5A Effects of receptor clustering and rebinding on intracellular signaling In the analysis of bacterial chemotaxis, we showed how receptor clustering can amplify a biochemical signal through cooperative effects (see Sect. 5.3.1). Here we explore another consequence of membrane protein clustering, namely, it can increase the likelihood of rapid rebinding of a ligand. In mammalian cells, clustering of receptors may by facilitated by lipid rafts, which are membrane microdomains rich in cholesterol and having diameters in the range 25-200 nm [7, 9]. One suggested role of lipid rafts is that they serve as mediators for several growth factors by localizing clusters of their corresponding receptors. (Growth factors acts as triggers for many cellular processes by binding to the extracellular domain of the receptors.) For example, in vitro experiments have shown that disruption of lipid rafts significantly effects the dissociation of fibroblast growth factor-2 (FGF-2) from heparin sulfate proteoglycans (HSPG), which is a co-receptor [5, 2]. Stochastic model of ligand rebinding We begin by describing a self-consistent stochastic mean-field model of ligand rebinding due to Gopalakrishnan et al. [3]. Consider a homogeneous distribution of receptors on a two-dimensional planar surface with mean surface density R 0 per unit area. Let R(t) denote the density of receptors bound to a ligand at time t, with dr dt = k R(t) + k + ρ(t)[r 0 R(t)], (5A.1) where k ± are the intrinsic association and dissociation rates, and ρ(t) is the ligand density in a neighborhood of the surface (a boundary layer of width that is comparable to the size of a ligand). Suppose that the initial density of ligand bound receptors is R(0) = R R 0 and the initial density of ligand in the bulk is zero. It follows that the only contribution to ρ(t) 0 when t > 0 is from ligands released from bound receptors at earlier times 0 < τ < t. Let G(r,,t) denote the probability density (per unit volume) that a ligand, which started in the boundary layer at time t = 0, is in contact with the surface at time t having shifted by r R 2 in the plane, with possible multiple visits to the semi-absorbing surface at intermediate times but without binding to a receptor. Since ρ(t) is independent of position in the plane, we have ] ρ(t) = k R(τ) G(r,,t τ)dr dτ. (5A.2) 0 R2 with k dτ the probability that a ligand is released from a bound receptor in the interval [τ,τ + dτ]. The next step is to calculate the Green s function G(r,,t). Let q(r,,t) denote the probability density (per unit volume) that a ligand released at time t = 0 first 2
returns to the surface at time t after shifting by r R 2 in the plane. The Green s function satisfies the integral equation ] dτ G(r,,t) = q(r,,t) + γ q(r,,τ)g(r r,,t τ)dr 0 R 2 δ. (5A.3) Here δ is the time interval for which a ligand resides in a volume element 3, before diffusing way into the bulk. The first term on the right-hand side is the probability that a ligand arrives back at the surface for the first time at t, whereas iterating the second term yields all contributions from multiple visits to the surcae (with binding to a receptor). The factor γ gives the probability of non-binding upon contact with the surface and takes the form γ = 1 k + R 0 δ/. (If R were comparable to R 0 then we would have to replace R 0 by the density of unbound receptors.) It turns out that one does not have to calculate q explicitly [4]. Let G 0 (r,,r) denote the Green s function in the case γ = 1, for which the planar surface is purely reflecting, and ] dτ G 0 (r,,t) = q(r,,t) + q(r,,τ)g 0 (r r,,t τ)dr 0 R 2 δ. (5A.4) We now observe that G 0 is the fundamental solution of the 3D diffusion equation in the domain (r,z),r R 2,z > 0 with a reflecting boundary at z = 0, which is wellknown. We can then express G in terms of G 0 by Fourier transforming equations (5A.3) and (5A.4) with respect to r and Laplace transforming with respect to t: Ĝ(k,,s) = q(k,,s) + γλ δ Ĝ(k,,s) q(k,,s) and Ĝ 0 (k,,s) = q(k,,s) + δ Ĝ0(k,,s) q(k,,s). It follows that Ĝ 0 (k,,s) Ĝ(k,,s) = 1 + ( /δ)ĝ 0 (k,,s) which can be rearranged to yield Ĝ(k,,s) = [1 + γ δ Ĝ(k,,s) ], Ĝ 0 (k,,s) 1 + k + R 0 Ĝ 0 (k,,s), (5A.5) where we have used (1 γ)( /δ) = k + R 0. Next, let p(t) = R(t)/R 0 denote the fraction of bound receptors. Laplace transforming equations (5A.1) and (5A.2), we find that and (k + s) p(s) p(0) = k + ρ(s) 3
Hence p(s) = ρ(s) = k R 0 p(s)ĝ(0,,s). p(0) s + k (1 Σ(s)), Σ(s) = k +R 0 Ĝ(0,,s). (5A.6) It can be shown that Ĝ 0 (0,,s) = 1/ Ds so that Σ(s) = k + R 0 Ds + k+ R 0. (5A.7) and, consequently, p(0) p(s) =. s + k Ds Ds + k+ R 0 (5A.8) Equation (5A.8) may be used to extract the short-time and long-time behavior of the bound receptor fraction p(t). At short times (large s), we have p(s) p(0)/(s+k ), which corresponds to exponential decay at the intrinsic rate k, that is, rebinding has no effect. On the other hand, in the late time regime (small s), we have p(s) p(0)/[s + k Ds/(k+ R 0 ]) which exhibits non-exponential behavior in time. More explicitly where c = D(k /k + R 0 ) 2. p(t) p(0)e k t, for t D (k + R 0 ) 2, (5A.9a) p(t) p(0)e ct erfc( ct), for t D (k + R 0 ) 2, (5A.9b) Extension to receptor clusters Gopalakrishnan et al. [3] used the above mean-field formalism to study the effects of receptor clustering on ligand rebinding. Suppose that we focus at a point on the surface within a cluster, which has increased receptor density R 1 compared to the background. The density of free ligand close to the surface at the given point is now given by ] ρ(t) = k p(τ) R 0( r )G(r,,t τ)dr dτ. (5A.10) 0 R 2 Here R 0 (r) = R 1 + (R 0 R 1 )Θ(r ξ ). That is, if the distance between the initial and final positions of a ligand in the plane is smaller than the size of a cluster ξ, then the receptor density is R 1, whereas if the ligand travels lateral distance greater than ξ then the receptor density is R 0, R 0 < R 1. 4
For the sake of illustration, suppose that the receptor distribution consists of dense isolated clusters so that R 0 0 and R 1 is large. Then ρ(t) k R 1 0 [ p(τ) G ] 0 (0,,t τ) 1 e ξ 2 /4D(t τ) dτ, (5A.11) where G(0,,t) is the inverse Laplace transform of Ĝ(0,,s) with R 0 R 1 : Ĝ(0,,t) = 1 πdt k +R 1 D e(k +R 1 ) 2 t/d erfc(k + R 1 t/d). (5A.12) The factor 1 e ξ 2 /4D(t τ) arises from integrating the 3D Green s function over a disc of radius ξ in the plane. The Laplace transform of the bound receptor fraction still has the form (5A.6) with [ ] Σ(s) = k + R 1 e st Ĝ(0,,t) 1 e ξ 2 /4Dt dt. (5A.13) 0 Performing an asymptotic analysis of p(s) in the limit s 0 shows that at large times t ξ 2 /D and densely packed clusters (large R 1 ) [3] p(t) p(0)e k ξ 0 t/2ξ, ξ ξ 0 2D k + R 1. (5A.14) Thus, in the case of dense isolated clusters of receptors, the effective dissociation rate of ligands at sufficiently long time-scales is reduced by a factor that is inversely proportional to the size of the cluster. Moreover, ξ 0 sets the critical cluster size below which clustering has negligible influence. It is interesting to relate the above result to a very different approach to determining an effective dissociation, which is rate based on the theory of Berg and Purcell [1], see Sect. 2.4. Recall that for a spherical cell of radius a, with N receptors on its surface and intrinsic ligand binding rate k +, the net ligand flux (in the absence of dissociation) is J = k + ρ 0, where ρ 0 is the far-field concentration of ligand and k+ = 4πaDNk + Nk + + 4πaD. (5A.15) Comparison with the diffusion-limited result in the limit k +, suggests that the effective absorption probability of a ligand in contact with the cell surface is γ = Nk + Nk + + 4πaD. (5A.16) It follows that the corresponding probability of non-absorption is 1 γ, indicating that the possible effect of rebinding can be accounted for by taking the effective dissociation rate to be [8] 5
b k = k (1 γ) = k 4πaD. Nk+ + 4πaD (5A.17) Now imagine that a receptor cluster of size ξ is sufficiently dense that it effectively acts like an absorbing disc of radius r. Assuming Nk+ 4πrD, we have 4k D 4πrD b, = k k Nk+ k+ rr1 after relating the receptor density within the cluster to the total number of receptors according to R1 = N/(πr2 ). This is consistent with the effective decay rate in equation (5A.14) under the identification ξ = r/4. The issue of protein clustering not only applies to receptors detecting extracellular signals (Sec. 5A), but also membrane-associated enzymes that play an important role in activating intracellular signaling pathways. A canonical motif is a signaling molecule in the cytoplasm being chemically modified by two antagonistic enzymes, with an activating enzyme such as a kinase located in the cell membrane and a deactivating enzyme such as a phosphatase distributed in the cytoplasm (see Sec. 9.1). The effectiveness of enzyme clustering on the substrate response depends on the number of substrate sites that have to be activated. If only one site of the cytoplasmic molecule has to be activated, then enzyme clustering reduces the response, since the effective target size presented to the substrate molecules is smaller. On the hand, if more than one site has to be activated, then clustering can enhance the response due to the rapid rebinding property of protein clusters [6]. This is shown schematically in Fig. 5A.1. Fig. 5A.1: (A) Clustering of enzyme molecules reduces their effective target size for substrate molecules in the bulk, which decreases substrate activation. (B) On the other hand, clustering also enhances rapid substrate rebinding, which increases substrate activation if multisite modification is required. Supplementary references 1. Berg, H. C., Purcell, E. M.: Physics of chemoreception. Biophys. J. 20, 93-219 (1977). 6
2. Chu, C. L., J. A. Buczek-Thomas, and M. A. Nugent. Heparan sulfate proteoglycans modulate fibroblast growth factor-2 binding through a lipid-raft-mediated mechanism. Biochem. J. 379:331-341 (2004). 3. Gopalakrishnan, M., Forsten-Williams, K., Nugent, M. A., Taubery, U. C. Effects of receptor clustering on ligand dissociation kinetics: theory and simulations. Biophys. J. 89, 3686-3700 (2005). 4. Ghosh, S., Gopalakrishnan, M., Forsten-Williams, K.: Self-consistent theory of reversible ligand binding to a spherical cell. Phys. Biol. 4 344-345 (2007). 5. Kramer, K. L., and H. J. Yost. 2003. Heparan sulfate core proteins in cell-cell signaling. Annu. Rev. Genet. 37:461-484. 6. Mugler, A., Bailey, A.G., Takahashi, K., ten Wolde, P.R., 2012. Membrane clustering and the role of rebinding in biochemical signaling. Biophys. J. 102, 1069-1078. 7. Munro, S. 2003. Lipid rafts: elusive or illusive? Cell. 115:377-388. 8. Shoup, D., Szabo, A.: Role of diffusion in ligand binding to macromolecules and cell-bound receptors Biophys. J. 40, 33-39 (1982) 9. Simons, K., and W. L. C. Vaz. 2004. Model systems, lipid rafts and cell membranes. Annu. Rev. Biophys. Biomol. Struct. 33:269-295 7