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Copyright © 2005 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 89, Issue 6, 3686-3700, 1 December 2005

doi:10.1529/biophysj.105.065300

Biophysical Theory and Modeling

Effects of Receptor Clustering on Ligand Dissociation Kinetics: Theory and Simulations

Manoj Gopalakrishnan^*, †, Kimberly Forsten-Williams^‡, ,  , Matthew A. Nugent^§ and Uwe C. Täuber^†

^* Department of Biological Physics, Max-Planck-Institut für Physik Komplexer Systeme, Dresden, Germany
^† Department of Physics and Center for Stochastic Processes in Science and Engineering, Boston University School of Medicine, Boston, Massachusetts
^‡ Department of Chemical Engineering and Virginia Tech-Wake Forest University School of Biomedical Engineering and Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia
^§ Department of Biochemistry, Boston University School of Medicine, Boston, Massachusetts

Address reprint requests to Kimberly Forsten-Williams, Tel.: 540-231-4851.

The cell membrane is composed of many different types of lipid species. This heterogeneity leads to the possibility of organization of different species into distinct domains 1. Such domains are especially suited and designed for specialized functions such as signal transduction, nutrient adsorption, and endocytosis. They can link specific cellular machinery and physical features and are equipped with mechanisms for maintenance (addition and removal of specific molecules) for a certain period of time, during which the domains may diffuse as single entities 2. Lipid rafts, which are microdomains rich in sphingolipids and cholesterol, represent one of the most interesting but insufficiently understood lipid domains 3. Various estimates are available for raft sizes, and diameters in the range 25–200nm have been reported using various methods 4. A limitation in this area remains that the definition of lipid rafts is rather broad and currently includes a wide range of what will likely prove to be distinct domains that may be distinguished by the particular protein and lipid compositions 2,4,5. Operational definitions of rafts based on resistance to detergent solubilization and sensitivity to cholesterol removal are limited by artifacts of the various procedures used to define rafts and on difficulties in relating model membranes to cell membranes. Nonetheless, it is clear that cell membranes are not homogeneous and that protein-protein, protein-lipid, and lipid-lipid interactions all participate in regulating raft size, dynamics, and function. Consequently, a myriad of functions have been prescribed to lipid rafts, one possibility being that lipid rafts may serve as mediators of signal transduction for several growth factors, including fibroblast growth factor-2 (FGF-2) 6,7,8.

Growth factors act as triggers for many cellular processes and their actions are typically mediated by binding of ligand to the extracellular domain of transmembrane receptor proteins. For many receptors, signal transduction requires dimerization or clustering whereby two or more receptors, after ligand binding, interact directly to facilitate signal transduction. Although ligand binding is generally specific to members of a family of transmembrane receptor proteins, heparin-binding growth factors such as FGF-2 interact with both specific members of the FGF receptor family and heparan sulfate glycosaminoglycan chains of cell surface proteoglycans (HSPGs). HSPGs represent a varied class of molecules, including the transmembrane syndecans, the glycosyl-phosphatidylinositol anchored glypicans, and extracellular proteoglycans such as perlecan (reviewed in Bernfield et al. 9 and Kramer and Yost 10). The interaction of FGF-2 with HSPGs is of a lower affinity than to the cell surface signaling receptor (CSR) but has been shown to stabilize FGF-2-CSR binding and activation of CSR 11,12. Moreover, HSPGs have recently been demonstrated to function directly as signaling receptors in response to FGF-2 binding, leading to the activation of protein kinase C-α12 and Erk1/2 6.

There is evidence that cell surface HSPGs are not distributed uniformly, but are instead localized in lipid rafts 6,14,15,16, and this association may be facilitated by FGF-2 binding and clustering 17. This localization and clustering may further have a dramatic influence on signaling through both persistence of signaling complexes and localization with intracellular signaling partners. For example, FGF-2 dissociation kinetics from HSPGs were significantly altered when cells were treated with the lipid raft-disrupting agents methyl-β-cyclodextrin (MβCD) (Fig. 1). Retention of FGF-2, even at long times, was significantly greater in the untreated state, suggesting that rafts regulate this process. These experiments suggest that clustering of HSPGs in lipid rafts effectively slows down dissociation by increasing the rebinding of released FGF-2. If this is indeed true, then the localization of binding sites to microdomains on the cell surface could be an important mechanism employed by receptors to boost signal transduction via increased persistence.

Display large version of this figure

Figure 1

Effect of the lipid raft disrupting agent MβCD and heparin on FGF-2 dissociation from HSPGs. Bovine vascular smooth muscle cells in tissue culture were treated with MβCD (0, i.e., untreated, or 10mM MβCD) for 2h at 37°C before cooling to 4°C. ¹²⁵I-FGF-2 (0.28nM) was added and allowed to bind to the cells for 2.5h before initiation of dissociation (t=0). After the binding period, unbound ¹²⁵I-FGF-2 was removed by washing the cells with cold binding buffer, and dissociation was initiated in binding buffer without FGF-2 (±heparin, 100μg/ml) at 4°C. The cells were allowed to incubate for the indicated time periods, at which point the amount of FGF-2 bound to HSPG sites was determined by extracting the cells with 2M NaCl and 20mM HEPES, pH 7.4 and counting the samples in a γ-counter. All data was normalized to the amount of ¹²⁵I-FGF-2 bound to HSPG sites at t=0 (100%) under each condition. Mean values of triplicate samples, mean±SE, are shown (data replotted from Chu et al. 6).

The relation between the apparent association and dissociation rates of ligands interacting with receptors on a (spherical) cell surface with the corresponding intrinsic rates has been studied previously by several authors 18,19,20,21,22,23. Berg and Purcell 18 demonstrated that for ligands irreversibly binding to N receptors on a spherical cell of radius a, the effective forward rate constant becomes a nonlinear function of N, assuming the form k_f=4πDa[Nk₊/(4πDa+Nk₊)], where k₊ is the association rate for a single receptor in close proximity to the ligands (i.e., the intrinsic binding rate). The quantity in brackets was termed the capture probability, γ, by Shoup and Szabo 19. The effective dissociation rate was analogously defined as the product of the intrinsic rate and the escape probability, 1–γ. This leads to 19,24

(1)

In general, the effective dissociation rate of ligands from a set of receptors depends on the frequency of rebinding, whereby a dissociated ligand wanders around in the solution for some time and reattaches to the binding surface upon contact. This is only implicitly included in the above approaches. A systematic mathematical study of the rebinding probability of a single ligand was undertaken by Lagerholm and Thompson 27. An independent self-consistent mean-field model of rebinding of ligands bound to receptors in an infinite two-dimensional plane was recently presented by us 28 in the context of analyzing surface plasmon resonance (SPR) experiments.

In this article, we generalize our earlier discrete model 28 to incorporate a continuum description for the receptor distribution as well as the ligand motion. The self-consistent stochastic mean-field theory of rebinding thus developed is then used as the basis for extending our investigation to include nonuniformity in the spatial distribution of receptors. In particular, we study how rebinding is affected by the presence of receptor clusters on the cell surface. Our broad conclusions from this study are as follows:

In the remainder of this article, we first develop the theoretical formalism to study rebinding of ligands to an infinite plane of uniformly distributed receptors. Motivated by recent experimental observations of the effect of lipid rafts on ligand rebinding 6, the formalism is then extended to include receptor clusters. Subsequently, our theoretical predictions are compared to Monte Carlo simulation data. Finally, we comment on possible applications, including a possible internal-diffusion model extension, and discuss consequences for the analysis of experimental results. In Table 1, we include a glossary of terms used.

Table 1 A glossary of the important quantities discussed in the article, along with the corresponding units (m=meter, s=seconds, M=mole)

	Quantity	Symbol	Typical units
	Microscopic length scale	λ	m
	Diffusion coefficient	D	m2 s−1
	Microscopic diffusion timescale	δt=λ2/4D	s
	Association rate	k+	M−1 s−1
	Dissociation rate	k−	s−1
	Equilibrium dissociation constant		M
	Mean surface density of receptors	R0	No. of molecules/m2
	Surface density of receptors inside clusters		No. of molecules/m2
	Bound receptor fraction at time t	p(t)	Dimensionless
	Ligand density profile close to the surface at time t	ρ(t)	No. of molecules/m3
	Return to origin probability density for a surface with R0 receptors per unit area		m−1
	Return to origin probability density for a perfectly absorbing surface	q(t)	m−1
	Probability of nonabsorption upon contact	γ	Dimensionless
	Timescale of exponential decay (Eq. (11a))	te	s

In this section, we present a generalization of our recently introduced lattice random-walk-based theory of rebinding to a continuum distribution of receptors on a two-dimensional infinite surface. Let us consider a homogeneous distribution of receptors on an infinite planar surface with constant mean surface density R₀ per unit area. The intrinsic dissociation and association rates are denoted by k₋ and k₊, respectively. We denote by R(t) the density of receptors bound to the ligand at any time t, and its dynamical equation has the form

(2)

(3)

1. denotes the (surface-integrated) one-dimensional probability density (with dimension of 1/length) of a random walk returning to its point of origin at time t, given that the origin constitutes a partially absorbing barrier with a density R₀−R(t) of absorbing points per unit area, and
2. G₂(r,t)=[4πDt]⁻¹ exp(−r²/4Dt) represents the (normalized) two-dimensional probability density for finding a diffusing particle at distance r from the origin at time t.

To eliminate the time-dependence of the boundary condition in 1, above, we choose R^* ≪ R₀. Let p(t)=R(t)/R₀ be the fraction of receptors bound to ligands at time t, so that p(0)=R^*/R₀≪1 (which also implies p(t) ≪ 1). When the spatial integration in Eq. (3) is extended to infinity, Eq. (2) is thus reduced to

(4)

The quantity is now calculated from the frequency of first passage events: Let q(τ) denote the probability density of ligands that at time τ return to the surface for the first time after dissociation. At this point in time, the ligands may be either absorbed or reflected back to the solution and subsequently return at a later time. The quantity could then be calculated by summing over of all such events.

To proceed with our formalism, it is useful to imagine the available space to be divided into cubic elements (i.e., coarse-grain the space), each with volume λ³. Here λ is a coarse-grained length scale, which we assume to be of the order of the size of a single ligand molecule. The ligand diffusion may now be viewed effectively as transfer of its center of mass between such elements. When a ligand occupies an element of volume adjacent to the surface, it may become bound to a receptor, and the probability for this to occur is denoted 1–γ, so that γ is the probability of nonabsorption of the ligand upon encounter. The equation for thus satisfies the integral equation

(5)

(6)

(7)

The Laplace-transformed version of Eq. (4) after all the above substitutions reads

(8)

After substitution of Eqs. (6) into 8, and employing the above result to substitute for γ, we arrive at

(9)

(10)

(11a)

(11b)

The nonexponential decay in Eq. (11a) was also predicted in a previous lattice model of the problem developed to model SPR experiments 28. Indeed, one can show that with the appropriate mapping, the time constants c of the continuum and the lattice models coincide. In the discrete variant, the receptors are distributed on a lattice (with unit length Δ) at a mean density θ_s, and upon hitting a receptor (the sizes of both ligand and receptor are assumed negligible in comparison with Δ), a ligand is absorbed with probability θ_a. The effective surface coverage is therefore given by θ=θ_sθ_a. These parameters are related to the continuum variables through the relations R₀=θ_s/Δ² and k₊=θ_aDΔ. Upon making these substitutions in Eq. (11a), we find that the expressions corresponding to the continuum and lattice formalisms match perfectly.

In this section, we adapt the stochastic self-consistent mean-field theory for ligand rebinding presented above to incorporate nonuniform spatial receptor distributions. We consider receptors distributed in clusters of radius r₀, such that the density of receptors inside the clusters is R′₀>R₀, where the latter represents the mean density of receptors on the surface.

To generalize the previous theory to incorporate receptor clusters, we adopt the following approximation: Any rebinding event where the originating and the final points are separated by a distance r<ξ is assumed to take place in a local environment with receptor density R′₀, whereas any ligand that travels a lateral distance r≥ξ to rebind is assumed to sense only a smaller receptor density that we assume to be of the order of the mean density R₀. For this approximation to be useful, we need to identify ξ with a physical length scale: here we simply assume that ξ∼r₀. It must be noted that no strict spatial segregation exists between the two classes of rebinding events in the actual system. However, it will be seen later in comparison with numerical results that this approximation is remarkably successful in predicting the different temporal decay regimes in the presence of receptor clusters.

We shall now quantify these ideas using the previously developed formalism as a basis. The complete expression describing the dynamics of the bound fraction, which obviously generalizes Eq. (4), becomes

(12)

Let us consider two special cases of interest.

This situation is realized when the clusters are tightly packed with receptors, but the number of clusters themselves is small, so that the mean surface coverage of receptors is a negligible fraction. In this case, the homogeneous part of the rebinding term in Eq. (12) is vanishingly small, and the equation reduces to

(13)

(14)

(15)

We now substitute this expression into Eq. (14), and use it to evaluate the ξ-dependent term in the brackets. (The first term gives see Eq. (9).) After inserting the result 29, we arrive at the expression

(16)

(17a)

(17b)

Let us now assume that the clusters are very densely packed with receptors, i.e., R′₀ is sufficiently large so that ξ ≫ ξ₀. In this case, the contributions in Eq. (17a) that involve incomplete gamma-functions are small (Γ(a,x) ∼ x^a−1e^−x for x ≫ 1, 30. Therefore, &Sigma;(0) ≈ 1-ξ₀/2ξ in Eq. (17a) when ξ ≫ ξ₀. After substitution in Eq. (8), we see that

(18)

(19)

We thus reach an intriguing conclusion: When the mean surface density is sufficiently small, clustering of receptors has (over sufficiently long timescales) the effect of reducing the effective dissociation rate of ligands by a factor that is inversely proportional to the size of the cluster. It should also be borne in mind that the very late time regime for any small but nonzero mean density should display the nonexponential behavior of Eq. (11b). However, the characteristic timescale for entry into this regime (for a uniform distribution) grows as , and is likely to be masked by other effects (e.g., finite-size effects, nonspecific binding) in experiments.

To view this result in the context of the previous findings of Berg and Purcell 18 and Shoup and Szabo 19, we may compare Eq. (19) with the analogous result in Eq. (1) obtained via very different arguments. Let us imagine that the density of receptors inside a cluster is so high (consistent with our own assumptions in reaching Eq. (19)) that the cluster effectively acts like an absorbing disk, for which the diffusion-limited onward rate constant is k_D=4Dr31 where r is the radius of the cluster. Let N be the number of receptors inside a cluster, which we assume to be so large that Nk₊ ≫ k_D in the denominator of Eq. (1). After re-expressing N in terms of the receptor surface density , we find that, within this approximation, the reduction factor for the association rate in Eq. (1) is identical to that in Eq. (19), with ξ=(π/4)r, an aesthetically pleasing result. It should, however, be emphasized that the framework of our theory is more general and provides a broader perspective.

When the radius ξ is sufficiently large (ξ ≫ ξ₀), there is also another (intermediate) time regime D(k₊R′₀)⁻² ≪ t ≪ ξ²/D, for which the last term in Eq. (16) is small, and the first term dominates (since again the incomplete gamma-functions vanish in the limit of large ξ specified above). In this regime, we hence recover the nonexponential dissociation encountered earlier in Rebinding on a Planar Surface in the context of a homogeneous receptor distribution, see Eq. (11b):

(20)

The preceding calculations, in particular Eqs. (17a), show that the clusters have to be of a minimum size (∼ξ₀=2D(k₊)⁻¹) if they are to produce a significant effect on the dissociation. It is, therefore, important to know how this cutoff size compares with independent estimates for the size of lipid rafts. The total number of proteins likely to be contained inside a raft of area 2100nm² has been estimated to be in the range 55–65 32, assuming very close packing, or close to 20 33, assuming the same density of packing inside the raft and the surrounding membrane. The number of specific proteins like HSPGs is possibly less. As a conservative estimate, we assume that there are n∼5–10 HSPGs inside a raft, which gives , where r is the raft radius. The condition that clusters affect dissociation substantially is ξ/ξ₀≥1, from our previous analysis. Let us now make the identification ξ ≈ r, which, combined with the previous estimate for the receptor density, gives the condition k₊≥2πDr/n. Let us use r∼25nm as a rough estimate for the size of a lipid raft 34, which then gives k₊≥10⁸–10⁹M⁻¹ s⁻¹, if we assume a diffusion coefficient D=10⁻¹⁰m² s⁻¹.

Our conclusion, therefore, is that rafts of extensions in the range 25–50nm should be capable of producing a measurable effect on ligand dissociation purely by a diffusion-controlled mechanism, provided the association rate of the specific protein is large enough. It must, however, be remarked that this conclusion strictly applies to monovalent ligands interacting with a monovalent single receptor only. If, as in the specific case of FGF-2, there is more than one receptor that can bind the ligand and the possibility of higher order complexes exists, then the inclusion of surface biochemical coupling reactions needs to be taken into account. In Comparison with Experiments, we provide a more detailed discussion of these aspects in the context of experiments with HSPGs.

When the mean surface density of receptors is high, one might expect that rebinding has significant effect on dissociation even without any additional clustering mechanisms and that any effect of rafts on dissociation would be confined to sufficiently small timescales. This argument is, in fact, supported by numerical simulations that we present below. Yet here we aim to quantify the effect of clustering on ligand rebinding in the case of high mean surface density. For this purpose, Eq. (12) is conveniently rewritten in the form

(21)

(22)

(23)

(24)

(25)

(26)

To summarize this section, the theoretical formalism we have presented predicts a number of interesting regimes for the effective dissociation of ligand from receptors on cell surfaces. For a uniformly distributed set of receptors on a plane, we find that the decay is exponential with the intrinsic dissociation rate initially (Eq. (11a)), but crosses over to a nonexponential decay at later times (Eq. (11b)) owing to multiple rebinding events. When the receptors are clustered, the effects of rebinding depend on the mean receptor density. When the mean density is low so that no appreciable rebinding occurs with a uniform distribution, clustering is predicted to have the effect of producing an exponential decay at intermediate times with a reduced decay coefficient that is a function of the cluster size and the other parameters (Eq. (19)). The very late time behavior is still presumably nonexponential, although a full characterization of this crossover has not yet been performed. When the mean density is sufficiently high, the effect of clustering was found to be nonmonotonic, small at early and late times and reaching a maximum at a certain intermediate time.

To check our analytical results, in particular Eqs. (19), we have performed lattice Monte Carlo simulations, which will be the subject of the next section.

The hopping-between-elements picture of ligand diffusion we presented in Rebinding on a Planar Surface is easily implemented in numerical simulations. The substrate surface is envisioned as a two-dimensional square lattice, with the length scale λ setting the lattice spacing. The unit timescale is set to δt=λ²/2D, the timescale of hopping between elements. (We use a different symbol here to distinguish from the more fundamental timescale δ introduced in Rebinding on a Planar Surface, above.) Using these units, all quantities we discussed above may be expressed in dimensionless form (see Table 2). The ligand motion is modeled as a three-dimensional random walk between elements in the space above the substrate.

Table 2 A list of the dimensionless forms of various quantities, scaled using the length scale λ and timescale δt=λ2/2D, respectively

		Quantity	Dimensionless form
	Surface density	R0	θ=R0λ2
	Association rate	k+
	Dissociation rate	k−
	Cluster size	r0
	Diffusion coefficient	D

Typical numerical values are λ≈1–5nm, D∼10−6cm2 s−1, k+∼105–108M−1 s−1, k−∼1–10−4s−1, and r0<100nm (estimates for lipid rafts, reviewed in Simons and Vaz 4).

In the simulations, we choose the association rate to be k₊=Dλ. With this choice, the binding rate of the ligand close to a receptor is p=λ⁻³k₊=Dλ⁻² and the probability of binding over a single Monte Carlo time step for a ligand close to the surface is , i.e., the binding is purely diffusion-limited. In real units, this choice corresponds to an association rate of ∼10⁻¹³cm³ s⁻¹∼10⁶M⁻¹ s⁻¹. A smaller value of k₊ involves only a trivial modification of the algorithm: The probability of binding is reduced to (in simulations, this factor may be simply absorbed into the dimensionless surface coverage, while keeping the binding purely diffusion-limited), but a larger association rate would require a more microscopic simulation, and is not addressed in this article.

We next discuss our choice for the dissociation rate. A realistic value of k₋ would fall in the range of 10–10⁻⁴s⁻¹, which means that the dimensionless rate would be a very small number (for λ≈5nm and D ∼ 10⁻¹⁰m² s⁻¹, we estimate δt∼10⁻⁷s), of the order of 10⁻⁶–10⁻¹¹. Since the timescale of measurement of dissociation would have to be at least of the order of this would require the simulation to be run over Monte Carlo steps. For computational efficiency, therefore, we choose in all the simulations.

The surface density of receptors R₀ is the next important parameter in the model, and its dimensionless version is denoted by θ=R₀λ². Assuming that the ligands and the receptor extracellular binding domains are not significantly different in size, the range of allowed values for this parameter is θ≤1. In the substrate lattice, therefore, θ simply represents the fraction of binding sites. Note that the simulations also could correspond to the case where the association rate k₊<Dλ, where we would maintain the binding to be diffusion-limited, but effectively reduce θ to θ′=θ(k₊/Dλ) in the simulation runs.

Our strategy is as follows: Keeping the overall density θ constant, we arrange the receptors into N clusters of (dimensionless) radius . Because of lattice constraints, it is not possible to ensure that all the receptors are contained in such clusters. Rather, our criterion is that, for a certain value of , N be selected such that the number of receptors outside clusters is kept at a minimum. The simulations are done with reasonably large lattices (10³×10³) so that small surface coverage could be explored. Fig. 2 shows two typical receptor configurations used in our simulations. All data were averaged over 100 different initial realizations of the receptor configuration.

Display large version of this figure

Figure 2

Two typical receptor configurations used in the Monte Carlo simulations. The mean receptor density (in dimensionless units) in A is 0.001 and in B it is 0.1. The cluster radius is in A and in B. The small dots are single receptors. The clusters are filled to saturation in both cases.

The ligand diffusion is governed by periodic boundary conditions on the four borders of the lattice so that a ligand that exits at one boundary reenters from the opposite side. The direction perpendicular to the plane of the lattice shall be referred to as the z-axis, and the surface itself is located at z=0. The ligand diffusion in the z-direction is not upper-bounded. We also neglect surface diffusion of the receptor proteins, irrespective of their being clustered or isolated, and treat them as static objects throughout this article (see, however, the discussion at the end of this section). At the beginning of the dynamics, a fraction p(0) of all the receptor sites are bound to a single ligand each. Although the precise value of p(0) is unlikely to have a large impact on the late time decay, we chose p(0)=0.25 in all the simulations so that we are not too far from the approximation p(0) ≪ 1 made in the setup of the theory.

There are three main dynamical processes in the simulation:

1. Dissociation of a ligand from a bound receptor takes place with probability per time step δt. This move updates the position of the ligand from z=0 to z=2, in units of the lattice spacing. (We use z=2 instead of z=1 to prevent immediate rebinding to the same receptor.)
2. Diffusion of the released ligands in solution: A free ligand moves a distance equal to one lattice spacing in one of the six directions with probability 1/6 per time step.
3. Readsorption of free ligands to free receptors: A free ligand at z=1 is absorbed by a free receptor below it, if there is one, with probability 1.

Display large version of this figure

Figure 3

(A) Receptor density impacts ligand dissociation for a uniform distribution of receptors. The value p(t) versus scaled time T=βt (t is measured in number of Monte Carlo time steps) is plotted for θ=0.001 and θ=0.1. The former displays exponential decay, whereas the latter is clearly nonexponential over the timescales shown. The lines are theoretical fits: in the former case and the function in Eq. (11b) in the latter case with c=0.08 (the theoretical value from Eq. (11a) is 0.06). The early time behavior of the high-density case (θ=0.1) plotted in the inset does indicate exponential decay (inset: t is the number of Monte Carlo steps), but the effective dissociation constant is ∼0.6k₋, less than the theoretical value k₋; see Eq. (11a), also Appendix C . (B) The high-density (θ=0.1) data plotted on a semilogarithmic scale, which shows more explicitly the strongly nonexponential nature of the decay.

We next addressed how clusters might impact dissociation, focusing first on the low-coverage regime. The coverage we chose was θ=10⁻³ (in terms of distribution over the cellular surface, this would roughly correspond to ∼10³ or 10⁴ receptors per cell for an association rate of ∼10⁹M⁻¹ min⁻¹ or 10⁸M⁻¹ min⁻¹, respectively) and we compared a homogeneous receptor distribution with a single cluster () and multiple clusters () (Fig. 4). We chose the clusters to be distributed randomly on the surface, but simulations with smaller lattices have shown that the dissociation curve is also not significantly different for a regular, periodic arrangement of clusters. In the real system, these clusters would have radii of ∼25–90nm, respectively. Simulations were carried out with two levels of receptor density inside clusters: In the first case, rafts were occupied by receptors to saturation (R′₀=1/λ²), and in the second case, the packing density was lowered to 0.1 (R′₀=0.1/λ²). Clear differences, despite each system having the same actual density of receptors and surface coverage, are evident when clustering is present. In both the cases, there is clear evidence of a significant intermediate exponential regime (Figure 4AB), which subsequently crosses over to a slower decay at later times. However, the effect of clustering on the dissociation rate is much more noticeable in the first case, where the packing density of receptors is high (Figure 4C). Moreover, we see that for the high packing density case, the dependence of the effective rate (defined in Fig. 4’s legend) on the cluster size observes the inverse linear relationship predicted by the theoretical analysis, Eq. (19) (Figure 4D).

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Figure 4

Dissociation is impacted by the degree of clustering when there is low mean surface density (θ=0.001). (A) Shown is p(t) versus scaled time where the curves correspond to uniform distribution, and (a single cluster in the last case), when the clusters are packed to saturation. The decay is exponential except for very late times. (B) Similar data as in A, but the packing density inside clusters is only 0.1, on a semilogarithmic scale. (C) Effective decay constant (exponential fit to the early portion, i.e., straight part, of the data) as a function of cluster radius for cases A and B. (D) Effective decay constant for A plotted against cluster radius on a logarithmic scale. The straight line is a fit function proportional to and the good agreement supports Eq. (19). The slope for the uniform case () in A and B is ∼0.67, which is less than the theoretical value 1, presumably due to (unavoidable) lattice effects in the simulations (for details, see Appendix C ).

The numerical results for the effective dissociation rates for the two cases discussed above may be put together in a single plot, by expressing the effective dissociation rate as a function of the ratio ξ/ξ₀. Clearly, for the same value of ξ(∼ raft radius), the threshold radius ξ₀ is different for the two cases (due to the inverse relationship to R′₀, Eq. (17b)) In fact, by substituting the numerical values of the simulation parameters (k₊=Dλ), it is easily seen that ξ₀=2λ for the case R′₀=1/λ² and ξ₀ =20λ for the case R′₀=0.1/λ². We may also use the equivalence with the Shoup-Szabo result (Eq. (1)) to express ξ in terms of the cluster radius from the previous discussion. In Fig. 5, we plot the ratio of the effective dissociation rate, defined as the exponential fit to the initial straight portion of the data (t>10), to the intrinsic rate (after correcting for the lattice effects), which shows that this ratio is a smooth, monotonically decreasing function of ξ/ξ₀. The theoretical prediction for the same is 1−Σ(0) (Eq. (8)), where Σ(0) is given by Eq. (17b), and is plotted as the smooth line in Fig. 5. It is clear that the data points agree very well with our theoretical prediction in the regime ξ/ξ₀≥2 (which is also the regime where clustering significantly alters the dissociation).

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Figure 5

The effective decay constants (defined in the legend of Fig. 4) plotted in C is plotted against the scaled cluster radius x=ξ/ξ₀. The smooth line is the theoretical fit function 1-&Sigma;(0), as defined by Eq. (17a).

Fig. 6 shows the effect of clustering in the high mean density case with θ=0.1 (∼10⁵ receptors per cell) and cluster radii of and A noticeable upward shift (decreased dissociation/increased binding retention) in the dissociation curve is observed, but the effect is nonmonotonic and vanishes for small and large times, in both cases. This is illustrated more clearly in Fig. 7, where we plot the difference between the bound fractions for clustered versus homogeneous receptor distributions as a function of time for the two values of the cluster radii. For the parameters used in the simulations β=k₋(λ²/2D)=10⁻⁴, R₀=0.1/λ², and k₊=0.1Dλ, the threshold cluster size is ξ₀≈20λ (i.e., in simulations) from Eq. (17b). For respectively, the parameter ω defined in Eq. (24) takes values 0.9 and 4.5. For the first case (since ω<1), therefore, we also compared the simulation results with the approximate theoretical prediction in Eq. (26) (smooth line in Fig. 7), expected to be valid in the early time regime. We observe that although the theoretical expression approximates the observed difference rather well at early times for small cluster size, it fails to capture the nonmonotonous behavior at somewhat late times. It is likely that this dense mean receptor regime lies outside the applicability range of the expression in Eq. (26). Clearly, a more systematic method to study the crossover from small to large receptor density would be desirable, but eludes us at this stage.

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Figure 6

(A) Effect of clustering on the dissociation rate for the high mean surface density case (θ=0.1). The value p(t) versus scaled time is shown for two values of association rates: k₊=0.1Dλ (main figure) and k₊=Dλ (inset) for uniform distribution (diamond), (plus symbol), and = 50.0 (square). The lower association rate in the main figure was used to increase the threshold cluster size (Eq. (17b)) to verify the theoretical predictions in Case 2 in Extension to Receptor Clusters. The axis labels are common to the main figure and the inset. (B) The data in the inset of A is plotted on a semilogarithmic scale to show the nonexponential nature of the decay more explicitly.

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Figure 7

The difference Δp(t) in the bound fraction between uniform distribution and clustered configurations when k₊=0.1Dλ, corresponding to (main figure) and (inset) is plotted against the scaled time T=k₋t, along with the theoretical prediction from Eq. (26) (thin line in the main figure). The theoretical curve agrees with the simulations in the very early regime, but deviates at later times. The impact of clusters vanishes at late times, in accordance with our arguments in Case 2 in Extension to Receptor Clusters. The axis labels are common to the main figure and the inset.

We now present a theoretical argument, which suggests that, over sufficiently long timescales, receptor clustering should have no effect on ligand dissociation, as found for the high-density receptor case. Let us consider two different scenarios:

The first case was already studied in the Theory section, above, where we showed that the dissociation is characterized by a single timescale Let us now map Case 2 into Case 1, and imagine the clusters as effectively single receptors with mean density Q₀, and effective association and dissociation rates and , respectively. The effective rates may be expressed in terms of the intrinsic rates using the Berg-Purcell-Shoup-Szabo relations, which give = nk₊(1–γ) and = k₋(1–γ), where the escape probability 1–γ has been defined earlier (see Eq. (1) and above). We now define the time constant for the clustered distribution as Upon substituting for the primed quantities and the cluster density, we see that c′=c, i.e., the clusters have no effect on the decay at all! This analysis, however, is not exact and numerical simulations did show a significant effect of clustering in the strong rebinding case, particularly at early times (inset, Fig. 6). Thus, for the simple one-to-one ligand-receptor binding case, it is conceivable that the effects of clustering are only transient but could still have a significant impact over a biologically relevant timescale.

Having compared the theoretical formulation in sufficient detail with lattice simulations, we turn to the question: How do the predictions of our simple model fit with experimental observations? We focus on the results of FGF-2 dissociation from HSPGs obtained by Chu et al. 6, shown in Fig. 1. FGF-2 binds to a high-affinity receptor FGFR as well as the HSPGs we discuss here, and higher-order clusters including both FGFR and HSPGs are possible 12. Therefore, any quantitative analysis of FGF-2 binding has to be done with care, because of the presence of competing interactions. Despite this and because of a lack of availability of experimental dissociation data with other raft proteins, we choose this system for our analysis.

The experiments reported in Chu et al. 6 were done with intact cells either in the absence or presence of the lipid raft-disrupting agents MβCD and filipin (filipin data is not shown in Fig. 1). Both lipid raft-disrupting agents were demonstrated to have a significant effect on the dissociation rate, but we focus here on the MβCD data set since the mechanism of action is simpler and more straightforward. Briefly, a k₋ value of 0.004±0.002min⁻¹ was obtained for the control cells, while treatment with MβCD increased the dissociation rate to ∼0.023min⁻¹ (with simple exponential fitting). If the MβCD treatment resulted in a completely homogeneous HSPG distribution, we arrive at a ratio of ∼5.75 for the reduction in the dissociation rate due to raft-associated clustering.

The first question, then, is whether the present estimates of the HSPG surface density in these cells would allow for a significant exponential regime for the temporal decay of the dissociation curve? Using Eq. (11b), we may compute the length of this time interval, t_e, where the decay is exponential. Let us use the following estimates: D≈10⁻¹¹–10⁻¹⁰m² s⁻¹, k₊∼1.5×10⁶M⁻¹ s⁻¹, and R₀∼10⁵–10⁶/l², where l∼5μm is a rough estimate for the cell radius. After substitution in the expression in Eq. (11a), these values give t_e ≈ 0.1–10s. This timescale is very small for typical dissociation measurements and suggests that the observed mode of decay in Fig. 1 is more likely to be the nonexponential function predicted in Eq. (11b). More evidence for the presence of strong rebinding in the experiments shown in Fig. 1 is seen when rebinding was prevented by the addition of heparin (Fig. 1), which acts as a solution receptor for the released FGF-2. The dissociation in the presence of heparin was found to be increased compared to both untreated and MβCD treated and essentially the same with and without lipid raft disruptors (Fig. 1). Further, although limited, the data points suggest that dissociation could be exponential. To summarize, the difference between MβCD treated and untreated without heparin indicates an effect on dissociation by clustering, and the heparin data suggests that rebinding is still an issue even in the absence of rafts.

It is important to note that because of the slow, nonexponential decay of the dissociation curve in the presence of strong rebinding, this function cannot be accurately characterized by a single rate valid over a well-defined time regime (unlike the weak-rebinding case). Rather, the effective rates obtained by fitting the experimental curves to exponential functions are only a simplified characterization of the decay valid over a limited timescale. Keeping these caveats in mind, we tried to see whether the observed experimental data, with and without raft disrupters, is reproduced by the theoretical functions of Eq. (11b) (homogeneous distribution) and Eq. (26) (raft-correction). The curves that were judged to be closest to the experimental data in Fig. 1 (by comparing with the exponential fit functions used to estimate the dissociation rates in Fig. 1) are shown in Fig. 8. The parameters c and k₋ω (Eq. (11b) and Eq. (26)) were tuned for the best fit, and the optimal numerical values found were c=1.1×10⁻⁴s⁻¹ and k₋ω=4×10⁻⁴s⁻¹. Let us now substitute for the following parameters: D=10⁻¹¹m² s⁻¹, k₋=0.25s⁻¹ (obtained from the heparin data in Fig. 1), and k₊=1.5×10⁶M⁻¹ s⁻¹11. We treat the surface densities R₀ ≈ N/l² (where l≈5×10⁻⁶m is the typical cellular dimension) and as unknowns, where N is the total number of HSPGs per cell and ξ is roughly the radius of a raft. Upon solving for the unknowns N and ξ, we find N ≈ 7.5×10⁵ and ξ ≈ 200nm. Both values are within reasonable limits of the known estimates of these parameters, and the resemblance between Fig. 8 and Fig. 1 supports the FGF2-HSPG system analysis under the strong rebinding category discussed in Theory, Case 2. The implications of this observation are:

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Figure 8

The three smooth curves in the figure represent the best exponential fits to the experimental data from Fig. 1, with time constants (in units of 1/s) 0.25 (MβCD+Heparin), 0.023 (MβCD), and 0.004 (untreated). The data points represent the best theoretical fits, using the function in Eq. (11b) for MβCD, and with the added raft correction (Eq. (26)) for untreated cells. The fit parameters are c=1.1×10⁻⁴s⁻¹ in Eq. (11b) and ω=1.6×10⁻³ in Eq. (26). The corresponding values of the intrinsic variables are discussed in the main text, in Results, Comparison with Experiments. Note that, at late times, some of the data points for the clustered configuration cross 1, which indicates that the limit of applicability of the perturbation theory has been reached, and that higher-order terms in the perturbation series have to be taken into account.

Suppose, however, as an aside, that Theory, Case 1 (low surface coverage) would have applied to this experimental system. From Fig. 5, we note that a reduction in the effective dissociation rate by a factor ∼5.75 (or a ratio of 0.17) for a low density system would require that the ratio ξ/ξ₀ should be ∼2.87. Let us now use Eq. (17b) to express this result in terms of the raft radius r by means of the substitutions r=(4/π)ξ and R′₀=n/πr² where n∼5–10 is a rough estimate of the number of HSPGs per raft. The condition that ξ/ξ₀≈2.87 now demands that, for r∼25nm, the association rate for FGF2-HSPGs should be nearly k₊ ∼ 3.44×10⁸(10⁹)M⁻¹