Cooperative protein transport in cellular organelles
CCooperative protein transport in cellular organelles
S. Dmitrieff ∗ and P. Sens † Laboratoire Gulliver (CNRS UMR 7083)ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05France (Dated: November 4, 2018)Compartmentalization into biochemically distinct organelles constantly exchanging material isone of the hallmarks of eukaryotic cells. In the most naive picture of inter-organelle transport drivenby concentration gradients, concentration differences between organelles should relax. We determinethe conditions under which cooperative transport, i.e. based on molecular recognition, allowsfor the existence and maintenance of distinct organelle identities. Cooperative transport is alsoshown to control the flux of material transiting through a compartmentalized system, dramaticallyincreasing the transit time under high incoming flux. By including chemical processing of thetransported species, we show that this property provides a strong functional advantage to a systemresponsible for protein maturation and sorting.
Eukaryotic cells contain many specialized compart-ments (organelles) constantly exchanging moleculesthrough complex energy-consuming transport processes.Along the secretory pathway for instance, proteinsand lipids synthesized in the endoplasmic reticulum(ER) are transported to the Golgi complex (itself sub-compartmentalized into stacked cisternae) for matura-tion and sorting, before being conveyed to particular cel-lular locations [1]. ER and Golgi (and even the differentGolgi cisternae) are characterized by fairly different lipidand protein compositions [2]. There is a widespread in-terest in understanding the molecular and physical basespermitting these organelles to maintain their identity ,namely their specific chemical composition, while con-stantly exchanging material[3].In a system where fluxes are linearly related to concen-tration differences ( i.e. satisfying Fickian diffusion), sta-tionary concentration gradients can only be maintainedby external fluxes. Fluctuations of the fluxes yield fluc-tuations of the local composition, and robust compart-ment identity (namely the existence of stable concentra-tion heterogeneity) is not to be expected. In the secre-tory pathway however, transport is heavily influencedby molecular recognition: transported molecules, carri-ers and recipient organelles cooperate through complexnetworks of molecular interactions [4, 5]. The goal ofthis paper is to study the consequences of such cooper-ative transport on the stationary state of a simple (2-compartment) system, and on its possible function inprotein maturation and sorting. Our model generalizesand extends the vesicular transport model proposed byHeinrich and Rapoport [6], where the rates of vesicleexchange between compartments are influenced by theircomposition. We discuss below both the case of a closedsystem and of an open system with incoming and outgo-ing fluxes. By including chemical transformation of the ∗ serge.dmitrieff@espci.fr † transported species, we show that cooperative transportcan strongly increase the accuracy of a system responsi-ble for protein maturation and sorting. I. STATIONARY COMPARTMENTDIFFERENTIATION IN A CLOSED SYSTEM
We first consider a single protein species distributedbetween two compartments constantly exchanging mate-rial, and indicate how these results might be extended toa multicomponent system.
A. One species system
We assume below that the total mass of the system(and the mass of each compartment) is maintained con-stant by an unspecified regulatory mechanism, so thatthe evolution of the concentration C in compartment 1can generically be described by the Master equation [7]: ∂ t C = I − J → ( C , C ) + J → ( C , C ) (1)with a similar expression for compartment 2 obtainedby the transformation 1 ↔
2. Here, J α → β ( C α , C β ) isthe mean flux from compartment α (with concentration C α ) to compartment β (of concentrations C β ). Compart-ments will naturally reach different concentrations if theyfollow different exchange rules. We focus on the more in-teresting case where J → ( C , C ) = J → ( C , C ). Thesource and sink term I in Eq.1 may include both ex-ternal fluxes in and out of compartment 1 and chemicaltransformation within this compartment.The transport of cargo between organelles may be sep-arated into three distinct steps; step
1: cargo packaginginside a membrane-based carrier, such as a small protein-coated vesicle or a membrane tubule [8], step
2: the ac-tual transport between secreting and receiving compart-ments, often involving molecular motors moving along a r X i v : . [ q - b i o . S C ] M a r ( a ) i n c r e a s i n g c o n c e n t r a t i o n C s /C tot C f /C tot d i ff e r e n t i d e n t i c a l C ∗ tot ( b ) C/C ∗ tot C C C tot /C ∗ tot C C J → J → J s FIG. 1. (a)
Location of the critical region in the parameter space { C s /C tot , C f /C tot } where stationary compartment differen-tiation occurs in a closed system (shaded blue, Eq.5). Increasing the total concentration C tot moves the system along the redline (arrow). (b) : Variation of the stationary compositions (black) and flux (red) with the total concentration, showing thebreaking of symmetry for C tot > C ∗ tot (Eq.5). cytoskeletal filaments[9, 10], and step
3: fusion of the car-rier with the receiving organelle (see [11] for a review).For step
1, we call J s the total flux of material secreted bya compartment, and S the fraction of this flux (a numberbetween 0 and 1) occupied by the species of interest. Letus assume step step
3) either withcompartment 2, with a probability P → , or back withcompartment 1 (with probability P → = 1 − P → ). Themean flux from compartment 1 to compartment 2 maythus be written: J → ( C , C ) = J s ( C ) S ( C ) P → ( C , C ) . (2)For a closed system with fixed concentration C tot = C + C (no source and sink term), the symmetricstate: C = C = C tot / ∂ t C = ∂ t C = 0). Linear stability analysis [7]shows that the symmetric solution is unstable provided( ∂ C J → ) C = C = C tot / <
0. The case of particles ran-domly entering transport vesicles which are secreted atconstant rate and fuse non-specifically with either com-partment corresponds to a linear flux ( J → ∼ C ) (akinto passive diffusion) and leads to identical compartments.Spontaneous compartment differentiation can only occurif the flux reaching the second compartment decreases with increasing concentration in the first one, and thisrequires non-linear transport ( cooperativity ).Let us assume the fluxes to have two rather univer-sal types of non-linearity as a function of concentration.Firstly, the out-going flux of a given species should sat-urate at high species concentrations. This can be dueto the limited capacity of transport vesicles, the limitedavailability of vesicle-coating proteins, or the formationof aggregates inapt for transport in a compartment be-yond a critical concentration. For simplicity, we chooseto keep the flux of secreted vesicles constant (and writeit J s ≡ K C s ), although direct interactions between car-goes and coat proteins are known to exist [8]. The pack- aging fraction S is assumed to saturate beyond a concen-tration C s following a Michaelis-Menten saturation [12]: S ( C ) = C C + C s (3)Secondly, vesicle fusion is known to be strongly reg-ulated by specific molecular interactions, including, butnot restricted to, interactions between matching pairs ofSNAREs[4]. Quantitative models have shown the impor-tance of this step for the generation and maintenance ofnon-identical compartments, using fairly detailed mathe-matical modeling of the different pairs of SNAREs [6, 13]and/or extensive numerical simulations [14]. Numerousfactors can however influence the delivery of transportvesicles, including specific interactions between the cargoand molecular motors [5]. Here, we adopt a very generictreatment of specific fusion, where the fusion probability P → deviates from its nonspecific value because of two-body interactions between constituents of the vesicle andthe receiving compartment: P → − / ∼ S ( C ) C . Af-ter normalization, the probability may be written: P → = C f + S ( C ) C C f + S ( C )( C + C ) (4)where C f is the typical concentration beyond whichspecific fusion becomes relevant. Within the descrip-tion outlined in Eqs.(3,4), linear transport correspondsto both characteristic concentrations being very large: C s , C f (cid:29) C tot .Spontaneous symmetry breaking (enrichment of onecompartment at the expense of the other) occurs when( ∂ C J → ) C tot / <
0. As shown in Fig.1a, this alwayshappens at high enough concentration C tot > C ∗ tot , with C ∗ tot3 = 4 C s C f ( C s + C ∗ tot ) (5)Beyond this threshold, any small perturbation from thesymmetric state brings the compartments into a stableasymmetric steady-state. As a consequence, the concen-tration of the least concentrated compartment (compart-ment 2, say) and the flux J → of material exchanged be-tween compartments both decrease with increasing con-centration when C tot > C ∗ tot , Fig.1b. At high concentra-tion, the asymptotic solution reads C ∼ C f C s /C tot .Although the actual location of the critical line de-fined by Eq.5 depends on the model (Eqs.(3,4)) for theexchange flux J → (Eq.2), its existence does not. Thiscritical behaviour is very general and stems from the pres-ence of two competing effects : cooperative fusion pro-motes protein enrichment (and increases with decreas-ing C f ), while saturation of protein packaging (beyonda composition C s ) limits transport. Including the pres-ence of different types of coat and fusion proteins doesnot fundamentally alter this picture [6]. B. Extension to a n-species system
Extending the analysis presented above to a n -component system is rather straightforward. Let us call C iα the concentration of the species i in the compartment α ( α = 1 , α can be defined as a vector C α = [ C α , C α , ..., C nα ],and satisfies the Master equation: ∂ t C iα = I iα − J iα → β + J iβ → α (6)where J iα → β is the mean flux of the species i from thecompartment α to the compartment β , and I iα is a netsource and sink term including both the presence of ex-ternal fluxes of species i in and out of compartment α ,and chemical transformation involving species i in com-partment α .For a closed system (no source and sink term), the totalconcentration for the i -th specie is fixed: C i tot = C iα + C iβ .All the equations may thus be written for the fractions φ iα = C iα /C i tot , satisfying φ i + φ i = 1. Then φ = − φ becomes implicit and the master equation is now writtenonly as a function of φ ≡ φ : ∂ t φ i = − j i → ( φ , − φ ) + j i → ( − φ , φ ) (7)with the normalized fluxes j iα → β = J iα → β /C i tot . Assum-ing as before that both compartments follow identicalexchange rules, φ / = 1 − φ / = [ , , ..., ] is a sta-tionary solution. The linear stability of the symmetricsolution is determined by the Jacobian matrix M : M i,k = − (cid:0) ∂ φ i j k → (cid:1) φ / (8)The symmetric state is unstable , and spontaneouslyevolves towards a non-symmetric state if If M has atleast one positive eigenvalue.In a multi-component system, the fluxes can be writtensimilarly to the main text: J iα → β = J α ( C α ) S iα ( C α ) P α → β ( C α , C β ) (9) The functions J α , S iα and P α → β may contain various non-linearities. In particular P α → β may involve any combi-nation of pair interactions { S iα , C jβ } which can lead toa very rich behaviour. One could in particular describein this way the transport of proteins directly interactingwith the secretion (coat proteins) or the fusion (SNAREs)machinery. C. Influence of a finite vesicle fusion time
If vesicular transport between secreting and receiv-ing compartment (the so-called step C v the resulting change of concentration in thecompartment. Allowing vesicles to dwell between com-partments for a finite time causes the total number ofmolecules in the compartments to decrease, and henceyields an effective total concentration C eff lower than theactual total concentration in the system C tot : C eff = C tot − N v (cid:88) i =1 C iv (10)where N v is the number of vesicles between compart-ments, and C iv the concentration carried by the i -th vesi-cle.This sum over all the vesicles is actually a randomvariable, but its mean can be computed analytically incertain cases. To know where the symmetric/asymmet-ric transition occurs, we may compute C eff in a sym-metric system, and the result will be valid up until thetransition. If the system is symmetric, vesicles have thesame average concentration ¯ C v and (cid:80) N v i =1 C iv ≈ N v ¯ C v .Let us assume each vesicle in the inter-cisternal mediumhas a rate of fusion W r towards any of the compart-ments. The average number of vesicles in the media isthen 2 K v /W r (where K v is the rate of individual vesi-cle secretion). Moreover, the average vesicle concen-tration ¯ C v can be written as the maximum concentra-tion C maxv a vesicle may carry, times the average vesiclesaturation fraction ¯ S (obtained from Eq.3), leading to C eff = C tot − K v W r C maxv ¯ S . Finally, the product K v C maxv is the number of vesicle leaving a compartment per unittime multiplied by the maximum concentration of eachvesicle, and can be identified with J s ≡ K C s . The crit-ical point of a system of total concentration C tot withvesicles staying a finite time between the compartmentscan thus be obtained from the critical point (Eq.5) of asystem with infinitely fast fusion, but with an effectivetotal concentration C symeff given by: C symeff = C tot − C s K W r ¯ S , ¯ S = C symeff C symeff + 2 C s (11)namely C symeff C tot = 12 − w r w r φ s + (cid:115) φ s + (cid:18) − φ s w r w r (cid:19) (12)with φ s = C s /C tot , w r = W r /K .Similarly, one can compute the effective concentrationin a fully asymmetric system, in which one compartmenthas a concentration C asymeff and the other has a concen-tration close to zero: C asymeff = C tot − C s K W r S (cid:48) , S (cid:48) = C asymeff C asymeff + C s (13)The difference between Eq.12 and Eq.13 being that in thelatter case, the empty compartment is sending out emptyvesicles, and only vesicles from the first compartmentcontribute to the depletion effect. We find : C asymeff ( φ s , w r ) = C symeff ( 12 φ s , w r ) (14)These predictions can be tested by numerical simu-lations (described in the Appendix). Comparison withthe solution of the infinitely fast fusion model are shownin Fig.2. Not only the location of the critical line, butalso the actual values of the concentrations in each com-partment in the asymmetric steady state, where found toagree very well, even for low vesicle fusion rate. II. COMPARTMENT DIFFERENTIATION INAN OPEN SYSTEM
The relative simplicity of the model presented inSec.I A, essentially characterized by two parameters( C s /C tot and C f /C tot , Fig.1.a), allows us to address is-sues of direct biological relevance, such as the presence ofexternal fluxes of material, and the possibility for chemi-cal transformation within the system. Organelles such asthe Golgi are strongly polarized, with distinct entry andexit faces. We investigate the consequences of coopera-tive transport in such open systems assuming that a par-ticular species enters the system through compartment1, and exits through compartment 2, while exchange be-tween the two compartments proceeds as described pre-viously. Mathematically, this amounts to including asource term I = J in and a sink term I = − J out inEq.1 yielding: J ≡ J → − J → = J in − ∂ t C = ∂ t C + K off C (15) . . . p < ( C − C ) > / C t o t .
25 0 . .
75 1 1 .
25 1 . φ f φ s =0.15 φ s =0.2 φ s =0.5 FIG. 2. Root mean square (RMS) of the difference of con-centration between the two compartments as a function of φ f = C f /C tot for various values of φ s = C s /C tot . Dash-dotted lines represent the mean-field values of the concen-tration (normalized by C asymeff /C tot with effective parameters φ effs = C s /C symeff and φ f = C f /C symeff . Solid lines representthe simulated results with vesicles in the inter-compartmentsmedium, with a vesicle fusion rate W r = K , i.e. up to 40%of the molecules are out of the compartments. The non-zerovalue of the RMS in the symmetric state is due to fluctuations. where a simple linear relationship was assumed for theout-flux: J out = K off C , and where the exchange fluxes( J → ) are still given by Eqs.2,3,4. At steady state, allfluxes must be balanced, including the net flux J betweenthe two compartments: J in = J out = J . A. Qualitative analysis
The dynamical behaviour of the set of equations Eq.15is discussed in some details in Sec.II B, but a qualita-tive understanding of the open system may be inferredfrom the results obtained for a closed system. We showed(Fig.1b) that the flux J → cannot exceed a maximumvalue and decreases upon increasing total concentrationbeyond a threshold. In an open system, this behaviourmay result in the absence of a steady state: if the in-flux into compartment 1 exceeds the maximum net fluxfrom 1 →
2, the concentration C of the entry compart-ment steadily increases with time, leading to a furtherdecrease of inter-compartment exchange. In the absenceof other compensatory mechanisms, C would divergeand C would vanish, leading to a vanishing exit flux.This divergence is probably not realistic, but it illus-trates the consequence of such non-linear transport foran open system: beyond a critical influx, the system isessentially blocked, filtering transit proteins at a very lowflux. While such feature has a negative impact on the rateof transport, it strongly increases the residency time ofmolecules and may prove advantageous to a system suchas the Golgi apparatus, whose function is to process andchemically modify proteins. C / C s C /C s C / C s C /C s FIG. 3. Phase-space trajectories of system with an exit flux J out = K off C ( K off = 0 . K ), and an input flux J in = 0 . K C s (left) and J in = 0 . K C s (right). Dash-dotted lines represent ˙ C = 0 and dashed lines ˙ C tot = 0. Red arrows represent initialcondition with convergent trajectories whereas blue arrows are for initial conditions yielding a divergence of C . B. Phase-space trajectories of an open system
We now discuss possible dynamical behaviours of anopen systems satisfying the kinetic equation Eq.15, wherethe fluxes between the two compartments are given byEqs.2,3,4. Although the exchange rules between the com-partments are symmetric, the existence of external fluxesbreaks the symmetry of the system, and different con-centrations should be expected in the two compartmentseven for low incoming flux. The critical behaviour at highincoming flux, as depicted in Fig.1 for a closed system,has nevertheless a profound impact on the steady states,or the absence thereof.As discussed above, one expects the flux exchangedbetween the two compartments to present a maximumvalue J max (necessarily smaller than the maximum pos-sible flux K C s , see Fig.1.b), theoretically leading to adiverging concentration in the first compartment and avanishing exchange flux if J in > J max . Depending oninitial conditions, a non-convergent behaviour might ac-tually also appear for values of J in a priori compatiblewith the existence of a steady-state. For instance, if theinitial concentration is very high in the first compart-ment, the divergent regime may occur for smaller in-flux J in < J max . This can be understood by considering thephase space trajectories of the system.The coordinates in phase space are the concentrations( C , C ), and the steady states (if any) are given by theintersections of the ˙ C = 0 and ˙ C tot = 0 curves. Since J out = K off C , the line ˙ C tot = 0 is obviously the line C = J in /K off , whereas the curve ˙ C = 0 has to becomputed numerically. If these two lines do not intersect,there is no steady state and C always diverges. If theydo intersect, the thus-defined fixed points may be linearlyunstable, or may be surrounded by a basin of attraction,as shown in Fig.3.The phase space representation show Fig.3 can be usedto study the consequences of a transient change of theinput flux (i.e. a pulse or a block of secretion). Letus consider a system which is in a stable steady state( C , C ) for an input flux J in . If the incoming flux is changed to J in at time t , the phase space trajectorieswill be changed, and the system will follow a new tra-jectory starting from ( C , C ). According to this newtrajectory, the system will reach a new position ( C , C )at a time t . If the flux is then switched back to itsoriginal value J in , ( C , C ) will not necessarily be in theattractive region of the stable steady state. Therefore,a transient change of the incoming flux may bring thesystem out of a stable steady state. In the case of astrong pulse ( J in >> J in ) the system may follow a di-vergent trajectory and the concentration C will increasestrongly with time. Formally, whatever the (finite) valueof ( C , C ) after a pulse, the system may reach a sta-tionary regime if the incoming flux J in after the pulse issmall enough. However, this may take a very long time.The approximation C → ∞ , J in = 0 shows this timegrows like ( C ) . III. CONSEQUENCE OF COOPERATIVETRANSPORT FOR PROTEIN MATURATIONAND SORTING
We now quantify the consequences of the kind of co-operative transport considered here on protein matura-tion and sorting. We investigate the situation sketchedin Fig.4, where a molecular species A enters the systemvia compartment 1 and is transformed into a species B by maturation enzymes, before leaving the system viacompartment 2. The processing accuracy is defined asthe total fraction of the input that exits the system asmature ( B ) molecules:Accuracy ≡ A (cid:90) ∞ J Bout dt (16)where A = (cid:82) ∞ J in dt is the total amount of A moleculesto have entered the system and J Bout is the out-flux of B molecules. The accuracy thus defined reaches unity whenno molecules exit the system without being processed( J Aout = 0). A B B A A A B A + B
FIG. 4. Sketch of an open system with protein maturation.Particle enter the system through compartment 1, undergomaturation A → B while in the system, are exchange betweencompartment via cooperative transport, and exit the systemvia compartment 2. A Michaelis-Menten maturation kinetics is chosen inorder to account for the limited amount of enzymes inthe system. Calling A and B the concentrations of A and B in the first compartment, we have: ∂ t B = R ( A ) A = R C m A A + C m (17)with an identical kinetics in compartment 2. Here, R is the maximal maturation rate and C m is the concen-tration of A beyond which enzymatic reaction saturates.For simplicity, we assume that the state ( A or B ) of amolecule influences neither its transport between com-partments nor its export from the system, so that Eq.15is still valid for the concentrations C , = A , + B , .Taking the weights of A and B in the fluxes to be theirrespective weights in the compartments : J A = A A + B J → − A A + B J → (18a) J B = B A + B J → − B A + B J → , (18b)the following set of kinetic equations is obtained:˙ A = J in − R ( A ) A − J A (19a)˙ B = R ( A ) A − J B (19b)˙ A = − R ( A ) A + J A − K off A (19c)˙ B = R ( A ) A + J B − K off B (19d)Normalizing rates with the vesiculation rate K andconcentrations with the concentration C s at which se-cretion saturates, Eq.19 is controlled by 5 parameters.These are: r = R /K and C m /C s , which compare theactivity of the maturation enzymes and of the secretionmachinery, C f /C s , which defines the threshold for domi-nant specific fusion (Eq.5), and K off /K , which comparesexit and exchange rates. The fifth parameter is the nor-malized amount of material going through the system: A /C s . For simplicity, we investigate a situation similarto the so-called pulse-chase procedure[15], where a fixedamount of material is delivered to the system in a finite amount of time (which we assume very small), and set A ( t = 0) = A and J in = 0 below.In order to focus on the role of cooperative transport,we further assume that particle export is not a rate-limiting step ( K off /K → ∞ ), and we analyze the pro-cessing accuracy in terms of a competition between thekinetics of maturation and transport (controlled by 4 pa-rameters). A. Processing accuracy for linear transport
In order to quantify the consequences of cooperativ-ity on the processing accuracy of a two-compartmentssystem, we compute the accuracy of a perfectly linearsystem by linearizing Eqs.2,3,4 when A (cid:28) C m , C s , C f ,yielding: J → = K C / J → = K C /
2. Choos-ing a linear exit flux for simplicity ( J out = K off C , forwhich we later take the limit K off → ∞ ), and the initialconditions C ( t = 0) = C (0) and C ( t = 0) = 0, the ki-netic evolution of the vector C = { C ( t ) , C ( t ) } is easilyobtained: C ( t ) = e M l t (cid:18) C (0)0 (cid:19) , M l = − K (cid:20) k off (cid:21) (20)where k off = K off K . The matrix M l can be diagonalized,and the matrix exponential becomes a regular exponen-tial, and the concentration in the second compartmentreads: C ( t ) = C (0)2 (cid:112) k (cid:0) e α + t − e α − t (cid:1) (21)with the eigenvalues : α ± = K (cid:18) ± (cid:113) k − (1 + k off ) (cid:19) (22)The (normalized) probability density that a particleexits the system from the second compartment at time t is P exit ( t ) = K off C ( t ) /C (0): P exit ( t ) = K k off (cid:112) k (cid:0) e α + t − e α − t (cid:1) (23)The mean residence time of a particle in the system isthus (cid:104) T (cid:105) ≡ (cid:82) ∞ dt ( tP exit ( t )) = 2(1 /K + 1 /K off ).The accuracy of protein maturation ( A → B ) and sort-ing is defined as the fraction of the total quantity ofmolecules that entered the system to leave as maturedmolecules (Eq.16). It may also be written as:Accuracy = (cid:90) + ∞ P exit ( t ) P ( B, t | A, dt (24)where which P ( B, t | A,
0) is the probability for a moleculeto be mature (state B ) at time t while starting imma-ture (state A ) at t = 0. At the linear level, the mat-uration kinetics (Eq.17) becomes: ∂ t B α = R A α , and P ( B, t | A,
0) = 1 − e − R t . The efficiency of the lin-ear system may then be computed analytically usingEqs.(22,23,24), yielding :Accuracy | linear = 2 r (1 + r + k off ) k off + 2 r (1 + r + k off ) (25)with r = R /K . Taking the limit k off → ∞ as dis-cussed, the benchmark to which more complex transportand maturation processes must be compared is thus thelinear accuracy: Accuracy | linear → r / (1 + 2 r ). B. Processing accuracy without specific vesicularfusion
Saturation of maturation enzymes and transport in-termediates ( C m , C s < A (cid:28) C f , with A the initialparticle concentration) has mixed effects on the systemsprocessing accuracy. Saturation of inter-compartmenttransport at high concentration (for A (cid:29) C s ) causes theparticle residency time of molecules to grow as A (Eq.3)while saturation of enzymatic reaction (for ( A (cid:29) C m )causes the mean maturation time increase linearly with A (Eq.17), so the net effect on processing accuracy de-pends on the precise values of the parameters.In order to get a fell for the role of the different param-eters, we compute the first order correction to the linearprocessing kinetics studied in Sec.III A, in the limit ofvery fast exit from the second compartment: k off → ∞ .In this case, the accuracy is controlled by the flux exitingthe first compartment, now written J out = K C s S ( C )and Eq.19 may be rewritten: C = A + B (26a)˙ A = − C s AC + C s − r C m AA + C m (26b)˙ B = − C s BC + C s + r C m AA + C m (26c)where the subscript 1 has been dropped in the con-centrations, time has been normalized by 1 /K , and r ≡ R /K . Taylor expansion of this set of equationfor A (cid:28) min ( C s , C m ) yields the first order correctionto the accuracy of the linear system (Eq.25):Accuracy = 2 r r + A C s C m (1 + 2 r ) − C s (1 + r ) C m (1 + r )(1 + 2 r ) + O (cid:34)(cid:18) A C s (cid:19) (cid:35) (27)An increase in processing accuracy at high concentra-tion requires that maturation saturates after secretionaccording to C m /C s > (1 + r ) / (1 + 2 r ). This can beseen in Fig.5, which shows the variation of the system’sprocessing accuracy as a function of the total amount ofmaterial to be processed, in the absence of cooperativefusion (dashed lines). Accuracy0 . . .
81 0 1 2 3 4 5 6 A / C s C m / C s =1.0 C m / C s =0.6 C m / C s =0.1 A /C s Accuracy
FIG. 5. Accuracy (Eq.16) as a function of the initial con-centration A , for different saturation ratios C m /C s of mat-uration and transport. At high concentration, specific vesiclefusion greatly enhance processing accuracy (solid lines, with C f /C s = 0 . C f → ∞ ,dashed lines) (with R /K = 2, and K off /K = 100). C. Processing accuracy and cooperative transport
Cooperative fusion, when combined with saturation ofthe transport, leads to a robust increase of the processingaccuracy of a compartmentalized organelle responsiblefor protein maturation and sorting (see Fig.5, solid lines).This increase can be understood as follows; at high con-centration ( A > C f , C s ), specific interactions promotebackward fusion of vesicles secreted by the highly concen-trated compartment. As the forward fusion probabilityis very low ( P → ∼ /A , Eq.4) the mean residency timeincreases as A , as compared to the linear increased ob-served in the absence of specific fusion (Sec.III B). On theother hand, the mean maturation time is still linear in A , so high concentrations lead to a more pronounced in-crease of the residency time compared to the maturationtime, resulting in an increased processing accuracy athigh concentration, even if the chemical transformationis performed by a limited amount of maturation enzymes( C m (cid:28) C s ). IV. CONCLUSION AND OUTLOOK
The predicted high processing accuracy displayed Fig.5essentially stems from the increase of the residency timeof molecules transiting through the system. In strik-ing contrast with the usual Fick’s law of gradient-driventransport, cooperative transport through the compart-mentalized system described here is strongly impaired bylarge concentration gradients. A strong prediction of ourmodel is that the transport time actually increases withan increasing incoming flux (above a threshold). Pulse-chase experiments on the Golgi seem to show this trend,but data are still too scarce for a direct comparison (seeFig.4. l in [15]). Although an apparent functional draw-back, slow transport through organelles is common. Forinstance, the typical transport time across the Golgi is oforder of 20 minutes [16], whereas diffusion of a membraneprotein over an area equal to that of the entire Golgi ap-paratus (of order 10 µ m ) should be of order one minute(with a diffusion coefficient ∼ . µ m / s [1]).In this paper, we showed that organelles constantlyexchanging material via transport vesicles may sponta-neously adopt different biochemical identities, provided: i) the flux of vesicles secreted by an organelle is bounded,and ii) there exists some level of specific vesicle-organellefusion directed by molecular recognition. In open sys-tems traversed by fluxes, these transport properties giverise to a dynamical switch from a linear to a low through-put kinetics above a critical influx. For compartmental-ized organelles whose function is to process and exportinfluxes of proteins, such as the Golgi apparatus, thisswitch allows the export rate to spontaneously adjust to the amount of material to be processed, a definitive func-tional advantage that may avoid the release of unpro-cessed material even under high influx. Future extensionof the present model to multi-component transport (seeS.I.) will allow to assess the importance of the exchangeof membrane area (lipids) between compartments (thecompartment sizes were assumed constant here). It willalso allow us to explore the role of specialized membranedomains; domains for protein processing and domains forprotein export, which have recently been reported [16]. ACKNOWLEDGMENTS
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To test the predictions of Eqs.13,14, in the case vesicle fusion occurs with a finite rate, we performed a numerical simulationof a system with finite vesicle fusion time and a total concentration C tot and compared the location of the critical line withthe infinitely fast fusion model, the equations of which we solved numerically. The numerical simulation consists of twocompartments of concentrations C and C from which vesicles may bud with a rate K v . Each vesicle budding from acompartment α has a saturation S ( C α ). At each timestep, each vesicle may merge with a compartment at a rate W r , and thecompartment is chosen according to the probability P described in the main text. The algorithm may be written as follows : def Pf1 ( Sv , C1 , C2)=(Sv ∗ C1+Cf ) / ( 2 ∗ Cf + Sv ∗ (C1+C2) ) def S (C) = C / (C+Cs ) − t h v e s i c l e while t < Tmax :t=t+dt i f rand ( 1 ) < Kv ∗ dt :Nves=Nves+1Sves [ Nves]=S (C1)C1=C1 − Cv ∗ Sves [ Nves ] i f rand ( 1 ) < Kv ∗ dt :Nves=Nves+1Sves [ Nves]=S (C2)C2=C2 − Cv ∗ Sves [ Nves ] for i =1 to Nves : i f rand ( 1 ) < Wr ∗ dt : i f rand ( 1 ) < Pf1 ( Sves [ i ] , C1 , C2) :C1=C1+Cv ∗ Sves [ i ] e l s e : C2=C2+Cv ∗∗