Kinetic regulation of coated vesicle secretion
aa r X i v : . [ q - b i o . S C ] J u l Kinetic regulation of coatedvesicle secretion
Lionel Foret ∗ and Pierre Sens † ∗ Max-Planck-Institute for the Physics of Complex Systems, Nothnitzer Strasse 38, 01187 Dresden, Germany, and † ESPCI, rue Vaucquelin, 75005 ParisSubmitted to Proceedings of the National Academy of Sciences of the United States of America
The secretion of vesicles for intracellular transport often rely onthe aggregation of specialized membrane-bound proteins into acoat able to curve cell membranes. The nucleation and growth ofa protein coat is a kinetic process that competes with the energy-consuming turnover of coat components between the membraneand the cytosol. We propose a generic kinetic description of coatassembly and the formation of coated vesicles, and discuss itsimplication to the dynamics of COP vesicles that traffic within theGolgi and with the Endoplasmic Reticulum. We show that station-ary coats of fixed area emerge from the competition between coatgrowth and the recycling of coat components, in a fashion resem-bling the treadmilling of cytoskeletal filaments. We further showthat the turnover of coat components allows for a highly sensitiveswitching mechanism between a quiescent and a vesicle produc-ing membrane, upon a slowing down of the exchange kinetics.We claim that the existence of this switching behaviour, also trig-gered by factors such as the presence of cargo and variation ofthe membrane mechanical tension, allows for efficient regulationof vesicle secretion. We propose a model, supported by differ-ent experimental observations, in which vesiculation of secretorymembranes is impaired by the energy consuming desorption ofcoat proteins, until the presence of cargo or other factors triggersa dynamical switch into a vesicle producing state.
Transport vesicle | protein Coat | COP vesicles | self- assembly | non-equilibrium phase transition
Introduction
The plasma membrane and the membrane of cell compartments suchas the ER and the Golgi continually produce vesicles for cargo trans-port. Vesicle formation generally involves specific proteins that ag-gregate into semi-rigid coats of dimensions in the 100nm range, wellvisible by electronic microscopy [1, 2, 3]. The process of vesicleformation is now rather well established [4], and is sketched Fig.1.First various cytosolic proteins assemble on the membrane into ele-mentary coat-building units, called monomers in the following. Themembrane-bound monomers then polymerize into coat structures thatlocally bend the membrane and recruit cargo molecules. As the coatexpands, the coated membrane invaginates until forming a nearlyspherical vesicle containing cargo [5, 6], that is eventually releasedfrom the membrane. The coat components soon disassemble and areready to participate to the formation of a new vesicle.The coats are classified in three major classes, COPII, COPI andClathrin. Although they involve distinct proteins, the three types ofcoat share many common features, from their size and shape to themechanism by which polymerization, cargo recruitment and mem-brane deformation is achieved [4]. Our approach is primarily aimedat studying the formation of COPI and COPII vesicles. However, thegenerality and robustness of its outcome suggest relevance for themore sophisticated Clathrin coats as well. The assembly of COPsand Clathrin, and the fission of COP vesicles can now be reconsti-tuted on purified liposomes with a restricted number of components[7, 8, 9]. Those experiments point-out the robustness of the coat for-mation process. They also confirm that coat polymerization is spon- taneous, only driven by weak short range attractions between themonomers, while coat disassembly requires the presence of an en-ergy source. More precisely, the assembly and disassembly of COPcoat components follow the cycle of activation - inactivation of a GT-Pase protein, Sar1 for COPII and Arf1 for COPI [4]. Once activated,the GTPases bind to the membrane and recruit individual coatomercomplexes (the monomers), that later polymerize into coats [10]. Theinactivation of the GTPase, triggered by the hydrolysis of its boundGTP, leads to its unbinding from the membrane and to the monomerdisassembly if the GTPase belongs to a monomer.Strikingly, FRAP experiments suggest that the exchange kinet-ics of coat components is much faster than the rate of vesicle secre-tion [10]. In other words, many futile monomers are released to thecytosol during the expansion of a coat. So, while new membrane-bound monomers polymerize at the coat periphery, others within thecoat disassemble and are expelled to the cytosol. Paradoxically, theconsumption of energy via
GTP hydrolysis seems to work againstcoat growth and to prevent vesicles formation. This resembles micro-tubules dynamics and by analogy to the treadmilling of microtubules,it has been suggested that the competition between growth and un-binding may produce stable coats of fixed area [10, 11, 12, 13].In this paper, we investigate theoretically the consequence of fu-tile release of coat components on the distribution of size and shapeof protein coats, and more practically on the amount of secreted vesi-cles. In our model (Fig.1), monomers are continuously "dropped"onto a membrane and proceed to aggregate into coats of growing ( ) p l j ( ) ffo l j l no J v j v l ffo J v J no J )a)b Fig. 1. a) A monomer cycle: activation/membrane binding ( J on ), aggregation( j p ( ℓ ) ), membrane unbinding v ia inactivation ( j off ( ℓ ) ) or vesiculation ( j v ). b) Coat population and the global fluxes in and out of the membrane.
The authors declare no conflict of interestThis paper was submitted directly to the PNAS office. c (cid:13) Issue Date
Volume Issue Number 1 – ize that curve the membrane. Monomers leave the membrane ei-ther individually after GTP hydrolysis or collectively as part of acompleted vesicle. Intuitively, one expects GTPase inactivation todecrease the rate of vesicle formation by reducing the lifetime ofmembrane-bound monomers. However, our generic approach revealsthat a deeper understanding of vesicle secretion requires a quantita-tive statistical model. Indeed, we report the existence of a discon-tinuous dynamical transition from a quiescent to a vesicle producingmembrane, upon variation of the rate of GTP hydrolysis. In otherwords, the apparently counter-productive energy consumption thatfavors the unbinding of coat components provides secretory mem-branes with a highly sensitive switch to regulate vesicle release, trig-gered for instance by a variation of the cargo concentration or themechanical tension of the membrane. Description of the model
Our goal is to describe the collective behaviour of a population ofevolving membrane domains (coats) formed by the aggregation ofidentical units (monomers), which are themselves continuously recy-cled between the membrane and a reservoir (the cytosol). Our startingpoint is the course of events depicted in Fig.1. We consider a patchof membrane much larger than the size of individual coats, which issubjected to a constant and homogeneous in-flux of monomers J on .The monomers have a finite lifetime on the membrane before beingrecycled to the cytosol, at a rate k off . While on the membrane, theydiffuse and eventually aggregate into curved protein coats. Coats thatmanage to reach a critical size leave the membrane as coated vesicles. Monomers : The formation of a new monomer on the membraneinvolves a succession of steps (GTPase binding on membrane and ac-tivation, and the recruitment of coat proteins) which are not individ-ually described in the present model. The rates associated with theseprocesses enter a unique parameter, the mean number of monomers J on formed on the membrane per units of time and area. Coat growth : Coat expansion proceeds by polymerization ofmonomers at the coat edge. Monomer-monomer binding is spon-taneous and results from weak short range interactions. The bindingenergy γ should be in the range of a few k B T ( k B T is the energyavailable from thermal fluctuations, with k B the Boltzmann constantand T the temperature in Kelvin), since the k B T provided by GTPhydrolysis is sufficient to break the bonds. The polymerization is thusthermally reversible and solely driven by the minimization of the freeenergy of the coat. Coat structure : Electron microscopy [1, 2] supports the assump-tion that the optimal area per monomer s ( ∼ nm ) and theoptimal radius of curvature of the coat R ( ∼ nm) are homo-geneous within the coat and remain constant during coat growth. Wethus adopt a model in which the state of a coat is fully characterizedby a single, slowly varying parameter: the number of polymerizedmonomers ℓ it contains, taken as a continuous variable for commod-ity. All other internal degrees of freedom in the coat (protein densityand coat shape) are considered to adjust to their optimal configura-tion faster than the typical rates of coat growth and GTP-hydrolysis.Under these assumptions, a given coat of size ℓ can be described asa spherical cap of constant curvature (defined as the dimensionlessquantity c = p s / (4 πR ) ∼ / ). A full spherical coat ( ℓc = 1 )with these properties contains several hundreds monomers. Monomer release : Contrary to the reversible monomer polymer-ization, monomer desorption is an energy consuming process drivenby GTP hydrolysis. It occurs at a rate k off , assumed constant herefor simplicity (see Supporting Information (SI) for a discussion ofthis assumption). This rate can be estimated from FRAP experiments[10, 12, 14, 15], k off ∼ . − s − . Under the assumption of con- stant protein density in the coat, the dissociation of monomer fromthe coat is followed by a rapid rearrangement of the coat structureand by a slight shrinkage of the coat; monomer inactivation thus op-poses coat expansion. Vesicle release : A mature coat containing a number ℓ v ( ℓ v c . ) of bound monomers forms a nearly closed sphere connected to therest of the membrane by a thin neck. At this stage, the coat cannotgrow further, and is eventually released into the cytosol as a coatedvesicle. The details of the scission mechanism vary between classesof coat, and may involve additional proteins ([4]). Here, we merelyassume that once a domain reachs the critical size ℓ = ℓ v , it leavesthe membrane as vesicles at a constant rate k v . Membrane properties : The curvature of the coat imposes a de-formation to the membrane which is opposed by membrane tension[16]. The membrane tension σ thus favors coat depolymerization,and can have a sizable effect on coat growth if it is larger than γ/ ( s √ ℓ v ) ≃ − J/ m (A detailed model for the elastic proper-ties of the protein-covered membrane is discussed in SI, section II.B).Tensions of the Golgi, the ER, and the plasma membranes are typicalin the range σ ∼ − − − J/ m [17], and may thus play a rolein vesicle secretion. Hereafter, membrane tension will be expressedin natural units: ¯ σ = σs ∼ − − k B T .The population of membrane coats is characterized by its sizedistribution n ( ℓ ) . The mean concentration of coats of size ℓ (between ℓ and ℓ + dℓ ) at a time t is n ( ℓ, t ) dℓ , and the mean concentration ofisolated monomers is n ( t ) . Our purpose is thus to compute, n ( ℓ ) and n at steady state, for given values of the parameters γ , ¯ σ , k off , J on , k v and ℓ v . More practically, we will compare the fluxes of coatelements leaving the membrane as inactive monomers J off and aspart of a vesicle J v (Fig.1). Theoretical framework
Monomer fluxes and conservation relations.
In this section,we derive the kinetic equations for the evolution of the coat size dis-tribution n ( ℓ ) . The monomer cycle can be divided into four steps, towhich correspond four different fluxes, as shown Fig.1.- J on ( t ) is the in-flux of single monomers binding to the membrane,taken as an input in our model.- j p ( ℓ, t ) is the flux of monomer joining domains of size ℓ , a bal-ance between polymerization and depolymerization for this domainsize. Integrated over the entire population, it gives the total flux ofmonomers incorporated into domains J p = R ℓ v dℓ j p ( ℓ, t ) .- j off ( ℓ, t ) is the flux of monomers expelled from domains of size ℓ into the cytosol after GTP hydrolysis. Under the assumption ofuniform release, it is given by j off ( ℓ ) = k off n ( ℓ ) ℓ . Integrated overthe entire population, it gives the total flux of individual inactivemonomers leaving the membrane J off = R ℓ v dℓ j off ( ℓ, t ) .- j v ( t ) is the flux of mature coats released as vesicles. Introducing therate of vesicle formation k v , we have j v = k v n ( ℓ v ) . The total flux ofmonomers leaving the membrane as part of a vesicle is J v ( t ) = j v ℓ v .The evolution of the coat size distribution satisfies (see SI): ∂ t n ( ℓ ) = − ∂ ℓ ( j p ( ℓ ) − j off ( ℓ )) , [1] and that the polymerization current j p reads: j p ( ℓ ) = − k p n ` ∂ ℓ n ( ℓ ) + n ( ℓ ) ∂ ℓ ∆ E ( ℓ ) ´ . [2] j p is proportional to the density of available monomers n , and tothe rate of monomer binding onto coats k p (see below). It containsa diffusive term accounting for random polymerizations and depoly-merization induced by thermal noise, and a convective term describ-ing the drift of monomers toward domains of lower energy, driven bythe “force” − ∂ ℓ ∆ E . The free energy difference ∆ E (in k B T units) etween a coat of size ℓ and ℓ isolated monomers diffusing on themembrane is obtained treating the coat as a rigid spherical cap [18](see also SI): ∆ E ( ℓ ) = γ p ℓ (1 − c ℓ ) + ¯ σc ℓ − µ ( n ) ℓ [3] where µ = ln n + γ √ − c + ¯ σc [4] is a chemical potential including the entropy of the freely diffusingmonomers, see SI.The net current j ( ℓ ) = j p ( ℓ ) − j off ( ℓ ) accounts for polymeriza-tion and desorption. It can be written in terms of an effective energy ˜ E ( ℓ ) ≡ ∆ E ( ℓ ) + k off k p n ( ℓ − : j = − k p n ( ∂ ℓ n + n∂ ℓ ˜ E ) , [5] with ˜ E ( ℓ ) = γ p ℓ (1 − c ℓ ) + Σ( n ) ℓ − µ ( n ) ℓ + const ., [6] Σ( n ) ≡ ¯ σc + k off k p n . [7] This equation introduces an effective tension Σ that illustrates thefact that desorption of inactive monomers and membrane tension for-mally play the same role in hindering coat maturation and vesiclesecretion. Note that more generally, the coatomer binding or inac-tivation rates may depend on the coat size, in which case monomerdesorption enters the effective energy as R dℓℓ ( k off ( ℓ ) /k p ( ℓ )) .Finally, the monomer influx J on and the flux of secreted vesicle j v are accounted for via the boundary conditions (see SI) j p (1) = J on − J p , [8] j p ( ℓ v ) − j off ( ℓ v ) = j v , [9] Steady state.
At steady state, all fluxes are balanced and ∂ t n = 0 .Eqs.(1-9) reduce to two conditions to be satisfied by n and n ( ℓ ) : j ( ℓ ) = j v = constant . [10] J on = J off + J v , [11] The former equation enforces that the size distribution is constant,and the latter that the flux of monomer binding to the membrane bal-ances the flux of monomer leaving the membrane, either after inacti-vation or by vesiculation.
Results
In this section, we focus on the steady state of a membrane receiv-ing a constant in-flux of monomer, each having a finite lifetime atthe membrane. The full characterization of the coat population andof vesicle secretion follows two steps. First, the stationary distribu-tion of coat size n ( ℓ ) is computed for a given concentration of freemonomers n with Eq.(10). Second, n is self-consistently derivedfor a given monomer in-flux J on by imposing that the in-flux matchesthe total monomer out-flux (Eq.(11)). While the first step relies en-tirely on the properties of the free energy landscape ˜ E ( ℓ ) , the secondintroduces collective effects emerging from the competitive growth ofmany domains, which ultimately give rise to the “secretory switch”. Effective energy landscape and steady-state distribution.
Apart from thermal fluctuations, a coat is driven toward growth orshrinkage by the effective “force” − ∂ ℓ ˜ E ( ℓ ) (Eq.(5)). Coat growthis thus formally analogue to thermal diffusion along the effective en-ergy landscape ˜ E ( ℓ ) (Eq.(6)). The analytical expression of the coat size distribution is given in the SI. Several different regimes can bedistinguished under increasing monomer concentration (Fig.2).The energy landscape illustrates the interplay between antago-nistic effects; short-range attractions between monomers promotespolymerization, while the entropy of the free monomer favors theirdispersion. The γ and µ terms in Eq.(6) reflect this competition.Furthermore, monomer inactivation and unbinding, and membranemechanical tension hinder coat growth ( k off and σ respectively, com-bined in the effective tension Σ , Eq.(7)). Depending on the monomerconcentration, the landscape may show a local maximum at a smallsize ℓ n (Fig.2 b - d ), which indicates a nucleation process. Coats mustreach the critical size ℓ n (through fluctuations in the pool of freemonomers) in order to consistently grow further, and the rate of nu-cleation is controlled by the height of the energy barrier. The effec-tive energy may also show an barrier to vesiculation for large coatsize ℓ ∼ ℓ v (Fig.2 b - c ), indicating that large coats are suppressed bythe effective membrane tension Σ . A local minimum then exist foran intermediate size ℓ ∗ , corresponding to kinetically stable coats. Low monomer concentration . At low monomer concentration, thelarge energy barrier to vesiculation at ℓ = ℓ v prevents coats to matureinto vesicles ( J v ≃ ) (Fig.2, states a and b ). The coat size distribu-tion resembles a distribution at thermal equilibrium: n ( ℓ ) ≃ e − ˜ E ( ℓ ) , a - b . Low concentration c . Intermediate concentration d . High concentration * l v ll n * l )(~ l E )( l n l l Fig. 2.
Effective energy landscape ˜ E ( ℓ ) (left column, in k B T units, fromEq.(6)) and the corresponding coat size distribution n ( ℓ ) (right column, in n cmc1 units, from Eqs.(5-10)), for different values of the free monomer density n . n /n cmc1 = 0 . ( a - first row green curve), . ( b - first row red), . ( c - second row) and . ( d - third row). The variation of ˜ E ( ℓ ) and n ( ℓ ) upona slight increase of n above the given value are shown in blue dashed lines.Other parameters are γ = 5 k B T , σ = 0 , k v = 0 . s − , k off = 0 . s − and, ℓ v = 500 . With those values, n cmc1 = 0 . .Footline Author PNAS Issue Date
Volume Issue Number nd the membrane follows a classical scheme common to many self-assembling systems ( e.g surfactants in solution [19]). The local en-ergy minimum at ℓ ∗ appears above a critical concentration n cmc1 , ana-logue to the “critical micellar concentration”, or “cmc” at which sur-factants in solution start forming micellar aggregates (see [19] andthe SI). For n < n cmc1 (Fig.2 a ), entropy dominates and the effectiveenergy increases monotonously with the coat size ℓ . Monomer ag-gregation is unfavorable, and the membrane contains mainly singlemonomers and few small transient domains formed by fluctuation.For n > n cmc1 (Fig.2 b ), long-lived coats can nucleate, at a ratefixed by the nucleation barrier, and grow up to the optimal size ℓ ∗ .Maturation into coated vesicles ( ℓ = ℓ v ) is prevented by monomerdesorption and membrane tension. Larger monomer concentration . The height of the energy barrier tovesiculation at ℓ = ℓ v decreases with increasing monomer concen-tration. When it falls below the nucleation energy barrier (Fig.2 c ),domains may mature into fully-formed vesicles and vesicle secretionbecomes increasingly probable. The optimal coat size ℓ ∗ increaseswith n , and eventually exceeds the critical size for vesiculation ℓ v at high monomer concentration (Fig.2 d ), under which conditions anynucleated domain matures into a fully formed vesicle.The growth of individual coats is controlled by the amount offree, active monomers n . On the other hand, the pool of freemonomer is depleted by their binding onto growing coats and is thusinfluenced by the coat population. As we shall see next, this feed-back induces remarkable collective effects within the coat population,which presents a discontinuous transition between a state of arrestedgrowth and a state of abundant vesiculation within a narrow range ofkinetic parameters. Vesicle secretion is controlled by collective effects.
The so-lution of the coupled Eqs.(10,11) is graphically represented on Fig.3as the intersection of the monomer in-flux J on and total out-flux J off + J v . It may fall in four different regimes ( a to d ), correspond-ing to the four distributions plotted in Fig.2. In a wide range of pa-rameters (see below), J off displays the remarkable property of beingnon-monotonous, with a sharp peak at a critical concentration of freemonomers. This behavior dramatically influences the membrane’sability to secrete vesicles. Indeed, a given monomer in-flux may cor-
1 1.21.10.9 0.08 0.06 0.04 0.02 a b c d J off + J v J v J on Fig. 3.
The fluxes of monomers leaving the membrane at steady state, as afunction of the density of active monomer n (in n cmc1 unit). J v (blue) is the vesic-ulation flux and J off (red) the sum of vesiculation and inactivation. At stationarystate, the total outgoing flux balances the incoming flux ( J off + J v = J on ) (hori-zontal dashed line). Fluxes are in k p ( n cmc1 ) unit and the black dots correspondto the three different states ( b , c , d ) depicted in Fig.2. respond to three distinct dynamical states of the membrane. We willshow below that regimes b and d represent respectively a quiescentmembrane and a membrane secreting large amount of vesicles, whileregime c is dynamically unstable. The secretory membrane thus con-stitutes a bistable dynamical system able to abruptly switch vesiclesecretion on and off at prescribed monomer turnover rates.Since all membrane-bound monomers are inactivated with thesame rate k off , the total flux of monomer leaving the membrane af-ter inactivation J off is directly proportional to the total amount ofmonomer on the membrane. The peak of J off in Fig.3 stems from thecomplex relationship between the concentration of isolated monomer n and the total amount of coat components on the membrane. No vesicle secretion regimes a and b . If no coat can form (lowmonomer concentration: state a ), the monomers out-flux is domi-nated by the desorption of free monomer: J off ≃ k off n . If coats canform, but do not mature into vesicles (state b ), the out-flux is domi-nated by the desorption of monomers belonging to coats of size ℓ ∗ : J off ≃ k off ℓ ∗ n ( ℓ ∗ ) . In this regime, monomers reaching the mem-brane tend to join a coat and the density of free monomer is almostinsensitive to the fluxes: ( n ∼ n ( ℓ ∗ ) /ℓ ∗ with ℓ ∗ ≫ , see SI).A small increase of n requires a pronounced increase of the to-tal amount of coat material on the membrane, which explains thesharp rise of the total monomer out-flux J off with n in Fig.3. Inthis regime, the optimal coat size is also insensitive to the monomerfluxes, and is obtained from the minimization of the effective energy ˜ E (Eq.(6)) ℓ ∗ ∝ „ γ Σ( n cmc1 ) « / , [12]The unstable regime c . For intermediate monomer density,metastable coated pits have a high probability to grow into fullyformed vesicle owing to the small barrier to vesiculation (Fig.2 c ).The rate of vesicle formation increases with n , so the total amountof membrane-bound material actually decreases with increasing con-centration of free monomer. This is shown by the dashed blue line inFig.2 c , and explain the decrease of J off in Fig.3. This situation can-not be maintained at steady state, and spontaneously evolves towardeither state b or d . Steady vesicle secretion regime d . If the monomer concentration onthe membrane is large, the effective coat energy exhibits a nucleationbarrier but no intermediate minimum (Fig.2 d ). After nucleation, acoat grows at nearly constant velocity until reaching the critical size ℓ v where it remains trapped for a time /k v before being released asa vesicle. In this regime, both J off and J v increase with n , Fig.3.The bistability exhibited by the coats dynamics relies on the ex-istence of an unstable steady state and holds as long as there ex-ist a (meta)stable coat of intermediate size. This feature is con-served even if the effective energy contains higher order terms, tobe expected if the ratio of monomer dissociation to binding rates( k off /k p n ) increases with the coat size (see SI). Furthermore, theswitch exists if there is a metastable-state within the accessible size-range: ℓ ∗ /ℓ v ( ∼ ℓ ∗ c ) < . From Eq.(12), this condition amountsto Σ > γc ( ∼ − ) . The effective tension Σ (Eq.(7)) accountsboth for the membrane mechanical tension ( ¯ σc ≃ − − − )and the ratio ( k off /k p n ). The binding rate is assumed to be lim-ited by monomer diffusion, and is expected to be of order the inversemonomer diffusion time over its own size ( k p ∼ D/s ∼ s − ,with D ∼ µm /s the membrane diffusion coefficient). With a dis-sociation rate k off ≃ s − , and a monomer density n ∼ (or k p n ∼ s − ), we find that secretory membranes are well into thebistable regime ( Σ ≃ − ≫ γc ), and should exhibit the secretoryswitch discussed below. iscussion The growth of coated pits and the secretion of coated vesicles resultfrom a kinetic balance between the polymerization and the inactiva-tion of coat components. It is thus to be expected that coat maturationcan only proceed if the coatomers turnover at the membrane is suffi-ciently slow [10, 11, 13]. Our model goes beyond this intuitive anal-ysis, and shows that secretory membranes are able to abruptly switchbetween a quiescent and a vesicle producing state upon a slowingdown of coatomer recycling. Switch-like behaviors are clearly ad-vantageous for biological systems. The highly non-linear nature ofa switch confers evident robustness with respect to the noisy envi-ronment, and allows for a precise regulation of the system’s activity.The biological consequences of the “secretory switch” are discussedbelow, together with the influence of important factors, such as thedensity of cargo or the membrane tension, in regulating the activity ofsecretory membranes. We also show how some apparently unrelatedobservations on COPs vesicles naturally fit into our global picture ofcoated vesicle secretion.
The “secretory switch”.
Consider a secretory membrane receiv-ing a fixed amount of coatomer per unit time. As coatomers accumu-late and aggregate, the membrane eventually reaches a steady statein which the flux of coatomer leaving the membrane (by inactivationor vesicle secretion) balances the in-flux. If the in-flux is low, themembrane is covered by monomers and stationary coated pits, thelatter being prevented to mature into fully-formed coated vesicles bythe combined effect of coatomer recycling and the energy associatedto membrane deformation. As the in-flux increases, the coated pitsto monomer ratio increases (Fig.2 a , dashed blue line), but as long asthe membrane remains in the “stationary pits" regime ( b - Fig.3) thesize and shape of the coated pits is weakly sensitive to the in-flux andno vesicle is produced. However, beyond a threshold value of the in-flux, state b disappears (Fig.3). Coated pits are not kinetically stable,and after a fast transient regime that sees the release of the previouslystable coated pits, the membrane settles into a state of steady vesiclesecretion (state d ).If the in-flux is now reduced, the membrane remains in the se-cretory state d , which possesses a stable branch for smaller in-flux(Fig.3). The secretory regime disappears below yet another critical V e s i c l e f l u x ( μ m - . s - ) secretion transition no secretion Low threshold High threshold )( − sk Increasing cargo density V e s i c l e f l u x ( μ m - . s - ) secretion transition no secretion Low threshold High threshold )( − sk Increasing cargo density secretion transition no secretion
Low threshold High threshold )( − sk Increasing cargo density
Fig. 4. number of secreted vesicle per µm .s ( = j v × µm/s ) as a func-tion of the rate of coatomer desorption from the membrane k off . The transitionto vesiculation is discontinuous, and is characterized by an hysteretic cycle (ar-rows), with high and low turnover thresholds. The parameters are the same asthose used for Fig.2 and 3. threshold and the systems jumps back into the quiescent state b .The coat population can thus undergo discontinuous dynamicaltransitions, characterized by an hysteretic cycle, between the twonon-equilibrium steady states b and d . In other words, the secretorymembrane works in an all-or-nothing fashion and can switch vesicleproduction on and off within a narrow range of control parameterssuch as the GTPase activation and inactivation rates, and membranetension. Fig.4 shows the rate of vesicle secretion as a function ofthe GTP hydrolysis rate k off . The discontinuous transition betweenquiescent and vesicle-producing membrane is clearly apparent, andis characterized by two hydrolysis rates (high and low thresholds).Between these two rates, the secretory membrane may be in eitherstate, depending on the system’s history (hysteresis). Regulation of vesicle secretion by cargo.
The adsorption flux J on , and the desorption rate k off , of coat components at the mem-brane tightly control vesicle secretion. Recent fluorescence experi-ments on COPs coat suggest that these rates vary with the amountof cargo present at the membrane. For COPI, the presence of extracargo leads to significant increase of the amount of coat componentsat the Golgi membrane, which could reflect either the increase of J on or the decrease of k off [12]. For COPII, FRAP experiments showthat the coatomer exchange rate between the ER membrane and thecytosol is doubled in the absence of cargo. This has been attributedto the increase of k off with decreasing cargo density [15].Our model predicts that vesicles can only be secreted if the re-cycling rate k off is below a critical value (Fig.4). By increasing thelifetime of the coatomers at the membrane, the presence of cargo isthus expected to promote vesicle secretion, and a minimal amount ofcargo at the membrane might actually be required for transport vesi-cle to be secreted. The two, high and low, recycling thresholds ofFig.4 would then corresponds to two critical cargo densities (low andhigh, respectively).Considering that newly synthesized cargo is brought to themembrane at a (slow) steady rate and is removed by vesicula-tion, membrane-bound cargo accumulates in the no-secretion regime,thereby decreasing k off and moving the system toward the secretionregime. Above the high cargo density threshold, the coat-machineryabruptly escapes the stationary pits regime and switches to vesicleproduction (Fig.4). This results in a decrease of membrane-boundcargo, which increases k off and moves the system toward the quies-cent state. Below the low cargo density threshold, vesicle secretionis switched off, letting the cargo accumulate until the high-densitythreshold is reached and vesicle production is resumed, starting anew cycle. Under constant cargo in-flux, the system should thus pe-riodically switch between quiescent phases and phases of vesicle se-cretion, following the hysteretic loop of Fig.4. If cargo synthesis isirregular, the membrane waits for sufficient accumulation of cargobetween 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-fluxover an extended patch of secretory membrane over time should thusbe of high interest. Oscillatory or pulsed vesicle secretion would be asignature of the secretory switch uncovered by our theoretical analy-sis (within our framework, steady secretion would indicate that cargosynthesis is sufficiently fast to compensate the secreted cargo).This accumulator mechanism would provide functional effi-ciency to the secretory membrane, as it would prevent the futile de-livery of empty vesicles. Strikingly, analysis of COPI vesicles inmutant cells where arf1 is unable to hydrolyse GTP have revealed amuch lower cargo content than in normal cells [20]. This observationsupports our prediction. Indeed, in the absence of GTP hydrolysis Footline Author PNAS
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Volume Issue Number k off = 0 ), the switch is gone and vesicle secretion remains “on”regardless of the available amount of cargo.The existence of an hysteretic cycle can also positively impacton the rate of cargo delivery. At steady state, the flux of cargo deliv-ered by a secretory membrane that would not possesses an unstableregime would automatically adjust to the flux of synthesized cargo.Here, and while the system is traveling along the secretion branch ofthe hysteretic cycle, the flux of secreted cargo is mainly controlledby the kinetics of coat formation, and can potentially be much largerthan the rate of cargo synthesis. Effects of the membrane tension: Regulation of vesicleformation and coat flattening.
Our calculation has shown thatmembrane tension plays essentially the same role as coatomer re-cycling in opposing vesicle secretion (Eq.(6)). Vesiculation is onlypossible below a high tension threshold, and the oscillatory behaviordescribed above may also result from variations of tension. Secre-tion removes membrane area from the organelle and may increase itstension, while the fusion of incoming vesicles [21] or other regula-tory mechanisms [18], dynamically relaxe tension. Vesicle secretionwould thus occur only when enough membrane area has been accu-mulated to relax the tension. Such a mechanism suggests a coordi-nation of purely mechanical origin between absorption and releaseof vesicles, and could prevent the uncontroled shrinking of secretorycompartments. Remarkably, the Golgi strikingly crumbles in cellswhere the GTP hydrolysis in COPI-coat is rendered inoperant [20].In Eq.(3), the effect of membrane tension was computed assum-ing that the coat rigidity κ (unit of energy) was sufficiently strongto impose the domain curvature, which remained constant regardlessof the coat size and the membrane properties. This simplificationceases to be valid under high membrane tension ( σ ∼ κ/R , where R is the radius of curvature of a tensionless coat). Higher tensionsresult in a flattening of the coat (see SI, section II.B). Beyond a ten-sion threshold σ > κ/ (2 R ) ( ≃ − J/ m for the coat rigidity κ = 100 k B T ), the formation of a closed sphere becomes impossibleand protein aggregates should grow as flat patches. High membranetension would thus favor the formation of flat coatomer aggregateswhich size is limited by coatomer recycling [22]. Such "flat lattice" are indeed observed for Clathrin coats at the basal membrane of ad-hered cells [23], where adhesive proteins are expected to generatehigh membrane tensions. Concluding remarks.
The model presented here is the simplestimplementation of a kinetic model of coated vesicle formation wherecoat growth competes with the inactivation of coat components. The secretory switch revealed by our work is very robust and relies solelyon the (non-equilibrium) Thermodynamics of coat formation. Be-yond the qualitative agreement with experimental findings discussedin the previous section, we hope that future experiments can fur-ther test some of our predictions, which include: i ) the existence ofmetastable domains of intermediate size, ii ) the role of membranetension in preventing the formation of curved protein coat, and even-tually its involvement in the formation of flat lattices, iii ) the oscilla-tory or pulsed secretion of vesicles in time.In this study, we have used the crudest possible description of thecoat structure, thus avoiding to deal with structural details of specificprotein coats. Further experimental observation, e.g on biomimeticsystems, could motivate the building of models focusing on the dy-namics of a single coat. Supplementary degrees of freedom for thecoat shape could be considered, allowing for the competitive growthof structures of various morphologies (tubules [6], spherical caps andflat lattices [23]), observed in living cells and biomimetic systems.In the same spirit, the heterogeneties of the coat structures maybe included in the model. The coupling beween the GTP hydrol-ysis rate and the curvature (COPI) or the degree of polymeriza-tion (COPII) revealed in recent experiments [14, 24], suggest thatmonomers inactivation may be enhanced at the coat center and lim-ited at the boundary. One could then imagine the formation of layerof active monomers preventing coat disassembly [14, 24]. Suchproperties suggest a strong analogy with microtubules [11], and it istempting to imagine exotic growth dynamics with shrinking cascadessuch as those observed for microtubules [25]. We thanks Jean-Baptiste Manneville (Institut Curie-Paris) for very stimulatingdiscussions and critical reading of the manuscript.
1. Matsuoka, K., Schekman, R., Orci, L. & Heuser, J.E. (2001) Surface structure of the COPII-coatedvesicle.
Proc. Natl. Acad. Sci USA , 13705-13709.2. Lederkremer, G.Z., Cheng, Y. Petre, B.M., Vogan, E., Springer, S., Schekman, R., Waltz, T. &Kirchhausen, T. (2001) Structure of the Sec23p/24p and Sec13p/31p complexes of COPII. Proc.Natl. Acad. Sci USA , 10704-10709.3. Barlowe, C. (2002) COPII-dependent transport from the endoplasmic reticulum. Current Opinionin Cell Biology , 417-422.4. Mc Mahon, H.T. & Mills, I.G. (2004) COP and clathrin-coated vesicle budding: different path-ways, common approaches. Current Opinion in Cell Biology , 379-391.5. Mc Mahon, H.T. & Gallop, J.L. (2005) Membrane curvature and mechanisms of dynamic cellmembrane remodelling. Nature , 590-595.6. Antonny, B. (2006) Membrane deformation by protein coat.
Current Opinion in Cell Biology ,386-394.7. Matsuoka, K., Orci, L., Amherdt, M., Bednarek, S. Y., Hamamoto, S., Schekman, R.& Yeung, T.(1998) COPII-Coated Vesicle Formation Reconstituted with Purified Coat Proteins and ChemicallyDefined Liposomes. Cell. , 253-275.8. Spang, A., Matsuoka, K., Hamamoto, S., Schekman, R.& Orci, L. (1998) Coatomer, Arf1p, andnucleotide are required to bud coat protein complex I-coated vesicles from large synthetic lipo-somes. Proc. Natl. Acad. Sci USA , 11199-11204.9. Takei, K., Haucke, V., Slepnev, V., Farsad, K., Salazar, M. Chen, H. & De Camilli, P .(1998) Gener-ation of Coated Intermediates of Clathrin-Mediated Endocytosis on Protein-Free Liposomes. Cell , 131-141.10. Presley, J.F., Ward, T.H., Pfeifer, A.C., Siggia, E., Phair, R.D. & Lippincott-Schwartz, J. (2002)Dissection of COPI and Arf1 dynamics in vivo and role in Golgi membrane transport. Nature ,187-193.11. Lippincott-Schwartz, J. & Liu, W. (2003) Coat control by curvature.
Nature , 507-508.12. Liu, W., Duden, R., Phair, R.D. & Lippincott- Schwartz, J. (2005) ArfGAP1 dynamics and its rolein COPI coat assembly on Golgi membranes of living cells.
J. Cell Bio. , 1053-1063. 13. Hinrichsen, L., Meyerholz, A., Groos, S. & Ungewickell, E.J. (2006) Bending a membrane: Howclathrin affects budding.
Proc. Natl. Acad. Sci. USA , 8715-8720.14. Bigay, J., Gounon, P., Robineau, S. & Antonny, B. (2003) Lipid packing sensed by ArfGAP1couples COPI coat disassembly to membrane bilayer curvature.
Nature , 563-566.15. Forster, R., Weiss, M., Zimmermann, T., Reynaud, E.G., Verissimo, F., Stephens, D.J. & Pep-perkok, R. (2006) Secretory Cargo Regulates the Turnover of COPII Subunits at Single ER ExitSites.
Curr. Biol. , 173-179.16. Sens, P. & Turner, M.S. (2004) Theoretical Model for the Formation of Caveolae and SimilarMembrane Invaginations. Biophys. J. , 1-917. Sheetz M. P. (2001) Cell control by membrane?cytoskeleton adhesion Nat. Rev. Mol. Cell Biol. ,392-395, Upadhyaya, A. & and Sheetz, M.P. (2004) Tension in Tubulovescilar Networks of Golgiand Endoplasmic Reticulum Membranes. Biophys. J. , 2923-292818. Sens, P. & Turner, M. (2006) Budded membrane microdomains as tension regulators. Phys. Rev. E , 031918.19. Israelachvili J. (1991) Intermolecular & Surface Forces , 2nd ed., Academic Press, San Diego20. Pepperkok, R., Whitney, J.A., Gomez, M. & Kreis, T.E. (2000) COPI vesicles accumulating in thepresence of a GTP restricted Arf1 mutant are depleted of anterograde and retrograde cargo.
J. CellScience , 135-144.21. Solon, J., Preceaux, J., Girard, P., Faur«e, M.-C., Prost, J. & Bassereau, P. (2006) Negative TensionInduced by Lipid Uptake.
Phys. Rev. Lett. , 098103.22. Matthew S. Turner, M. S., Sens, P. & Socci, N. D. (2004) Nonequilibrium Raftlike MembraneDomains under Continuous Recycling. Phys. Rev. Let. , 16830123. Benmerah, A. & Lamaze, C. (2007) Clathrin Coated Pits: Vive la Difference? Traffic , 970-98224. Antonny, B., Bigay, J., Casella, J.-F., Drin, G., Mesmin, B. & Gounon, P. (2005) Membrane cur-vature and the control of GTP hydrolysis in Arf1 during COPI vesicle formation. BiochemicalSociety Transactions , 619-622.25. Dogterom, M. & Leibler, S. (1993) Physical aspects of the growth and regulation of microtubulestructures. Phys. Rev. Lett. , 1347-1350.6