Topological damping in an ultrafast giant cell

Abstract

Cellular systems are known to exhibit some of the fastest movements in biology, but little is known as to how single cells can dissipate this energy rapidly and adapt to such large accelerations without disrupting internal architecture. To address this, we investigate Spirostomum ambiguum-a giant cell (1-4 mm in length) well-known to exhibit ultrafast contractions (50% of body length) within 5 ms with a peak acceleration of 15[Formula: see text]. Utilizing transmitted electron microscopy and confocal imaging, we identify an association of rough endoplasmic reticulum (RER) and vacuoles throughout the cell-forming a contiguous fenestrated membrane architecture that topologically entangles these two organelles. A nearly uniform interorganelle spacing of 60 nm is observed between RER and vacuoles, closely packing the entire cell. Inspired by the entangled organelle structure, we study the mechanical properties of entangled deformable particles using a vertex-based model, with all simulation parameters matching 10 dimensionless numbers to ensure dynamic similarity. We demonstrate how entangled deformable particles respond to external loads by an increased viscosity against squeezing and help preserve spatial relationships. Because this enhanced damping arises from the entanglement of two networks incurring a strain-induced jamming transition at subcritical volume fractions, which is demonstrated through the spatial correlation of velocity direction, we term this phenomenon "topological damping." Our findings suggest a mechanical role of RER-vacuolar meshwork as a metamaterial capable of damping an ultrafast contraction event.

Keywords: organelle topology; ultrafast biophysics; vertex model.

Fig. 1.

Vacuolar meshwork inside a giant cell S. ambiguum undergoing ultrafast contractions, leading to organelle deformations. (A) Color differential interference contrast (DIC) images of the same S. ambiguum cell before and after an ultrafast contraction event, depicting $\sim$50% length shortening in less than 10 ms while maintaining cell volume. The cytoplasm is filled with vacuoles packed at high density (Scale bar, 200 $\mathrm{\mu }$m). (B) Single-cell contraction under a high-speed camera reveals large-scale deformation (40% deformation from spherical to ellipsoidal) in vacuoles under compressive strain in the first 3.2 ms of the contractile process. The red box marks one vacuole before contraction, while the blue box marks the same vacuole after a single contraction event. The image sequence of the detailed shape change under this compressive load for the same vacuole is shown in (C) at a 400 $\mathrm{\mu }$sec, or 0.4 msec interval. Also see Movies S1 and S2 (Scale bar, 20 $\mathrm{\mu }$m). (D) The histogram of the aspect ratio of deforming vacuoles in relaxed and contracted organisms quantitatively establishes that the vacuoles are indeed deformed due to the ultrafast contraction process and relax back to minimum energy spherical shape at a longer time scale. The vertical dashed line indicates 40% deformation (relaxed: mean = 1.22, std = 0.19, 1,902 measurements from three organisms; contracted: mean = 1.55, std = 0.35, 344 measurements from three organisms). (E) The histogram of the angle between the long axis of the vacuoles and the long axis of the cell body before and after the contraction depicts that the deformation is normal to the compressive stress. Here, we define the direction of the long axis of the cell body as the $x$-direction, and the direction of the short axis as the $y$-direction (relaxed: mean = 96.1${}^{°}$, std = 50.8${}^{°}$, 1,902 measurements from three organisms; contracted: mean = 87.6${}^{°}$, std = 28.9${}^{°}$, 344 measurements from three organisms).

Fig. 2.

Intimate spatial relationship between RER and vacuolar meshwork. (A) TEM imaging of S. ambiguum before (Upper) and after (Lower) ultrafast contraction shows the vacuolated cytoplasm, and the RER is intimately wrapping around the vacuoles in both cases. Refer to SI Appendix, Fig. S1 for low magnification images. (B) Cross-section montage of a relaxed organism and its segmentation of RER (green), vacuoles (gray), and mitochondria (yellow). The entangled topology between RER and vacuoles spans the entire cross-section, not limited to a specific region. The inset on the upper right corner shows the original image. Width of original image: 65.287 $\mathrm{\mu }$m. The histogram on the lower left corner shows the distance between RER and nearby vacuoles in contracted and relaxed organisms under TEM. In the relaxed organism, the distance ranges from 30 to 70 nm (343 measurements, mean $\pm$s td: 51.9 $\pm$ 14.6 nm), while in the contracted organism, this distance becomes more variable but remains below 150 nm (269 measurements, mean $\pm$ std: 82.1 $\pm$ 52.9 nm). (C) TEM occasionally reveals the “bridging” RER (arrows) that connects the RER of two nearby vacuoles. This is expected as RER is a continuous lumen. RER is false colored in red and vacuoles in yellow on the Top figure. (D) Extreme tram-tracking of RER along the contour of vacuoles was observed, even when cytoplasm invaginated inward to the vacuoles (arrows). (E) At higher magnification, we can see some materials (white arrows) that might be the protein connections between RER and vacuoles. (F) Confocal imaging of endoplasmic reticulum (ER) and vacuolar meshwork of a contracted organism (Left) and a cropped magnified region (Upper right). Note that the image is a longitudinal section. The image shows a fenestrated web-like structure of ER wrapping around vacuolar meshwork throughout the entire organism. Refer to Movies S3–S5 for complete z-stack video. (G) 3D schematic drawing illustrating the entangled topology between fenestrated ER and vacuoles.

Fig. 3.

A 2D vertex-based entangled soft particle model allows us to study the effect of topological constraints on the mechanical properties of packed vacuoles in response to large shape changes. (A) We define filament fraction (${\varphi }_{\mathrm{f}}$) as the ratio between the number of strings ($N_{\text{f}}$) and the number of edges in the Delaunay diagram ($N_{\text{D}}$), which we use to describe the degree of topological constraints in the system. The volume fraction ($\varphi$) is defined as the ratio between the total area of particles and the area enclosed by the boundary. See SI Appendix, Section A.7.2. (B) We consider 10 energy terms in the model following the recommendation from Boromand et al. (33) with appropriate extensions to include the entangled strings. The system evolves according to an overdamped molecular dynamic scheme. See SI Appendix, Fig. S3. (C) We curated 10 dimensionless numbers from the known contraction kinematics, geometry, fluid properties, and mechanical properties of lipid bilayer membranes, and all simulation parameters are determined to match all 10 dimensionless numbers. The process ensures dynamic similarity and allows us to convert simulation results back to actual physical units and compare them with experiments. (D) Three snapshots of a simulated contraction process. The system displayed here has $\varphi =0.79$ and ${\varphi }_{\mathrm{f}}=0.268$. We report this particular simulation as it is the one which matches the experimental observation the best (Fig. 4). A constant boundary force is applied to deform the system during the contraction phase, with $\varphi$ kept constant. The red box marks one particle before contraction, while the blue box marks the same particle after contraction. The image sequence of the shape changes under this load for the sample particle is shown in (E) at a 625 $\mathrm{\mu }$s, or 0.625 ms time interval. Scale bar = 100 $\mathrm{\mu }$m in (D), while scale bar = 20 $\mathrm{\mu }$m in (E). Also see Movies S6 and S7.

Fig. 4.

Geometric constraints turn entangled particle systems into topological dampers in constant force simulation. Systems with the same $\varphi$ of 0.79 but varying ${\varphi }_{\mathrm{f}}$ were compressed by a constant boundary force. (A) Normalized body length over time. The gray zone indicates the experimentally observed final body length after contraction (40%–50%) (7). The system with ${\varphi }_{\mathrm{f}}=0.268$ falls within the range. (B) Normalized contraction rate in $x$-direction over time. The dashed line indicates the experimentally observed peak velocity (0.2 m/s) (7). Systems with ${\varphi }_{\mathrm{f}}\ge 0.268$ can rapidly dampen the velocity, while systems with lower ${\varphi }_{\mathrm{f}}$ have their contraction velocity at peak level for roughly 3 ms. (C) Normalized reactive forces in $x$-direction over time. The dashed line indicates the magnitude of external load, normalized to 1. Increasing topological constraints increases the ability of the system to counteract external forces, which explains the rapid slowdown in contraction kinematics. (D) Squeeze flow viscosity of the system as a function of time. The dotted line is the estimation of the actual organism based on the peak strain rate and normal stress in $x$-direction (0.25 Pa-s, see SI Appendix, Section A.7.8). Systems with ${\varphi }_{\mathrm{f}}$ of 0.268 and 0.322 maintain comparable squeeze flow viscosities to experimental estimations, while systems with ${\varphi }_{\mathrm{f}}<0.268$ fail to maintain their viscosities despite the initial peak. (E and F) Energy budget of two systems with the same $\varphi$ of 0.79 but ${\varphi }_{\mathrm{f}}$ of 0.0 and 0.268. The cumulative net energy input is plotted in the black line, while other colored lines indicate different energy terms. The system with no entanglement has a larger cumulative net energy input. It rapidly responds to the energy input but is not able to store the energy for greater than 2 ms. Systems with entanglement reduce the cumulative net energy input to one-third and also have a better ability to buffer the energy input. Note that the majority of the energy contribution comes from the particle interfacial tension term, and the contributions from strings or string rupture are minimal.

Fig. 5.

The phenomenon of topologically assisted strain-induced jamming. The entangled strings create more contact with neighboring particles, making the particles overly constrained and jammed more easily. (A) Preserved neighboring relationship after contraction for two systems. The neighbors are identified as the edges on the Delaunay diagram. Newly formed neighbors are pink, while preserved neighbors are gray. The left figure (${\varphi }_{\mathrm{f}}=0$) preserves 19.6% of neighbors, while the right figure (${\varphi }_{\mathrm{f}}=0.268$) preserves 54.0%. $\varphi =0.79$ for both systems. Scale bar = 100 $\mathrm{\mu }$m. (B) The preserved neighboring relationship of the system over time. Increasing topological constraints helps preserve the neighboring relationship after contraction. (C) Averaged total contact number of the particles over time for systems with different ${\varphi }_{\mathrm{f}}$. Systems with ${\varphi }_{\mathrm{f}}\ge 0.161$ undergo strain-induced jamming transition as the averaged total contact number is above 4 (39). (D and E) Velocity unit vectors of vacuoles in relaxed (D) and contracting (E) organisms. In relaxed organisms, vacuole movements are disorganized, indicating poor spatial correlation. In the contracting organisms, the vacuoles on both ends are directed toward the center, and the vacuoles near the middle are directed away from the center in perpendicular direction, showing a strong spatial correlation. Scale bar = 100 $\mathrm{\mu }$m for both. (F) Velocity unit vectors of particles for the same system as in (A) at the end of contraction. The velocity unit vectors for the system with no entanglement (Left) show less spatial correlation, while the ones for the system with entanglement (Right) show more spatial correlation. Scale bar = 100 $\mathrm{\mu }$m. (G) Spatial correlation of velocity direction as a function of normalized intervacuole distance (mean $\pm$ SE). Jammed cases in simulations and contracting organisms (three organisms, total 118 vacuoles) exhibit slower decay in spatial correlation, with negative correlation when normalized distance exceeds 0.3 due to confinement. The unjammed case shows a faster decay in spatial correlation and a negative correlation at larger distances. Relaxed organisms (three organisms, total 182 vacuoles) show fast decay in spatial correlation at short distances and nearly zero correlation as the normalized distance goes beyond 0.1. These indicate that the vacuolar meshwork is not jammed in relaxed organisms but jammed in contracting organisms, constituting a strain-induced jamming transition (Normalized distance where spatial correlation = 0.25 (mean $\pm$ SE): ${\varphi }_{\mathrm{f}}=0.268$: 0.148 $\pm$ 0.006; ${\varphi }_{\mathrm{f}}=0$: 0.071 $\pm$ 0.007; experiment (contracting): 0.147 $\pm$ 0.013; experiment(relaxed): 0.040 $\pm$ 0.007) (SI Appendix, sections A.9 and C, and Fig. S12).

Fig. 6.

We used 45 table tennis balls and fabrics to demonstrate the concept of topologically assisted strain-induced jamming. Two systems were compared, with an equal number of table tennis balls and fabrics. One system was arranged in an entangled topology [lower in (A)], while the other was not [upper in (A)] (see Materials and Methods for details). (B) After applying a 2,450-g external load, the entangled system showed superior resistance to the load. Refer to Movies S8 and S9. (C) Quantitative differences between the entangled and untangled systems are demonstrated through compression testing. Mean curves of three samples in each group are shown as solid lines, while individual experimental data are transparently displayed. The system’s reactive forces are normalized by the reactive forces of an individual table tennis ball when compressed by 5 mm (mean $\pm$ SE = $152.1\pm 0.7$ N). At a compression of 95 mm (dashed vertical line), corresponding to the onset of unavoidable table tennis ball deformation, a rapid increase in reactive forces was observed in all traces. Refer to Movies S10 and S11.