Figures and data
Sketch of the cellular geometry with nomenclature of the sub-cellular structures discussed in the paper. a. Cross section of cell showing nucleus and Endoplasmic Reticulum (ER) (adapted from image in public domain [40]). b. Cut through cross section of the tubular ER network at the edge of cell. c. Sketch of the contraction and expansion of the tubular junctions (3D view and cross section); contractions leads to flow leaving the junction into the network while expansions lead to flow leaving the network and entering the junction. d. Contraction and expansion of the peripheral sheets. e. Contraction and expansion of the tubules driven by pinching (3D view and cross section). f. Contraction and expansion of the perinuclear sheets.
A quantitative test of the pinching-tubule hypothesis. a. Cross-sectionally averaged flow velocities in a typical edge as obtained in our simulations. b. c. Histograms of instantaneous speeds (b) and edge traversal speeds (c) using data from simulations in the C0 network with flow (blue) and with just diffusion (red). The insets in (b) and (c) illustrate the distributions of instantaneous speeds and average edge traversal speeds respectively as experimentally measured in Ref. [11]. The symbols indicate the values taken by the probability mass function and the curves are log-normal distributions fitted to all average edge traversal speeds obtained. d-f. Histograms of average edge traversal speeds obtained from simulations in networks C1-C4 from Fig. 9d with flow (d) and only diffusion (e) and from simulations in the regular honeycomb network with active flows (f). The inset in (f) illustrates the honeycomb geometry. Points indicate mean ± one standard deviation over the four networks (C1-C4) of normalised frequencies in each speed range; curves are log-normal (d-f) or normal (f) distributions fitted to all average edge traversal speeds for each set of pinch parameters. The means of the original simulation results and of the fitted distributions are indicated in the legends in each of (c-f).
Impact of non-zero slip length on transport and flow. (a) Distributions of average edge traversals speeds in simulations of a C1 network pinching with the original pinch parameters as measured in Ref. [11] and used in Fig. 2, for different slip lengths λ = 0, 3, 30 and 300 nm. (b) Longitudinal flow profile u(r) inside a cylindrical tubule for different slip lengths, all with the same volume flux Q; an increase of the slip length leads to a redistribution of the flow in the cross section.
Mixing by active pinching flows. a. Initial configuration of blue and red particles in honeycomb network. The strips used to quantify mixing are illustrated in black dotted line. b-d. The configuration after t = 3 s of mixing in a passive network with no flow (b), an active network pinching with the original pinch parameters (c), and an active network pinching with maximally long pinches at 10 times the original rates (d). e. The measure of mixing Var(ϕ(t)) against t for the passive network (blue), the network pinching with the original parameters (red), and the network pinching with maximally long and 10x faster pinches (yellow).
Illustration of coordination mechanism allowing the interactions between two pinch in series to induce the net transport of a suspended particle; the mechanism is akin to small-scale peristalsis.
Histograms of instantaneous speeds (top) and average edge traversal speeds (bottom), for a-b. an active honeycomb network (a) and the reconstructed C0 network from Fig. 9a-c (b) with pinch parameters 
Contour plots of the mean values of the average edge traversal speeds obtained from simulations of our model in a junction- and tubules-driven C1 network from Fig. 9 with different values of (α, β) and with contraction volumes ΔV expelled during each contraction drawn from: a. the normal distribution estimated for the junction volumes N(0.0045, 0.0021) (in μm3); b. two thirds the estimated normal distribution for the junction volume; and c. half the estimated distribution for the junction volume. Thick solid black lines indicate the mean of the average edge traversal speed distribution reported in Ref. [11] (45.01 μm/s) and thick dotted black lines indicate mean ± standard deviation (45.01 ± 12.75 μm/s).
text on the next page FIG. 8: a-b. Distributions of instantaneous speeds (a), and average edge traversal speeds (b) obtained from simulations of the C1 network from Fig. 9 driven by the contraction of a perinuclear sheet. In these simulations the sheet undergoes one contraction+relaxation lasting 2T = 5 s, and expels a volume Vsheet = 10 μm3 of fluid during a contraction. c-d. Colour maps of normalised average edge traversal speeds obtained from simulations of the C1 network from Fig. 9 driven by contraction of tubules + sheets (c) and junctions + tubules (d) respectively. e. The speeds averaged along the y direction of the network, V (x), are plotted against x, to effectively project the information onto one dimension from c. (blue solid line) and d. (red dashed line). f-g. Histograms of average edge traversal speeds (dots) and normal fits (lines) and mean AETS (inset) in C1 networks with parameters adjusted as follows to approximate a network with peripheral sheets: Junction k expels a volume Vk/6 of fluid in each pinch, the pinches are α−1 = 2.5 times slower than the original tubule pinches, and all nodes actively pinch (f); and node k expels a volume Vk/2 of fluid in each pinch, the pinches are α−1 = 5 times slower than the original tubule pinches, and only a third of the nodes actively pinch (g).
a. Skeleton image of COS7 ER reproduced from Ref. [11]. b. Model ER graph (blue solid lines) reconstructed from (a) using ImageJ. c. Experimental images in (a) superimposed with mathematical model from (b). d. Microscopy images of four different COS7 ER networks (labelled C1-C4) with reconstructed model networks (blue solid lines) superimposed. e. Distributions of edge lengths in the C0-C4 networks. Bottom right: mean edge lengths, mean degrees (i.e. number of edges connected to a node) and number of nodes of the C0-C4 networks.
Mathematical model of a pinching tubule. The tubule has a radius of R outside the pinch, a(z, t) in the pinch (where z is the axial coordinate), and b(t) at its narrowest point i.e. the centre of the pinch. The portion of the tubule before the tubule has length L1 while that after has length L2; the pinch is symmetric and has a length 2L. Q1, Q2, Q3, and Q4 denote the volume fluxes through the tubule in the four different regions as indicated. The pressures at the end of the tubule are p = p1 at z = 0 and p = p2 at z = L1 + 2L + L2.
Elements of graph theory required to model the ER network. a. A graph G (black solid lines) and its spanning tree T (black dots). b. The unique cycle Ce (red) formed by adding an edge e ∈ G\T to T. c. A breadth-first search (BFS) starting at the rightmost node; the graph is explored in the order red, green, blue.
Illustration of the C1 network from Fig. 9 with M1 = 13 exit nodes (red squares) and M2 = 9 perinuclear sheet nodes (blue asterisks).
Estimation of areas of peripheral sheets, taken as the regions encircled in yellow.
Mathematical idealisation of two contracting peripheral sheets as two paraboloids for the purpose of computing an order-of-magnitude estimate of the energy expended to contract a peripheral sheet.
