Active unidirectional network flow generates a packet molecular transport in cells
aa r X i v : . [ q - b i o . S C ] O c t Active unidirectional network flow generates moleculartransport in packets
Matteo Dora and David Holcman ∗ Abstract
Internet, social media, neuronal or blood vessel are organized in complex net-works. These networks are characterized by several quantities such as the underlyinggraph connectivity (topology), how they grow in time, scaling laws or by the meantime a random search can cover a network and also by the shortest paths betweentwo nodes. We present here a novel type of network property based on a unidi-rectional transport mechanism occurring in the Endoplasmic Reticulum networkpresent in the cell cytoplasm. This mechanism is an active-waiting transportation,where molecules have to wait a random time before being transported from onenode to the next one. We find that a consequence of this unusual network trans-portation is that molecules travel together by recurrent packets, which is quite alarge deviation behavior compared to classical propagation in graphs. To conclude,this form of transportation is an efficient and robust molecular redistribution insidecells.
Contrary to many graphs such as internet, small-world [1, 2], or other complex networks,the Endoplasmic Reticulum (ER), which is a fundamental organelle located inside thecell cytoplasm, consists at steady-state of a network of interconnected tubules which of-ten present three-way junctions (Fig. 1a) [3, 4], where each vertex (node or sheet) isconnected in average by three edges to three neighboring vertices, with no preferentialconnectivity. What defines the topology of the ER remains unclear, but the edges aremade of small tubules, that could appear and disappear transiently [5]. The role of theER is to redistribute proteins, but it is not clear how this is actually performed and whatis the associated time scale [7]. Large amount of single particle trajectories data revealedthat the direction of the flow in each tubule (network edge) alternates at random time [6],leading to an usual redistribution across nodes. Indeed, under this condition, it happensthat all the edges incident to a node can have an inward flow, and thus the material inthat specific node cannot escape and is trapped until the flow changes direction in at leastone of the edges. We call this temporal trapping situation a capture state (Fig. 1b). Aswe shall, transport in this situation is very different from flows or diffusion in classical ∗ ´Ecole Normale Sup´erieure, 75005 Paris, France
10 20 30 40Distance from source d(S, T)
Mean First Passage Time(C) S M F P T BL Linear region Exponential region
Effect of switching rate λ (D) Switching time scale M F P T Trapped trajectory Untrappedtrajectory (B) Active graph(A) ER network capture state
Figure 1: (A)
ER network reconstructed from SIM data [6]. (B)
Active graph model:At a given moment of time, molecular transportation is unidirectional, but this directionswitches at random time. (C)
Heatmap of the mean first passage time (MFPT) (left)and scatter plot of the MFPT for each node ranked by distance from the source (right).Dashed lines separate the boundary layer, linear and exponential regions. (D)
MFPT vsthe switching time τ switch for a node located at distance 25 from the source. A minimumis observed for a time compatible with the one reported in the ER.2etworks [8]. Although this vertex asymmetry is only transient here, this unidirectionalproperty is reminiscent of the diode networks, introduced in percolation problems byconstructing neighbouring lattice sites that transmit connectivity or information in onedirection only [9, 10].This property makes the ER network quite different from other types of networks andone key parameter is the edge directional switching rate λ . How protein or moleculartrafficking depends on such a rate? A measure of material redistribution is characterizedby the mean time it takes for a molecule located in node A to arrive for the first timeto node B. We found three regimes (Fig. 1c): when the node initial node A is close B(boundary layer), the time is then a fast increasing function of the (graph) distance. Atintermediate distances, the mean time increases linearly and finally, for regions locatedfar away and connected by very few nodes, the mean time increases drastically. The firstregion corresponds to ≈
10 nodes, while the last one is of the order of the diameter ofthe graph. Interestingly, the mean first arrival time has a minimum with respect to theswitching rate λ (Fig. 1d).To conclude, for a switching rate λ = 30 s − , it takes about 25 minutes for a moleculeto arrive in average to any node. This time scale associated to the mean first arrivaltime accounts for the total number of nodes of the graph, because on average each trajec-tory will visit a large portion of the graph before arriving for the first time to the targetnode, a situation similar to the classical escape through a narrow window [11]. Anotherpossible measure of the time scale in such active graph is the arrival time of the fastestparticles among many to a given node starting from an initial node. When there is noedge directionality, particles simply disperse by diffusion with a time scale given by thefirst eigenvalue (of the Laplace operator on the graph [12]). The transient regime followsfrom the classical diffusion rules, where in the end, the density in all nodes is uniform,equals to the ratio of the number of particles to the total number of nodes N particle /N node .However when the edge direction switching is considered, we observed an unexpectedphenomena: first, the time scale reduces by two order of magnitude, and does not varymuch with the initial number of particles when N > h τ i ∼ δ min D log( N particle ) , where δ min is the graph distance between the source andthe target node and D = λ switch a is the effective diffusion coefficient [11] (a is the meandistance between two nodes and λ switch is the switching rate), as confirmed by stochasticsimulations (Fig. 2a-b). This arrival time for the fastest is quite different from the meantime of arrival for one particle given by h τ i ≈ √ N net πλ ln N net + O (1), where the rate λ isthe reciprocal of the time to switch between two node and N net is the size of hexagonallattice.Second, due to the capture effect, particles can be trapped many times before reaching thetarget node. Interestingly, the particles travel by packets that form and deform, arrivingto the target node at random times (Fig. 2c). This situation contrasts with the case of no3 A)(C)
N = 100N = 1000N = 10000
Distance from source d(S, T) T i m e f o r t h e f i r s t t o a rr i v e ( s ) Extreme statistics
S T
N = 100N = 1000N = 10000
Optimal trajectories(B)Packet trafficking = 300 ms
Time for the first particle to arrive at a target node T, when thereare initially N particles starting from the source S. The stochastic simulations dashed iscompared to fit the law c δ c +log N ,( c = 0 .
07 and c = − .
38) where δ is the Graph distance. (B) Gastest trajectories from S to T for different number N of particles released from thesource. (C) Number of particles in the target node colored by edge where they are comingfrom. A classical behavior leading to constant equilibrium distribution is reached for asmall switching time scale τ switch = 30 ms . However, a new form of trafficking emerges inpackets of particles for a time scale of τ switch = 3 s . This transportation is associated topeaks of density arriving to a node: large groups of particles arrive synchronously fromthe same edges to a node. This mode of transportation differs significantly from classicaltransient or steady-state. 4witching, because the redistribution of material does not converge for long-time regimeto the uniform steady-state where all material is shared uniformly by the nodes.To conclude we showed here that the ER dynamics generates an atypical protein redistri-bution: proteins travel in recurrent packets to any point. This property allows proteins tobe delivered in groups, which is probably a more stable delivery process than the arrivalof individuals. Moreover the arrival of the fastest particles or packets occurs along theshortest paths (Fig. 2b), which is the most efficient mode of redistribution. We predictedthat the time scale of packets redistribution takes 20-30 seconds.Active networks allow the redistribution of particles by packets, that can arrive to a ver-tex at different moments of time. This form of redistribution is very different from bloodin capillary networks or spikes dispersion in neuronal networks. This mode of delivery,modulated by the switching rate, could have applications to other situations, but alreadyshows that the ER network has evolved to redistribute packets of proteins at a time scaleof few seconds, across most of the cytoplasm.Acknowledgements: This research was supported by a FRM grant to D. Holcman. Wethank P. Parutto, E. Avezov and D. Ron for discussions. References [1] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, “Classes of small-world networks,”
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