Active flow network generates molecular transport by packets: case of the endoplasmic reticulum

Abstract

Biological networks are characterized by their connectivity and topology but also by their ability to transport materials. In the case of random transportation, the efficacy is measured by the time it takes to travel between two nodes of the network. We study here the consequences of a unidirectional transport mechanism occurring in the endoplasmic reticulum (ER) network, a structure present in the cell cytoplasm. This unidirectional transport mechanism is an active-waiting transportation, where molecules have to wait a random time before being transported from one node to the next one. We develop here a general theory of transport in an active network and find an unusual network transportation, where molecules group together in redundant packets instead of being disperse. Finally, the mean time to travel between two nodes of the ER is of the order of 20 min, but is reduced to 30 s when we consider the fastest particles because it uses optimal paths. To conclude, the present theory shows that unidirectional transport is an efficient and robust mechanism for fast molecular redistribution inside the ER.

1. Introduction

The endoplasmic reticulum (ER) is a fundamental organelle located inside the cytoplasm, involved in synthesis and export of proteins and membrane lipids across the cell [1]. The general structure of the ER is represented by a network of membranes called cisternae. The ER network deforms constantly, but at steady state, it consists of a network of interconnected tubules, presenting in most cases three-way junctions (figure 1a) [2,3], where each vertex (node or sheet) is connected on average to three neighbouring vertices with no preferential connectivity. What defines the topology of the ER remains unclear, but the edges are made of small tubules, that could appear and disappear transiently [4].

Figure 1.

First passage time simulations in an active network. (a) ER network reconstructed from SIM data [6]. The average length of an edge is 0.5 μm. (b) Schematic of an active network model: the flow in a tubule is unidirectional and alternates at random times. A capture state (red box) is defined as a node with all converging (blue) arrows. (c) Heat-map of the mean first passage time (MFPT) (i) and scatter plot of the MFPT for each node versus the distance from the source S (ii), for a switching time scale τ_switch = 30 ms. A boundary separates (dashed lines) the linear and exponential regions. (d) MFPT (computed at distance 25) versus the switching time τ_switch for a node located at distance 25 from the source. The minimum is obtained τ_switch ≈ 180 ms. The MFPT for diffusion (dashed line) is ≈19 min. The network size is 1920 nodes. (Online version in colour.)

One fundamental role of the ER is to redistribute proteins, but the exact mechanism and the associated time scales remain controversial. First evidence [5] suggested that molecules are simply moving by diffusion. Indeed, at the molecular population level, fluorescence recovery after photobleaching (FRAP) experiments, which consist of a transient high-intensity illumination of fluorescent probes, monitored the recovery of the moving material [5]. These experiments were analysed under the assumption that transport is diffusive in the entire network. Then the average diffusion coefficient was estimated D_diff ≈ 1.4 μm² s⁻¹ [5]. As we shall see below, this coefficient is sufficient to estimate the time scale of redistribution in a passive network. However, more recently, a large amount of single-particle trajectories data revealed that there is a directional flow in each tubule (network edge). In addition, molecules are confined in nodes [6]. Indeed, to analyse super-resolution single particle trajectories, a stochastic processes approach [7] was used to reveal that a pure diffusion model is not sufficient to characterize the ER lumen transportation: while the motion in nodes is characterized as confined diffusion, with an effective confined diffusion coefficient of D_node ≈ 0.4 μm² s⁻¹, an active component was revealed inside tubules, characterized by a mean velocity v_eff ≈ 30 μm s⁻¹. Finally, recent photoactivation experiments, which consist of activating a local region and in tracking how the material is redistributed [6] revealed a fast propagation of material together at a certain distance away from the source immediately after a region has been photoactivated.

To reconcile these experiments, we introduce here a network model to study how independent particles are redistributed inside the ER. Our aim will be to define and quantify the time scale for material to redistribute inside the ER lumen. Thus in the present manuscript, we introduce a new transport mechanism in the network, in which the direction of the flow alternates at random times. This model accounts for single molecular trajectories that could alternate randomly inside tubules connecting two neighbouring nodes [6]. We call such a model an active flow network (AFN). Using the AFN model, we will study the consequences of the physical constraints coming from an alternating flow on the redistribution of particles such as ions, molecules or proteins inside the ER.

Characterizing the redistribution of material inside a network falls into the branch of random walk processes in network science, except that here, we will need to develop a theory to account for the AFN. Briefly, the time scale of redistribution can be studied by computing the mean time it takes for molecules to reach a target T located at a given distance d(S, T) from a source S. This time is relevant to quantify how long it takes for proteins to reach an exit site of the ER [8,9]. In passive networks, the mean time computed for one [10,11] or for many particles [12] depends on the distance d(S, T) and the exact topology of the network. However, the mean time properties are not known in the case of a network, the topology of which changes in time (active network). Indeed, active networks are mostly studied in the context of transportation (e.g. railways, vehicular traffic) or communication (e.g. internet, telephony [13]). In these situations, unidirectional flow often emerges owing to the limited capacity of the channel, which does not allow simultaneous bidirectional flow. Such constraint is not encountered inside the ER, instead, the topological changes (precisely the switching in edge directions) is an intrinsic characteristic of the network, and we thus focus on phenomena which arise from the creation of unidirectional flows in the absence of limiting capacity.

The manuscript is organized as follows: we first present the ER network reconstructed from live cell microscopy. Using this reconstructed network and the AFN properties, we study the redistribution of molecules across the network using the classical notion of first passage time [14]. We find that the time of redistribution increases drastically for sparsely connected regions, located at the end of the network. In the second part, we study a synthetic AFN honeycomb and ER networks with similar properties, and we estimate the time scale of material redistribution for the mean and also the fastest. In the third part, we show that the AFN can exhibit a novel type of transportation: under an alternating flow, all edges incident to a node could be inward directed, and thus molecules present are trapped. When the flow changes direction in at least one of the edges, particles can escape. We call this temporal trapping situation a capture state. In the final section, we study the steady-state and transient properties of the AFN, confirming persistent transport in packets, that we suggest is a possible novel mechanism of transport inside the ER. Thus transport in the ER modelled as an AFN is very different from flows or diffusion in classical networks [15]. This unidirectional property of the network is reminiscent of diode networks, introduced in percolation problems [16,17].

2. Results

(a). First passage simulations in a reconstructed endoplasmic reticulum network

To study the AFN, we used a graph (figure 1a) of the ER, reconstructed from structured illumination microscopy (SIM) [18]: nodes (red) are connected by junctions (grey) [6]. The AFN consists of a network with N nodes and a Poissonian rate constant ${\lambda }_{\mathrm{switch}}\equiv {\tau }_{\mathrm{switch}}^{-1}$ that controls the switching direction of the flow in each tubule. This property makes the ER network quite different from non-active networks (figure 1b). When arrows in tubules connecting a node to the rest of the network are converging (blue arrows), the particles located inside such a node are trapped.

To evaluate the redistribution efficiency of an AFN, we shall focus on estimating the mean time for a particle starting from a node S to visit the network. We start by running stochastic simulations as follows. Each particle attempts to jump through an outward directed edge with a Poissonian rate ${\lambda }_{\mathrm{wait}}\equiv {\tau }_{\mathrm{wait}}^{-1}$. When more than one outward edges are available, a particle can randomly select one of them with equal probability and move to the connected node in a time τ_tubule. In the following, we will consider that the time spent crossing the tubule is negligible with respect to τ_wait and sets to 0. If the edges are all inward directed, a moving particle is stopped and will thus wait the next Poissonian event to attempt a new jump.

We estimated numerically the mean time for a random particle starting in node S to arrive for the first time at node T on the ER-reconstructed network (figure 1c). Interestingly, the simulations reveal that the ER network can be divided into three regions (figure 1c): (i) a first subregion consists of nodes close to the source S (boundary layer). For those nodes, the mean first passage time (MFPT) increases quickly with the graph distance d(S, T); (ii) at intermediate distances, the mean time increases linearly; and (iii) finally, for nodes located far away, making very sparse connections with the rest of the network, the mean time increases drastically, because it is a rare event to enter these regions. We call these unaccessible regions exponential subregions. These regions are quite inaccessible at the end of the ER network connected by very few nodes (figure 1c). They could be further classified with respect to other parameters such as the fractal and or the anomalous diffusion exponents [10,19].

We also studied the effect of the switching time scale τ_switch on the MFPT. In principle, owing to the presence of capture states, the MFPT in the active network will increase compared to a undirected network. Instead, we found that the MFPT has a minimum at τ_switch = 0.25 s (figure 1d), making the MFPT time scale very similar to the undirected case for a large range of switching time τ_switch values.

To conclude, we found that an active graph (τ_switch = 30 ms and τ_wait = 100 ms) can redistribute material with a time scale similar to a passive network, about 25 min for a particle to arrive on average to any node, except for the far extreme nodes located in an exponential region. The mean first arrival time depends on the total number of nodes in the network, because on average each trajectory will visit a large portion of them, before arriving for the first time to the target node, a situation similar to the classical escape through a narrow window [20]. Although the MFPT measures how molecules redistribute in the ER, we next investigate how this property would be generalized to any honeycomb networks.

(b). Transport in a honeycomb network

To assess the role of the ER topology on transportation, we tested whether the MFPT could be similar between the SIM imaging reconstructed ER network (figure 1) and a three-way junction honeycomb network. We chose a honeycomb network because on average a node of the ER network has three neighbours [6]. We evaluate the arrival time in two networks: one with a periodic boundary condition and the other one with reflecting nodes at the boundary (figure 2). We found that with similar AFN dynamics properties, as the one reconstructed in figure 1, the first network can be decomposed into three subregions, with the last one corresponding to the four corners, which are difficult to access. However, for the periodic network, exponential regions are absent as there are no corners. We conclude that a three-way junction in a finite network could recapitulate the properties of a AFN ER network.

Figure 2.

MFPT in three-way junction network. Hexagonal lattice for a finite (a) and periodic (b) network. The MFPT is plotted versus the distance from the source S to a target T located at distance d(S, T). Notations are similar to figure 1, with τ_switch = 30 ms and τ_wait = 100 ms. The network size is N_net = 2000 nodes. (Online version in colour.)

(c). Transportation of the fastest particles: extreme arrival time

Various fast molecular signallings occur in the ER, such as transport of protein, unfolded protein response (UPR) or calcium release. In this context, where many particles are involved, the statistic of the mean as given by the MFPT may become irrelevant, because extreme events [21] such as the arrival time of the fastest particle to their target is sufficient to activate the cellular process. For example, the fastest folded proteins reaching exit sites involving COPII-mediated trafficking [22], will be transported to the next organelles. We thus focus here on the arrival time of the fastest particles among many, starting at a source node S and arriving at a target node T. We recall that in the absence of edge directionality, an ensemble of particles starting at S will simply disperse by diffusion with a time scale given by the first eigenvalue of the Laplace operator on the graph [23]. After a transient regime, the density of particles converges in long-time to the uniform distribution: at steady state, the number of particles per node equals the ratio of the total number of particles to the total number of nodes N_particle/N_node. However, for the AFN, the flow in tubules (edges) switches leading to capture state (figure 1b). We run numerical simulations in the honeycomb network (figure 2a) and found that these capture states could create a synchrony between independent particles.

The trajectories of the fastest particles concentrate towards the shortest paths between S and T, as the total number of particles increases (figure 3a,b; see also the electronic supplementary material, figure S1), where we plotted the lengths of the fastest trajectories that concentrate as the number of released particles increases. The MFPT for the fastest among N ≫ 1 particles can be derived from first principles (electronic supplementary material and [24]): indeed, considering an effective diffusion process in the network, the distribution of arrival time of the first particle is computed from the arrival distribution of a single one:

$$Pr\{{\tau }^1>t\}={Pr}^N\{t_1>t\}={\int }_{\mathrm{\Omega }}p(x,t)\hspace{0.17em}\text{d}x,$$ 1.1

where the density is solution of

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}p=\frac{{\lambda }_{\mathrm{eff}}{a}^2}{2}(\frac{3}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}p+\frac{3}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}p+\sqrt{3}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }x\mathrm{\partial }y}p)=\mathcal{L}(p),$$ 1.2

and the effective per-tubule transition rate in the limit of fast switching is ${\lambda }_{\mathrm{eff}}\sim \frac{1}{3\hspace{0.17em}{\tau }_{\mathrm{wait}}}$ see the electronic supplementary material for the details). The mean time for the fastest among N is (computations in the electronic supplementary material)

$${\overline{\tau }}^N={\int }_0^{\mathrm{\infty }}Pr\{{\tau }^1>t\}\hspace{0.17em}\text{d}t={\int }_0^{\mathrm{\infty }}{[Pr\{t_1>t\}]}^N\hspace{0.17em}\text{d}t,$$ 1.3

leading to the asymptotic expression

$${\overline{\tau }}^{N_{\mathrm{particle}}}\approx c_N\frac{{\delta }_{min}^2}{D\log (N_{\mathrm{particle}})},$$ 1.4

where δ_min is the graph distance between the source S and the target node T and D = λ_eff (a²/3) is the effective diffusion coefficient [20] (a is the tubule length) and c_N is a constant that only depends on the network topology. We use formula 0.4 to fit the simulations and we obtain good agreement in figure 3b. This approximation, which is only valid when τ_switch ≪ τ_wait, captures well the statistics of the fastest in the AFN.

Figure 3.

(a) Trajectories of the fastest particles released at S for different number N. (b) Mean arrival time for the first particles at a target node T, for N particles. The stochastic simulations (dashed) is compared to the fit (continuous) c₁δ²/(c₂ + log N), (c₁ = 0.07 and c₂ = −2.38) where δ is the network geodesic distance. (c) Number of particles in the target node coloured by edge where they are coming from; (i) Diffusion-like behaviour (small switching time scale τ_switch = 30 ms ≪ τ_wait = 100 ms) leads to a uniform steady=state distribution; (ii) τ_switch = 300 ms; and (iii) τ_switch = 3 s: groups of particles arriving synchronously to a node. (Online version in colour.)

The arrival time for the fastest is quite different from the mean time of arrival for a single particle, it is given by

$$⟨\tau ⟩\approx \frac{3\sqrt{15}\hspace{0.17em}{N}_{\mathrm{net}}}{2\pi {\lambda }_{\mathrm{eff}}}\ln N_{\mathrm{net}}+O(1),$$ 1.5

where N_net is the number of nodes in the network. At this stage we conclude that when the number of particles N ≥ 1000, the mean time for the fastest is two orders of magnitude faster ≈20 s (figure 2b), compared to 25 min for the mean time (here, N_net ≈ 2000). However, this fast time does not diminish that much when increasing by a factor 10 the number of particles to 10 000. The time scale of redistribution of the fastest is associated with the extreme statistics [25–27] leading to the extreme behaviour [21,24,28–30].

(d). Active flow network can generate a flow of particles travelling in packets

Another interesting effect of the AFN is the correlation induced by the flow on independent particles when the condition on the times satisfies τ_switch ≫ τ_wait. Indeed, owing to the capture state (figure 1b), particles accumulate in nodes. This accumulation creates a synchronization of the particle motion. Indeed, as soon as one of the edges reverses its direction, the particles start flowing together towards the accessible neighbouring node. This behaviour is generic across consecutive trapping nodes, thus a large fraction of the particles travel together in synchronized packets (figure 3c, with τ_switch = 3 s). When τ_switch ≪ τ_wait, capture states disappear and we recover an almost classical diffusion regime, where particles are uncorrelated. Interestingly, for the AFN, the distribution of material does not converge for long-time to the uniform steady state, where all material is shared uniformly by the nodes, but aggregates and keep appearing and disappearing (figure 4).

Figure 4.

(a) Example of three transient transportation regimes: (i) three snapshots at time 0; 30 s; 60 s. All particles start at the centre (red ball. For τ_switch = 30 ms, we recover a classical diffusion dispersion, as described in figure 3c; (ii) τ_switch = 300 ms shows already the appearance of packets (red); (iii) τ_switch = 3 s: packets of particles (red) appear and disappear as time increases. In all cases, we use τ_wait = 100 ms, (see the three associated movies in the electronic supplementary material). (Online version in colour.)

(e). Redistribution time scale of the active flow network

In this section, we investigate the redistribution time scale and the MFPT of the AFN and compare it to an equivalent undirected network. In particular, we study two competing mechanisms. First, particles can be trapped in capture states and thus we expect the exploration of the network to be slower when compared to the undirected network, where such a situation does not occur. Second, the presence of directionality in the flow drives particles farther away because they cannot linger going back and forth along the same edge. We formalize this latter effect by evaluating the probability that a particle moves to a new node and successively jumps back to the node it came from (we call this situation backtracking), thus slowing down the exploration of new nodes.

To compute the consequences of capture states on the time it takes to travel along a trajectory, we shall estimate the steady-state probability p_trap that a particle is trapped in a capture state. For the AFN, the edges switch at a Poissonian rate ${\lambda }_{\mathrm{switch}}\equiv {\tau }_{\mathrm{switch}}^{-1}$ and the waiting time in a node follows an exponential distribution $p_{\mathrm{wait}}(t)=(1/{\tau }_{\mathrm{wait}})\hspace{0.17em}{\text{e}}^{-t/{\tau }_{\mathrm{wait}}}$. For a node connectivity d, the steady-state probability that a particle is in a trapping state can be computed by considering two cases: either the particle was already in the node and one edge switches an even number of times between the instant of transitions, or the particle was previously in the node and had switched to a neighbouring node before returning to the considered node. These two cases lead at steady state to the Markov equation (see the electronic supplementary material, S1 for the details of the derivation):

$$\begin{array}{rl}\hspace{0.17em}{p}_{\mathrm{trap}}& =(1-p_{\mathrm{trap}}){\int }_0^{\mathrm{\infty }}(\frac{1+{\text{e}}^{-2{\lambda }_{\mathrm{switch}}t}}{2})(\frac{1}{2}{)}^{d-1}{p}_{\mathrm{wait}}(t)\hspace{0.17em}\text{d}t\\ & +p_{\mathrm{trap}}{\int }_0^{\mathrm{\infty }}(\frac{1+{\text{e}}^{-2{\lambda }_{\mathrm{switch}}t}}{2}{)}^d{p}_{\mathrm{wait}}(t)\hspace{0.17em}\text{d}t,\end{array}$$ 1.6

where the sum of the first term is made over a single node where the flow direction could have only switched an even number of times in the trapping state. For the second term, only one edge among d could switch an even number of times, while the state of the other one does not matter. A direct integration for d = 3 gives

$$p_{\mathrm{trap}}=\frac{24q^3+34q^2+11q+1}{192q^3+130q^2+23q+1},$$ 1.7

where q = τ_wait/τ_switch. We compare the approximated solution equation (1.7) (green dashed line) with the simulations (green solid line) in figure 5, where trapping in the diffusion network is shown in blue. Interestingly, in the limit of fast switching τ_switch ≪ τ_wait, the trapping probability converges to (1/2)^d, as there is no correlation between the state of edges. For τ_switch ≫ τ_wait, the Markovian approximation breaks down as the transition events are not exponentially distributed anymore. To conclude, in the limit of fast switching, the AFN converges to a diffusing network, but the total time that a particle spends in a node owing to trapping is computed by summing over all possibilities to be trapped n times:

$${\tau }_{\mathrm{tot}}\sim \sum _{n=1}^{\mathrm{\infty }}{(p_{\mathrm{trap}})}^{n-1}(1-p_{\mathrm{trap}})\hspace{0.17em}\mathit{\Gamma }(n,{\lambda }_{\mathrm{wait}}),$$ 1.8

where Γ(n, λ_wait) is the gamma distribution for the sum of n independent exponential waiting times with rate λ_wait ≡ 1/τ_wait. Finally, the mean time is

$$⟨{\tau }_{\mathrm{tot}}⟩=\sum _{n=1}^{\mathrm{\infty }}{(p_{\mathrm{trap}})}^{n-1}(1-p_{\mathrm{trap}})\hspace{0.17em}n\hspace{0.17em}{\tau }_{\mathrm{wait}}=\frac{1}{1-p_{\mathrm{trap}}}\hspace{0.17em}{\tau }_{\mathrm{wait}}.$$

To conclude this first part, owing to trapping, an AFN can retain particles in nodes much longer than the waiting time scale τ_wait. At a coarse-grained level (observation time scale much larger than each time τ_wait and τ_switch), transport in the AFN is described as an effective diffusion-like model where 〈τ_tot〉 is the effective waiting time in a node.

Figure 5.

Effect of the switching time τ_switch on the trapping and the backtracking probability in the undirected network model (blue), active flow network. The simulation (solid green line) is compared to the analytical approximation (dashed green line). (a) Probability of trapping p_trap(τ_switch). (b) Probability of backtracking p_back(τ_switch). (Online version in colour.)

We now quantify the phenomenon of backtracking, which consists of a particle jumping back to the node it came from, thus wasting time by visiting again the previous node. For undirected networks, the probability for a particle to jump back to the node it came from depends only on the degree of connectivity: if the current node has d incident edges, the probability of going back is $p_{\mathrm{back}}^{\mathrm{undirected}}=\frac{1}{d}$. However, in the AFN, this probability is affected by the direction of the edge, which flips at a rate ${\lambda }_{\mathrm{switch}}\equiv {\tau }_{\mathrm{switch}}^{-1}$. To compute the probability of a backtracking event in the AFN, we study the state of a particle starting in a node N that comes back to N after visiting an adjacent node A. To go back to node N after jumping from A, it is necessary that, after the waiting time t spent in A, the flow from N to A has reverted its direction to be oriented now from A to N. Moreover, the particle has to choose the edge that goes back to N among the other outward edges. To compute the probability of selecting this edge, we fix the $\overrightarrow{NA}$ edge and consider the configurations with other k edges in an outward directed state: the probability of choosing the backward one is 1/(k + 1). Because the state of the backward edge is fixed (outward flow), the total number of configurations is 2^d−1. The probability of a configuration with k other edges being outward directed is ${(1/2)}^{d-1}\left(\begin{array}{c}d-1\\ k\end{array}\right)$. Thus, the probability of choosing the specific path from A to N is (electronic supplementary material, S1 for the derivation)

$$\begin{array}{rl}P[\text{choosing}\hspace{0.17em}\overrightarrow{NA}\mid \overrightarrow{NA}\hspace{0.17em}\text{outward}]& =\frac{1}{2^{d-1}}\sum _{k=0}^{d-1}\frac{1}{k+1}\left(\begin{array}{c}d-1\\ k\end{array}\right)\\ & =\frac{2^d-1}{d\hspace{0.17em}{2}^{d-1}}.\end{array}$$ 1.9

The conditional probability of going back to N through node A, when the previous transition of the particle was made from N to A is

$$\begin{array}{rl}P[(N,A)\mid (A,N)]& ={\int }_0^{\mathrm{\infty }}\frac{1-{\text{e}}^{-2{\lambda }_{\mathrm{switch}}t}}{2}{p}_{\mathrm{wait}}(t)\hspace{0.17em}\text{d}t\\ & \times P[\text{choose edge }\overrightarrow{NA}\mid \overrightarrow{NA}\hspace{0.17em}\text{outward}],\end{array}$$ 1.10

where we have integrated over the switching time t with respect to the waiting time distribution p_wait(t) in a node and $(1-{\text{e}}^{-2{\lambda }_{\mathrm{switch}}t})/2$ which represents a switching event for the edge direction during time t. Finally, at steady state, the backtracking probability is the conditional probability to switch in the absence of trapping:

$$p_{\mathrm{back}}=(1-p_{\mathrm{trap}})\frac{2^d-1}{d\hspace{0.17em}{2}^{d-1}}{\int }_0^{\mathrm{\infty }}\frac{1-{\text{e}}^{-2{\lambda }_{\mathrm{switch}}t}}{2}{p}_{\mathrm{wait}}(t)\hspace{0.17em}\text{d}t.$$ 1.11

For a connectivity of d = 3, a direct integration leads to expression (see the electronic supplementary material for further details)

$$p_{\mathrm{back}}=(1-p_{\mathrm{trap}})\hspace{0.17em}\frac{7}{12}\hspace{0.17em}\frac{q}{2q+1},$$ 1.12

where the trapping probability p_trap is defined in equation (1.7) and q = τ_wait/τ_switch. We found a good agreement between this analytical expression (1.12) for p_back and numerical simulations (figure 5).

(f). Transient material redistribution of the active flow network

To study how particles are quickly redistributed following a local transient release at the source location S, we generated numerical simulations in the honeycomb network. We monitor the transient for the number of particles with respect to the distance of the source along a single path (figure 6a) and also by averaging over a circle centred around the source S (figure 6b, blue graph circle). Interestingly, the kymograph (distance versus time) of the diffusion network is characterized by a propagation front r² = at, where a is a constant that depends on the network topology and the diffusion coefficient (figure 6c). Under the front, the material is distributed following the law of diffusion on the network. However, for the AFN, numerical simulations reveal that the propagation front deviates significantly (figure 6d) from the diffusion one. In particular, as the switching time τ_switch increases, we observe that the material gets more fragmented, confirming the predictions of travelling in packets. Interestingly, packets can appear at long-distance quite rapidly owing to the motion of the fastest packet [21,31]. By looking at kymographs representing the time evolution of the average density of particles versus distance from the source (figure 6, right column), we found that the AFN (figure 6d) behaves similarly to the undirected network (figure 6c) for values of τ_switch in the range 30–3000 ms. Instead, looking at the density along a single path, we observe the anisotropic behaviour of the AFN, as shown in figure 6d, τ_switch = 300 ms. For an undirected network (figure 6c), representing a diffusive situation, there is almost no difference between the single path density and the average density, confirming that the redistribution is isotropic. To conclude, we predict that following a transient release at one location, stable hot spots of materials could appear quickly far away from the source, as observed along single paths of the network. These hot spots are owing to the coordinated travel of many particles until they are trapped in a node. This prediction could be tested using photoactivation experiments, where materials are transiently illuminated and followed using live-cell microscopy [6,32].

Figure 6.

Kymograph of particle distribution released at a single point S. The density of particles is computed along a single path (a) and on a disk of fixed radius (b). Results were obtained simulating N = 10 000 particles. (c) Distance to the source versus time for the undirected (diffusion) network in the two cases (a,b). (d) same as (c) for the active network with τ_switch = 30; 300; 3000 ms. (Online version in colour.)

3. Discussion and Conclusion

We studied here the properties of the AFN, a transport model of material inside the ER of living cells. The AFN can generate atypical protein redistribution, where molecules travel in packets that form and deform while visiting any accessible node of the network. This property allows proteins to be delivered in groups, which is a more reliable mode of transportation compared to individual uncorrelated motion occurring for pure diffusion.

Moreover, the arrival of the fastest particles or packets occurs along the shortest paths (figure 2b). The present approach predicts that the time scale of packets redistribution (in most of the network, except at corners, for τ_switch in the range 30–3000 ms and τ_wait = 100 ms) takes on average 20 min, yet if N = 1000 particles are simultaneously released, the time for the first to arrive is on average 20–30 s (table 1). These values are compatible with motion on an undirected network (diffusion).

Table 1.

Parameters and results of the simulations on the reconstructed ER network. (τ_wait is the exponential waiting time in nodes, τ_switch the time for edge to switch, τ_tubule the time needed to cross an edge. The results include the mean first passage time (MFPT) and the mean extreme first passage time (MEFPT) (i.e. the time for the first particles among N to arrive) computed for d = 25, N = 1000. In both cases, the target node is located at a distance of 25 edges from the source node. The edges of the reconstructed ER have a mean length of 0.5 μm.)

model type	${\tau }_{\mathrm{wait}}(\hspace{0.17em}\mathrm{ms})$	τswitch	τtubule	MFPT d = 25 (min)	MEFPT (s)
undirected	100	NA	0	19	30
AFN	100	30	0	20	33
AFN	100	300	0	19	15
AFN	100	3000	0	45	47

(a). Packet transportation guarantees a reliable delivery

The AFN allows the redistribution of many particles by redundant packets, a mechanism that guarantees a reliable delivery of enough proteins to their final destination. Indeed, when enough molecules are transported at the same time, the biological function that has required this protein synthesis can be performed with a high probability, as opposed to a transport where molecules would arrive one at a time and thus could be lost during the non-targeted transportation. Moreover, proteins undergo maturation while travelling in the ER, a process which requires a variable time. Delivering proteins in groups ensures the presence of at least some correctly folded and matured copies. Delivery in groups can also play an important role at ER exit sites, where ensembles of particles are segregated and exported by COPII-coated vesicles [22].

Particle synchronization is achieved by transiently freezing their motion in capture states. This mechanism impacts the transport performance because it effectively slows down the particle redistribution. Yet, numerical simulations (figure 5) show that with the help of the unidirectional properties of the network nodes, particles actually explore the network faster because backtracking probability is reduced (as discussed in the electronic supplementary material). This faster exploration of the network can thus compensate for the slowdown caused by capture states. In a wide range of parameter values, the active network can exhibit packet transportation and at the same time have a redistribution time scale comparable to that of an undirected diffusive network. Even if packet transportation does not imply a performance hit, this active transport requires energy in the form of ATP consumption [6]. The exact mode of transportation in the ER remains open, although the range of switching time scale of tubule flow has been estimated to be within 30 ms to 3 s [6]. Such range should be further explored.

Moreover, the present analysis predicts (electronic supplementary material, S2 and figure S4–S5) that packets persist at steady state. This implies that after waiting a long time after the release of material from a source node, it should still be possible to see a non-uniform distribution in nodes. In particular, most of the nodes of the ER network should be almost empty and only a small fraction of nodes (attractor points, electronic supplementary material, figure S4) should contain most of the particles.

(b). Active flow network: a theory to reconcile different types of experiments

Under the classical FRAP experiments, material redistribution in the ER was mostly viewed as driven by diffusion [5,32] and the dynamics are characterized by the overall diffusion coefficient D_ER. This model is relevant at coarse spatio-temporal scale, where the dynamics in tubules can be neglected. At a smaller scale, single-particle tracking experiments have revealed the presence of an active flow in tubules. Moreover, photoactivation experiments have shown a fast and unexplained propagation at large distances [6]. These two statements seem in contradiction with the diffusion framework. However, based on the single-particle trajectories observations [6], the present model exhibits at a small time scale the behaviour of an active flow, and at a large time scale, it can reproduce the properties of diffusion, thus effectively reconciling the FRAP diffusion-based approach with the presence of an active flow. Moreover, we have shown here that switching directionality generates a novel type of dynamics, characterized by motion in synchronous packets. This type of motion cannot be observed at the coarse spatio-temporal scale of FRAP experiments, nor it can be measured within a small set of single-particle trajectories. We suggest that high-resolution photoactivation experiments should be able to resolve the appearance of hot spots where material is concentrated, as we predicted here by the AFN model.

(c). Transport in other active networks

The network redistribution we explored here share some similarities with other networks. For example, the one of the slime mould Physarum polycephalum [33], which grows as a random network of tubes, where peristalsis is used to drive internal cytoplasmic flows. However, the scale here is very different: for the slime mould, the scale is of a few millimetres and the fluid in tubules is modelled by continuous Stokes equations to link velocity and pressure. The fluid oscillates back and forth across the slime mould network with a time scale of 100 s. Monitoring the phase of contraction in each tubule revealed that contractions are cycling over time: interestingly, simulation results [33] revealed that the phase spatial gradient of this contraction is linear across the organism. Thus a peristaltic wave can redistribute materials within the entire organism. However, in the ER, peristalsis could only be used to generate the fast motion in tubules. The physical mechanism of retaining particles in the ER node remains to be understood.

Another ER related subcellular network is the one generated by mitochondria [34,35]. By fusing and dividing, they can form tubular networks in eukaryotic cells. How material gets redistributed in such network remains unclear, but it is possible that the different electrochemical membrane potential between each individual could contribute to creating a local electric field, influencing the redistribution of ions and electrons and any charged proteins.

To conclude, the present study suggests that in the ER, the switching rate of tubule orientation is a key modulator for material redistribution. The strategy of the ER network invading most of the cytoplasm is to redistribute packets of proteins at a time scale of a few seconds. It would be interesting in future works to analyse this redistribution with respect to the exit site distribution [22], a key structure connecting ER with other organelles.

Data accessibility

The codes are available upon reasonable request to the corresponding author. Simulations are performed in Python. Codes will be available in the future from our repository website: http://bionewmetrics.org/ and https://doi.org/10.5281/zenodo.3893116.

Competing interests

We declare we have no competing interests.

Funding

M.D.'s PhD thesis is funded by ANR-18-NEUC-0001-01. This research is part of ERC-Adv OrganellenanoComp 2019.