Nanodomains in Biomembranes with Recycling - The Journal of

Article pubs.acs.org/JPCB

Nanodomains in Biomembranes with Recycling Mareike Berger, Manoel Manghi, and Nicolas Destainville* Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 118 route de Narbonne, F-31062 Toulouse, France ABSTRACT: Cell membranes are out of thermodynamic equilibrium notably because of membrane recycling, i.e., active exchange of material with the cytosol. We propose an analytically tractable model of biomembrane predicting the effects of recycling on the size of protein nanodomains also called protein clusters. The model includes a short-range attraction between proteins and a weaker long-range repulsion which ensures the existence of so-called cluster phases in equilibrium, where monomeric proteins coexist with finite-size domains. Our main finding is that, when taking recycling into account, the typical cluster size at steady state increases logarithmically with the recycling rate at fixed protein concentration. Using physically realistic model parameters, the predicted 2fold increase due to recycling in living cells is most likely experimentally measurable with the help of super-resolution microscopy. serving as adhesive foci in adherens junctions.12,22 Grouping identical receptors in a same cluster can also optimize and make more reliable their response to external stimuli.11,15,23−25 Clusters could also result in the inactivation or storage of proteins that would otherwise interact with their environment in a possibly unfavorable way. Forming domains could function as a means of regulating the concentration of the proteins on the cell membrane, as proteins in the reserve pool would not be active.13 This Article aims at developing a simplified biophysical model of the cell membrane describing the lateral distribution of embedded proteins and giving a realistic explanation of the formation of nanoclusters.6,26 In order for the proteins to be condensed in a cluster phase, a generic attractive interaction at short range is required, the origin of which has been discussed in many works (refs 6 and 11 and references therein). In a few words, attraction between membrane proteins, either integral or peripheral, is a combination of electrostatic, hydrophobic, depletion, and wetting forces involving proteins and the surrounding lipids. Above a critical protein concentration, attraction should in principle lead to phase separation between a protein-rich phase and a protein-poor (lipid-rich) one. Without any mechanism preventing it, the condensation into one big protein macrophase in thermodynamic equilibrium should ensue. Two types of theoretical approaches have then been developed to account for the observed finite size of protein clusters: (1) in equilibrium, an effective long-range repulsive potential between the proteins can prevent the formation of the big macrophase;25,27−29 (2) inherent far-f rom-equilibrium recycling of the cell membrane due to material exchange with

I. INTRODUCTION Being the interface between the cellular and extracellular media, the cell membrane has plenty of important biological functions such as signal transduction or transport of solutes.1,2 In the original 1972 cell membrane model by Singer and Nicolson,3 the plasma membrane was visioned as a homogeneous mixture of lipids and proteins in which the proteins represent about 50% of the membrane mass. Since then, this basic model has known regular improvements, showing an increasing organizational complexity. The organization in nanodomains had been suspected for long, but a variety of super-resolution light microscopy techniques and atomic-force microscopy (AFM) have recently improved further our understanding of membrane supramolecular organization. Even though their physicochemical origin remains a matter of debate,4−6 there is growing evidence, both experimental2,7−17 and numerical (see ref 18 for a recent review), of the generic existence of dynamic, nanometer sized functional domains on the cell membrane. From a biological perspective, many ideas on the functioning and roles of protein clusters have been proposed.2 First of all, clusters can be composed of different types of proteins and lipids that perform a certain biological task together and therefore need to be closely segregated in a tiny membrane region. A biologically important example for these specialized clusters is the G-protein clustered with receptors and effectors in signaling platforms.19 Gprotein coupled receptors (GPCR) have been shown to be involved in a wide variety of biological processes by activating cellular signal transduction pathways.19,20 For example, the MOR-GPCR was examined by particle-tracking and was found to be locally confined in nanodomains, presumably because of interprotein long-range forces.21 E-cadherins constitute another example for which nanopatterning of the membrane has been shown to be involved in its biological function, with clusters © 2016 American Chemical Society

Received: July 30, 2016 Revised: September 12, 2016 Published: September 21, 2016 10588

DOI: 10.1021/acs.jpcb.6b07631 J. Phys. Chem. B 2016, 120, 10588−10602

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The Journal of Physical Chemistry B

particularly true for peripheral proteins on the internal leaflet.1 Alternatively, they can be carried by vesicles, for example, during exocytosis. There is no reason to anticipate that they arrive in clusters as large as those observed on the membrane, and we assume for simplicity sake that they are principally carried as monomers, as in refs 12, 22, and 40. Conversely, endocytosis and related processes remove material from the membrane. Without better insight, we assume that the removed membrane patches have the same composition as the average membrane one. The off-rate dcn/dt of any cluster type is thus assumed to be proportional to its area fraction cn. Our recycling scheme is thus of “monomer deposition/raft removal” (MDRR) type as defined in ref 31. These processes are assumed to be active, that is to say, driven by energy consumption. The membrane is thus maintained out of equilibrium, and we study its steady state. The goal of the present work is to quantify the respective contributions of equilibrium and far-from-equilibrium mechanisms in setting the typical cluster size. Using a formalism first proposed by Smoluchowski in a different but related context (see, e.g., ref 42), we will write below a master equation that embraces both equilibrium and nonequilibrium features. We adopt an approach where the plasma membrane is modeled as a two-dimensional fluid consisting of two types of particles. For the sake of simplicity, we indeed assume all proteins to be identical and the surrounding lipid phase to be homogeneous. We are interested in the distribution of the proteins in the lipid phase. Since proteins represent about onehalf of the membrane mass, the protein area fraction ϕ is on the same order of magnitude, even though certainly slightly smaller because many proteins protrude out of the membrane plane. Accepted values lie around ϕ = 0.2.6,15 We shall explore the range ϕ = 0.01−0.3 in this work and the Appendices. The proteins are clustered in n-particle multimers (Figure 1) with n = 1 for

the cytosol, that continually mixes the plasma membrane components, can also break large assemblies.4,30−33 The novelty of this paper consists of combining two such approaches in a single, analytically tractable model. We demonstrate that taking out-of-equilibrium considerations into account does not change the qualitative properties of the system as compared to the equilibrium model. However, switching recycling on while maintaining the protein area fraction constant increases the typical cluster sizes logarithmically with the recycling rate, which should be experimentally measurable. The apparently counterintuitive fact that the typical cluster size grows with increasing recycling will be thoroughly discussed.

II. STATE OF THE ART AND MODEL If a condensed phase of proteins exists below the critical temperature (or above the critical concentration), minimization of the interfacial free energy in equilibrium should lead to a large macro-phase in coexistence with a low density “gas” phase of protein monomers. This is in contradiction with the experimental observation of nanodomains in cells. Therefore, a mechanism is required which explains why condensation stops at a given domain size. Before presenting our own model in detail, we rapidly review useful anterior approaches tackling the finite size of nanoclusters (see refs 6, 26, and 32 for more details). In equilibrium, competition between the short-range attraction discussed above and a weaker but longer-range repulsion can lead to nanoclustering by destabilizing too large protein assemblies. This is the so-called cluster phase mechanism.27 The origin of the repulsion can be many-fold.6 An efficient mechanism comes from the up−down symmetry breaking that a vast majority of proteins are supposed to impose to the membrane, leading to a local spontaneous membrane curvature.6 The coupling between inclusion concentration and membrane curvature has recently received a striking experimental demonstration.34,35 When grouped in a cluster, asymmetric proteins collectively shape the membrane and lead to buds with a spherical cap shape, the elastic energy of which grows faster than the number of proteins.36 This is equivalent to the existence of a long-range repulsion of elastic origin which destabilizes too large clusters, because the energetic cost would overcome the gain in terms of line tension if macroscopic clusters grew.25,27,28 A typical finite cluster size emerges,29,36 controlled by the attraction/repulsion competition. Out-of-equilibrium active processes are also able to explain the finite size of nanodomains.4,30 The coalescence of domains by diffusion, ultimately leading to a macro-phase in equilibrium, is in fact in competition with the traffic of membrane material to and from the membrane, for example, by endocytosis and exocytosis of membrane patches. This membrane recycling likely breaks too large assemblies into smaller pieces. The growth of the condensed phase is thus stopped at a given typical size.12,31,37−40 A simple dynamic scaling argument has, for instance, been proposed in ref 37: when starting from a random configuration, one shows that the typical domain size grows with time as L ≈ (Dat)1/3, where D and a are the protein diffusion coefficient and typical diameter.41 Additionally, protein traffic sets a typical recycling time τr, itself depending on off- and onrates, from and to the membrane. Recycling prevents equilibration beyond the time scale τr and the coarsening should stop at a domain size L ≈ (Daτr)1/3. Beyond this simplistic viewpoint, there are several ways to implement recycling.31,32 Proteins can arrive at the membrane as monomers through direct exchange with the cell cytosol. This is

Figure 1. Sketch of our out-of-equilibrium model of protein cluster dynamics: inside the membrane (in gray), clusters can eject and capture monomers (small blue discs), with respective reaction rates kn (for a cluster of size n) and k′ (chosen as independent of n, see text). Monomers are delivered to the system from the cytosol (black arrow on the left) with rate jon, whereas vesicles (gray spheres) are taking material from the membrane to the cell interior (gray arrows). Thus, clusters are extracted from the membrane with a rate joff independent of their size.

monomers. The cluster-size distribution is described by the area fraction cn = number of n-mers × s/S, where S is the system area. The area of a monomer is s = πb2 with an approximate monomer diameter 2b = 5 nm.1,43 If N is the total number of proteins, their total area fraction is ∞

ϕ ≡ Ns /S =

∑ cnn n=1

10589

(1) DOI: 10.1021/acs.jpcb.6b07631 J. Phys. Chem. B 2016, 120, 10588−10602

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The Journal of Physical Chemistry B The area fractions of protein domains of all sizes (n ≥ 1) and of multimers (n ≥ 2) are respectively M = ∑cn and M* = M − c1. The mean aggregation numbers of all sizes and of multimers are, respectively, n̅ =

ϕ , M

n* =

ϕ − c1 M*

The value of μ (or equivalently of c1) is fixed by the value of ϕ through the constraint ϕ = ∑cnn.29 The value of f 0 + μ rather than f 0 alone sets the shape of G(n) and thus of the cluster-size distribution. Figure 2 shows G(n) for different values of ϕ. For a

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III. THERMODYNAMICAL EQUILIBRIUM IN THE ABSENCE OF RECYCLING We first recall some known facts in equilibrium. As already explored in ref 29, for instance (see also ref 44), the equilibrium cluster-size distribution (cn) is found by minimizing the free energy per particle F 1 = f̅ = N ϕ

∞

∑ cn ln(cn) − cn + cnF(n) n=1

(3)

where the two first terms in the sum account for the mixing entropy. Here and in the following, all energies are measured in units of the thermal energy kBT and the temperature T is held fixed. The chemical potential conjugated to N is μ = (∂F/∂N)S,T. The concentration ensues:29,44 cn = eμn−F(n). The cluster-size distribution depends only on the chemical potential μ and the free energy of a domain F(n). Following refs 29 and 40, we use the following form F(n) = −f0 (n − 1) + γ n − 1 + χ (n − 1)α

(4)

As explained in Appendix A (eq A3), it combines a classical liquid droplet approximation in two dimensions (the first two terms) and a repulsive contribution, leading to the finite size of equilibrium clusters (the last term).6 The value of α has been thoroughly explored36 and typically lies between 1.5 and 2, the exact value depending on the membrane tension. Below, we first choose α = 2, the low tension case where analytical calculations can be performed entirely. The case α = 3/2 is tackled numerically in Appendix E, and we arrive to similar qualitative conclusions in this case. The orders of magnitude of the parameters have already been discussed in connection with biophysical data.27,29,36 The value of f 0 > 0 is in the 10−30 kBT range, and γ has the same order of magnitude as f 0. If we choose α = 2, χ appears to be of the order 0.1 to get equilibrium clusters containing few dozens of particles as observed experimentally. This value of χ is consistent with the one that can be obtained on the basis of an elastic description of the biomembrane, as proposed in ref 36. Applying the procedure detailed in this reference for a moderate membrane tension σ ∼ 10−4 J/m2,60 one obtains α = 1.7 and χ = 0.07. Consistently, we choose α = 2, f 0 = 20, γ = 20, and χ = 0.1 (in units of kBT) as a reference parameter set. Alternative parameter sets are studied in Appendix E, with very similar qualitative conclusions. In order to compute cn, we switch now to the grand-canonical ensemble. It can be shown44 that the adequate thermodynamical potential is now G̃ (n) = F(n) − μn. Using cn = eμn−F(n) and the fact that F(1) = 0, we get ̃

cn = e−G(n) = c1e−G(n)

Figure 2. Top: G(n) for the reference parameter set α = 2, f 0 = 20, γ = 20, and χ = 0.1 (all energies are in units of kBT). Curves from top to bottom: Before the phase transition, here for ϕ = 10−7 (μ = −16.12, in red), the energy for monomers is minimal; for ϕ = ϕc = 3.48 × 10−7 (μ = μc = −14.87, in gray), G(n) has an inflection point near n = 10; above the transition, here for ϕ = 10−4 (μ = −13.57, in blue) and ϕ = 0.1 (μ = −13.27, in purple), there are two local minima of comparable energy, one for monomers and one for higher aggregation numbers. Bottom: area fractions cn of clusters of size n (or n-mers) in equilibrium, for two different area fractions ϕ = 10−4 (μ = −13.57, blue dots) and 10−3 (μ = −13.47, greens dots). Monomer concentrations (n = 1) are nearly superimposed. Note that the multimer peak position increases very slowly with ϕ, as well as the monomer area fraction c1. Gaussian fits of the multimer peak are superimposed (continuous lines).

very low ϕ (or very low μ), the energy for forming a monomer is minimal. At the critical concentration ϕc, G(n) has an inflection point indicating the existence of a local minimum for higher values of ϕ > ϕc where clusters nucleate. Above ϕc, the cluster-size distribution is bimodal. Monomers, and very rare dimers or trimers, coexist with a condensed phase of larger clusters of average size n*. From the bottom panel of Figure 2, it can be seen that the multimer peak is well fitted by a Gaussian centered on n* (see eq 2) ⎡ (n − n*)2 ⎤ ⎥ cn = A 0 exp⎢ − 2σ0 2 ⎦ ⎣

(5)

with μ = ln(c1) < 0 (since c1 < 1), after introducing G(n) = G̃ (n) + μ, i.e.,

(7)

for n close to n*. Note that n* is nearly insensitive to ϕ because the second minimum of G̃ (n) itself is slowly shifted to the right when μ is in its physically relevant interval. Note also that ϕ grows rapidly with μ because of the exponential in eq 5. The range of values of μ is thus narrow (few kBT) for values of ϕ of 29

G(n) = −(f0 + μ)(n − 1) + γ n − 1 + χ (n − 1)α (6) 10590


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the repulsion is pairwise additive, with a range larger than the typical cluster diameter, the repulsive barrier when clusters are getting close grows like χ[(n + m)2 − n2 − m2] = 2χnm > 50 kBT as soon as n, m > 15. By comparison, a monomer approaching an n-cluster feels a lower repulsive barrier ≈ F′rep(n) ≈ 2χn before feeling the attraction, and can thus fuse with an n-cluster much more easily. This whole argument in favor of fragmentation and fusion events involving monomers is corroborated by the absence of observation in kinetic Monte Carlo simulations of fragmentation and fusion events involving two large multimers.27 In the case where fusion and fragmentation events involve at least one monomer and where k′ is assumed to be independent of n (see below), the master equation simplifies to

interest (Figure 2). More quantitatively, we demonstrate in Appendix B that, when monomers and clusters coexist, μ ≃ μ0 +

1 ln ϕ n*

(8)

where μ0 is a constant. The chemical potential μ depends weakly on ϕ because n* ≫ 1.

IV. STEADY STATE IN THE PRESENCE OF RECYCLING The equilibrium approach recalled in the previous section gives one explanation of the occurrence of finite-size clusters appearing on the cell membrane. Still the cell membrane constantly exchanges material with the cytosol, which we now take into account. The following master equation describes the time evolution of the domain size distribution12,31,42 dcn = 1(n) + dt +

1 2

dcn = 1(n) + kncn + 1 − k′cnc1 + k′cn − 1c1 − kn − 1cn dt

∞

∑ (kn,mcn+ m − kn′,mcncm)

for n ≥ 2 where kn ≡ kn,1 = because these kinetic constants are assumed to be equal to their equilibrium counterparts.45 For monomers,

m=1

n−1

∑ (km′ ,n− mcn− mcm − km,n− mcn) m=1

(10)

(0) (0) k′c(0) n c1 /cn+1

(9)

dc1 = 1(1) − k′Mc1 + dt

where 1(n) is the n-cluster net flux from the cytosol to the membrane (1 = 0 in equilibrium). For monomers, the second sum on the r.h.s. vanishes because monomers cannot result from the fusion of smaller clusters. This mean-field approach is presumably valid in dimension 2.42 It is strictly equivalent to the equilibrium approach presented in the previous section when recycling vanishes (1 = 0). Consistently, equilibrium concentrations will be denoted by c(0) n from now on. The coefficients kn,m and kn,m ′ respectively control the rates of cluster fragmentation, in which one domain of n + m particles breaks into two smaller ones of size n and m, and domain fusion, in which two domains containing n and m monomers fuse to form a single domain of size n + m.31 In equilibrium, the system makes in average equally often the transition from one state to the other, which translates into the detailed balance condition kn,mcn+m = kn,m ′ cncm.45 We assume from now on that only the fragmentation and fusion terms involving at least one monomer are taken into account, as in ref 40. Indeed, the energy barrier to be overcome by a dimer or a small multimer to escape a cluster of size n ≫ 1 is significantly larger than the one for a monomer because the binding energy between a small m-mer and a large n-mer is roughly proportionnal to m. Owing to Kramers’ theory, m-mers with m > 1 can hardly escape from a cluster. In the extreme case where an n-cluster would break into two roughly equal parts, say of size n/2 for simplicity sake, the free-energy barrier to be overcome can be estimated as the interface energy that needs to be spent in order to separate the two clusters (the repulsive term is essentially unchanged at this stage due to its long-range character). The length of the newly created interface is roughly twice as long as the original cluster diameter. The height of the barrier is ≈(2/π)γ n > 50 kBT for n as small as 15 particles (the smallest multimer size for the reference parameter set, see Figure 2). This is much larger than the free energy needed for a monomer to escape a cluster, ≈F(n − 1) − F(n) ∼ 10 kBT, which makes the former event much less probable. Conversely, small multimers are very rare in the “gas” phase, and their fusion with clusters is therefore exceptional. As for the fusion of large multimers of sizes m and n much larger than 1, the long-range character of the repulsion also makes them improbable. In the simplest case evoked in Appendix A where

∞

∑ kmcm+ 1 m=1

(11)

where we recall that M = ∑n≥1 cn. We have chosen to consider k′ as independent of n because it measures the capture rate of a monomeric particle by an n-cluster through a diffusive process. Either the capture process is reactionlimited or it is diffusion-limited. In the former case, the capture reaction is limited by a repulsive energy barrier between the cluster and the monomer that the monomer has to overcome, as already discussed above. Since a vast majority of clusters have a size n close to n*, the energy barrier can be considered as essentially independent of n. However, as evoked above, the barrier scales like 2χn and is not higher than a few kBT with our reference parameter set. The capture process is most likely diffusion-limited. In this case, solving the 2D backward Smoluchowski equation shows that the capture time is τ ≃ d2/ (2D) ln(d/a), where a is the cluster diameter and d > a the typical distance between clusters; D is the diffusion coefficient of a free monomer in the bilayer.46 For the same reason as above, d and a can be considered as essentially independent of n. In both cases, k′ thus weakly depends on n, and will be considered as constant from now on. If the capture is diffusion-limited, taking D ∼ 10−12 m2 s−1,43 a ≃ 50 nm (ref 6 and references therein), and d ≃ 200 nm (so that ϕ ≈ a2/d2 ≈ 0.1), we get τ ∼ 0.01 s. Now owing to the master equation (11), τ is related to k′ through τ−1 = k′M0,61 −3 where M 0 = ∑n ≥ 1 cn(0) ≈ 2π σcn(0) is the cluster * ≃ 10 concentration (estimated by the integration of the Gaussian). We get k′ ∼ 105 s−1, and we shall use this value in the following. Reference 31 proposes the same estimate of k′ by using a related argument. As justified above, we consider in this work the “monomer deposition/raft removal” (MDRR) scheme of ref 31 (Figure 1). Endocytosis takes clusters from the membrane with a rate joff independent of their size,62 and proteins are assumed to be transported to the membrane in a monomeric state with a rate jon. Thus, the recycling term reads31 1(n) = jon δn ,1 − joff cn

(12)

At steady state, the total recycling term is ∞ dϕ/dt = 0 = ∑n = 1 1(n)n and it ensues that 10591


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Figure 3. Results obtained when recycling is switched on by numerically solving the descending recurrence in eq 15. The parameters are the reference ones (α = 2, f 0 = 20, γ = 20, and χ = 0.1), and ϕ = 0.1 is held fixed. Left: area fractions cn of n-clusters, from the thermodynamical equilibrium state to high recycling rates. The recycling rates (in s−1) are given by joff = 0 (equilibrium, in purple, corresponding to G(n) given in Figure 2, also in purple), 10−10 (blue), 10−7 (green), 10−4 (orange), and 10−2 (red). The large-n cutoff was nmax = 150 in these numerical calculations. Middle: Exact values of n* as calculated through the descending recurrence (red dots) as well as the approximate value inferred from the values of c1 and eq 17 (gray dots, superimposed with the red ones for low values of joff). The gray rectangle shows the biologically relevant values of joff. Inset: c1/c(0) 1 as a function of joff (in s−1). Log−linear (main panel) and log−log (inset) coordinates. Right: Comparison between the monomer concentration c1 vs the reduced recycling rate j found numerically (dots; same data as in the inset of the middle figure) and analytically through the small j expansion (continuous line). From the notations defined in the text, we expect c1/c1(0) − 1 = [c1(1)/c1(0)]j + 6(j 2 ). This scaling ceases to be valid above a limiting value j ∼ 10−9. The dashed line is then a power-law fit of the six last dots c1 ∝ jν, with ν ≃ 0.32. Log−log coordinates.

ϕ=

jon joff

membrane endocytosed per hour is equivalent to the total surface of the cell plasma membrane of a fibroblast.47 The turnover time of plasma membranes is thus typically of 1 h. Now 1/joff is precisely the typical time during which all clusters of any size have been recycled. Thus, 1/joff ∼ 1 h and joff ∼ 10−4−10−3 s−1.40 The numerical results then show that the average cluster size n* increases with joff, by a factor ≃2 between equilibrium and these biological recycling rates. Figure 3 (inset of the middle panel and right panel) shows how c1/c(0) 1 also grows with recycling. Qualitatively, when jon (and joff) grows at fixed ϕ, more monomers are added to the system, which tends to increase the value of c1: c1 > c(0) 1 . The reactions (n) + (1) ⇆ (n + 1) are displaced to their right, because the flux from the left to the right is proportional to c1. This tends to increase the relative fraction of larger multimers as compared to their equilibrium counterparts. The Gaussian peak is shifted to the larger values of n. Owing to eqs 7 and 14, approximating g(n) by a constant near n0*, we get cn ≃ Const exp[−(n − n0*)2/2σ02 + n ln(c1/c(0) 1 )], where n* 0 is the equilibrium mean multimer size. By completing the square in the exponential, it follows that the new maximum of cn is at

(13)

The area fraction ϕ is now fixed by this constraint and not by the equilibrium chemical potential μ anymore. At steady state, all concentrations satisfy dcn/dt = 0. The master equation (10) thus provides an order-2 recurrence relation relating cn+1, cn, and cn−1. Solving this recurrence requires the knowledge of c1 and c2, and in addition, the master equation for n = 1 involves all the cn. This is not a practical way to tackle the problem. Taking a maximum aggregation number nmax much larger than the typical cluster size, an order-2 descending recurrence can be built up instead. In the joff → 0 limit, the detailed balance condition is close to be respected. It ensues that kn ≃ k′cnc1/cn+1. Comparing this relation to its equilibrium counterpart, we get cn+1/c(0) n+1 ≃ (c1/ (0) c(0) 1 )(cn/cn ). We are thus led to write ⎡

cn =

⎤n

c1 ⎥ cn(0)⎢ (0) ⎢⎣ c1 ⎥⎦

g (n) (14)

where g(n) is some factor to be determined, satisfying g(1) ≡ 1 by definition and g(n) → 1 when joff → 0 for any n. Injecting this identity in eq 10, the descending recurrence relation g (n − 1) =

n* = n0* + σ0 2 ln(c1/c1(0))

Numerical values are consistent with this calculation, while joff < 10−8 s−1 (see Figure 3, middle). When joff gets larger, this approximation becomes less reliable. One can also explore the consequences of reducing the uptake rate joff at fixed delivery rate jon. Indeed, the previous choice of fixing ϕ, while varying joff and jon concomitantly, dwells on the idea that “killing” in some way the cell would freeze it in a state where ϕ is kept constant because of the rapidity of the process. We observed above that this would result in a typical 2-fold decrease of the cluster size when using realistic parameter values. However, there also exist drugs lowering the endocytosis rate, which amounts to reducing joff without significantly changing jon. What would be the consequences on the typical cluster size? Reducing joff at fixed jon results in an increase of ϕwhich cannot, however, become larger than 1; this limits the range of values that joff can take. We choose a reference state joff = 10−4 s−1, a biologically relevant value jon = 5 × 10−7 s−1, and consequently

[joff̂ + c1 + c1(0)Γ(n)]g (n) − c1g (n + 1) c1(0)Γ(n) (15)

̂ ensues at steady state for n ≥ 2, after introducing the notations joff = joff/k′ and Γ(n) ≡

cn(0) −1 cn(0)

(17)

= eG(n) − G(n − 1) ≃ eG ′ (n) (16)

V. NUMERICAL RESULTS The numerical method used to solve eq 15 is given in Appendix C. The so-obtained distributions (cn) are displayed in Figure 3 (left), and show that the multimer distributions remain Gaussian. The recycling rate in a real cell plasma membrane is estimated as follows. Experiments indicate that the typical amount of plasma 10592


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The Journal of Physical Chemistry B ϕ = jon/joff = 0.005. Then, we reduce joff to 10−5 and 10−6 s−1 so that ϕ respectively grows to 0.05 and 0.5. Surprisingly, we observe by solving numerically the recurrence relation very little changes of the different parameters characterizing the distribution (cn), except the amplitude A of the Gaussian peak. While ϕ undergoes a 100-fold increase, n* shows a modest increase of 0.4‰, c1 grows by 2‰, and the width σ of the Gaussian does not change. As for A, it logically grows proportionally to ϕ. However, one can see in Appendix E that, for a different parameter set, the dependency of n* upon ϕ at fixed jon is weak but actual.

−1/2 ⎡ γ 1 ⎤ ⎥ σ ≃ ⎢2χ − 4 (n*)3/2 ⎦ ⎣

The next step is to find an expression for the amplitude A. We use the condition ϕ = ∑cnn and eq 18: 2 2 ∞ ϕ = c1 + A ∑n = 2 ne−(n − n*) /2σ ≃ c1 + 2π An*σ . It follows that

A=

n − n − 1 ) − f0 + χ (nα − (n − 1)α ) − ((1 + 2(n − n*))/2σ 2)

+ c1e−((1 − 2(n − n*))/2σ − eγ (

2

(23)

(18)

We consider in priority the case where ϕ is held fixed and joff and jon vary before tackling the fixed jon case at the end of the section. As seen in Figure 3 (left), the transition from the equilibrium state to low recycling rates at fixed ϕ is performed by smoothly shifting the Gaussian peak toward higher aggregation numbers while at the same time the number of monomers is increasing because more and more monomers are injected in the system through recycling. Furthermore, we observe that the width σ is very slowly depending on joff. We focus on the concentrations cn at aggregation numbers n close to n*. Inserting eq 18 in the master equation (10) at steady state gives for any n close to n* joff̂ = eγ(

ϕ − c1 2π σn*

At low or moderate recycling, c1 ≪ ϕ can be omitted in the numerator. In Figures 4, 5, and 6, we compare the analytical

VI. ANALYTICAL SOLUTION In the following, we propose analytic identities relating the four parameters that characterize the distributions (cn), namely, the monomer fraction c1, the average multimer size n*, the width σ of the Gaussian peak, and its amplitude cn* ≡ A. We indeed assume that the distribution can be written as a Gaussian distribution plus a delta peak at n = 1 accounting for monomers: ⎡ −(n − n*)2 ⎤ cn ≃ c1δn ,1 + A exp⎢ ⎥ ⎣ ⎦ 2σ 2

(22)

Figure 4. Comparison of the monomer concentration c1 vs the multimer peak position n* found by calculation (eq 21, continuous line) and by the recurrence (dots). The parameters are the reference ones with ϕ = 0.1. Both σ and n* depend on the recycling rate joff, and the red dots from the left to the right correspond to the recycling rates joff = 0, 10−11, 10−10, 10−9, 10−8, 10−7, 10−6, 10−5, 10−4, 10−3, and 10−2 s−1. Linear−log coordinates.

− c1

)

n − 1 − n − 2 ) − f0 + χ ((n − 1)α − (n − 2)α )

(19)

The expressions are linearized in n − n*, assuming that n − n* ≪1 n*

and

n − n* ≪1 σ2

Figure 5. Standard deviation σ of the Gaussian vs the multimer peak position n*, found by analytical calculation (eq 22 of the main text, continuous line) and by fitting the numerical Gaussians (dots). The parameters are the reference ones with ϕ = 0.1. We have chosen the same recycling rates joff as in Figure 4.

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These approximations can be justified close to n* by the observation that n* ≫ 1 in general and especially for high recycling rates and that the standard deviation was found in the range σ ≃ 2.5−6 for the various parameter sets explored in this study. By equating the so-obtained expansion to 0, both order-0 and order-1 coefficients must vanish simultaneously. As detailed in Appendix D (see eq D1 and below), it follows from the order-1 coefficient that c1 ≃ exp[F ′(n*)] = c1(0)Γ(n*)

values of the monomer concentration c1, the standard deviation σ, and the amplitude A with the values found by directly fitting the numerical distributions of Figure 3 (left) with a Gaussian. The very good agreement shows the validity of our approximations. Our next step will consist of deriving an analytical expression for these different quantities as a function of ̂ . We begin with small joff ̂ expansions before dealing with joff ̂ , in particular biologically relevant ones. intermediate values of joff ̂ Limit at Fixed ϕ. In equilibrium (joff = 0), eq VI.A. Small joff 14 implies that geq(n) = 1 for all n. Our small parameter will be j ≡ 2 joff/(k′c(0) 1 ). We naturally write g (n) = 1 − wn j + 6(j ) where the sequence (wn) is to be characterized. Expanding eq 15 at order 1 in j leads to the recurrence relation

(21)

The second equality ensues from F′(n) = G′(n) + μ and exp(μ) = c(0) 1 . Note that it can be checked that eq 17 derives simply from the expansion of this relation at order 1 in the small variable c1/ c(0) 1 − 1, that is to say, close to equilibrium. In the same way, we find in Appendix D an expression for σ by equating the order-0 term to zero: 10593


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The Journal of Physical Chemistry B ν ln j 2χ

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j ν ln off ≃ n0* + 2χ jcross

(30)

n* ≃ Const +

with ⎡ ⎤−1 ⎢ ⎥ 3γ K 1 ln ν = ⎢1 + 3/2 ⎥ * γ χ 32 1/4 3/2 n ⎡χ (n *) − ⎤ ⎥ 0 (n *) ⎢⎣ ⎣ 0 0 ⎦ 8⎦

Figure 6. Comparison of the amplitude A of the Gaussian vs the multimer peak position n* found by analytical calculation (lines) and by the recurrence (dots). The analytical curve comes from eq 23, having omitted c1 ≪ ϕ in the numerator. The parameters are again the reference ones with ϕ = 0.1. We have chosen the same recycling rates joff as in Figure 4.

⎛ wn + 1 1 ⎞ 1 wn − 1 = wn⎜1 + + ⎟− Γ ( n ) Γ ( n ) Γ ( n) ⎝ ⎠

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We have also introduced jcross, the crossover value of joff where n* starts becoming different from n0*. For the reference parameter set and ϕ = 0.1, jcross ≃ 10−11 s−1 (Figure 3, middle), well below the physiological recycling rate. Owing to eq 21, c1 ∝ Γ(n*), with Γ(n*) ≃ e−(f 0+μ)e2χn*. One eventually gets the expected scaling law

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c1 ∝ e 2χn * ∝ j ν

If one also expands c1 = c1(0) + c1(1)j + 6(j 2 ), we show in Appendix F that, if w∞ is the limit of wn, then c1(1) ≃

c1(0) n0*

w∞

The searched exponent ν (see Figure 3, right) is given by eq 31. With the reference parameter set, νanalytic ≃ 0.49, whereas the numerical solution gave νnum ≃ 0.32. This overestimate by 50% is due to the various approximations we used. In the different parameter sets tested in Appendix E, νnum remains close to 0.3. Note that the exponent ν would be difficult to measure experimentally because it would require one to have access to the monomer density in the dilute phase, whereas only clusters are visible with super-resolution microscopy. In this respect, our main finding is that the typical cluster size n* grows logarithmically with j (i.e., with joff) in this intermediate regime as well (eq 29). ̂ The crossover between the small and intermediate values of joff (Figure 3, right) ensues from eq 27, where ñ must be positive. ̂ , both n* and ln j decrease. Thus, When decreasing the value of joff ̂ and one eq 27 ceases to be valid below a limiting value of joff ̂ limit. crosses over the small joff VI.C. Varying ϕ at Fixed jon. We now explore analytically the consequences of varying the concentration ϕ at fixed delivery rate jon. We have observed that in this case the typical cluster size n* depends very weakly on ϕ. −f 0+2χn0* We start from the approximate relation c(0) deriving 1 ≃e itself from eq 21. In equilibrium, this relation leads to

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where the value of w∞ only depends on G(n). With our parameters and ϕ = 0.1, one gets w∞ ≃ 3.09 × 1010, c(0) 1 = 1.72 × 3 10−6, and n0* ≃ 24.2. The obtained value c(1) 1 ≃ 2.19 × 10 coincides within 1% with the value extracted from numerical calculations at very small recycling rates, joff = 10−15−10−12 s−1, as illustrated in Figure 3 (right). This again validates our approximations. Once c1 is known, the other quantities of interest, n*, A, and σ can be expressed as a function of j by the three relations in eqs 21−23, and the problem is solved. This linear approximation remains valid while the correction c(1) 1 j remains small as compared to c(0) 1 , that is to say, while joff ≪ −10 k′c(0) , as observed in Figure 3 (right). 1 n0*/w∞ ∼ 10 ̂ at Fixed ϕ. We start from VI.B. Intermediate Values of joff the following relation, ensuing from eq 14 taken at n = n*: ⎡

A=

⎤n *

c ⎢ 1 ⎥ cn(0) * ⎢⎣ c1(0) ⎥⎦

g (n*) (26)

We make the substitution c1/c(0) = Γ(n*). We get A = 1 n* (0) (0) −G(n*) c(0) . In Appendix n* [Γ(n*)] g(n*). Next, we use cn* = c1 e G, we also study analytically the sequence g(n) and we prove (see eq G23) that g(n*) ≃ g(ñ) ≃ κ[Γ(n*)]−ñeG(ñ), where n ̃ ≃ n1̃ + (1 − χ )n* + (ln j)/2

n0* ≃

1 (f 2χ 0

(

1 f + μ0 2χ 0 ln c(0) 1 = μ and

+ ln c1(0)) ≃

+

1 n0*

)

ln ϕ , t h e l a s t

equality coming from eq 8. At dominant order, we assume n0* on the right-hand side to be constant (it varies very slowly with ϕ) and we note ψ = (n*0 )−1. Injecting this relation in eq 30, we get

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Here κ > 0 and ñ1 < 0 are constants independent of n* and j. Furthermore, eqs 22 and 23 provide the approximation A ≃ (ϕ χ )/( π n*) at large values of n*. Altogether, these relations lead to K ≃ n*[Γ(n*)]n *−ñ eG(n)̃ − G(n*)

(32)

n* ≃

⎤ j 1 1 ⎡⎢ (f0 + μ0 ) + ν ln off + ψ ln ϕ⎥ ⎥⎦ 2χ 2χ ⎢⎣ jcross

(33)

We have observed that ν ≃ 0.3 depends very weakly on the parameters and the concentration (Appendix E). To finish this calculation, we need a relation between the crossover value jcross of the uptake rate joff and ϕ. Numerically, we observe the scaling law jcross(ϕ) ∝ ϕω to be very well verified, with ω ≃ −0.77 (for the reference parameter set). We have not been able to derive this scaling law analytically so far. We finally getA

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where K = (ϕ χ )/( π κc(0) 1 ) is a constant. Its value is K ≃ 5 × 106 with our reference set of parameters and ϕ = 0.1. The independent eqs 27 and 28 relate ñ and n*, which are both unknown. Solving this two-equation system gives both ñ and n*. This calculation is done in Appendix H, and we get 10594


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The Journal of Physical Chemistry B n* ≃ Const +

1 [ν ln joff + (ψ − ων) ln ϕ] 2χ

roughly constant multimer size n* in this case, that is to say, an increase of the number of multimers. This prediction should be testable by super-resolution fluorescence microscopy or AFM as well. Owing to eq 13, fixing the value of jon amounts to imposing joff ∝ 1/ϕ. One could conceivably imagine alternative functional forms joff(ϕ) of the removal rate, reflecting how the cell regulates the membrane protein fraction or how specific drugs perturb the cell metabolism. Another extreme would be joff(ϕ) = Const, thus imposing jon ∝ ϕ. We have not examined this regime because it would deserve a specific treatment, as the two ones examined above (ϕ = Const in sections VI.A and VI.B and jon = Const in section VI.C). However, we anticipate that the typical multimer size n* will significantly increase with ϕ because of the conjugate effects of the increase of n* with ϕ at equilibrium and of the increase of c1 with jon. This case will deserve future investigation to be compared to potential experimental situations of interest. From an experimental perspective, it can be argued that macrophase separation is in fact observed on giant vesicles resulting from blebs extracted from plasma membranes.48 At first sight, this seems to contradict our above statements. However, as already pointed out in ref 49, these macrophase separations are observed only when these membranes are cooled to temperatures significantly below the physiological temperature. This increases the relative strength of the attraction f 0 and of the line tension γ, and thus potentially leads to larger domains,29 in a way that will need to be quantified in the future. At 37 °C, these membranes become “almost exclusively [at >99%] optically homogenous”,48 which shows that, if domains exist, their size is below the diffraction limit (≈300 nm). The low membrane tension σ of these objects as compared to cells could also explain why domains are larger (their diameter is expected to scale as 1/ σ 36). Our study relies on some assumptions, among which the “monomer deposition/raft removal” recycling scheme, as in refs 12, 22, and 40. Alternative recycling schemes have been discussed elsewhere.31,32,37,38 However, we have argued that the chosen recycling scheme is realistic. In this context, we have also assumed the independence of the removal rate joff on the cluster size n, thus assuming that endocytosis vesicles are not smaller than the typical cluster size. Hence, preferentially breaking large clusters into smaller pieces is not the mechanism by which the macroscopic phase separation is prevented in this work. This assumption can be questionable in the case of large clusters with a diameter typically larger than 100 nm. In this case, one would have to appeal to more elaborated models where the off-rate joff depends on the cluster size,12,50 which might modify our conclusions by impacting the area fraction of large clusters. We have also supposed that the cluster fragmentation and fusion events involve at least one monomer (and never two multimers) because: (i) fragmentation of a multimer into two roughly equal parts would require a large barrier related to interfacial energy to be overcome; (ii) fusion of two large multimers would require a large barrier due to the long-range repulsive forces to overcome. We have seen that one of the important consequences of this choice is that the typical cluster size n* grows with recycling because reactions (n) + (1) ⇆ (n + 1) are displaced to their right. This can seem counterintuitive because one would again expect recycling to disfavor large clusters, as in the arguments explaining the finite size of clusters on the sole basis of nonequilibrium arguments: “recycling reduces the lifetime of membrane components and therefore the likelihood that they meet preexisting clusters or nucleate new ones”.50 Here, in contrast, f inite size is already ensured in

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The numerical values are ν ≃ 0.30 and ψ − ων ≃ 0.27. The proximity of these two numerical values renders n* nearly insensitive to jon = joffϕ, as observed. To conclude, this property is the result of a subtle balance between antagonist effects: increasing joff increases n*, whereas concomitantly reducing ϕ weakly reduces n*0 while significantly increasing jcross.

VII. CONCLUSION AND DISCUSSION In the case where the protein concentration ϕ is held fixed, we have characterized several dynamical regimes depending on the strength of the recycling rates joff or jon. (i) In the low recycling regime, where c1 is close to its equilibrium value c(0) 1 , i.e., where (c1 (0) − c(0) )/c ≲ 1, this quantity grows linearly with joff and the 1 1 proportionality coefficient can be calculated perturbatively. (ii) In the experimentally relevant intermediate regime c(0) 1 ≪ c1 ≪ ϕ, we observed a power-law c1 ∝ jνoff where the exponent ν ≈ 0.3 could be correctly estimated by an analytical argument. The biologically relevant range of recycling rates belongs to this regime. In these two regimes, the typical multimer size n* grows logarithmically with the recycling. (iii) When increasing further the recycling rate, c1 saturates to ϕ and proteins are essentially present in their monomeric form, thus destroying the cluster phase. Alternatively, the uptake rate joff can be varied at fixed delivery rate jon. We have observed in this case that the typical multimer size n* is nearly insensitive to joff in the intermediate regime (ii) above. We have given a, so far partial, analytical proof of this observation. The starting point of the present study was the existence of two types of physically relevant arguments accounting for the finite size of clusters in the condensed phase. Some works argue that, even though a membrane is maintained out of equilibrium by active processes, equilibrium statistical mechanics is sufficient to account for the finite size of membrane nanodomains.25,27−29 For example, the observation of nanodomains of rhodopsin (a GPCR) in cadavers’ retina cells,15 very similar to those observed in fixed cells or membrane sheets, suggests that active processes are not essential. Alternatively, some other works propose that nonequilibrium mechanisms are necessarily at play, driven by active membrane recycling.22,31,32,37−40 We explored whether both approaches could be discriminated from an experimental perspective. Our conclusion is that the sole observation of cluster size distributions, as extracted from super-resolution microscopy experiments, cannot bring a definitive answer if one does not compare directly live cells and systems where recycling has been switched off. Within the realistic parameter sets that we have explored in this work, it appears that the typical cluster size n* is about twice smaller when switching recycling off, from a realistic value joff ∼ 10−3 s−1 to 0, assuming the membrane protein concentration ϕ to remain constant. Thus, our prediction is that stopping active recycling processes in a living cell in some way at constant ϕ would result in twice smaller and consequently twice more numerous clusters at fixed protein concentration. As compared to the alternative models cited above where stopping active recycling results in a macrophase separation because they lack long-range repulsion, this prediction should be testable by super-resolution fluorescence microscopy or AFM. Alternatively, one can conceivably slow down endocytosis in some way at constant delivery rate jon. This leads naturally to an increase in the membrane concentration ϕ. Our model predicts a 10595


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using the same notations as in ref 36 with f 0 = −(fatt + f rep) > 0. The value of α has been thoroughly explored in this reference and typically lies between 1.5 and 2 in the case of coupling to the membrane curvature. The exact value depends on the membrane surface tension, going from α = 2 at low tensions to lower values at biological tensions. Note that an exponent α = 2 also corresponds to the case where the repulsion is pairwise additive with a range larger than the typical cluster diameter. Indeed, in this case, Frep(n) ∝ n(n − 1)/2 ∝ n2.

equilibrium by the repulsive interaction and we explore how switching recycling on perturbs cluster size distributions. To finish, note also that the proximity of a phase transition can be relevant in the present context.51 We have chosen here to focus on the simplest case of a system that is far from this transition (and below the transition temperature, Tc, or above the critical concentration, ϕc) in order not to obscure the interplay between equilibrium and nonequilibrium processes of interest. Combining equilibrium and nonequilibrium considerations close to a phase transition potentially leads to an even richer phenomenology.

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APPENDIX B: CLUSTER PHASE THERMODYNAMICS In equilibrium, the cluster phase can alternatively be described as an ideal mixture of an ideal gas of monomers with area fraction c1 and an ideal gas of multimers with area fraction M*. Monomers and multimers are subject to the formal chemical reactions

APPENDIX A: CLUSTER FREE ENERGY

A.1. Classic Liquid Droplet Approximation

Below the critical temperature Tc, the line tension along the contact line between two phases (protein clusters and surrounding lipids) results from the fact that it is energetically unfavorable for the different types of particles to be directly next to each other.44 The line tension γ̂ > 0 acts on the cluster boundaries and results on a free-energetic cost Fline n = γ̂(2πrn) ≃ γ n , with rn the radius of a roundish cluster of aggregation number n. The value of γ is typically in the range of a few tens of kBT. For γ = 20 kBT, this yields γ̂ ≃ 1.3 kBT/nm ≃ 5 pN for a monomer diameter 2b = 5 nm (see refs 31 and 36). Note that this kBT/nm ∼ pN order of magnitude is also comparable to the measured line tensions along the phase boundaries between different lipid phases in lipidic membranes,52,53 in the alternative case where the phases of solute and solvent correspond in fact to two lipid phases. The roundish character of a droplet results from the minimization of its interfacial energy. Its energy is composed of the above boundary term plus a bulk contribution:29 Fatt(n) = fatt (n − 1) + γ(n − 1)1/2

(n) + (1) ⇆ (n + 1)

We can associate chemical potentials to monomers and n-mers, respectively denoted by μ1 and μn. In equilibrium when monomers and multimers coexist, eq B1 implies μn+1 = μn + μ1 from which it follows that μn = nμ1. The average chemical potential of multimers is thus μ* = n*μ1

ϕ = c1 + n*M *

(B3)

Now n*M* ≃ ϕ except very close to the transition. It follows from eq B2 and the expressions of μ* and μ1 that

(A1)

μ = μ0 +

1 ln ϕ n*

(B4)

where μ0 = (μ*,0 − ln n*)/n* − μ01 is a new constant. We have assumed n* to be essentially independent of ϕ, which can be checked consistently afterward as done in section III. This relation can be numerically checked with a very good accuracy.

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APPENDIX C: NUMERICAL CALCULATION OF G(N)RESOLUTION OF EQ 15 We assume that, for n sufficiently large, cn is vanishingly small. We thus set g(nmax) > 0 and g(nmax + 1) = 0. After calculating g(1) through the recurrence relation, which only depends on c1 and g(nmax), we find their respective values by solving numerically, with the help of the MATHEMATICA software, the two equations

A.2. Repulsive Potential

In addition to the energy of a liquid droplet, other energies can be involved. The existence of an effective long-range repulsion between the solute particles has been evoked in the main text. Contrary to the macro-phase separation above, this repulsion results in the existence of a stable phase of intermediate sized clusters in equilibrium. The long-range repulsive energy can be approximated by the combination of a linear term and a power law

1 = g (1) nmax

ϕ=

(C1)

⎡

⎤n

c1 ⎥ ncn(0)⎢ (0) ⎢ ⎥⎦ c ⎣ 1 n=1

∑

g (n) (C2)

with the latter equation deriving from ∑ncn = ϕ through eq 14. Thanks to the linear character of the recurrence, this is equivalent to setting g(nmax) = 1 and solving the unique equation (0) n max ∑nn=1 nc(0) n (c1/c1 ) g(n) = g(1)ϕ, where c1 only is unknown. Once c1 is known, the whole sequence (g(n))n≤nmax can be calculated, from which the cn derive, as desired (see Figure 9). We shall see below that the bimodal character of the cluster-size distribution is preserved out of equilibrium. In practice, g(n) is

(A2)

with an effective exponent α > 1 and where χ gives the strength of the repulsive potential. Combining the attractive and repulsive contributions in F(n) = Fatt(n) + Frep(n) leads to F(n) = −f0 (n − 1) + γ n − 1 + χ (n − 1)α

(B2)

where we recall that n* is the average equilibrium multimer size (excluding monomers). Since monomers and multimers are both ideal mixtures, we can write μ* = ln M* + μ*,0 and μ1 = ln c1 + μ01 = μ + μ01, in units of kBT; μ*,0 and μ01 are constants. Equation 2 reads

The free energy per particle fatt < 0 accounts for the short-range attractive potential between adjacent particles. We have chosen to express Fatt(n) as a function of n − 1 (rather than n) so that Fatt(1) = 0, as there is no binding energy in a monomeric cluster. When starting from a dilute gas phase and increasing the monomer concentration, such a free-energy leads to a macrophase separation above a critical concentration. Then, the essentially monomeric, low-density phase coexists with a large condensed phase, a giant cluster minimizing the surface energy term.

Frep(n) = frep (n − 1) + χ (n − 1)α

(B1)

(A3) 10596


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The Journal of Physical Chemistry B decreasing with n and is essentially constant around the multimer peak and beyond. We denote by g(∞) its limit.

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2

e−(1 ± 2(n − n*))/2σ = 1 −

1 n − n* ∓ + 6[(n − n*)2 ] 2 2σ σ2 (D1)

In addition, at order 1 in (n − n*),

APPENDIX D: TAYLOR EXPANSIONS IN THE GAUSSIAN-PEAK APPROXIMATION

n −

In this appendix, we derive eq 21. The approximations used here are validated by the agreement between numerical and analytical calculations displayed in Figures 4, 5, 6, and 7. We start from eq 19, itself deriving from the master equation (10) for n ≥ 2 and the approximation of the distribution (cn) proposed in eq 18 and justified in the main text. We use several Taylor expansions:

n−1 ≃

≃

n* − n* − 1 ⎛ 1 ⎞ 1 +⎜ − ⎟(n − n*) ⎝ 2 n* 2 n* − 1 ⎠ 3/2 3/2 1 1⎛ 1 ⎞ 1⎛ 1 ⎞ + ⎜ ⎟ − ⎜ ⎟ 8 ⎝ n* ⎠ 4 ⎝ n* ⎠ 2 n* × (n − n*)

(D2)

where the coefficients of the expansion have been themselves expanded up to order 3/2 in powers of 1/n*. Similarly, n−1 − ≃

n−2

3/2 3/2 1 3⎛ 1 ⎞ 1⎛ 1 ⎞ + ⎜ ⎟ − ⎜ ⎟ (n − n*) 8 ⎝ n* ⎠ 4 ⎝ n* ⎠ 2 n*

(D3)

Furthermore, for any 1 < α ≤ 2, α(α − 1) *(α − 2) n 2 + α(α − 1)n*(α − 2)(n − n*)

nα − (n − 1)α ≃ αn*(α − 1) −

(D4)

and (n − 1)α − (n − 2)α 3α(α − 1) *(α − 2) n ≃ αn*(α − 1) − 2 + α(α − 1)n*(α − 2)(n − n*)

(D5)

at order 1 in (n − n*) and order (α − 2) in n*. Plugging these expansions in eq 19, several terms cancel and we get ⎡ 1 α(α − 1) *(α − 2) γ joff̂ ≃ e F ′ (n*)⎢ − 2 − χn + 2 ⎣ 2σ 4(n*)3/2 ⎛ 1 n − n* ⎤ n − n* ⎞ ⎟ ⎥ + c1⎜ − 2 + − 2 ⎝ 2σ σ σ2 ⎠ ⎦ (D6)

where we have used γ /(2 n* ) − f0 + χαn*(α− 1) ≃ F ′(n*) at large n* (see eq 4). We have kept F′(n*) in the exponential because it does not tend to zero at large n* (or large σ), contrary to the other terms, which justifies expanding the exponentials. The relation (D6) holds for all n close to n*. Equating order-1 terms in (n − n*), we get c1 = eF′(n*), i.e., eq 21. Replacing c1 in the above equation and equating order-0 terms, we obtain an expression for 1/σ2, leading in turn to −1/2 ⎡ ĵ ⎤ γ σ ≃ ⎢α(α − 1)χ (n*)α− 2 − (n*)−3/2 − off ⎥ 4 c1(n*) ⎥⎦ ⎢⎣

(D7)

Figure 7. Same as Figures 4, 5, and 6 and Figure 3 of the main text (right) with the parameter set α = 3/2, f 0 = 20, γ = 27, χ = 1, and ϕ = 0.1. In the three top panels, joff takes the values 0 and from 10−10 to 1 s−1. 5 Here w∞ ≃ 1.76 × 1010, c(1) 1 ≃ 2.86 × 10 , and ν ≃ 0.29. In the third panel, we have shown the analytical curves coming from eq 23 without (blue curve) or with (orange curve) the term c1 in the numerator.

In the special case where α = 2 and the recycling joff is low, this relation becomes eq 22 in the main text. Strictly speaking, to infer two equations from the relation in eq D6 valid for any n close to n*, we only need two different values of n, e.g., n* and n* + 1. This justifies the validity of the 10597


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To finish, we have explored two different area fractions for the reference parameter set (α = 2, f 0 = γ = 20, and χ = 0.1), namely, ϕ = 0.01 and ϕ = 0.3. We recall that biological protein area fractions are probably between 0.1 and 0.3. Our conclusions also remain identical on a qualitative level. In particular, the measured exponent ν at intermediate recycling remains equal to 0.32 for ϕ = 0.3, whereas it is slightly lower (ν ≃ 0.31) for ϕ = 0.01 (and ν ≃ 0.30 for ϕ = 0.001).

conditions in eq 20 even though σ is not very large. To finish, note that no more than two relations can be found using the above expansion in powers of (n − n*), even if going to higher orders, because a Gaussian probability distribution and its moments are fully characterized by only two quantities, its expectation value n* and its standard deviation σ. For example, we have checked that going to the order 2 leads to the same relation as compared to the order 0.

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APPENDIX F: CALCULATION OF C(1) 1 The recurrence (24) cannot be solved exactly because it depends on the choice of the potential G(n) through Γ(n). Like g(n), this recurrence relation appears to have a stable solution when solved in a descending way with the condition that w1 = 0. Equation 14 then yields

APPENDIX E: ADDITIONAL DATA AND PARAMETER SETS

E.1. Graphs for the Reference Parameter Set

We provide some additional graphs showing the very good agreement between our analytical approach and numerical results, displayed in Figures 4, 5, and 6.

⎡

E.2. Additional Parameter Sets

A=

In order to check the robustness of our results with respect to the values of the different parameters and the concentration ϕ, we have tested different parameter sets compatible with experimental facts. The problem is analytically tractable for α = 2 only, because one ends with an hypergeometric ODE, an analytical solution of which is known. However, α probably lies between 1.5 and 2,36 and we have therefore tackled numerically two parameter sets with α = 3/2. In Figure 7, we have plotted the soobtained results for one such parameter set: f 0 = 20, γ = 27, χ = 1, and ϕ = 0.1. Note that the choice χ = 1 is consistent with the value given in ref 36 on the basis of an elastic model of the biomembrane: χ = 0.75 for α = 1.48. Our qualitative conclusions remain unchanged, even though some details can be different, such as the monotonicity of σ(n*), owing to eq 22. The exponent ν at intermediate recycling is measured to be ν ≃ 0.29 in this case. We have also tested a parameter set initially designed to fit the cluster sizes of ref 27 resulting from Monte Carlo simulations: α = 3/2, f 0 = 19, γ = 27.295, and χ = 0.6073 (again in units of kBT). We have also focused on ϕ = 0.1. Again, our qualitative conclusions remain identical (data not shown) even though the clusters are typically bigger (n0* ≃ 47 in equilibrium and n* ≃ 187 if joff = 10−3; nmax = 300 is required in this case). The agreement between numerical and analytical calculations remains excellent. The measured exponent ν at intermediate recycling is also ν ≃ 0.29. We have also examined with this parameter set the dependency of the equilibrium cluster size n0* upon joff at fixed delivery rate jon and variable ϕ. As displayed in Figure 8, there is a slight discrepancy between plots, especially at high recycling.

⎤n *

c ⎢ 1 ⎥ cn(0) * ⎢⎣ c1(0) ⎥⎦

g (n*) (F1)

when taken at n = n*. We set c1 = c1(0) + c1(1)j + 6(j 2 ). Since c(0) 1 is known from equilibrium, it is the coefficient c(1) 1 that we are looking for. We also set A = A 0 − A1 j + 6(j 2 ). Close to equilibrium, n* is given by eq 17. After taking its logarithm and keeping the 6(j) terms only, eq F1 leads to c1(1) =

c1(0) ⎛ A ⎞ ⎜wn * − 1 ⎟ A0 ⎠ n0* ⎝

(F2)

2 *2 Notably, we have used ln cn(0) * ≃ ln A 0 − (n* − n0 ) /(2σ0 ) ≃

ln A 0 − σ0 2 ln 2(c1/c1(0))/2 = ln A 0 + 6(j 2 ) owing to the Gaussian character of cn close to n0*. In practice, wn* appears to be very close to the limit w∞ of wn and this quantity is much larger than A1/A0, which leads to the simpler approximate relation c(1) 1 ≃ c(0) 1 /n* 0 w∞ used in the main text.

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APPENDIX G: STUDY OF THE SEQUENCE G(N)

G.1. Numerical Example

An example of numerical solution of the descending recurrence relation (15) is given in Figure 9 for the reference parameter set. An exponential decay follows a short plateau near n = 1, before a sudden saturation to g(∞). This typical behavior, observed for all the parameter sets explored in this work, is fully exploited in the analytical approach, and it is also given an analytical explanation below in terms of hypergeometric functions.

Figure 8. Average multimer size n* as a function of the delivery rate jon for the parameter set α = 3/2, f 0 = 19, γ = 27.295, and χ = 0.6073; ϕ = 0.1 (red dots) and ϕ = 0.01 (gray dots). Log−linear coordinates.

Figure 9. Numerical solution of the recurrence relation (15) for the following parameter set: α = 2, f 0 = 20, γ = 20, χ = 0.1, ϕ = 0.1, and joff = 10−5 s−1. Linear−log coordinates. 10598


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The Journal of Physical Chemistry B G.2. Analytical Study in the Continuous Limit

The two independent solutions of ODE are then given by hypergeometric functions54 (expressed in the variable u):

Our goal here is to give an analytical solution to the recurrence relation (15) on g(n). We again consider it in the ascending way. For any initial conditions g(1) and g(2), the space of solutions is a two-dimensional vector space : . However, one of the eigenvalues associated with the recurrence relation is always smaller than 1 and the second one is close to 2. Therefore, we expect that only a 1D subspace :1 of the solutions of : does not diverge at large n, as expected from physical solutions of the recurrence relation. In order to characterize :1, we consider the continuous limit of the recurrence relation, obtained by developing g(n ± 1) up to order 2: 1 g (n ± 1) ≃ g (n) ± g ′(n) + 2 g ″(n). We thus get an order 2 ordinary differential equation (ODE) on g, and we shall select the solutions having a finite limit when n (0) ̂ → ∞. Using c1/c(0) 1 = Γ(n*) and j = joff/k′/c1 , the ODE more precisely reads [Γ(n*) − Γ(n)]g ′(n) +

g±(u) = u λ±2F1(λ± , λ± + a − 1; 2λ± + au1; −u)

In the present context, au1 − 1 = 1/χ = 1 − a and b ≪ 1/χ so that λ− ≃ −1/χ and λ+ ≃ j/(2χΓ0t*) ≪ λ−. Hence, g−(u) ≃ u 2−1/ χ2F1( −1/χ , −2/χ ; 1 − 1/χ ; −u)

g+(u) ≃ u j /(2χ Γ0t *)2F1(j /(2χ Γ0t *), −1/χ ; 1 + 1/χ ; −u) (G10)

Any solution g can be written as a linear combination of g− and g+. We shall now see that the requirement that g(u) does not diverge when u → ∞ restricts the accessible solutions. The additional condition g(1) = 1 will set the unique solution to our initial problem. G.4. Asymptotic Behaviors

Given that z ∈ − *, we can use the relation (ref 54, eq 15.3.7)

(G1)

z A2F1(A , B ; C ; z) Γ̂(C)Γ̂(B − A) ⎛ 1 ⎟⎞ ⎜ = 2F1 A , 1 − C + A ; 1 − B + A ; ̂ ̂ ⎝ z⎠ Γ(B)Γ(C − A) Γ̂(C)Γ̂(A − B) ⎛ 1 ⎞⎟ ⎜ + z A−B 2F1 B , 1 − C + B ; 1 − A + B ; z⎠ Γ̂(A)Γ̂(C − B) ⎝

The second approximation will consist of keeping only the two leading terms in the expression of G′(n) in eq 16: G′(n) ≃ −(f 0 + μ) + αχnα−1. This is valid in the large n limit where 1/ n goes to 0. In order to go further into the analytical investigation, we now set α = 2 because an analytical solution of the ODE exists in this case. We recall that α = 2 is of physical relevance for a biomembrane in the low-tension limit.36 We have previously explored numerically the value α = 3/2 in Appendix E. Defining Γ0 = e−(f 0+μ) and the new variable t = e 2χn ∈ + *, we obtain the new ODE

(G11)

Here Γ̂(z) denotes Euler’s gamma function (not to be confused with Γ(n) as defined in eq 16). The divergence at −∞ comes from the second term of the r.h.s. because in our case A − B = λ± − (λ± + a − 1) = 1 − a = 1/χ > 0. The hypergeometric function appearing in this second term appears to be identical for both functions g+ and g− [and equal to 2F1(λ+ − 1/χ, λ− − 1/χ; a; 1/ z)]. Defining 54

2χ Γ0[t *(χ + 1) + t(χ − 1)]tg ′(t ) + 2χ 2 Γ0(t * + t )t 2g ″(t )

with t* = e

(G2)

2χn*

, which can in turn be written as

t 2(t + t *)g ″(t ) + at(t + t *u1)g ′(t ) = bt *g (t )

g (u) = p− g−(u) − p+ g+(u)

(G3)

j 2t *χ 2 Γ0

=

(G12)

with

by setting a = 1 − 1/χ, b=

(G9)

at the lowest order in j and

1 [Γ(n*) + Γ(n)]g ″(n) = jg (n) 2

= jg (t )

(G8) 2

p− = Γ̂(λ−)Γ̂(2λ+ + 1 + 1/χ )Γ̂(λ− + 1 + 2/χ )

joff 2c1k′χ 2

p+ = Γ̂(λ+)Γ̂(2λ− + 1 + 1/χ )Γ̂(λ+ + 1 + 2/χ )

(G4)

and u1 = (χ + 1)/(χ − 1). If we define the new variable u = t /t * = e 2χ(n − n*) ∈ + *, this ODE simplifies to u 2(u + 1)g ″(u) + au(u + u1)g ′(u) = bg (u)

divergences cancel and we obtain a solution of our original ODE that does not diverge at infinity, as requested. This solution is represented (after normalization to fulfill g(1) = 1) as a function of the original variable n in Figure 10, with the same parameters as in Figure 9. We observe an exponential decay at small n and a saturation at large n, as in the numerical solution of Figure 9. The crossover between both regimes appears at a value ñ that agrees remarkably well between the numerical and analytical solutions, as expected from the fact that the analytical approximation becomes exact in the large n limit. By contrast, the fall between n = 1 and ñ is clearly underestimated by the analytical approach because we have neglected the n -term in G(n) when approximating G′(n) above. We shall return to this issue below. From eq G12 and the behavior of hypergeometric functions close to 0 and −∞,54 we obtain

(G5)

It can be shown to be equivalent to a hypergeometric equation54 by setting g(u) = uλh(u) and suitably choosing λ, as discussed now. G.3. Solution of the Hypergeometric Equation

Setting z = −u ∈ − *, h(u) = y(z), and choosing λ = λ± = (1 − au1 ±

(au1 − 1)2 + 4b )/2

(G6)

the previous ODE becomes the hypergeometric equation in its usual form: z(1 − z)y″(z) + [2(1 − z)λ± + a(u1 − z)]y′(z) − [b + a(1 − u1)λ±]y(z) = 0

(G13)

g (n) ≃ p− e 2χλ−(n − n*) ≃ p− e−2(n − n*) (G7)

(G14)

for small values of n and 10599


Article


Figure 11. Comparison of the numerical solution of the recurrence relation (15) as in Figure 9 (red dots) and of the approximate solution n G(n) , with κ ≃ 2 × 10−3 being a suitably chosen g(n) = κ[c(0) 1 /c1] e constant (gray dots). The dots are almost superimposed for 2 ≤ n ≤ ñ. Linear−log coordinates.

Figure 10. Analytic solution of the hypergeometric ODE (G2) for the same set of parameters as that in Figure 9. The full red line is the solution given by eq G12, whereas the orange and gray dashed lines are the small and large n approximations given by eqs G14 and G15, respectively. The crossover between both regimes, defined as the intersection between both approximations, occurs at n = ñ < n*. Linear−log coordinates.

This equality is verified because Γ(n)eG(n−1) ≡ eG(n) and Γ(n)eG(n) ≃ eG(n+1)/eG″(n) with eG″(n) ≃ e2χ ≃ 1 if χ ≪ 1. In conclusion, we have argued so far that, when n < ñ, g(n) = n G(n) κ[c(0) = κeG(n)Γ(n*)−n and, when n > ñ, g(n) ≃ g(ñ) = 1 /c1] e G(ñ) −ñ κe Γ(n*) = g(∞) is almost constant. Note that the recycling rate joff does not play significantly on the value of g(n) for n < ñ, but it plays on the value of ñ and consequently on the value of g(∞).

g (n) → S(λ− , λ+ , χ ) ⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞ ≡ Γ̂⎜ − ⎟Γ̂⎜2λ− + 1 + ⎟Γ̂⎜2λ+ + 1 + ⎟ χ⎠ ⎝ χ⎠ ⎝ χ⎠ ⎝ ⎡ 2 ̂ ̂ ⎢ Γ(λ−)Γ λ− + 1 + χ ×⎢ ⎢ Γ̂ λ− − 1χ Γ̂ λ− + 1 + 1χ ⎣ ⎤ 2 Γ̂(λ+)Γ̂ λ+ + 1 + χ ⎥ − ⎥ 1 1 Γ̂ λ+ − χ Γ̂ λ+ + 1 + χ ⎥⎦

(

(

( )( ( )(

)

)

G.6. Study of ñ(b)

)

The crossover occurs at n = ñ, a function of the different parameters through eq G16. In particular, it is a function of j and n* through the ratio b = j/(2t*χ2Γ0), i.e., through the ratio j/ e2χn*. We now study the function ñ(b) (all other parameters being fixed) in order to understand the observed scalings of c1(j), as displayed in Figure 3 (right) for example. More precisely, ñ depends on b via λ− and λ+ (see eq G6): 1 λ± = ( −1 ± 1 + 4χ 2 b ) 2χ (G19)

)

(G15)

at large n. Finally, this function has to be divided by g(1) to fulfill the condition g(1) = 1, as displayed in Figure 10. Note that the previous expression is not defined when 1/χ is an integer because Γ̂(−1/χ) diverges. However, g(n) has a finite limit in this case owing to the properties of the function Γ̂, so this is not a true singularity. From these asymptotic behaviors, we obtain the value ñ of n at the crossover between both regimes n ̃ ≃ n* −

1 S ln 2 p−

We first study the function f(b) = b|S (b)/p−(b)|. After a tedious but straightforward calculation, one shows that f(b) has a finite limit (a function of χ only) when b → 0. It ensues that ln|S /p−| = C(χ) − ln b at small b, with ⎡ π sin χ ⎢ C(χ ) = ln⎢ ⎢ χ sin 2χπ B πχ , − 2χπ ⎣

() ( )(

(G16)

by equating eqs G14 and G15. G.5. Small Values of n (n < ñ)

(G20)

where B is Euler’s beta function. C(χ) has a finite value even if 1/χ is an integer. The above behavior of ln|S /p−| has been established in the b → 0 limit. In practice, it appears to remain valid up to b ∼ 1 for the reference parameter set used so far, that is to say, for all the values of n* and j of interest in this work. Using eqs G4 and G16, one gets 1 n ̃ = n1̃ + (1 − χ )n* + ln j (G21) 2 where 54

We have seen that the ODE fails to characterize precisely the function g(n) on the interval [1, ñ] because we have neglected the n -term in G(n) when approximating G′(n). We now propose a different approach on this interval. We start from eq 15, rewritten as Γ(n)[g (n − 1) − g (n)] = Γ(n*)[g (n) − g (n + 1)] (G17)

̂ because it is always much smaller where we have neglected joff than c1. We now show that g(n) = κeG(n)Γ(n*)−n, with κ being a constant, is an approximate solution of this recurrence relation, as illustrated in Figure 11. Indeed, replacing g(n) by this expression in the equation above, we get ⎡ eG(n) ⎤ ⎡ G(n) eG(n + 1) ⎤ ⎥ = ⎢e ⎥ Γ(n)⎢eG(n − 1) − − Γ(n*) ⎦ ⎣ Γ(n*) ⎦ ⎣

)

⎤ ⎥ ⎥ ⎥ ⎦

f0 + μ

1 − [C(χ ) + ln 2] − ln χ (G22) 2 2 is independent of n* and j. Equation G21 is used in the main text. Its validity can also be tested with the help of our numerical solution for g(n). If ñ is measured as the position of the maximum n1̃ =

(G18) 10600



■

of g″(n), then fitting the numerical values gives 0.88 and 0.59 instead of 1 − χ = 0.9 and 1/2 for the reference parameter set. This is quite satisfactory and again validates our approximations. In addition, for our reference parameter set and ϕ = 0.1, one gets ñ1 ≃ −3.0. Whenever ñ1 < 0 and ln j < 0, eq G21 implies that ñ < n*. It follows that g (n*) ≃ g (n)̃ = κ eG(n)̃ Γ(n*)−ñ

■

APPENDIX H: CALCULATION OF THE EXPONENT ν We start from eq 28 (H1)

with K = (ϕ χ )/( π κc(0) 1 ). Taking the logarithm of this relation, we obtain ln

K = (n* − n)̃ ln Γ(n*) + G(n)̃ − G(n*) n*

(H2)

Owing to eq 16 and to G(ñ) − G(n*) ≃ G′(n*)(ñ − n*) + 1/ 2G″(n*)(ñ − n*)2, we obtain the simple relation ln

K 1 ≃ G″(n*)(n ̃ − n*)2 n* 2

Using n ̃ ≃ n1̃ + (1 − χ )n* + ln

1 2

(H3)

ln j and the definition of G,

⎛ ⎞2 γ K 1 ⎞⎜⎛ 1 ⎟ ⎟ n n j ≃ ⎜χ − χ ln − * + ̃ 1 ⎠ n* ⎝ 8 (n*)3/2 ⎠⎝ 2

(H4)

Since j < 1 in the cases of interest, ln j < 0 and the expression in the square is negative. Thus, 1 ln j ≃ −n1̃ + χn* − 2

ln χ−

K n*

γ 1 8 (n*)3/2

(H5)

Approximating n* by n*0 in the slowly varying logarithm and expanding the square-root around n0*, we get 3γ 1 K 1 ln j ≃ Const + χn* + ln 2 32 n0* (n0*)1/4 (n* − n0*) ⎡χ (n *)3/2 − γ ⎤3/2 ⎣ 0 8⎦

(H6)

This eventually yields n* ≃ Const′ +

ν ln j 2χ

(H7)

with ⎡ ⎤−1 ⎢ ⎥ 3γ K 1 ln ν = ⎢1 + ⎥ 3/2 32χ n0* (n *)1/4 ⎡χ (n *)3/2 − γ ⎤ ⎥ ⎢⎣ ⎣ 0 0 ⎦ 8⎦

■

REFERENCES

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(G23)

K ≃ n*[Γ(n*)]n *−ñ eG(n)̃ − G(n*)

Article

(H8)

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest. 10601


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