dimensionless form of the surface position, and  is a dimensionless form of the time (Eqs. 13 and 14). QC(, ) was calculated by numerical integration of Eq. 15, for a particle radius, a, of 30 Å, and a solution diffusion coefficient, D, of 106 cm2s1. In (a-d), kd = 100 s1, implying  = 1 µm and  = 3 × 103, and in (e-h) kd = 0.01 s1, implying  = 100 µm and  = 3 × 105 (Eqs. 13 and 14). The rebinding parameter b (Eq. 19, Fig. 2) equals (a and e) 0.01, (b and f) 1, (c and g) 100, or (d and h) 104. The probability density was plotted for  equal to 0 (line), 0.1 (dash), 0.5 (dot), 1 (dot-dash), and 5 (dot-dot-dash). The surface density probability, QC(, ), was normalized by the surface density probability at the origin at time zero, [QC(, )],=0.

View larger version (24K): [in this window] [in a new window]   FIGURE 6   Solution probability QA(, , ). PA(r, z, t) = (D/kd)3/2 QA(, , ) is the probability of finding the tagged molecule in solution a distance r from the origin and a distance z from the surface, at time t, where kd is the surface dissociation rate,  is a dimensionless form of the surface position,  is a dimensionless form of the distance from the surface, and  is a dimensionless form of the time (Eqs. 13 and 14). QA(, , ) was calculated by numerical integration of Eq. 16, for a particle radius, a, of 30 Å and a solution diffusion coefficient, D, of 106 cm2s1. In (a-c) kd = 100 s1, implying  = 1 µm and  = 3 × 103, and in (d and e) kd = 0.01 s1, implying  = 100 µm and  = 3 × 105 (Eqs. 13 and 14). The rebinding parameter b (Eq. 19, Fig. 2) equals (a and d) 0.01, (a and d) 1, (b and d) 100, or (c and e) 104. The probability density was plotted for  equal to 0.1 (dash), 0.5 (dot), 1 (dot-dash), and 5 (dot-dot-dash). The solution probability was equivalent within plot resolution for (a) kd = 100 s1 and b = 0.01 or b = 1, and for (d) kd = 0.01 s1 and b = 0.01, b = 1, or b = 100. The solution probability, QA(, , ), was normalized by the surface density probability at the origin at time zero, [QC(, )],=0.

At time zero, the surface probability function Q_C(, ) is a spatial Gaussian with width  (Eq. 4). Near the origin, this Gaussian shape is retained approximately as time proceeds (Fig. 4). The primary feature in this spatial region is that the peak at  = 0 decreases in magnitude with time . For low values of b, this decrease is exponential in  (Eq. 25) and reflects the desorption, without rebinding, of the molecule from its initial position on the surface. For higher values of b, the peak decreases more slowly with . This characteristic describes dissociation followed by rebinding at the same or a nearby position.

The behavior of the surface probability function is more complex further from the origin. Fig. 5 shows the values of Q_C(, ) for values of   100  (or r  100 a = 0.3 µm). At these positions, the probability (as plotted) that the molecule has been present at a given position since the initial time is given by (Eq. 25) Q_C(, )/[Q_C(, )]_,=0  exp[] exp[10⁴]. However, the magnitudes of Q_C(, )/[Q_C(, )]_,=0 are much larger, on the order of 10⁹ to 10³. Therefore, these relatively high values of the surface probability functions represent the probability that the molecule has dissociated and rebound to the surface.

For small values of b (Fig. 5, a, e, and f), at large values of , the probability of rebinding after a given time is very small. For larger values of b (Fig. 5, b, c, g, and h), molecules travel large distances in solution before rebinding, resulting in a spread of molecules away from the origin. In the case of even larger values of b (Fig. 5 d), the higher probability of rebinding after a short time results in slower spread as molecules rebind closer to the origin. At very large values of b, rebinding occurs at, or very near, the origin as the probability of a molecule being in the solution approaches zero. In this case there is very little spread of molecules along the surface. The maximum spread occurs at large values of b for smaller values of  (Fig. 5, c and h). A final feature (apparent in Fig. 5, a, b, f, and g) is that, at a given position, the surface probability increases, peaks, and then decreases with time. This feature results from the combination of increased spreading at larger times along with a lower probability of finding the molecule anywhere on the surface at these later times.

The plots for the solution probability, Q_A(, , ), (Fig. 6) demonstrate the b-dependence of rebinding well. The solution concentration is at a maximum for lower values of b, while it approaches zero as b becomes very large. Furthermore, it is demonstrated that once a molecule is in the solution it rapidly diffuses away from the surface so that for longer times, , there is a very low probability that the molecule is in solution next to the surface. For larger values of b, the molecule rebinds faster than it diffuses away in solution, again illustrating the low spread of molecules along the surface at large b.

Spatially integrated surface probability S()

The probability of locating a molecule anywhere on the surface at time  is found by integrating Q_C(, ) over all space:

(29)

In the limit of no rebinding, b  0, and the spatial integral gives a simple exponential with rate k_d; in the limit of extreme rebinding, b  , and the spatial integral gives a w function with rate k_t (Abramowitz and Stegun, 1974):

(30)

The function S() is shown in Fig. 7 and has been previously described (Thompson et al., 1981).

View larger version (19K): [in this window] [in a new window]   FIGURE 7   Spatially integrated surface probability S(). The function S() gives the probability of finding the tagged molecule on the surface at time t = /kd, where kd is the surface dissociation rate. This plot shows S() as calculated from Eq. 29 for the rebinding parameter b equal to 0 (line), 0.01 (long dash), 1 (intermediate dash), 10 (short dash), 100 (dot), and  (dot-dash).

Temporal rebinding probability Y()

The function S() describes the overall probability of finding the tagged molecule on the surface at time . Of interest also is the individual probability that a molecule rebinds at time , given initial release at time zero. This individual rebinding probability, Y(), can be found by writing S() as an infinite sum of functions S_n()

(31)

where each successive population represents one additional dissociation and rebinding event. S₁() is the probability that the tagged molecule has not yet dissociated at time . S₂() is the probability that the molecule dissociated at time ₁, rebound at time ₂, and remains bound at time ; averaged over all possible values of ₁ and ₂. S₃() is the probability that the molecule dissociated at time ₁, rebound at time ₂, dissociated at time ₃, rebound at time ₄, and remains bound at time ; averaged over all possible values of ₁, ₂, ₃, and ₄. Subsequent functions S_n() are found by adding more dissociation and rebinding events.

The functions S_n() are

(32)

and so forth. Y() is the probability that the molecule rebinds to the surface at time  given initial release at time zero and is the function of interest. In the limit of no rebinding, Y() = 0, S_n() = 0 for n  2, and

(33)

In the limit of extreme rebinding, Y() = (), and

(34)

These limits are consistent with Eqs. 30 (Fig. 7).

As shown in Appendix B, the function Y() can be found by Laplace transforming S() along with the S_n(), carrying out the infinite sum (Eq. 31) in Laplace transform space, and then inverse Laplace transforming. The simple result is

(35)

This function is shown in Fig. 8. When rebinding is not favored (low b), Y() has a low magnitude and low slope, giving a small but finite probability of rebinding over a long time range. As rebinding becomes more favored (higher b), the magnitude is higher at low values of  but the slope becomes more negative and at longer times the probability of rebinding is low. At very high values of b, rebinding occurs at very short times after surface dissociation. The integral of Y() over all time equals one. The molecule always eventually rebinds to the surface.

View larger version (17K): [in this window] [in a new window]   FIGURE 8   Temporal rebinding probability Y(). The function Y() gives the probability that the molecule rebinds to the surface at time t = /kd, where kd is the surface dissociation rate, given initial release at time zero. This plot shows Y() as calculated from Eq. 35 for the rebinding parameter b equal to 0.01 (line), 1 (dash), 10 (dot), and 100 (dot-dash).

The surface probability, S(), is plotted in Fig. 9 in terms of the subpopulations S₁(), S₂(), S₃(), and so forth. It is seen that as b becomes larger an increasing number of populations are required to describe the total surface probability. In the case of b = 0.01, the total surface probability, S(), may be described as consisting of two populations: those molecules that have not yet been released at time , S₁(), and those molecules that were released at time ₁, rebound at time ₂, and remained bound at time , S₂(). However, for b = 1, more populations than S₁() and S₂() are required to describe the total surface probability, S(). Here we have calculated the terms S₂() and S₃() by numerical integration of Eqs. 32 (with Eq. 35). In the case of b = , the expression for each population can be calculated exactly according to Eq. 34. Here it is seen that the 10 first populations, S₁()-S₁₀(), are sufficient to yield S() for short times , but that many more populations are required at longer times .

View larger version (15K): [in this window] [in a new window]   FIGURE 9   Components Sn() of spatially integrated surface probability. These plots illustrate the manner in which the functions Sn() sum to give S() (Eq. 31). Three cases are shown corresponding to (a) b = 0.01, (b) b = 1, and (c) b = . In all three plots, S() (line) was calculated directly from Eq. 29, S1() (long dash) was calculated from Eq. 32, and S2() (short dash) was calculated by numerical integration of Eq. 32. In (a), the sum of S1() and S2() is equivalent to S() within plot resolution. In (b), S3() (intermediate dash) was calculated by numerical integration of Eq. 32. The sum of S1(), S2(), and S3() (dot) is equivalent to S() only at small times. In (c), S2()-S10() (intermediate dash) were calculated from Eq. 34; the sum of Sn() (dot), for n = 1 to 10, is equivalent to S() only at small times.

Spatial and temporal rebinding probability W(, )

The probability density Q_C(, ) describes the overall spatial and temporal probability of finding the tagged molecule on the surface following successive rebinding events. Of interest also is the individual probability that the molecule rebinds at a position, , and a later time, , given initial release at the origin at time zero. As for S() and Y(), this individual rebinding probability may be found by first writing the probability density Q_C(, ) as the sum of an infinite number of functions

(36)

where each successive population represents one additional dissociation and rebinding event. Here, Q₁(, ) is the probability that the molecule started at position  and has not yet dissociated at time . Q₂(, ) is the probability that the molecule started at position ₁, dissociated at time ₁, rebound at position and time ₂, and remained bound until time ; averaged over all possible values of ₁, ₁, and ₂. Q₃(, ) is the probability that the molecule started at position ₁, dissociated at time ₁, rebound at position ₂ and time ₂, dissociated at time ₃, rebound at position  and time ₄, and remained bound until time ; averaged over all possible values of ₁, ₂, ₁, ₂, ₃, and ₄. Subsequent Q_n(, ) functions are found by adding more dissociation and rebinding events.

The first three functions Q_n(, ) are