arXiv:0807.4511v1 [q-bio.SC] 28 Jul 2008 w.nsog—— — www.pnas.org inlForet Lionel secretion vesicle coated of regulation Kinetic ∗ ainpoes hyas ofimta otplmrzto i polymerization coat that confirm the also of robustness They the process. compo point-out recons mation experiments of be Those number now restricted 9]. can 8, a [7, vesicles with C liposomes COP of purified of assembly on fission The tuted the and well. , as relevance and coats suggest Clathrin outcome sophisticated its more of Howeve robustness vesicles. 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Fig. otpplto n h lblflxsi n u ftemembran the of out and in fluxes global the and population Coat ( otdssebeadaeeple otectsl Paradoxic cytosol. energy the of to membran consumption expelled new are wit while and others periphery, disassemble So, coat coat the coat. at polymerize a released monomers of are bound expansion monomers vesicle the futile of many during rate words, cytosol the other than In faster [10]. much tion is components coat of ics monomer. monom a the to to belongs and GTPase membrane the the if i disassembly from of unbinding hydrolysis its the to by leads GTP, triggered GTPase, the coat of coats individual into inactivation polymerize recruit later and that monomers), membrane activ (the Once the complexes [4]. to COPI bind for GTPases Arf1 and the COPII for Sar1 inactivati - protein, activation o Pase of disassembly cycle the and follow a assembly components of the coat precisely, presence betwee More the attractions source. requires range ergy disassembly short coat weak while by monomers, driven only taneous, noammrn n rce oageaeit ot fgrowin of coats into aggregate "drop to continuously proceed are and monomers membrane (Fig.1), a model onto secre our of In amount a the on size cles. practically of more distribution and the coats, protein on of components coat of 13] release 12, tile 11, [10, an area fixed growth of between coats competition stable produce the may that binding suggested micro been of has treadmilling the it to analogy by resembl and This dynamics tubules formation. vesicles prevent to and growth coat office. PNAS the to directly submitted was paper This interest of conflict no declare authors The j c p 06b h ainlAaeyo cecso h USA the of Sciences of Academy National The by 2006 ( ℓ tiigy RPeprmnssgetta h xhnekin exchange the that suggest experiments FRAP Strikingly, nti ae,w netgt hoeial h consequenc the theoretically investigate we paper, this In ) ,mmrn unbinding membrane ), )a )b a) ooe yl:atvto/ebaebnig( binding activation/membrane cycle: monomer A J j no p ) ( l PNAS J j no ffo via ) ( † l l SC,reVuqei,705Paris 75005 Vaucquelin, rue ESPCI, v su Date Issue aiatvto ( inactivation ia T yrlsssest okagainst work to seems hydrolysis GTP J ffo J Volume l j v off v ( ℓ ) rvsclto ( vesiculation or ) j su Number Issue v J on e. ,aggregation ), no GT- a of on 1] The [10]. smicro- es dshape nd sbound ts e vesi- ted ly the ally, tubules, ffu- of e i the hin j COP f . secre- othe to v un- d the n en- n omer ated, ped" ). 1 et- – b) e- er g 7 size that curve the membrane. Monomers leave the membrane ei- stant protein density in the coat, the dissociation of monomer from ther individually after GTP hydrolysis or collectively as part of a the coat is followed by a rapid rearrangement of the coat structure completed vesicle. Intuitively, one expects GTPase inactivation to and by a slight shrinkage of the coat; monomer inactivation thus op- decrease the rate of vesicle formation by reducing the lifetime of poses coat expansion. 2 membrane-bound monomers. However, our generic approach reveals Vesicle release: A mature coat containing a number ℓv (ℓvc . that a deeper understanding of vesicle secretion requires a quantita- 1) of bound monomers forms a nearly closed sphere connected to the tive statistical model. Indeed, we report the existence of a discon- rest of the membrane by a thin neck. At this stage, the coat cannot tinuous dynamical transition from a quiescent to a vesicle producing grow further, and is eventually released into the cytosol as a coated membrane, upon variation of the rate of GTP hydrolysis. In other vesicle. The details of the scission mechanism vary between classes words, the apparently counter-productive energy consumption that of coat, and may involve additional proteins ([4]). Here, we merely favors the unbinding of coat components provides secretory mem- assume that once a domain reachs the critical size ℓ = ℓv, it leaves branes with a highly sensitive switch to regulate vesicle release, trig- the membrane as vesicles at a constant rate kv. gered for instance by a variation of the cargo concentration or the Membrane properties: The curvature of the coat imposes a de- mechanical tension of the membrane. formation to the membrane which is opposed by membrane tension [16]. The membrane tension σ thus favors coat depolymerization, Description of the model and can have a sizable effect on coat growth if it is larger than Our goal is to describe the collective behaviour of a population of −5 2 γ/(s0√ℓv) 10 J/m (A detailed model for the elastic proper- evolving membrane domains (coats) formed by the aggregation of ties of the protein-covered≃ membrane is discussed in SI, section II.B). identical units (monomers), which are themselves continuously recy- Tensions of the Golgi, the ER, and the plasma membranes are typical cled between the membrane and a reservoir (the cytosol). Our starting in the range σ 10−6 10−4J/m2 [17], and may thus play a role point is the course of events depicted in Fig.1. We consider a patch in vesicle secretion.∼ Hereafter,− membrane tension will be expressed of membrane much larger than the size of individual coats, which is −2 in natural units: σ¯ = σs0 10 1k T . ∼ − B subjected to a constant and homogeneous in-flux of monomers Jon. The population of membrane coats is characterized by its size The monomers have a finite lifetime on the membrane before being distribution n(ℓ). The mean concentration of coats of size ℓ (between recycled to the cytosol, at a rate koff . While on the membrane, they ℓ and ℓ + dℓ) at a time t is n(ℓ,t)dℓ, and the mean concentration of diffuse and eventually aggregate into curved protein coats. Coats that isolated monomers is n1(t). Our purpose is thus to compute, n(ℓ) manage to reach a critical size leave the membrane as coated vesicles. and n1 at steady state, for given values of the parameters γ, σ¯, koff , Monomers: The formation of a new monomer on the membrane Jon, kv and ℓv. More practically, we will compare the fluxes of coat involves a succession of steps (GTPase binding on membrane and ac- elements leaving the membrane as inactive monomers Joff and as tivation, and the recruitment of coat proteins) which are not individ- part of a vesicle Jv (Fig.1). ually described in the present model. The rates associated with these processes enter a unique parameter, the mean number of monomers Theoretical framework J formed on the membrane per units of time and area. on Monomer fluxes and conservation relations. Coat growth: Coat expansion proceeds by polymerization of In this section, we derive the kinetic equations for the evolution of the coat size dis- monomers at the coat edge. Monomer-monomer binding is spon- taneous and results from weak short range interactions. The binding tribution n(ℓ). The monomer cycle can be divided into four steps, to which correspond four different fluxes, as shown Fig.1. energy γ should be in the range of a few k T (k T is the energy B B - is the in-flux of single monomers binding to the membrane, available from thermal fluctuations, with k the Boltzmann constant Jon(t) B taken as an input in our model. and T the temperature in Kelvin), since the 10kB T provided by GTP hydrolysis is sufficient to break the bonds. The polymerization is thus - jp(ℓ,t) is the flux of monomer joining domains of size ℓ, a bal- thermally reversible and solely driven by the minimization of the free ance between polymerization and depolymerization for this domain energy of the coat. size. Integrated over the entire population, it gives the total flux of monomers incorporated into domains J = ℓv dℓ j (ℓ,t). Coat structure: Electron microscopy [1, 2] supports the assump- p 1 p 2 - joff (ℓ,t) is the flux of monomers expelledR from domains of size tion that the optimal area per monomer s0 ( 100 nm ) and the ∼ into the cytosol after GTP hydrolysis. Under the assumption of optimal radius of curvature of the coat R ( 50 nm) are homo- ℓ 0 uniform release, it is given by . Integrated over geneous within the coat and remain constant during∼ coat growth. We joff (ℓ) = koff n(ℓ)ℓ thus adopt a model in which the state of a coat is fully characterized the entire population, it gives the total flux of individual inactive monomers leaving the membrane J = ℓv dℓ j (ℓ,t). by a single, slowly varying parameter: the number of polymerized off 1 off - is the flux of mature coats released as vesicles. Introducing the monomers ℓ it contains, taken as a continuous variable for commod- jv(t) R ity. All other internal degrees of freedom in the coat (protein density rate of vesicle formation kv, we have jv = kvn(ℓv). The total flux of and coat shape) are considered to adjust to their optimal configura- monomers leaving the membrane as part of a vesicle is Jv(t)= jvℓv. tion faster than the typical rates of coat growth and GTP-hydrolysis. The evolution of the coat size distribution satisfies (see SI): Under these assumptions, a given coat of size ℓ can be described as ∂tn(ℓ)= ∂ℓ (jp(ℓ) joff (ℓ)) , [1] − − a spherical cap of constant curvature (defined as the dimensionless and that the polymerization current jp reads: 2 2 quantity c = s0/(4πR0) 1/20). A full spherical coat (ℓc = 1) ∼ jp(ℓ)= kpn1 ∂ℓn(ℓ)+ n(ℓ)∂ℓ∆E(ℓ) . [2] with these propertiesp contains several hundreds monomers. − ` ´ Monomer release: Contrary to the reversible monomer polymer- jp is proportional to the density of available monomers n1, and to ization, monomer desorption is an energy consuming process driven the rate of monomer binding onto coats kp (see below). It contains by GTP hydrolysis. It occurs at a rate koff , assumed constant here a diffusive term accounting for random polymerizations and depoly- for simplicity (see Supporting Information (SI) for a discussion of merization induced by thermal noise, and a convective term describ- this assumption). This rate can be estimated from FRAP experiments ing the drift of monomers toward domains of lower energy, driven by −1 [10, 12, 14, 15], koff 0.1 10 s . Under the assumption of con- the “force” ∂ℓ∆E. The free energy difference ∆E (in k T units) ∼ − − B

2 www.pnas.org — — Footline Author between a coat of size ℓ and ℓ isolated monomers diffusing on the size distribution is given in the SI. Several different regimes can be membrane is obtained treating the coat as a rigid spherical cap [18] distinguished under increasing monomer concentration (Fig.2). (see also SI): The energy landscape illustrates the interplay between antago- nistic effects; short-range attractions between monomers promotes 2 2 2 ∆E(ℓ)= γ ℓ(1 c ℓ) +σc ¯ ℓ µ(n1) ℓ [3] − − polymerization, while the entropy of the free monomer favors their 2 2 where pµ = ln n1 + γ√1 c +σc ¯ [4] dispersion. The γ and µ terms in Eq.(6) reflect this competition. − Furthermore, monomer inactivation and unbinding, and membrane is a chemical potential including the entropy of the freely diffusing mechanical tension hinder coat growth (koff and σ respectively, com- monomers, see SI. bined in the effective tension Σ, Eq.(7)). Depending on the monomer The net current j(ℓ)= jp(ℓ) joff (ℓ) accounts for polymeriza- concentration, the landscape may show a local maximum at a small − tion and desorption. It can be written in terms of an effective energy size ℓn (Fig.2b-d), which indicates a nucleation process. Coats must k 2 E˜(ℓ) ∆E(ℓ)+ off (ℓ 1) : reach the critical size ℓ (through fluctuations in the pool of free ≡ 2kpn1 − n monomers) in order to consistently grow further, and the rate of nu- j = kpn1(∂ℓn + n∂ℓE˜), [5] cleation is controlled by the height of the energy barrier. The effec- − tive energy may also show an barrier to vesiculation for large coat with size ℓ ℓv (Fig.2b-c), indicating that large coats are suppressed by ∼ 2 2 the effective membrane tension Σ. A local minimum then exist for E˜(ℓ)= γ ℓ(1 c ℓ)+Σ(n1)ℓ µ(n1) ℓ + const., [6] − − an intermediate size ℓ∗, corresponding to kinetically stable coats. p 2 koff 1 Σ(n1) σc¯ + . [7] Low monomer concentration. At low monomer concentration, the ≡ 2kp n1 large energy barrier to vesiculation at ℓ = ℓv prevents coats to mature into vesicles (Jv 0) (Fig.2, states a and b). The coat size distribu- This equation introduces an effective tension Σ that illustrates the ≃ ˜ tion resembles a distribution at thermal equilibrium: n(ℓ) e−E(ℓ), fact that desorption of inactive monomers and membrane tension for- ≃ mally play the same role in hindering coat maturation and vesicle secretion. Note that more generally, the binding or inac- ~ tivation rates may depend on the coat size, in which case monomer E(l) n(l) desorption enters the effective energy as dℓℓ(koff (ℓ)/kp(ℓ)). 40 0.001 Finally, the monomer influx Jon andR the flux of secreted vesicle a - b. jv are accounted for via the boundary conditions (see SI) Low concentration

jp(1) = Jon Jp, [8] − 20 jp(ℓv) joff (ℓv)= jv, [9] −

Steady state. At steady state, all fluxes are balanced and ∂tn = 0. Eqs.(1-9) reduce to two conditions to be satisfied by n1 and n(ℓ): l l* l l* 0 n v 0

j(ℓ) = j = constant. [10] 20 0.001 v c. Intermediate concentration Jon = Joff + Jv, [11]

The former equation enforces that the size distribution is constant, and the latter that the flux of monomer binding to the membrane bal- 10 ances the flux of monomer leaving the membrane, either after inacti- vation or by vesiculation.

0 0 Results 0.001 In this section, we focus on the steady state of a membrane receiv- d. High concentration ing a constant in-flux of monomer, each having a finite lifetime at 0 the membrane. The full characterization of the coat population and of vesicle secretion follows two steps. First, the stationary distribu- tion of coat size n(ℓ) is computed for a given concentration of free monomers n1 with Eq.(10). Second, n1 is self-consistently derived -40 for a given monomer in-flux Jon by imposing that the in-flux matches the total monomer out-flux (Eq.(11)). While the first step relies en- 0 tirely on the properties of the free energy landscape E˜(ℓ), the second 0 500 0 500 introduces collective effects emerging from the competitive growth of l l many domains, which ultimately give rise to the “secretory switch”. ˜ Fig. 2. Effective energy landscape E(ℓ) (left column, in kB T units, from cmc Eq.(6)) and the corresponding coat size distribution n(ℓ) (right column, in n1 Effective energy landscape and steady-state distribution. units, from Eqs.(5-10)), for different values of the free monomer density n1. cmc (a - first row green curve), (b - first row red), Apart from thermal fluctuations, a coat is driven toward growth or n1/n1 = 0.986 1.039 1.054 (c - second row) and 1.160 ( d - third row). The variation of E˜(ℓ) and n(ℓ) upon shrinkage by the effective “force” ∂ℓE˜(ℓ) (Eq.(5)). Coat growth − a slight increase of n1 above the given value are shown in blue dashed lines. is thus formally analogue to thermal diffusion along the effective en- −1 −1 Other parameters are γ = 5 kB T , σ = 0, kv = 0.13s , koff = 0.12s ergy landscape ˜ (Eq.(6)). The analytical expression of the coat cmc E(ℓ) and, ℓv = 500. With those values, n1 = 0.009.

Footline Author PNAS Issue Date Volume Issue Number 3 and the membrane follows a classical scheme common to many self- respond to three distinct dynamical states of the membrane. We will assembling systems (e.g surfactants in solution [19]). The local en- show below that regimes b and d represent respectively a quiescent ∗ cmc ergy minimum at ℓ appears above a critical concentration n1 , ana- membrane and a membrane secreting large amount of vesicles, while logue to the “critical micellar concentration”, or “cmc” at which sur- regime c is dynamically unstable. The secretory membrane thus con- factants in solution start forming micellar aggregates (see [19] and stitutes a bistable dynamical system able to abruptly switch vesicle cmc the SI). For n1 < n1 (Fig.2a), entropy dominates and the effective secretion on and off at prescribed monomer turnover rates. energy increases monotonously with the coat size ℓ. Monomer ag- Since all membrane-bound monomers are inactivated with the gregation is unfavorable, and the membrane contains mainly single same rate koff , the total flux of monomer leaving the membrane af- monomers and few small transient domains formed by fluctuation. ter inactivation Joff is directly proportional to the total amount of cmc For n1 > n1 (Fig.2b), long-lived coats can nucleate, at a rate monomer on the membrane. The peak of Joff in Fig.3 stems from the fixed by the nucleation barrier, and grow up to the optimal size ℓ∗. complex relationship between the concentration of isolated monomer Maturation into coated vesicles (ℓ = ℓv) is prevented by monomer n1 and the total amount of coat components on the membrane. desorption and membrane tension. No vesicle secretion regimes a and b. If no coat can form (low Larger monomer concentration. The height of the energy barrier to monomer concentration: state a), the monomers out-flux is domi- vesiculation at ℓ = ℓv decreases with increasing monomer concen- nated by the desorption of free monomer: Joff koff n1. If coats can tration. When it falls below the nucleation energy barrier (Fig.2c), form, but do not mature into vesicles (state b),≃ the out-flux is domi- domains may mature into fully-formed vesicles and vesicle secretion nated by the desorption of monomers belonging to coats of size ℓ∗: ∗ ∗ ∗ becomes increasingly probable. The optimal coat size ℓ increases Joff koff ℓ n(ℓ ). In this regime, monomers reaching the mem- ≃ with n1, and eventually exceeds the critical size for vesiculation ℓv brane tend to join a coat and the density of free monomer is almost ∗ 1/ℓ∗ ∗ at high monomer concentration (Fig.2d), under which conditions any insensitive to the fluxes: (n1 n(ℓ ) with ℓ 1, see SI). ∼ ≫ nucleated domain matures into a fully formed vesicle. A small increase of n1 requires a pronounced increase of the to- The growth of individual coats is controlled by the amount of tal amount of coat material on the membrane, which explains the free, active monomers n1. On the other hand, the pool of free sharp rise of the total monomer out-flux Joff with n1 in Fig.3. In monomer is depleted by their binding onto growing coats and is thus this regime, the optimal coat size is also insensitive to the monomer influenced by the coat population. As we shall see next, this feed- fluxes, and is obtained from the minimization of the effective energy back induces remarkable collective effects within the coat population, E˜ (Eq.(6)) 2/3 which presents a discontinuous transition between a state of arrested ∗ γ growth and a state of abundant vesiculation within a narrow range of ℓ cmc , [12] ∝ „ Σ(n1 ) « kinetic parameters. The unstable regime c. For intermediate monomer density, metastable coated pits have a high probability to grow into fully Vesicle secretion is controlled by collective effects. The so- formed vesicle owing to the small barrier to vesiculation (Fig.2c). lution of the coupled Eqs.(10,11) is graphically represented on Fig.3 The rate of vesicle formation increases with n1, so the total amount as the intersection of the monomer in-flux Jon and total out-flux of membrane-bound material actually decreases with increasing con- Joff + Jv. It may fall in four different regimes (a to d), correspond- centration of free monomer. This is shown by the dashed blue line in ing to the four distributions plotted in Fig.2. In a wide range of pa- Fig.2c, and explain the decrease of Joff in Fig.3. This situation can- rameters (see below), Joff displays the remarkable property of being not be maintained at steady state, and spontaneously evolves toward non-monotonous, with a sharp peak at a critical concentration of free either state b or d. monomers. This behavior dramatically influences the membrane’s Steady vesicle secretion regime d. If the monomer concentration on ability to secrete vesicles. Indeed, a given monomer in-flux may cor- the membrane is large, the effective coat energy exhibits a nucleation barrier but no intermediate minimum (Fig.2d). After nucleation, a 0.1 a b c d coat grows at nearly constant velocity until reaching the critical size ℓv where it remains trapped for a time 1/kv before being released as a vesicle. In this regime, both Joff and Jv increase with n1, Fig.3. 0.08 Joff+ Jv The bistability exhibited by the coats dynamics relies on the ex- istence of an unstable steady state and holds as long as there ex- 0.06 ist a (meta)stable coat of intermediate size. This feature is con- served even if the effective energy contains higher order terms, to be expected if the ratio of monomer dissociation to binding rates 0.04 (koff /kpn1) increases with the coat size (see SI). Furthermore, the switch exists if there is a metastable-state within the accessible size- J ∗ ∗ 2 0.02 on range: ℓ /ℓv( ℓ c ) < 1. From Eq.(12), this condition amounts 3 ∼ −4 to Σ > γc ( 10 ). The effective tension Σ (Eq.(7)) accounts Jv ∼ 2 −5 −3 both for the membrane mechanical tension (σc¯ 10 10 ) 0 ≃ − and the ratio (koff /kpn1). The binding rate is assumed to be lim- 0.9 11.1 1.2 ited by monomer diffusion, and is expected to be of order the inverse 4 −1 monomer diffusion time over its own size (kp D/s0 10 s , 2 ∼ ∼ Fig. 3. The fluxes of monomers leaving the membrane at steady state, as a with D µm /s the membrane diffusion coefficient). With a dis- cmc ∼ −1 function of the density of active monomer n1 (in n1 unit). Jv (blue) is the vesic- sociation rate koff 1s , and a monomer density n1 1% (or ulation flux and J (red) the sum of vesiculation and inactivation. At stationary 2 −1 ≃ ∼ off kpn1 10 s ), we find that secretory membranes are well into the state, the total outgoing flux balances the incoming flux (Joff +Jv = Jon) (hori- ∼ −2 3 cmc 2 bistable regime (Σ 10 γc ), and should exhibit the secretory zontal dashed line). Fluxes are in kp(n1 ) unit and the black dots correspond ≃ ≫ to the three different states (b, c, d) depicted in Fig.2. switch discussed below.

4 www.pnas.org — — Footline Author Discussion threshold and the systems jumps back into the quiescent state b. The growth of coated pits and the secretion of coated vesicles result The coat population can thus undergo discontinuous dynamical from a kinetic balance between the polymerization and the inactiva- transitions, characterized by an hysteretic cycle, between the two tion of coat components. It is thus to be expected that coat maturation non-equilibrium steady states b and d. In other words, the secretory can only proceed if the turnover at the membrane is suffi- membrane works in an all-or-nothing fashion and can switch vesicle ciently slow [10, 11, 13]. Our model goes beyond this intuitive anal- production on and off within a narrow range of control parameters ysis, and shows that secretory membranes are able to abruptly switch such as the GTPase activation and inactivation rates, and membrane between a quiescent and a vesicle producing state upon a slowing tension. Fig.4 shows the rate of vesicle secretion as a function of down of coatomer recycling. Switch-like behaviors are clearly ad- the GTP hydrolysis rate koff . The discontinuous transition between vantageous for biological systems. The highly non-linear nature of quiescent and vesicle-producing membrane is clearly apparent, and a switch confers evident robustness with respect to the noisy envi- is characterized by two hydrolysis rates (high and low thresholds). ronment, and allows for a precise regulation of the system’s activity. Between these two rates, the secretory membrane may be in either The biological consequences of the “secretory switch” are discussed state, depending on the system’s history (hysteresis). below, together with the influence of important factors, such as the Regulation of vesicle secretion by cargo. The adsorption flux density of cargo or the membrane tension, in regulating the activity of J , and the desorption rate k , of coat components at the mem- secretory membranes. We also show how some apparently unrelated on off brane tightly control vesicle secretion. Recent fluorescence experi- observations on COPs vesicles naturally fit into our global picture of ments on COPs coat suggest that these rates vary with the amount coated vesicle secretion. of cargo present at the membrane. For COPI, the presence of extra cargo leads to significant increase of the amount of coat components The “secretory switch”. Consider a secretory membrane receiv- at the Golgi membrane, which could reflect either the increase of Jon ing a fixed amount of coatomer per unit time. As coatomers accumu- or the decrease of koff [12]. For COPII, FRAP experiments show late and aggregate, the membrane eventually reaches a steady state that the coatomer exchange rate between the ER membrane and the in which the flux of coatomer leaving the membrane (by inactivation cytosol is doubled in the absence of cargo. This has been attributed or vesicle secretion) balances the in-flux. If the in-flux is low, the to the increase of koff with decreasing cargo density [15]. membrane is covered by monomers and stationary coated pits, the Our model predicts that vesicles can only be secreted if the re- latter being prevented to mature into fully-formed coated vesicles by cycling rate koff is below a critical value (Fig.4). By increasing the the combined effect of coatomer recycling and the energy associated lifetime of the coatomers at the membrane, the presence of cargo is to membrane deformation. As the in-flux increases, the coated pits thus expected to promote vesicle secretion, and a minimal amount of to monomer ratio increases (Fig.2a, dashed blue line), but as long as cargo at the membrane might actually be required for transport vesi- the membrane remains in the “stationary pits" regime (b - Fig.3) the cle to be secreted. The two, high and low, recycling thresholds of size and shape of the coated pits is weakly sensitive to the in-flux and Fig.4 would then corresponds to two critical cargo densities (low and no vesicle is produced. However, beyond a threshold value of the in- high, respectively). flux, state b disappears (Fig.3). Coated pits are not kinetically stable, Considering that newly synthesized cargo is brought to the and after a fast transient regime that sees the release of the previously membrane at a (slow) steady rate and is removed by vesicula- stable coated pits, the membrane settles into a state of steady vesicle tion, membrane-bound cargo accumulates in the no-secretion regime, secretion (state d). thereby decreasing koff and moving the system toward the secretion If the in-flux is now reduced, the membrane remains in the se- regime. Above the high cargo density threshold, the coat-machinery cretory state d, which possesses a stable branch for smaller in-flux abruptly escapes the stationary pits regime and switches to vesicle (Fig.3). The secretory regime disappears below yet another critical production (Fig.4). This results in a decrease of membrane-bound cargo, which increases koff and moves the system toward the quies- cent state. Below the low cargo density threshold, vesicle secretion Low threshold High threshold

) is switched off, letting the cargo accumulate until the high-density -1 threshold is reached and vesicle production is resumed, starting a .s Increasing cargo density -2 new cycle. Under constant cargo in-flux, the system should thus pe- riodically switch between quiescent phases and phases of vesicle se-

μm ( secretion transition no secretion cretion, following the hysteretic loop of Fig.4. If cargo synthesis is irregular, the membrane waits for sufficient accumulation of cargo between transient residences in the secreting state, where the accu- mulated cargo is released. The vesicle flux over time should then ap- pear as an irregular pulsed signal. Recording the total vesicle out-flux over an extended patch of secretory membrane over time should thus Vesicleflux be of high interest. Oscillatory or pulsed vesicle secretion would be a signature of the secretory switch uncovered by our theoretical analy- sis (within our framework, steady secretion would indicate that cargo synthesis is sufficiently fast to compensate the secreted cargo). −1 This accumulator mechanism would provide functional effi- koff (s ) ciency to the secretory membrane, as it would prevent the futile de- 2 Fig. 4. number of secreted vesicle per µm .s (= jv × 1µm/s0) as a func- livery of empty vesicles. Strikingly, analysis of COPI vesicles in tion of the rate of coatomer desorption from the membrane koff . The transition mutant cells where arf1 is unable to hydrolyse GTP have revealed a to vesiculation is discontinuous, and is characterized by an hysteretic cycle (ar- rows), with high and low turnover thresholds. The parameters are the same as much lower cargo content than in normal cells [20]. This observation those used for Fig.2 and 3. supports our prediction. Indeed, in the absence of GTP hydrolysis

Footline Author PNAS Issue Date Volume Issue Number 5 (koff = 0), the switch is gone and vesicle secretion remains “on” are indeed observed for Clathrin coats at the basal membrane of ad- regardless of the available amount of cargo. hered cells [23], where adhesive proteins are expected to generate The existence of an hysteretic cycle can also positively impact high membrane tensions. on the rate of cargo delivery. At steady state, the flux of cargo deliv- ered by a secretory membrane that would not possesses an unstable Concluding remarks. The model presented here is the simplest regime would automatically adjust to the flux of synthesized cargo. implementation of a kinetic model of coated vesicle formation where Here, and while the system is traveling along the secretion branch of coat growth competes with the inactivation of coat components. The the hysteretic cycle, the flux of secreted cargo is mainly controlled secretory switch revealed by our work is very robust and relies solely by the kinetics of coat formation, and can potentially be much larger on the (non-equilibrium) Thermodynamics of coat formation. Be- than the rate of cargo synthesis. yond the qualitative agreement with experimental findings discussed Effects of the membrane tension: Regulation of vesicle in the previous section, we hope that future experiments can fur- formation and coat flattening. Our calculation has shown that ther test some of our predictions, which include: i) the existence of membrane tension plays essentially the same role as coatomer re- metastable domains of intermediate size, ii) the role of membrane cycling in opposing vesicle secretion (Eq.(6)). Vesiculation is only tension in preventing the formation of curved protein coat, and even- possible below a high tension threshold, and the oscillatory behavior tually its involvement in the formation of flat lattices, iii) the oscilla- described above may also result from variations of tension. Secre- tory or pulsed secretion of vesicles in time. tion removes membrane area from the organelle and may increase its In this study, we have used the crudest possible description of the tension, while the fusion of incoming vesicles [21] or other regula- coat structure, thus avoiding to deal with structural details of specific tory mechanisms [18], dynamically relaxe tension. Vesicle secretion protein coats. Further experimental observation, e.g on biomimetic would thus occur only when enough membrane area has been accu- systems, could motivate the building of models focusing on the dy- mulated to relax the tension. Such a mechanism suggests a coordi- namics of a single coat. Supplementary degrees of freedom for the nation of purely mechanical origin between absorption and release coat shape could be considered, allowing for the competitive growth of vesicles, and could prevent the uncontroled shrinking of secretory of structures of various morphologies (tubules [6], spherical caps and compartments. Remarkably, the Golgi strikingly crumbles in cells flat lattices [23]), observed in living cells and biomimetic systems. where the GTP hydrolysis in COPI-coat is rendered inoperant [20]. In the same spirit, the heterogeneties of the coat structures may In Eq.(3), the effect of membrane tension was computed assum- be included in the model. The coupling beween the GTP hydrol- ing that the coat rigidity κ (unit of energy) was sufficiently strong ysis rate and the curvature (COPI) or the degree of polymeriza- to impose the domain curvature, which remained constant regardless tion (COPII) revealed in recent experiments [14, 24], suggest that of the coat size and the membrane properties. This simplification monomers inactivation may be enhanced at the coat center and lim- ceases to be valid under high membrane tension (σ κ/R2, where ited at the boundary. One could then imagine the formation of layer ∼ 0 R0 is the radius of curvature of a tensionless coat). Higher tensions of active monomers preventing coat disassembly [14, 24]. Such result in a flattening of the coat (see SI, section II.B). Beyond a ten- properties suggest a strong analogy with microtubules [11], and it is sion threshold σ > κ/(2R2) ( 10−4J/m2 for the coat rigidity tempting to imagine exotic growth dynamics with shrinking cascades 0 ≃ κ = 100kB T ), the formation of a closed sphere becomes impossible such as those observed for microtubules [25]. and protein aggregates should grow as flat patches. High membrane tension would thus favor the formation of flat coatomer aggregates We thanks Jean-Baptiste Manneville (Institut Curie-Paris) for very stimulating which size is limited by coatomer recycling [22]. Such "flat lattice" discussions and critical reading of the manuscript.

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