Trapping by Clusters of Channels, Receptors, and Transporters: Quantitative Description

Abstract

Various membrane functional units such as receptors, transporters, and channels, whose action necessarily involves capturing diffusing molecules, are often organized into multimeric complexes forming clusters on the cell and organelle membranes. These functional units themselves are usually oligomers of several integral proteins, which have their own symmetry. Depending on the symmetry, they form clusters on different packing lattices. Moreover, local membrane inhomogeneities, e.g., the so-called membrane domains, rafts, stalks, etc., lead to different patterns even within the structures on the same packing lattice. Units in the cluster compete for diffusing molecules and screen each other. Here we propose a general approach that allows one to quantify the screening effects. The approach is used to derive simple approximate formulas giving the trapping rates of diffusing molecules by clusters of absorbers on lattices of different packing symmetries. The obtained results describe smooth variation of the trapping rate from the sum of the rates of individual absorbers forming the cluster to the effective collective rate. The latter shows how the trapping efficiency of an individual absorber decreases as the number of absorbers in the cluster increases and/or the inter-absorber distance decreases. Numerical tests demonstrate good agreement between the rates predicted by the theory and obtained from Brownian dynamics simulations for clusters of different shapes and sizes.

Introduction

It is well known that many cellular processes are initiated by binding of diffusing molecules to specific sites on membrane surfaces. The extensively explored examples include binding of food molecules to cell surface receptors in chemotaxis, neurotransmitter interactions with the nicotinic acetylcholine and other receptors, and ligand binding to transporter proteins, to mention just a few. Similarly, in channel-facilitated transport, diffusing metabolites first have to be trapped by the channel entrance. When these traps or binding sites are sufficiently far from each other, they do not interact in the sense that trapping by a site is independent of the presence of other sites. The situation changes when the intersite distance decreases and the sites form a cluster on the membrane surface. In such a case, sites start to compete for diffusing molecules and screen each other. As a result, the binding capability of a cluster could be significantly lower than the sum of the capabilities of the noninteracting sites forming the cluster.

It has been shown that channels and receptors are frequently clustered on membrane surfaces (1–6). As an example, in Fig. 1 we present a high-resolution AFM image (6) that demonstrates voltage-dependent anion channel clustering on the mitochondrial outer membrane. Moreover, there is evidence that cells actively control spatial organization of its receptors and transporters to modulate the efficiency of ligand-protein interaction and regulate transport rate of certain solute molecules (7–9). Clustering of proteins is complex and dynamic (10,11) with crosstalk to the cytoskeleton (12). Although the mechanisms and physiological role of cluster formation remain mostly unclear, analysis of the screening effects pioneered by Goldstein and Wiegel (13–16) is important for understanding the effects of clustering in different processes.

Figure 1.

High-resolution AFM image of the supramolecular organization of voltage-dependent anion channels in high protein density regions of mitochondrial outer membranes demonstrates pronounced channel clustering (from Gonçalves et al. (6) with permission).

In this article we discuss screening effects in the framework of the simplest model assuming that diffusing molecules are point particles and binding sites are perfectly absorbing circular disks located on the otherwise reflecting flat surface. From a mathematical point of view, this is a complicated many-body problem, which has a highly nontrivial exact solution only in the case of a two-disk cluster (17,18). One can learn about sophisticated formalisms developed to analyze the problem in the literature ((17–23) and references therein).

The functional units that form clusters—channels, transporters, receptors—are usually oligomeric protein structures themselves, which have different inherent symmetries. Correspondingly, they cluster on lattices of different packing symmetries. Examples of compact clusters on triangular and square lattices are shown in panels A and B of Fig. 2. Due to inhomogeneities of biological membranes such as local lipid demixing, stalks, and intermembrane contacts, the clustering of the functional units is not necessarily compact. Various lipid domains, sometimes with quite unexpected composition (24), are clearly abundant on the cell surface (25). Various extended cluster structures are also plausible. Examples of noncompact clusters on triangular and square lattices are shown in panels C and D of Fig. 2. The goal of this article is to review a recently proposed general approach (26,27) to an approximate quantitative description of the screening effects in such structures, applicable to both compact and noncompact clusters.

Figure 2.

(A and B) Examples of compact and (C and D) noncompact clusters of absorbing disks arranged on triangular (panels A and C) and square (panels B and D) lattices and corresponding effective spots, studied in Berezhkovskii et al. (26,27).

The suggested approach allows one to derive simple formulas for the effective rate constants that characterize the cluster trapping rates (26,27). The formulas show how the rate constants depend on the disk radius, the number of disks in the cluster, and the cluster shape and size. We have applied this approach to study trapping by clusters assuming that the disks occupy neighboring sites of triangular (26) and square (27) lattices; some examples of studied clusters are shown in Fig. 2. Here we discuss the general approach (Main Idea), briefly summarize the results obtained previously (26,27) (Clusters on Triangular Lattices and Clusters on Square Lattices, respectively), and present results for clusters formed by the disks occupying neighboring sites on a hexagonal lattice (Clusters on Hexagonal Lattices). Before discussing our approach, in Preliminaries, we offer some useful formulas for the rate constant and the concept of boundary homogenization, which plays the central role in our analysis.

Preliminaries

Useful formulas for the rate constant

Perfectly absorbing sphere

Consider trapping of diffusing particles by a perfectly absorbing sphere of radius R centered at the origin. In steady state the particle concentration c(r), where r is the distance from the origin, satisfies the diffusion equation

$$\begin{array}{cc}\frac{D}{r^2}\frac{d}{dr}\left(r^2\frac{dc\left(r\right)}{dr}\right)=0,& r>R\end{array},$$ (1)

absorbing boundary conditions on the surface of the sphere, c(R) = 0, and the requirement c(r)|_r→∞ = c_∞, where D and c_∞ are the particle diffusion coefficient and concentration at infinity, respectively. Solution to this equation is given by

$$\begin{array}{cc}c\left(r\right)=\left(1-\frac{R}{r}\right)c_{\infty },& r>R\end{array}.$$ (2)

One can use this solution to find the steady-state flux J, which is the number of particles trapped by the sphere per unit time,

$$J=4\pi R^{2}D{\frac{dc\left(r\right)}{dr}|}_{r=R}=4\pi DR{c}_{\infty }.$$ (3)

The ratio of this flux to the particle concentration at infinity is the Smoluchowski (Sm) rate constant for a perfectly absorbing sphere (sph),

$$k_{\text{sph}}^{\text{Sm}}=\frac{J}{c_{\infty }}=4\pi DR.$$ (4)

Perfectly absorbing circular disk

Next, consider trapping of diffusing particles by a perfectly absorbing circular disk of radius a located on a flat reflecting surface. In this case, the rate constant (defined as the ratio of the steady-state flux of the particles to their concentration at infinity) was first obtained by Hill (H) (28) and Berg and Purcell (BP) (29). This rate constant is given by

$$k_{\text{disk}}^{\text{HBP}}=4Da.$$ (5)

Perfectly absorbing noncircular spot

A generalization of the formula in Eq. 5 to the case of a noncircular perfectly absorbing spot on a flat reflecting surface was proposed by Dudko et al. (30),

$$k_{\text{spot}}={\left(\frac{32}{{\pi }^2}AP\right)}^{1/3}D,$$ (6)

where A and P are the area and perimeter of the spot. For a round spot of radius a, we have A = πa², P = 2πa, and k_spot = $k_{\text{disk}}^{\text{HBP}}$. Whereas Eq. 5 is an exact result, Eq. 6 provides an approximate formula, which was suggested based on dimensional analysis and Brownian dynamics simulation results.

Partially absorbing sphere

When the sphere of radius R centered at the origin is partially absorbing, the boundary condition on its surface takes the form

$$D{\frac{dc\left(r\right)}{dr}|}_{r=R}=\kappa c\left(R\right),$$ (7)

where κ is the rate constant that characterizes the trapping efficiency, and κ = ∞ and κ = 0 correspond to perfectly absorbing and reflecting surfaces, respectively. The rate constant in this case was first obtained by Collins and Kimball (CK) (31). This rate constant, defined as the ratio of the steady-state flux to c_∞, is given by

$$k_{\text{sph}}^{\text{CK}}=\frac{k_{\text{sph}}^{\text{Sm}}\kappa A_{\text{sph}}}{k_{\text{sph}}^{\text{Sm}}+\kappa A_{\text{sph}}}=\frac{4\pi R^2\kappa D}{D+\kappa R},$$ (8)

where A_sph = 4πR² is the surface area of the sphere. The expression above has a simple physical interpretation: it describes the decrease of the Smoluchowski rate constant by the factor

$$\kappa A_{\text{sph}}/\left(k_{\text{sph}}^{\text{Sm}}+\kappa A_{\text{sph}}\right)=\kappa R/\left(D+\kappa R\right),$$

which is the trapping probability of a particle that starts from the surface of the sphere.

Partially absorbing circular disk

A similar formula for the rate constant for trapping by a partially absorbing circular disk of radius a on a flat reflecting surface was suggested by Zwanzig and Szabo (ZS) (32),

$$k_{\text{disk}}^{\text{ZS}}=\frac{k_{\text{disk}}^{\text{HBP}}\kappa A_{\text{disk}}}{k_{\text{disk}}^{\text{HBP}}+\kappa A_{\text{disk}}}=\frac{4\pi a^2\kappa D}{4D+\kappa \pi a}.$$ (9)

In contrast to Eq. 8, which is an exact result, the formula in Eq. 9 provides a very good approximation for the rate constant over the entire range of κ.

Boundary homogenization

Berg and Purcell (29), in their classical article on chemotaxis, considered trapping of diffusing particles (ligands) by N small perfectly absorbing circular disks (receptors) of radius a randomly distributed over the surface of a perfectly reflecting sphere (cell) of radius R, a << R. Based on the definition of the rate constant as the ratio of the steady-state flux of the particles to their concentration at infinity, they obtained the following approximate expression for the rate constant:

$$k_{\text{BP}}=\frac{k_{\text{sph}}^{\text{Sm}}{k}_{\text{disk}}^{\text{HBP}}N}{k_{\text{sph}}^{\text{Sm}}+k_{\text{disk}}^{\text{HBP}}N}=\frac{4\pi DRaN}{\pi R+aN}.$$ (10)

This formula shows that when N is small so that $aN<<R$, the rate constant is the sum of the rate constants of N noninteracting perfectly absorbing disks, k_BP = 4πDaN = $k_{\text{disk}}^{\text{HBP}}$N. In the opposite limiting case, R << aN, k_BP = 4πDR = $k_{\text{sph}}^{\text{Sm}}$, i.e., the patchy sphere traps diffusing particles as if it is perfectly absorbing. This happens despite the fact that the surface fraction occupied by the disks,

$$\sigma =A_{\text{disk}}N/A_{\text{sph}}=a^{2}N/\left(4R^2\right),$$

can be small, σ << 1.

Shoup and Szabo (33) pointed out that the Berg-Purcell formula for the rate constant, Eq. 10, can be interpreted as the Collins-Kimball formula, Eq. 8. Indeed, Eq. 10 becomes identical to Eq. 8 if we take

$$\kappa ={\kappa }_{\text{BP}}=\frac{k_{\text{disk}}^{\text{HBP}}N}{A_{\text{sph}}}=\frac{DaN}{\pi R^2}=\frac{4D}{\pi a}\sigma .$$ (11)

This is an example of the so-called boundary homogenization, which is the replacement of a patchy surface by an effective uniform partially absorbing surface with κ chosen so that the steady-state flux remains unchanged. Such a replacement is possible because the memory of a local configuration of the absorbing disks on the surface decays with the distance from the surface. Sufficiently far away from a patchy surface, the steady-state fields of the particle fluxes and concentrations are indistinguishable from the corresponding fields in the case of uniform partially absorbing surface with correctly chosen κ. One can learn more about boundary homogenization in recent articles (34–36) and references therein.

According to Eq. 11, κ_BP being linear in σ does not depend on the disk arrangement on the surface. In fact, Eq. 11 provides only the leading term of the small-σ expansion of the dependence κ(σ) (34–36). In our further analysis we use a more general approximate formula for κ(σ) (36),

$$\begin{array}{cc}\kappa \left(\sigma \right)=\frac{4D}{\pi a}\sigma f\left(\sigma \right),& f\left(\sigma \right)=\frac{1+\alpha \sqrt{\sigma }-\beta {\sigma }^2}{{\left(1-\sigma \right)}^2}\end{array},$$ (12)

where α = 1.62, 1.75, 1.37 and β = 1.36, 2.02, 2.59 for triangular, square, and hexagonal lattices of perfectly absorbing disks on the otherwise reflecting flat surface, respectively. Because σ → 0, f(σ) approaches unity and κ(σ) in Eq. 12 reduces to κ_BP in Eq. 11, because screening effects can be neglected in the small-σ limit. The formula in Eq. 12 was obtained by fitting the dependences κ(σ) found numerically for lattices of different types (36). This formula shows that the trapping rate constant increases with σ much faster than κ_BP. Note that numerical results for the disks randomly arranged on the flat surface can also be well fitted by Eq. 12 with α = 0.34 and β = −0.58 (36).

Main Idea

The main idea of our approach to the problem of trapping of diffusing particles by a cluster of perfectly absorbing disks is to replace the cluster by an effective uniform spot, which is partially absorbing, and then to find the rate constant using the analog of the Collins-Kimball-Zwanzig-Szabo formula. The rate constant k obtained in this way is given by

$$k=\frac{k_{\text{spot}}\kappa A}{k_{\text{spot}}+\kappa A},$$ (13)

where k_spot is the rate constant for the perfectly absorbing spot given in Eq. 6, A is the spot area, and κ is its effective trapping rate. To apply this formula, one has to

• 1.Construct the effective spot, i.e., to define the spot geometry; and
• 2.Determine the effective trapping rate κ.

In this section we discuss both issues in general. Applications of the general approach to particular cases are discussed in Clusters on Triangular Lattices, Clusters on Square Lattices, and Clusters on Hexagonal Lattices.

Consider a cluster formed by N perfectly absorbing disks of radius a whose centers occupy neighboring sites of a regular lattice. We introduce an adjoint lattice that contains sites of the initial lattice at the centers of its elementary cells, and use it to define the effective spot as a union of elementary cells of the adjoint lattice containing the disks. Examples of effective spots replacing a two-disk cluster on the triangular, square, and hexagonal lattices are shown in Fig. 3 . The area A_N of the spot is

$$A_N=N{A}_1,$$ (14)

where A₁ is the area of the elementary cell of the adjoint lattice.

Figure 3.

Effective spots replacing a cluster formed by two disks of radius a on the (A) triangular, (B) square, and (C) hexagonal lattices of period l. (Dashed lines) Elementary cells of the adjoint lattices forming the spots.

To prescribe a trapping rate to the effective spot, we assume that κ is the same as that of an infinite plane covered by perfectly absorbing disks of radius a arranged in the same regular lattice. Using Eqs. 12 and 14, we find that

$$\kappa A_N=4DaNf\left(\sigma \right)=k_{\text{disk}}^{\text{HBP}}Nf\left(\sigma \right),$$ (15)

where we have used the fact that σA₁ = πa². Substituting this expression for κA_N into Eq. 13, we can write the rate constant of the N-disk cluster as

$$k_N=\frac{4DaN}{f{\left(\sigma \right)}^{-1}+4DaN/k_{\text{spot}}}.$$ (16)

As follows from Eq. 6, the ratio 4DaN/k_spot is given by

$$\frac{4DaN}{k_{\text{spot}}}={\left(2{\pi }^2\right)}^{1/3}\frac{aN}{{\left(A_N{P}_N\right)}^{1/3}},$$ (17)

where P_N is the spot perimeter, which is a function of N and the shape of the cluster. Substituting the ratio in Eq. 17 into Eq. 16, we arrive at

$$k_N=\frac{4DaN}{f{\left(\sigma \right)}^{-1}+{\left(2{\pi }^2\right)}^{1/3}aN/{\left(A_N{P}_N\right)}^{1/3}},$$ (18)

which is the key result of our approach.

As the distance l between the centers of neighboring disks increases, the disk surface fraction σ decreases, and function f(σ), defined in Eq. 12, approaches unity. The spot area and perimeter are proportional to l² and l, respectively. Therefore, the second term in the denominator is proportional to a/l and, hence, vanishes as l → ∞. Thus, at large $l$ the rate constant, Eq. 18, reduces to the sum of the rate constants of N noninteracting disks, 4DaN, as it must be. The ratio k_N/(4DaN) can be considered as the dimensionless absorbing efficacy of a single disk of the cluster. Thus, as the intersite distance l increases, the efficacy approaches its upper limit of unity.

To discuss the large-N behavior of k_N, we note that the product A_NP_N never grows with N faster than N². Therefore, the ratio N/(A_NP_N)^1/3 monotonically grows with N. At a fixed value of the ratio a/l (and, hence, σ) and sufficiently large N, the denominator in Eq. 18 is determined by the second term. In this limiting case, the rate constant takes the form

$$k_N={\left(\frac{32}{{\pi }^2}{A}_N{P}_N\right)}^{1/3}D=k_{\text{spot}},$$ (19)

which shows that the cluster becomes perfectly absorbing as N → ∞. One can also see this from Eq. 13, in which k_spot ∝ (A_NP_N)^1/3, A = A_N ∝ N, and the ratio κA/(k_spot + κA) tends to unity as N → ∞.

Below we use the results of this section to discuss the dependence of the rate constant k_N on the number N of the disks in the cluster as well as on the cluster size and shape for clusters on the triangular, square, and hexagonal lattices. Because the results for clusters on the first two lattices have been published (26,27), we discuss them briefly in the next two sections. After that we give a more detailed discussion and new results for clusters on the hexagonal lattice.

Clusters on Triangular Lattices

When the disk centers occupy neighboring sites of a triangular (tr) lattice, the adjoint lattice is hexagonal. The equivalent spot is a union of the two hexagons surrounding the sites of the initial lattice occupied by the disk centers. In Fig. 3 A we show a disk dimer and the spot formed by the two surrounding hexagons. When neighboring sites of the initial lattice are separated by distance l, l ≥ 2a, the edge length b of the hexagon (h) (Fig. 3 A) and its area, respectively, are $b=l/\sqrt{3}$ and $A_h=2\sqrt{3}{b}^2=\sqrt{3}{l}^2/2$. Using these relations, we can write k_N in Eq. 18 as

$$k_{N,tr}=\frac{4DaN}{f_{tr}{\left(\sigma \right)}^{-1}+{\left[4{\pi }^2{N}^2/\left(P_N/b\right)\right]}^{1/3}\left(a/l\right)},$$ (20)

where the disk surface fraction is given by

$$\sigma =\pi a^2/A_h=\left(2\pi /\sqrt{3}\right){\left(a/l\right)}^2,$$

and function f_tr(σ) is

$$f_{tr}\left(\sigma \right)=\left(1+1.62\sqrt{\sigma }-1.36{\sigma }^2\right)/{\left(1-\sigma \right)}^2$$

(see Boundary Homogenization).

The expression for k_N,tr was checked numerically (26) by comparing its predictions with the values of the rate constants obtained from Brownian dynamics simulations for 14 clusters of different shape and size (2a ≤ l ≤ 20a) with N = 2, 3, 4, 5, 6, 7, 13, and 19. We define the relative error as |k_theory − k_sim|/k_sim, where k_theory and k_sim are the values of the rate constant predicted by the theory and obtained from simulations, respectively. It was found that for most clusters the maximum relative error is within 3%.

The rate constant dependence on the cluster shape is due to the shape dependence of the perimeter P_N. To discuss this dependence we compare the rate constants of linear (l) and large compact (c) clusters, for which the perimeters, respectively, are

$$P_{N,tr}^{\left(l\right)}=2\left(2N+1\right)b$$

and

$$P_{N,tr}^{\left(c\right)}=4\sqrt{\left(2\pi /\sqrt{3}\right)N}b,N\text{>>}1\text{.}$$

We find the rate constants by substituting the perimeters into Eq. 20. As a result, we obtain

$$k_{N,tr}^{\left(l\right)}=\frac{4DaN}{f_{tr}{\left(\sigma \right)}^{-1}+{\left[2{\pi }^2{N}^2/\left(2N+1\right)\right]}^{1/3}\left(a/l\right)}$$ (21)

and

$$\begin{array}{cc}{k}_{N,tr}^{\left(c\right)}=\frac{4DaN}{f_{tr}{\left(\sigma \right)}^{-1}+{\left(\sqrt{3}/2\right)}^{1/6}\sqrt{\pi N}\left(a/l\right)},& N>>1\end{array}.$$ (22)

At large N where the screening effects are most pronounced, the ratio of the rate constants is

$$\begin{array}{cc}\frac{k_{N,tr}^{\left(l\right)}}{k_{N,tr}^{\left(c\right)}}\simeq {\left(\frac{P_{N,tr}^{\left(l\right)}}{P_{N,tr}^{\left(c\right)}}\right)}^{1/3}\simeq {\left(\frac{3N}{2\pi }\right)}^{1/6},& N>>1\end{array}.$$ (23)

This shows that $k_{N,tr}^{\left(l\right)}$ significantly exceeds $k_{N,tr}^{\left(c\right)}$ at large N, as might be expected, because screening in compact clusters is much more efficient. It should be pointed out that Eq. 21 fails as N → ∞. In this limiting case, Eq. 21 predicts $k_{N,tr}^{\left(l\right)}$ ∝ N^2/3, whereas the correct asymptotic behavior of the rate constant of a linear cluster is $k_{N,tr}^{\left(l\right)}$ ∝ N/ln N. Fig. 4 shows the reduction of the efficacy, $k_{N,tr}^{\left(c\right)}$/(4DaN), for compact clusters as a function of N at different values of the interdisk distance l. As N → ∞, the efficacy tends to zero as N^−1/2, as follows from Eq. 22.

Figure 4.

Efficacy of compact clusters of absorbing disks positioned on the triangular lattice, as functions of the disk number in the cluster at different interdisk distances. To see this figure in color, go online.

Clusters on Square Lattices

When an N-disk cluster is formed by perfectly absorbing disks whose centers occupy neighboring sites of a square (sq) lattice of period l, the adjoint lattice is also the square one of the same period. In Fig. 3 B we show a disk dimer and the effective spot in this case. The formula for the rate constant in Eq. 18 in the case of the square (sq) lattice reduces to (27)

$$k_{N,sq}=\frac{4DaN}{f_{sq}{\left(\sigma \right)}^{-1}+{\left[2{\pi }^2{N}^2/\left(P_N/l\right)\right]}^{1/3}a/l},$$ (24)

where the disk surface fraction is given by σ = πa²/l², and function f_sq(σ) is

$$f_{sq}\left(\sigma \right)=\left(1+1.75\sqrt{\sigma }-2.02{\sigma }^2\right)/{\left(1-\sigma \right)}^2$$

(see Boundary Homogenization). In (27), we compared the rate constants predicted by Eq. 24 with those found in Brownian dynamics simulations for 18 clusters of different shape and size (2a ≤ l ≤ 20a) with N = 2, 3, 4, 5, 6, 7, 8, 9, 10, and 13. The comparison showed good agreement between the theoretical predictions and numerical results.

Trapping by a straight linear cluster with equally spaced disk centers can be analyzed using either a triangular or a square lattice. The resulting formulas for the rate constant are

$$k_{N,tr}^{\left(l\right)}=\frac{4DaN}{f_{tr}{\left({\sigma }_{tr}\right)}^{-1}+{\left[2{\pi }^2{N}^2/\left(2N+1\right)\right]}^{1/3}a/l}$$ (25)

and

$$k_{N,sq}^{\left(l\right)}=\frac{4DaN}{f_{sq}{\left({\sigma }_{sq}\right)}^{-1}+{\left[{\pi }^2{N}^2/\left(N+1\right)\right]}^{1/3}a/l},$$ (26)

where

$${\sigma }_{tr}=\left(2\pi /\sqrt{3}\right)a^2/l^2$$

and

$${\sigma }_{sq}=\pi a^2/l^2$$

are the disk surface fractions for the two lattices, and we have used the following formulas for the spot perimeters:

$$P_{N,tr}^{\left(l\right)}=2\left(2N+1\right)l/\sqrt{3}$$

and

$$P_{N,sq}^{\left(l\right)}=2\left(N+1\right)l.$$

Although the approximate formulas in Eqs. 25 and 26 are different, they give practically the same values of the rate constant. Maximum deviation of the ratio $k_{N,sq}^{\left(l\right)}/k_{N,tr}^{\left(l\right)}$ from unity does not exceed 3.7% for N = 2, 2.5% for N = 3, 1.7% for N = 4, 1.2% for N = 5, and is less than 1% for longer clusters, N ≥ 6.

Concluding this section, we consider trapping by a rectangular N-disk cluster with the goal to analyze the rate constant dependence on the cluster shape. The perimeter of a m × n = N rectangular (r) cluster is $P_{m\times n}^{\left(r\right)}$ = 2(m + n)l, m, n ≥ 2. Using this, we can find the rate constant $k_{m\times n}^{\left(r\right)}$ by Eq. 18,

$$k_{m\times n}^{\left(r\right)}=\frac{4DaN}{f_{sq}{\left(\sigma \right)}^{-1}+{\left[{\pi }^2{m}^2{n}^2/\left(m+n\right)\right]}^{1/3}\left(a/l\right)}.$$ (27)

As found in (27), this formula accurately predicts the rate constant over the entire range of l, l ≥ 2a, even for the smallest rectangular cluster of size 2 × 2. At a fixed value of the product mn = N, the sum m + n has a minimum at

$$m\ast =n\ast =\sqrt{N}$$

and a maximum at m = N, n = 1 (or m = 1, n = N). Therefore, $k_{m\times n}^{\left(r\right)}$ has a maximum when the cluster is linear, and a minimum when the rectangular cluster is almost symmetric and its shape is close to square. This is a consequence of the fact that screening is less efficient for linear clusters and more efficient for compact ones.

Clusters on Hexagonal Lattices

In the last two sections we discussed the earlier obtained results on trapping by clusters on triangular and square lattices (26,27). This section presents results for clusters on hexagonal (h) lattices. In this case, the adjoint lattice is the triangular (tr) one of the period $\sqrt{3}l$, where l is the distance between neighboring sites of the initial hexagonal lattice. A disk dimer and the corresponding effective spot formed by two elementary cells of the triangular lattice containing the disks are shown in Fig. 3 C. The area of a unit cell of the adjoint lattice is

$$A_{tr}=3\sqrt{3}{l}^2/4.$$

Using this, we can write the rate constant in Eq. 18 as

$$k_{N,h}=\frac{4DaN}{f_h{\left(\sigma \right)}^{-1}+\left(2/\sqrt{3}\right){\left[{\pi }^2{N}^2/\left(P_N/l\right)\right]}^{1/3}\left(a/l\right)},$$ (28)

where the disk surface fraction is

$$\sigma =\pi a^2/A_{tr}=\left[4\pi /\left(3\sqrt{3}\right)\right]a^2/l^2,$$

and function f_h(σ) is given by

$$f_h\left(\sigma \right)=\left(1+1.37\sqrt{\sigma }-2.59{\sigma }^2\right)/{\left(1-\sigma \right)}^2$$

(see Boundary Homogenization).

To check the accuracy of the formula in Eq. 28, we compare its predictions with the values of the rate constants obtained from Brownian dynamics simulations. In simulations we find the mean lifetime τ of a particle diffusing in a cubic cavity of volume V containing a cluster of perfectly absorbing disks in the center of one of its walls, which are otherwise perfectly reflecting. The particle starting point is uniformly distributed over the cavity volume. The rate constant of interest is given by the ratio of the cavity volume to the mean lifetime, k = V/τ. We used the following set of dimensionless parameters: disk radius a = 5 × 10⁻²; time step Δt = 10⁻⁶; diffusion coefficient D = 1; and the cavity side and its volume were 8 and 512, respectively. The mean lifetime was found by averaging over 5 × 10⁴ trajectories.

Simulations were run for 14 clusters of different shape and size with N = 2, 3, 4, 5, 6, 10, 12, 13, 14, 16, and 18, shown in Figs. 5 and 6 . For each cluster, the distance between the centers of neighboring disks varied from l = 2a (disks in contact) to l = 20a. The rate constants obtained from simulations and predicted by Eq. 28 are shown in Fig. 5 by symbols and solid curves, respectively. One can see good agreement between the two.

Figure 5.

Comparison of the rate constants predicted by Eq. 28 (solid curves) with the rate constant values obtained from Brownian dynamics simulations (symbols). (A, Curves, bottom to top) Clusters containing N = 2, 3, 4, and 5 disks. Simulation results for the four- and five-disk clusters in square brackets are not shown (see more in the text). (B, Curves, bottom to top) Clusters containing N = 6, 10, 12, and 13 disks. (C, Curves, bottom to top) Clusters containing N = 14, 16, and 18 disks. To see this figure in color, go online.

Figure 6.

Maximum relative errors in predictions of the rate constants k_N, Eq. 28, and k_spot, Eq. 6. To see this figure in color, go online.

Note that the effective spots corresponding to the two four-disk clusters of different shapes shown in panel A of Fig. 5 have not only equal areas but also equal perimeters. Consequently, Eq. 28 predicts the same rate constants for both clusters despite the fact that they have different shapes. The same is true for the three five-disk clusters shown in the same panel of Fig. 5. Both predictions are corroborated by the results of Brownian dynamics simulations, which yielded maximum difference between the rate constants not exceeding 1.6% of k_4,h for the four-disk clusters and 2.1% of k_5,h for the five-disk clusters. Simulation results for the clusters given in Fig. 5 A in square brackets are not shown.

Maximum relative errors of the rate constants predicted by Eq. 28 for clusters of different shape are summarized in Fig. 6, where using simulation results we also give the relative errors of k_spot predicted by Eq. 6 for effective spots of different shapes. The relative errors show that both formulas for the rate constant, Eqs. 6 and 28, work reasonably well.

Concluding Remarks

Clustering of functional units—channels, transporters, and receptors—is a ubiquitous phenomenon described for various cell and organelle membranes (1–12). Because of the membrane inhomogeneities and different inherent symmetries of the units, they might cluster into structures that are different in both the degree of compactness and symmetry of the packing lattices. If they do, these structural features manifest themselves in different mutual screening of the units which, especially in the case of tight packing and/or large clusters, can significantly reduce the unit trapping efficiency.

Here, we offer a simple general analytical approach for an approximate quantitative description of this screening, which allows one to treat clusters of perfectly absorbing circular disks of complex structures. As the surface fraction of absorbing disks increases, the cluster becomes perfectly absorbing whereas the efficacy of an individual disk decreases. Of course, the proposed approach is not universal; nevertheless, as shown above it works reasonably well, permitting calculation of the efficacy of trapping for the clusters of various shape and size on lattices of different packing symmetries. The proposed approach provides a tool for analyzing the effects of channel/receptor/transporter clustering in different intracellular and intercellular processes featuring domains of various packing.