J. Phys. Chem. B 2000, 104, 863-868
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Temporal Dependence of Ligand Dissociation and Rebinding at Planar Surfaces B. Christoffer Lagerholm† and Nancy L. Thompson* Department of Chemistry, UniVersity of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290 ReceiVed: September 21, 1999
In a previous work, conditions for which a ligand reversibly bound to a planar surface dissociates and then rebinds to the surface were theoretically examined (Lagerholm, B. C.; Thompson, N. L. Biophys. J. 1998, 74, 1215). The coupled 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. An expression was also found for the probability that a ligand rebinds to the surface at a given position and time after its release. In this work, the formalism is extended to calculate analytical, closed form 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. These functions depend only on the intrinsic dissociation rate and a “rebinding parameter,” which depends on a group of constants including the intrinsic association and dissociation rates, the density of surface binding sites, and the diffusion coefficient in solution. The results are interpreted in terms of typical conditions for biologically relevant ligands and their receptors.
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;1-5 neurological processes in which soluble transmitters such as serotonin stimulate cells by binding to specific receptors;6-8 regulation of cellular growth and proliferation by interactions between specific growth factors and their cell-surface receptors;9,10 and blood hemostasis, which is mediated in large part by soluble proteins such as fibrinogen which associate with specific receptors on platelet surfaces.11,12 A series of theoretical investigations during the past 30 years has addressed the thermodynamic, kinetic, and transport characteristics of interactions between soluble ligands in threedimensional solution and specific receptor sites on planar or spherical surfaces.13-39 A key feature of these processes is that they occur at the interface of three-dimensional solution and a two-dimensional surface. This physical feature implies that the mechanisms by which ligand-receptor interactions proceed may be complicated by the interplay between the intrinsic chemistry of the ligand-receptor 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 understanding 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 technique40-42 or total internal * To whom correspondence should be addressed. Campus Box 3290 (paper mail), (919) 962-0328 (telephone), (919) 962-2388 (fax),
[email protected] (e-mail). † Current address: Center for Light Microscope Imaging and Biotechnology, Carnegie Mellon University, Pittsburgh, PA 15213.
reflection fluorescence microscopy,18,43-45 and can play a major role in the performance of surface-based biosensors.46,47 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.37 Analytical solutions for the probabilities of finding a molecule on the surface or in the solution, given initial placement at the origin, were derived. 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 expressions37 provide fundamental equations which form the basis for subsequent modeling of the effects of rebinding on ligand-receptor interactions. In the work described herein, we extend the previous results to calculate the average number of rebinding events that have occurred at a given time after placing a molecule on the surface as well as the time-dependent average rebinding rate. The results are interpreted in terms of typical conditions for biologically relevant ligands and their receptors. Results Ligand Dissociation and Rebinding at Planar Surfaces. We consider a reversible bimolecular reaction at a surface (the xy-plane) coupled with diffusion in solution (Figure 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 surface binding sites, B. We imagine the case in which 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 ka
A + B {\ }C k
10.1021/jp9933830 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/12/2000
d
(1)
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Figure 1. Schematic of rebinding phenomena. Molecules in solution, 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 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 the same or a different position and later time.
where ka and kd are the kinetic association and dissociation rate constants, respectively. Of interest is the subsequent time course of the molecule as it dissociates from and rebinds to the surface. Temporal Probability Densities. The probabilities that the tagged molecule dissociates from the surface between times t and t + dt, U(t) dt, and that the tagged molecule has not dissociated from the surface by time t, V(t), are given by
U(t) ) kde-kdt V(t) ) 1 -
(2a)
∫01U(t′) dt′ ) e-k t d
(2b)
The probability that the tagged molecule rebinds to the surface between times t and t + dt, given that it dissociated at time zero, is denoted here as Y(t) dt. The probability that the tagged molecule has not rebound by time t, given that it dissociated at time zero, is denoted as Z(t). As shown previously,37 these functions are determined solely by the intrinsic dissociation rate, kd, and a “rebinding parameter”, b (see below). The functions Y(t) and Z(t) are
[x
Y(t) ) kd
b
πkdt
]
- b2w(ibxkdt) ) -
Z(t) ) 1 -
∂ w(ibxkdt) ∂t
∫0tY(t′) dt′ ) w(ibxkdt)
(3a)
(3b)
In eqs 3a and 3b48 2
w(iη) ) eη erfc(η)
(4)
Y(t) and Z(t) are real and decay monotonically with time. The functions Y(t) and Z(t) are shown in Figure 2. The parameter b indicates the degree of rapid binding to the surface after dissociation and is related to the fundamental parameters of the system as
b)
kaN k d + k aA
x
kd D
(5)
where N is the total density of surface binding sites (unoccupied plus occupied), A is the concentration of identical competing ligands in solution, and D is the ligand diffusion coefficient in solution. When b f 0, rebinding does not occur. In this limit, Y(t) ) 0 and Z(t) ) 1. When rebinding is not favored (low b),
Figure 2. Temporal rebinding probabilities Y(t) and Z(t). (a) The function Y(t) dt gives the probability that a molecule rebinds to the surface between times t and t + dt, given that the molecule dissociated at time zero. (b) The function Z(t) gives the probability that a molecule has not rebound by time t, given that the molecule dissociated at time zero. Curves were calculated from eqs 3. The rebinding parameter b equals (line) 100; (long dash) 10; (short dash) 1; and (dot dash) 0.1. When b f ∞, Y(t) ) δ(t) and Z(t) ) 0; when b f 0, Y(t) f 0 and Z(t) ) 1.
Y(t) has a low magnitude and low slope, giving a small but finite probability of rebinding over a long time range. When rebinding is more favored (higher b), the magnitude of Y(t) is higher at low values of t but the slope becomes more negative and at longer times the probability of rebinding is very low. At very high values of b, rebinding occurs at very short times after surface dissociation. When b f ∞, molecules immediately rebind to the sites from which they dissociate. In this limit, Y(t) ) δ(t) and Z(t) ) 0. 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. It is of interest to define a 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. We define P0(t) as the probability that the tagged molecule has not dissociated by time t; P1(t) as the probability that the tagged molecule dissociated at time t1 and has not rebound by time t; P2(t) as the probability that the tagged molecule dissociated at time t1, rebound at time t2, and has not dissociated by time t; P3(t) as the probability that the tagged molecule dissociated at time t1, rebound at time t2, dissociated at a time t3, and has not
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rebound by time t; and so forth. Then
Therefore
P0(t) ) V(t) P1(t) ) P2(t) ) P3(t) )
P4(t) )
∫0 dt1 U(t1)Z(t - t1) t
∫0 dt2∫0 t
∫0t dt3∫0t
3
∫0t dt4∫0t
4
dt2
dt3
t2
∫0t ∫0t
2
dt1 U(t1)Y(t2 - t1)V(t - t2)
(6c)
dt1 U(t1)Y(t2 - t1) ×
3
∫0t
dt2
2
b 2 {(1 - ib)e-kdt + ibw(xkdt) - w(ibxkdt)} + 2 1+b kdt i 1 b (i + b)kdt + e-kdt - i + kdt w(xkdt) 2 2 2 π 1+b (7c)
]
(
x }
)
These functions are shown in Figure 3. When b f 0, P1(t) ) 1 - exp(-kdt) and Pn(t) ) 0 for n > 1. When b f ∞, Pn(t) ) (kdt)n/2 exp(-kdt)/(n/2)! for even n and Pn(t) ) 0 for odd n. In general, for n > 0 and b > 0, Pn(0) ) Pn(∞) ) 0. For odd n, Pn(t) is larger for smaller values of b where rebinding is less prominent, whereas for even n, Pn(t) is larger for higher values of b where rebinding is favored. In all cases, the time at which Pn(t) is maximized decreases with increasing b. For all values of b, the sum over all n of Pn(t) equals 1. This result may be demonstrated by using Laplace transforms. The transforms of Pn(t), denoted here as Pˆ n(ω), are found by using eqs 6 with the relationship that the transform of a convolution of two functions is the product of the two transforms:48
ˆ (ω)Yˆ (ω)]n/2 Pˆ n(ω) ) Vˆ (ω)[U
n ) 0, 2, 4, ...
Pˆ n(ω) ) Zˆ (ω)[U ˆ (ω)](n+1)/2[Yˆ (ω)](n-1)/2
(8a)
n ) 1, 3, 5, ... (8b)
From eqs 2 and 348
{U ˆ (ω), Vˆ (ω)} ) {Yˆ (ω), Zˆ (ω)} )
{kd, 1} ω + kd
{bxkdω, 1}
xω(xω + bxkd)
(9a)
(9b)
1
ω
)1
{
kd3/2b
ω (ω + bxkdω + kd) 3/2
(10)
}
(11)
The inverse transform of the last expression in eq 11 is
N(t) )
b {u 2w(-iu1xkdt) u1 - u2 2 u12w(-iu2xkdt)} + 2b
(7b)
{}
∑n{P2n(t) + P2n+1(t)} ) -1 Lωft
(7a)
1 {w(ibxkdt) - ibw(xkdt) + (ib - 1)e-kdt} P1(t) ) 1 + b2
{[
-1 Pˆ n(ω)} ) Lωft ∑ n)0
n)1
U(t3 - t2)Y(t4 - t3)V(t - t4) (6e)
)
∞
∞
N(t) )
dt1 U(t1)Y(t2 - t1) ×
-1 Pn(t) ) Lωft {
Equation 10 expresses the notion that the probabilities of all possible molecular histories, at an arbitrary time t after placing the tagged ligand on the surface, must sum to 1. Temporal Dependence of Rebinding. The average number of rebinding events that have occurred at a time t, after placing a ligand on the surface may be calculated as (eqs 8 and 9)
U(t3 - t2)Z(t - t3) (6d)
P0(t) ) e-kdt
(
∑ n)0
(6b)
Functions Pn(t) with values of n > 4 may be found by including additional integrals to the functions shown in eqs 6. By using eqs 2-4 in eqs 6a-c, direct integration produces closed form expressions for the first several functions Pn(t) (see Appendix):
P2(t) )
∞
(6a)
x
kdt - b2 (12a) π
where
b u1,2 ) - ( 2
x
b2 -1 4
(12b)
The function N(t) is shown in Figure 4a. The average number of rebinding events that have occurred at time t after placing the ligand on the surface increases from N(0) ) 0 to N(∞) f ∞. For a given time t, N(t) increases with the parameter b. When b f ∞, N(t) ≈ kdt and when b f 0, N(t) ≈ 0. The large b limit can be understood as the case in which each dissociation, after an average time of kd-1, is immediately followed by rebinding to the surface. The average rebinding rate, R(t), is the time derivative of N(t), or
R(t) )
k db {w(-iu1xkdt) - w(-iu2xkdt)} u1 - u2
(13)
The function R(t) is shown in Figure 4b. The function is always positive, R(0) ) 0, and R(∞) f 0. The time at which the rebinding rate peaks decreases with the parameter b, and the maximum rebinding rate increases with the parameter b. When b f ∞, R(t) ≈ kd and when b f 0, R(t) ≈ 0. The average time at which an initial rebinding event has occurred, denoted here as τ1, is shown in Figure 5a. This time was calculated by solving the transcendental equation for which N(τ1) ) 1 (eqs 12). The time τ1 is always greater than the inverse of the intrinsic dissociation rate, kd-1. This limit is found when the propensity for rebinding is high (b f ∞). Lower values of b imply larger values for the time τ1. Figure 5b shows the average time after which 10 rebinding events have occurred, τ10, which was calculated numerically by finding the time at which N(τ10) ) 10 (eq 12). The parameter τ10 has properties similar to those of τ1. The value of τ10 decreases from infinity
866 J. Phys. Chem. B, Vol. 104, No. 4, 2000
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Figure 4. Number of rebinding events N(t) and rebinding rate R(t). (a) The function N(t) gives the number of rebinding events that have occurred on the average at time t after placing a molecule on a surface site. Curves were calculated from eqs 12. (b) The function R(t) gives the average rebinding rate at a time t after placing a molecule on a surface site. Curves were calculated from eq 13. The rebinding parameter b equals (line) 100; (long dash) 10; (short dash) 1; (dot dash) 0.1; and (dot) 0. Figure 3. Probability functions Pn(t). The functions Pn(t) describe the probabilities of possible molecular histories of dissociation and rebinding as described in the text. Curves were calculated from eqs 7 and are shown for (a) P0(t); (b) P1(t); and (c) P2(t). The rebinding parameter b equals (line) 100; (long dash) 10; (short dash) 1; (dot dash) 0.1; and (dot) 0.
to 10kd-1 as the rebinding parameter b ranges from zero to infinity. The time at which the rebinding rate peaks, denoted by τm, was found numerically from the expression for R(t) (eq 13). As shown in Figure 6a, τm ≈ kd-1 for a wide range of b values. Therefore, the maximum rebinding rate occurs between the first and second dissociation events. The peak rebinding rate, Rm ) R(τm), is shown in Figure 6b. As expected, this maximum rebinding rate increases monotonically with the parameter b from zero to kd. The average time between dissociation and rebinding (i.e., the average time that a molecule spends free in solution between two surface binding events) is given by (eq 3a)
τf )
∫0∞tY(t) dt
(14)
Because rebinding always eventually occurs, some dissociation
and rebinding events involve very long intermediate times, and τf as defined above is not finite. A quantity that is finite is the average fraction of time spent free in solution, which depends on the time that has elapsed after placing a ligand on the surface. The fraction of elapsed time spent free may be calculated by inserting factors such as (t2 - t1) and (t4 - t3) in eqs 6 and using the Laplace transform techniques outlined above. This fractional time can be used to predict the apparent surface diffusion coefficient arising from sequential dissociation and rebinding. These calculations will be the subject of a separate work.49 Rebinding Propensities. As shown in Figure 5, kdτ1 e 100 or kdτ10 e 104 if b g 0.1. Therefore, in systems with b values that are small but not negligible, some rebinding is still predicted to occur. For example, if kd ) 1 s-1 and b ) 0.1, the average time between placing the molecule on the surface and the initial rebinding event is 100 s and the average time between placing the molecule on the surface and the first 10 rebinding events is 104 s. Significant rebinding is expected to occur in systems where kdτ1 e 2 or kdτ10 e 20. As shown in Figure 5, these inequalities occur for b approximately equal to 1. For example, the average time between placing the molecule on the surface and the initial rebinding event is 2kd-1 when b ) 1.9, and the
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Figure 5. Average times for 1 and 10 rebinding events, τ1 and τ10. The times (a) τ1 and (b) τ10 give the average times that have elapsed, after a molecule is placed on a surface site, when the molecule has dissociated and rebound 1 or 10 times. Curves were calculated numerically from eqs 12 as described in the text. The intrinsic dissociation rate kd equals (line) 0.01 s-1; (dash) 1 s-1; and (dot) 100 s-1.
Discussion
average time between placing the molecule on the surface and the first 10 rebinding events is 20kd-1 when b ) 3.8, for all values of kd. In this case, the molecule initially spends approximately half of the elapsed time in solution and half on the surface. Very extensive rebinding is predicted to occur in systems where kdτ1 e 1.1 or kdτ10 e 11 (b ) 18.7 and b ) 28.5, respectively). Therefore, for values of b greater than ≈20, molecules spend at least 90% of the initially elapsed time on the surface, while they experience repeated rebinding after dissociation. In the absence of competing molecules where rebinding is more prominent, A ) 0 and b ) kaN(kdD)-1/2 (eq 5). If one considers a typical condition for which significant rebinding occurs (b g 1), then N g ka-1(kdD)1/2. Typical association rates at membrane surfaces are ka ≈ 106 M-1 s-1 and typical ligand diffusion coefficients in solution are 10-6 cm2/s. Therefore, for kd ) 1 s-1, rebinding will be significant for N g 6000 molecules/ µm2. This value is not much larger than typical receptor densities on cell surfaces. Higher values of N are expected in regions of cell surface receptor clusters. For weaker binding (kd ) 100 s-1), very 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 densities is less stringent (N g 600 molecules/µm2).
In a previous work, we presented a new theoretical treatment of the rebinding process for the case in which monovalent ligands reversibly bind to monovalent sites on planar membrane surfaces.37 Analytical solutions for the probabilities of finding a molecule on the surface or in the solution, given initial placement at the origin, were derived. These functions 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 from the origin, denoted by Y(t) (eq 3a and Figure 2). In the work described here, we have used the functional form of Y(t) along with the probabilities describing possible molecular histories, Pn(t) (eqs 6 and Figure 3), to find an analytical expression for the average number of rebinding events that have occurred at a given time after a molecule is placed on the surface, N(t) (eqs 12 and Figure 4a). A functional form for the time derivative of N(t), denoted by R(t), was also found (eq 13 and Figure 4b). The functions N(t) and R(t) allow numerical calculation of the average times that have elapsed after a molecule is placed on the surface before the initial (τ1, Figure 5a) and first 10 (τ10, Figure 5b) rebinding events; as well as the time at which the rebinding rate peaks (τm, Figure 6a) and the peak rebinding rate (Rm, Figure 6b). For a given experimental system, the extent of rebinding can be determined by examining the behaviors of N(t), R(t), τ1, τ10, τm, and Rm. These functions depend only on the intrinsic dissociation rate
Figure 6. Time for maximum rebinding rate, τm, and maximum rebinding rate, Rm. The rebinding rate R(t) reaches its maximum value, Rm, at time τm. These quantities were calculated numerically from eq 13: (a) τm; (b) Rm. The intrinsic dissociation rate kd equals (line) 0.01 s-1; (dash) 1 s-1; and (dot) 100 s-1.
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kd and the value of the rebinding parameter b, which depends only on the intrinsic association and dissociation rate constants, the total density of surface binding sites, the ligand diffusion coefficient in solution, and the concentration of identical competing ligands in solution (eq 5). Acknowledgment. This work was supported by NSF grant MCB-9728116 and NIH grant GM-37145. Appendix The derivation of eqs 7 requires the following indefinite integrals:
∫erfc(ixkdη) dη )
(
η+
)
1 erfc(ixkdη) + i 2kd
x
η kdη e πkd
∫e(1+b )k η erfc(bxkdη) dη 2
)
d
2 1 ib e(1+b )kdη erfc(bxkdη) + erfc(ixkdη) 2 kd(1 + b ) kd(1 + b2)
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