Fluid mechanics of luminal transport in actively contracting endoplasmic reticulum
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
- UK Dementia Research Institute at University of Cambridge, Department of Clinical Neurosciences, University of Cambridge, United Kingdom
Figures
Figure 1
Sketch of the cellular geometry with nomenclature of the subcellular structures discussed in the paper.
(a) Cross-section of cell showing nucleus and endoplasmic reticulum (ER) (adapted from image in public domain). (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.
Panel A adapted from image in public domain (Ruiz Villarreal, 2006).
Figure 2
A quantitative test of the pinching-tubule hypothesis.
(a) Cross-sectionally averaged flow velocities in a typical edge as obtained in our simulations. 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 Holcman et al., 2018. 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. Histograms of average edge traversal speeds obtained from simulations in networks C1–C4 from Figure 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 ± 1 SD 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).
Figure 3
Impact of non-zero slip length on transport and flow.
(a) Distributions of average edge traversal speeds in simulations of a C1 network pinching with the original pinch parameters as measured in Holcman et al., 2018 and used in Figure 2, for different slip lengths , and 300 nm. (b) Longitudinal flow profile inside a cylindrical tubule for different slip lengths, all with the same volume flux ; an increase of the slip length leads to a redistribution of the flow in the cross-section.
Figure 4
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. The configuration after 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 against for the passive network (blue), the network pinching with the original parameters (red), and the network pinching with maximally long and 10× faster pinches (yellow).
Figure 5
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. Red dot indicates the position of a particle on the tubule’s centreline at each step of the coordination mechanism.
Figure 6
Impact of spatiotemporal pinch parameters on transport.
Histograms of instantaneous speeds (top) and average edge traversal speeds (bottom), for an active honeycomb network (a) and the reconstructed C0 network from Figure 9a–c (b) with pinch parameters and decreased to times the original values from Holcman et al., 2018, and the same measured diffusivity . Histograms of instantaneous speeds (top) and average edge traversal speeds (bottom) for the C1–C4 networks from Figure 9d with varying pinch parameters: original parameters from Holcman et al., 2018 (c); pinch length increased to the total length of the tubule (d); a fivefold increase in the rate of pinching and pinch length set to the total length of the tubule (e); a tenfold increase in the rate of pinching and pinch length set to the total length of the tubule (f). Bottom rows: points indicate mean ± 1 SD over the four networks (C1–C4) of normalised frequencies in each speed range; curves are log-normal distributions fitted to all average edge traversal speeds for each set of pinch parameters; insets show means of original simulation results and of fitted distributions.
Figure 7
Transport driven by contracting tubular junctions.
Contour plots of the mean values of the average edge traversal speeds obtained from simulations of our model in a junction- and tubule-driven C1 network from Figure 9 with different values of and with contraction volumes expelled during each contraction drawn from (a) the normal distribution estimated for the junction volumes (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 Holcman et al., 2018 (45.01 µm/s) and thick dotted black lines indicate mean ± SD (45.01 ± 12.75 µm/s).
Figure 8
Transport driven by contracting perinuclear sheets or peripheral sheets.
Distributions of instantaneous speeds (a) and average edge traversal speeds (b) obtained from simulations of the C1 network from Figure 9 driven by the contraction of a perinuclear sheet. In these simulations, the sheet undergoes one contraction + relaxation lasting s, and expels a volume µm3 of fluid during a contraction. Colour maps of normalised average edge traversal speeds obtained from simulations of the C1 network from Figure 9 driven by contraction of tubules + sheets (c) and junctions + tubules (d), respectively. (e) The speeds averaged along the direction of the network, , are plotted against , 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 expels a volume of fluid in each pinch, the pinches are times slower than the original tubule pinches, and all nodes actively pinch (f); and node expels a volume of fluid in each pinch, the pinches are times slower than the original tubule pinches, and only a third of the nodes actively pinch (g).
Figure 9
Model ER networks and their statistics.
(a) Skeleton image of COS7 ER reproduced from Supplementary figure 3 of Holcman et al., 2018. (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.
Figure 10
Mathematical model of a pinching tubule.
The tubule has a radius of outside the pinch, in the pinch (where is the axial coordinate), and at its narrowest point that is the centre of the pinch. The portion of the tubule before the tubule has length while that after has length ; the pinch is symmetric and has a length . and denote the volume fluxes through the tubule in the four different regions as indicated. The pressures at the end of the tubule are at and at .
Figure 11
Elements of graph theory required to model the ER network.
(a) A graph (black solid lines) and its spanning tree (black dots). (b) The unique cycle (red) formed by adding an edge to . (c) A breadth-first search (BFS) starting at the rightmost node; the graph is explored in the order red, green, blue.
Figure 12
Illustration of the C1 network from Figure 9 with exit nodes (red squares) and perinuclear sheet nodes (blue asterisks).
Figure 13
Estimation of areas of peripheral sheets, taken as the regions encircled in yellow.
Figure 14
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.
Videos
Video 1
Flows in an active C0 network pinching with the original pinch parameters.
Edges are colour-coded with magnitude of instantaneous flow.
Video 2
Flows in an active honeycomb network pinching with the original pinch parameters.
Edges are colour-coded with magnitude of instantaneous flow.
Video 3
Mixing in time in a passive honeycomb network with no flow.
Video 4
Mixing in time in an active honeycomb network pinching with the original pinch parameters.
Video 5
Mixing in time in an active honeycomb network pinching with maximally long pinches at 10 times the original rates.
Additional files
Download links
A two-part list of links to download the article, or parts of the article, in various formats.
Downloads (link to download the article as PDF)
Open citations (links to open the citations from this article in various online reference manager services)
Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)
Fluid mechanics of luminal transport in actively contracting endoplasmic reticulum
eLife 13:RP93518.
https://doi.org/10.7554/eLife.93518.3
