Our research complies with all relevant ethical regulations and is approved by IFOM ETS and the University of California San Diego.Cell cultureImmortalized MEFs derived from RPTP α + /+ murine background58 were grown in complete media composed of DMEM (Lonza) supplemented with 10% Fetal Bovine Serum (FBS South America, Euroclone) and 2 mM L-glutamine at 37 °C and 5% CO2. For imaging experiments, MEFs were seeded on borosilicate glass coverslips of 1½ thickness (Corning) or Nunc Glass Base Dishes (Thermo Fischer Scientific) coated with sterile 10 µg ml−1 fibronectin (Roche). The supplier and identifier for all reagents are listed in Supplementary Table 5. Cytoskeletal drug perturbations were performed by supplementing complete media with 10 µM Blebbistatin and 100 nM Jasplakinolide (Merck) for 3-4 h. Other drugs were tested at different concentrations for the same incubation time, specifically: CK666 (100 µM), SMIFH2 (20 µM), Latrunculin A (1 µM), Cytochalasin B (1 µM), and Cucurbitacin E (5 nM). Transient expression of the plasmids listed in Supplementary Table 5 was achieved by Neon Transfection System Microporator (Thermo Fischer). Briefly, 1 × 106 cells were trypsinized, washed once with PBS, and mixed with a total of 10 μg of recombinant DNA in electroporation buffer R (Invitrogen). Cells were singularly electroshocked at 1600 mV for 20 ms by placing the electroporation tip into the column filled by E2 buffer (following the manufacturer’s specifications). Cells were seeded and allowed to recover for 24 h on plastic dishes or glass coverslips.Genome-edited MEF cell lines were generated by CRISPR-Cas9 technology. Specifically, cells were transiently transfected with the plasmid pSpCas9(BB)−2A-GFP (PX458) (Addgene) encoding Cas9 and GFP reporter for positive transfection. After 24 h, cells expressing Cas9 were electroporated with 1 nmol of synthetic single-guide RNA directed against exon 3 of the murine gene sptbn1 (Invitrogen). With the help of IFOM cell sorting and cell culture facilities, after 48 h from the initial transfection GFP positive cells were sorted by arbitrarily choosing an intermediate level of fluorescent intensity and cultured at clonal dilution in 96-well plates. Clonal populations were expanded and successful KO screened by western blots for protein expression. Among the 50 clones screened, we selected clones 8, 9, and 10 as sptbn1 KO and clone 15 as sptbn1+/+ control for Cas9 off-target effects.Erythrocytes and hiPSC-derived cortical neuronsErythrocytes were obtained from 6-12 weeks old C57/J male mice (Charles River), immediately after euthanasia in accordance with the guidelines established in the Principles of Laboratory Animal Care (directive 86/609/EEC) and were approved by the Italian Ministry of Health. Erythrocytes were washed in PBS supplemented with 10 mg ml−1 glucose, harvested by centrifugation at 300 × g for 5’, resuspended at the desired concentration, immobilized, and fixed with 4% PFA in PBS on PLL-coated glass coverslips for immunofluorescence investigations as reported27.hiPSC-derived cortical neurons were generated by the IFOM Cell Culture facility following the NGN2-inducible system described in ref. 59. Briefly, NGN2 hiPSC cells were cultured on Matrigel-coated glass coverslips in Induction Media (DMEM, 2 mM L-glutamine, 10 mM NEAA) supplemented with 2 µg ml−1 Doxycycline and 10 µM Y27632. On day 1, after seeding, Y27632 was withdrawn from the media. At day 3, hiPSC were detached and seeded again on Matrigel-coated coverslips in Cortical Neuron Culture Media (Neurobasal (Thermo Fischer), 2 mM L-glutamine, 10 ng ml−1 NT-3, 10 ng ml−1 BDNF, 10 µg ml−1 laminin) containing 2 µg ml−1 Doxycycline and cultured up to 21 days by exchanging the media every 48 h. At the desired time points, coverslips were washed in PBS and fixed with 4% PFA for immunofluorescence investigations.MicropatterningBorosilicate glass coverslips (Corning) were washed for 1 h with 20% Acetic Acid, rinsed in Milli-Q water, and stored in 90% ethanol. When needed, coverslips were air-dried and activated by a plasma cleaner (Harrick Plasma) for at least 3 min. The surface was passivated by incubation with 0.1 mg ml−1 poly(ethylene glycol)-b-poly(l-lysine) (PEG–PLL, Ruixibio) for 1 h at room temperature to prevent fibronectin coating. A quartz mask (Delta mask B.V.) was washed with isopropanol and activated under UV light for 7 min (UVO Cleaner, Jelight). PEGylated coverslips were aligned to the desired pattern in the mask, and illuminated under UV light for 7 min. The quartz layer prevents UV illumination of the passivated surface while the photolithography-made pattern allows the light to pass, burning the PEG–PLL. Patterned coverslips were then coated with 10 μg ml−1 fibronectin for 1 h at room temperature, while the passivated surface prevented fibronectin adherence. After rinsing the coverslips several times with PBS, MEFs were seeded at the desired cell density and cultured at 37 °C in the same media described before.Membrane fragility assay96-well black bottom plates were coated with 10 µg ml−1 fibronectin, cells seeded at 2 × 103 cells per well density and left undisturbed for 24 h. Membrane fragility was evaluated by performing CellToxTM Green cytotoxicity assay (Promega) under different osmotic conditions in Ringer’s buffer (see Supplementary Table 5 for buffer composition). This assay records the increase in fluorescence of the dye that can penetrate the cell and bind to the DNA only in the presence of a damaged plasma membrane. Otherwise, the dye remains extracellular and at low fluorescence emission. Before the beginning of the experiment, all wells and cells were equilibrated in Ringer’s 1x for 30 min. A replica 96-well plate was prepared by dissolving CellToxTM Green dye 1:1000 in Ringer’s 1x, 1.5x, and 0.5x solutions, including positive control conditions where media was supplemented with 30 µg ml−1 digitonin, and negative controls to record no-cell background. Immediately before the experiment, all the wells in the experimental plate were emptied and filled with the respective ones of the replica plate with the help of a multichannel pipette to limit the delay between the beginning of the osmotic shocks and the recording of fluorescence intensities. Fluorescence was measured at 485–500 nmEx/520–530 nmEm using a 96-well fluorescence plate reader equilibrated at 37 °C, recording fluorescence reads every 2 min for 3 h. Each experimental condition was recorded in triplicate, experiments were qualitatively assessed and discarded when positive controls treated with digitonin failed to show an immediate steep increase in fluorescence before reaching a plateau. The average of the three wells per condition was calculated and normalized to the initial time point. Values were analyzed and plotted with the software GraphPad Prism. Given the noisy nature of the single record, to avoid over-estimating or under-estimating plateau values at given time points, fluorescence intensity curves were fitted to the one-phase association equation to obtain estimates of the plateau fluorescence for each experimental condition.ImmunofluorescenceThe antibodies used in this study were the following: mouse anti-βII-spectrin (dilution 1:200, BD-bioscience), rabbit anti-βII-spectrin (1:200, Abcam), mouse anti-βI-spectrin (dilution 1:200, NeuroMab), mouse anti-αII-spectrin (1:200, Invitrogen), and rabbit anti-β-actin (1:100, Cell Signaling), rabbit anti-adducin (1:100, Abcam). Before fixation, cells were seeded on 10 μg ml−1 fibronectin-coated coverslips/glass base dishes. Cells were fixed in 4% paraformaldehyde for 10 min, then neutralized using 10 mM NH4Cl in PBS for 10 min. Alternatively, fixation was performed in ice-cold pure methanol for 2 min at − 20 °C. Cells were subsequently washed three times with PBS, permeabilized for 2–5 min using PBS containing 0.1% Triton X-100, and blocked with 3–5% BSA for 10 min at room temperature. Cells were incubated with primary antibody overnight at 4 °C. After 3 washing steps in PBS, cells were incubated with CF568/AttoN647-conjugated goat anti-mouse or anti-rabbit (1:100–1:400, Thermo Fischer Scientific) and Alexa 488/568-conjugated phalloidin (1:200, Sigma-Aldrich) for 1 h at room temperature. After three washes in PBS, cells were mounted with anti-fade glycerol-based media (for confocal microscopy) or PBS (for TIRFM and ExM) and stored at 4 °C. All primary antibodies and fluorophore-conjugated secondary antibodies are listed in Supplementary Table 5.Expansion microscopy (ExM)The procedure for ExM was adapted from the original report26. For completeness of information and reproducibility all sensible steps are described. The immunofluorescence procedure was performed as described, except for the fluorophores used: CF568-conjugated goat anti-mouse (1:100 Sigma-Aldrich) and AttoN647-conjugated goat anti-rabbit (1:100, Sigma-Aldrich). When ExM was performed on micropatterned coverslips before the immunostaining procedure, a quenching step of 2 min at − 20° with ice-cold methanol was required to prevent reactivity of the PEG-PLL surface coating with the amino-reactive Anchoring buffer.AnchoringUpon completion of the immunostaining procedure, the specimen on coverslips were incubated with the Anchoring buffer, consisting of PBS supplemented with 1 mM of the amino-reactive MA-NHS (Methacrylic Acid N-Hydroxysuccinimide Ester, Sigma-Aldrich). Specimens were incubated for 1-1.5 h on gentle rocking at room temperature.GelationThe gelation solution was composed of the Monomer stock (sodium acrylate 33 wt%, acrylamide 50 wt%, bis-acrylamide 1 wt%, 1.8 M NaCl, 1x PBS, Sigma-Aldrich), supplemented before gelation with 0.2 wt% of TEMED and Ammonium Persulfate (APS, Sigma-Aldrich). The gelation solution was supplemented with 0.01 wt% of 4-Hydroxy-TEMPO (Sigma-Aldrich) to slow down the reaction and allow complete diffusion inside the cells. Anchoring buffer and excess of MA-NHS was removed by 2x brief washes with PBS. Gelation was performed in a custom-made gelation chamber: 2 microscopy slides were coated with parafilm and spaced by silicone isolator gaskets of 0.5 mm thickness. Coverslips were placed on the bottom side of the chambers, biological specimens facing the inner side of the chamber, and excess of PBS buffer was removed. A Complete Gelation Solution was added and chambers were placed flat on ice for 30 min to favor homogeneous diffusion of acrylamide and avoid gelation artefacts. After this step, chambers were placed for 1 h at 37°.DigestionGelation chambers were carefully dismounted by removing the top slide, and the gel was trimmed with a razor blade. Gel shapes with no-mirror symmetry are useful to recognize the surface of the gel with the biological specimens located at the interface. Coverslips with trimmed gels still attached were placed in an opportune plastic dish and incubated with Digestion Buffer (50 mM Tris-HCl, 125 mM NaCl, 2.5 mM EDTA, 0.5% Triton X-100) supplemented with Proteinase K (1:100, New England Biolabs). Specimens were incubated on an orbital shaker at 37 °C per 2 h and 60 rpm. hiPSC-derived cortical neurons cultivated on Matrigel were digested at 30 °C Over Night and 60 rpm orbital shaking.ExpansionFor the physical expansion step, gels were transferred into bigger plastic dishes to ensure undisturbed planar expansion. The digestion buffer was replaced by Milli-Q water and incubated with gentle shacking. At least 4 washing steps of 30 min were performed. Expanded gels were kept at 4 °C in the dark.MountingAcrylamide hydrogels required immobilization on glass coverslips to avoid drift during volumetric imaging. Coverslips were plasma activated (Harrick Plasma) for 3 min and then coated with poly-L-Lysine for 1 h at room temperature, washed in Milli-Q water, and dried before letting the gel settle.TIRFM and confocal microscopyConfocal microscopy was performed on a Leica TCS SP8 laser-scanning confocal module mounted on a Leica DMi 8 inverted microscope, equipped with a motorized stage, and controlled by the software Leica Application Suite X (ver. 3.5.2.18963). For image acquisition, a HC PL APO CS2 63 × /1.40 oil immersion objective was used. DIC, epifluorescence (EPI), and total internal reflection fluorescence microscopy (TIRFM) of fixed specimens, live time lapses, and drug treatments were performed on a Leica AM TIRF MC system. Two different TIRFM-grade objectives were used: HCX PL APO 63 × /1.47NA oil immersion and HCX PL APO 100 × /1.47NA oil immersion. Three different laser lines were used for fluorochrome excitation: 488 nm, 561 nm, and 635 nm. A specific dichroic and emission filter set for each wavelength has been used. The microscope was controlled by Leica Application Suite AF software (ver. 2.6.1.7314), and images were acquired with an Andor iXon DU-8285_VP camera and 100–400 msec exposure times depending on signal intensity. For live imaging experiments, environmental conditions were maintained by an Okolab temperature and CO2 control system.STED microscopySTED super-resolution microscopy was performed on a Leica TCS SP8 laser-scanning confocal module mounted on a Leica DMi 8 inverted microscope, equipped with a 775 nm pulsed depletion laser, and controlled by the software Leica Application Suite X (ver. 3.5.2.18963). For image acquisition, a HC PL APO CS2 100 × /1.40 oil immersion STED WHITE objective was used. Biological samples were immunolabelled with the STED compatible secondary antibodies conjugated with Atto-594 and Atto-647N (Sigma-Aldrich), while F-actin was labeled with Phalloidin-Abberior STAR 635 (Merck). Samples were mounted in Mowiol anti-fade. Imaging was performed with a tunable pulsed White Light Laser set to the two excitation wavelengths of the fluorophores, and signal depletion was performed with a 770 nm pulsed depletion laser at 50–60% output power. Images were deconvolved by the software Huygens Professional, considering a theoretical PSF.Transferrin uptake and cell area analysisClonal cell lines were seeded on fibronectin-coated glass-bottom 24-well plates at the density of 1 × 104 cells per well and let them undisturbed for 24 h. Cells were serum-starved in DMEM for 30 min before supplementing them with DMEM, 10% FBS, 2 mM L-glutamine, and 50 µg ml−1 of rhodamine-labeled transferrin (Invitrogen). At different time points (0, 10, 20, and 30 min), excess of transferrin from the cell surface were stripped by a fast wash in 10x PBS, and fixed in 4% PFA. Cells were incubated in PBS containing 0.1% Triton X-100, 3–5% BSA, 1:100 Alexa 488-Phalloidin, and 1:5000 DAPI for 30 min at room temperature. After three washing steps of 5 min in PBS, imaging was performed on a Leica DMi8 Thunder Widefield microscope equipped with HC PL APO 40 × /0.95NA air objective, motorized stage, and camera at constant settings. Image analysis was performed in Fiji by creating two independent binary masks of the DAPI and phalloidin channels. Particles adjacent to the image border or that did not contain nuclei were excluded. Clusters of cells were manually segmented when possible and a final Analyze Particle tool was run in Fiji to segment objects comprised between 300 and 20000 µm2. Individual object area values were exported to GraphPad Prism for final analysis. Mean rhodamine-labeled transferrin intensity signals were obtained by first subtracting the background with the sliding paraboloid method in Fiji (rolling ball radius 50 pixels) and lastly by applying a Gaussian blur filter (radius 2 pixels). The image was multiplied by the binary mask generated to extract cell area values, and the mean rhodamine intensity for each individual cell was calculated. Values were exported to GraphPad Prism for statistical analysis and graphical representation.Fluorescence intensity distributionThe TIRFM dataset was analyzed to extract Intensity distribution in Fiji. Briefly, images were acquired at constant laser intensity, exposure time and no electronic gain between independent channels. Images were converted to 8-bit format and were down-sampled to the pixel size of 1 × 1- µm by interpolating the bilinear average. A binary cell mask was generated by the “Analyze Particle” tool, and a cell outline was added to the ROI manager. For each channel, the Histogram of intensity distribution was generated, and the count value matched the projected cell area in µm2. Extracted values were normalized by the total count to obtain a relative frequency and allow averaging between multiple cells.Clusters segmentationTIRFM images were processed by the tool “Subtract Background” in Fiji, and the method Sliding paraboloid with a rolling ball radius of 50.0 pixels was applied. Single-channel images of the 16-bit format were independently thresholded with the default method by imposing a cutoff value of 5%. This approach sets the 95% of the intensity distribution curve to 0, while the remaining 5% (P0.95) is set to max. The Despeckle and Smooth tools were run to homogenize the signal, and the Analyze Particle was used to obtain Area and Shape descriptors for each individual cluster. To extract the Normalized Cluster Area from TIRFM time lapses, the same approach was applied over the entire stack. The binary mask of the clusters was obtained by applying the threshold method with a cutoff value of 5% referred to the initial frames of the time lapses, and not by calculating the 5% threshold for each individual frame. The Despeckle and Smooth tools were run, and finally, the Analyze Particle tool (Fiji) was applied to create a binary mask of objects with size 0-Infinity. The resulting mean value of the entire 8-bit binary image represents a proxy of the cluster area and can be measured over the entire stack to determine temporal variations with respect to the initial frames (an increase in the mean equals an increase in the cumulative clusters area over the entire cell).Orientation analysisTo analyze signal orientation on the TIRFM dataset, the Fiji plugin OrientationJ was used to quantify the local orientation properties of an image, based on the structure tensor of a defined local neighborhood60,61. Briefly, images were background subtracted by the Sliding paraboloid method with a rolling ball radius of 50.0 pixels. The vector field was independently calculated for each channel by applying a Gaussian gradient of 5 pixels. Coherency maps, as well as overlaid vector images, were generated. The distribution of orientations was independently calculated for each channel by the Distribution tool of the plugin. The same Gaussian gradient of 5 pixels was applied, with no Min coherency (0%) set as default. Dominant Direction was instead calculated only for the actin channel, the resulting value for the single cell under investigation was set as 0°. All the other channels were aligned according to actin dominant direction. Coherency analysis during time lapses was performed with similar parameters. The sum of the two fluorescent channels was used to create a binary mask of cells over time by the Analyze Particle tool in Fiji. ROI for each frame was added to the ROI manager and used to calculate mean coherency values for both channels over the entire stack. Values were extracted, normalized to the initial frames and plotted using the software GraphPad Prism.Colocalization analysisThe Fiji plugin JACoP (Just Another Co-localization Plugin) was used to determine co-localization parameters between channels, which included Pearson’s correlation coefficient and Cross-Correlation coefficient with the 50-pixel shift. In particular, this function returns a series of Correlation coefficients when the two channels are shifted by ± 50 pixels between each other. Costes’ automatic threshold function was used to obtain the correlation coefficient in the function of the mask stringency (mCherry-MLC channel was used). Raw data were then plotted by using the software GraphPad Prism.Osmotic shocks and drugs washoutAssays were performed on custom-designed 2-way aluminum slides, sealed on the two planar faces by 22 × 22-mm glass coverslips welded by high vacuum grease (Sigma-Aldrich). Coverslips were acid-washed with a 20% HNO3 solution before being dried and coated with 10 µg ml−1 fibronectin. The chamber was rinsed with CO2-independent 1 × Ringer’s solution and equilibrated on the microscope stage at 37 °C. If required, Ringer’s solution was supplemented with the drug under investigation (see Supplementary Table 5 for buffer compositions). The top and bottom glass surfaces of the chamber allow simultaneous fluorescence and DIC illumination during media exchange. MEFs were transfected 24 h before the experiment with the opportune plasmid combinations; in case of drug perturbations, cells were treated for 3-4 h before the beginning of the imaging experiments. During time lapse acquisition, the custom-designed slides allowed the exchange of media, marking the beginning of the hypo-osmotic shock or the washout of the drugs.FRAP experimentsMEFs expressing GFP-βII-spectrin constructs were imaged 24 h after transfection with a confocal spinning disk microscope (Olympus) equipped with iXon 897 Ultra camera (ANDOR) and a FRAP module furnished with a 405- nm laser. The environmental control was maintained with an OKOlab incubator. Images were acquired using a 100 × /1.35Sil silicone oil immersion objective. MEFs were trypsinized and seeded on glass base dishes (Matek, Sigma-Aldrich) coated with 10 µg ml−1 fibronectin. Before imaging, CO2-independent 1 × Ringer’s solution was exchanged. Circular regions of interest of 3-5 µm diameter were photobleached with the 405- nm laser at 100% intensity, and post-bleach images were acquired with 15–20% laser intensity for 100 frames (1 frame every 3 s for full-length and truncated GFP-βII-spectrin constructs and every 0.5 s for PE/ANKbs only). FRAP data were analyzed, and curves fitted to the one-exponential recovery equations (one-phase association) by the software GraphPad Prism:$$I={I}_{0}+{I}_{\max }\left[1-{e}^{-{kt}}\right]$$
(1)
where I represent the relative intensity compared to the prebleach value, k is the association rate, and the half-time recovery is expressed in seconds.Sensitized FRET emissionImaging settingsFRET recordings were performed on cells transfected with cpst-βII-spectrin-FL, or cpst-βII-spectrin-ΔABD imaged through a Leica SP8 Confocal microscope. Briefly, three channels were sequentially recorded by a HyD detector at constant electronic gain between channels: donor, FRET, and acceptor channels. Donor excitation was achieved with an Argon laser at 458 wavelengths, and spectral detection bandwidths were set at 475–485 nm; FRET excitation with the same 458 nm laser and spectral detection bandwidths 515–525 nm; acceptor excitation with 514 nm laser and spectral detection bandwidths 520–600 nm. Images were acquired with a pinhole set to 1 AU in 8-bit and 512 × 512-pixel format.Inverted FRET index calculationRaw images were processed by applying a Gaussian blur of sigma radius of 1 µm, ensuring this spatial resolution in the resulting FRET values. A binary mask of the cell is created and used to set all the values outside the cell to NaN. According to the original report22, the inverted FRET ratio was calculated by dividing the Donor image by the FRET image. Floating values were normalized between 0 and 1. For whole-cell inverted FRET calculations, the Analyze particle tool (Fiji) was used to get the Mean gray values. Since outside the cell mask pixels were set to NaN, mean gray values only resulted from the cell projected area. For pixel-by-pixel analysis a stack of 2 channels was generated: the first channel represented the inverted FRET ratio image (mean gray values between 0 and 1), and the second channel the Acceptor image (mean gray value between 0 and 255, 8-bit format). A custom-written plugin allowed the retrieval of values in both channels for corresponding pixels, creating a.csv file to be analyzed in R. Scatter plots at each experimental condition were generated by excluding saturated pixels from the analysis.MLC trackingTIRFM time lapses of transfected MEFs were acquired at a 10 s frame rate for > 15 min. Raw images were processed in FiJi: the background was subtracted (Sliding paraboloid method with rolling ball radius of 100 pixels), bleach correction was performed by the built-in Histogram matching method, and a Gaussian blur filter applied (1 µm radius). Time lapse corresponding to the mCherry-MLC channel were processed with the Particle Tracker function of the Mosaic plugin with the following settings: Cutoff 0.001, Per/Abs 0.6, Link Range 2, and Displacement 10. Only tracks shorter than 300 s were considered. A.csv file with tracks data, in particular track lenghts, was created and plotted by using the software GraphPad Prism.Western blottingFor western blot analysis, cells were lysed directly on the plate by adding the opportune amount of modified Laemmli sample buffer composed of Tris-HCl 135 mM (pH 6.8), sodium dodecyl sulfate (SDS) 5%, urea 3.5 M, NP-40 2.3%, β-mercaptoethanol 4.5%, glycerol 4%, and traces of bromophenol blue. Total protein content was normalized by seeding cells at equal densities; this lysis buffer does not allow total protein quantification but prevents membrane-bound proteins from being degraded during trypsinization. The equal volume between samples was then loaded onto 12–8% SDS polyacrylamide gels and transferred after electrophoretic separation onto a nitrocellulose membrane (Amersham GE-Healthcare). After the transfer, membranes were blocked in PBS supplemented with 0.1–0.3% Tween20 and 5% milk for 1 h at room temperature, then incubated overnight at 4° with primary antibodies at the following dilutions: mouse anti-βII-spectrin 1:2000 (BD-bioscience), rabbit anti-βII-spectrin 1:2000 (Abcam), and mouse anti-β-tubulin 1:5000 (Sigma-Aldrich). After three washing steps in PBS–Tween20 (0.1–0.3%), membranes were incubated for 1 h at room temperature with HRP-conjugated secondary antibodies (BioRad). Three washing steps of 5 min in PBS–Tween20 (0.1–0.3%) were performed between primary and secondary antibody incubation. Proteins were detected by ECL Western blotting reagents (Amersham GE-Healthcare), using the digital Chemidoc XRS + system run by the software Image Lab (Biorad).ModelWe proposed a model to investigate whether the spectrin periodic pattern seen between stress fibers in the cell emerges from a periodic hexagonal pattern, like that of the red blood cell. This hexagonal pattern better tils the 2D space and hence, facilitates the study of topological transitions. We hypothesized that this change in the cytoskeleton configuration is possible due to the interaction of the forces generated by its components. Therefore, to investigate the mechanical forces of the cell cytoskeleton, we modeled it as a network of springs and cables, as in refs. 34,35,36,37,38,39, where the edges and nodes correspond to filament bundles and cross-linkers, respectively. This mesoscopic description gives a good approximation for the spatio-temporal evolution of the cytoskeletal components under investigation. Unlike molecular dynamics and dissipative particle dynamics models30,40, this approach allows the exploration of a larger cytoskeletal meshwork without computationally expensive simulations. Furthermore, we used a 2D network assuming that the out-of-plane deformations are negligible in the cells under consideration.In the initial model, spectrin bundles were represented by the edges of a triangular mesh connected by short actin filaments (nodes). Spectrin bundles behave like springs, in the sense that they return to a resting length after stretching or shrinking. Hence, the \({N}_{e,S}\) spectrin bundles in our model generate a spring potential energy \({U}_{s,S}\) when they diverge from the resting length \({d}_{0,S}\), that is given by$${U}_{s,S}={\sum}_{j=1}^{{N}_{e,S}}\frac{{k}_{s,S}{\left({d}_{j}-{d}_{0,S}\right)}^{2}}{2},$$
(2)
where \({k}_{s,S}\) is the spectrin spring constant and \({d}_{j}\) is the length of the edge \(j\) expanding between the \(l\) and \({l}^{{\prime} }\) nodes.To simulate the evolution of the spectrin mesh to a relaxed configuration where the potential energy generated by its edges minimizes, we assumed, as in refs. 35,36, that the position \({\bar{r}}_{l}\in {{\mathbb{R}}}^{2},l\in \left\{1,\ldots,{N}_{n,A}\right\}\) of the nodes representing the actin short filaments connecting the spectrin bundles freely moves at a velocity \({\bar{v}}_{l}\), and that adhesion complexes generate a viscous resistance to cytoskeleton network deformations. In the model, the viscous resistance is given by \(\xi {\bar{v}}_{l}\), where \(\xi\) is a drag coefficient. This resistive force is balanced by the force generated by the spectrin potential energy at each node$$\bar{F}\left({\bar{r}}_{l}\right)=-\frac{\partial {U}_{s,S}}{\partial {\bar{r}}_{l}}.$$
(3)
Thus, \({\bar{r}}_{l}\) evolves according to$$\frac{\partial {\bar{r}}_{l}}{\partial t}=\frac{{\bar{F}}_{l}}{\xi }.$$
(4)
The simulations were run in MATLAB_R2021a on a desktop computer. We used the delaunayTriangulation.m function to initialize the spectrin triangular mesh. Then, we solved Eq. 4 with the Euler method using small time-steps \(\varDelta t\) to ensure numerical stability. We evolved the system until there was no significant change in its configuration, i.e., until the system was “equilibrated”. For Supplementary Fig. 8F the equilibration time is 60 s, for Fig. 6D–F 120 s, and for Fig. 6G–I 600 s. Since we were interested in the evolution between two different configurations of the spectrin mesh, instead of taking the mechanical parameters of the model from the literature related to one configuration or another, we fitted these parameters to qualitatively match the experimental observations. Other parameters, such as the length of the spectrin bundles, were taken from the literature. The units of the parameters were set to match the scales in the experiments. All the model parameters are in Supplementary Table 4.In Fig. 6D, we introduced edges with a cable element in the model that produces a potential energy given by$${U}_{c}={\sum}_{j=1}^{{N}_{e,c}}\frac{{k}_{c}{d}_{j}^{2}}{2},$$
(5)
where \({k}_{c}\) is the cable constant. Note that the cable elements generate a force that shrinks the corresponding edges. Since they connect to the spectrin network on one end and points representing focal adhesions with zero velocity to the other end, these cable elements stretch the spectrin network. Note that the evolution of the network is now given by the balance of all the forces acting in the cell cytoskeleton, i.e.,$$\frac{\partial {\bar{r}}_{l}}{\partial t}=\frac{{\bar{F}}_{l}}{\xi }=-\frac{\partial \left({U}_{s,S}+{U}_{c}\right)}{\partial {\bar{r}}_{l}}$$
(6)
A recent study using molecular dynamics simulations40 shows that an actin-spectrin model under a low strain rate is more prone to exhibit detachment of the actin-spectrin interface rather than fragmentation of the spectrin bundle. We included this observation by detaching the edges corresponding to spectrin bundles when an expanding force generated by their spring potential \({U}_{s,S}\) is greater than a force threshold \({F}_{{th}}\). We simulated the spectrin detachment from short actin filament by eliminating the corresponding edge from the network.However, in the experimental data, the spectrin mesh was constrained by stress fibers. Therefore, we included stress fibers in the model by adding the corresponding edges to the top and the bottom of the spectrin mesh. As in ref. 36, these edges have a spring and a cable element with constants \({k}_{s,F}\) and \({k}_{c,F}\), respectively, creating a potential energy \({U}_{F}={U}_{s,F}+{U}_{c,F}\). The contraction generated by the cable element resembles the contraction generated by the myosin rods along the long actin filaments of the stress fibers. The stress fiber resting length \({d}_{0,F}\) corresponds to the calculated edge length in a relaxed state. Focal adhesions delimit stress fibers. The length of the edge connecting to focal adhesions was initially larger than the other edges in the stress fibers because we assumed that actin polymerization at the focal adhesions and actin exchange along the stress fiber also generate forces that affect the stress fibers resting state, as in ref. 41. Thus, this initial arrangement produces an instability that increases the length of all the stress fiber edges, thereby increasing their generated force and pulling the spectrin mesh to the focal adhesions. Moreover, during experiments, the stress fibers actively moved toward each other. To mimic this motion, the focal adhesion points moved vertically toward each other at a constant velocity \({v}_{A}\) during the first half of the simulation (300 s) and at zero velocity afterward.We introduced myosin linkers connecting the spectrin mesh to the stress fibers in the model. The myosin linkers initially span from an extreme node of the stress fibers to the center of the closest triangle formed by spectrin bundles in the spectrin mesh. These edges have a cable element with constant \({k}_{c,L}\). Thus, they generate a shrinking force that affects the corresponding node in the stress fiber and the nodes of the spectrin triangle (the force is equally distributed among the three nodes of the spectrin triangle). We assumed that these myosin linkers do not shrink indefinitely, instead, they detach from the network if their lengths are less than \({d}_{\min }\). When one of the sides of the spectrin triangle where a myosin linker connects detaches, then the myosin linker detaches from that spectrin triangle and attaches to a randomly selected free spectrin triangle within a radius \(r\), \({d}_{\min } < \,r < \,{d}_{\max }\). If there are no free spectrin tringles, the myosin linker is removed from the network.We also added myosin rods inside the spectrin mesh. These rods attach to spectrin triangles in both extremes and have a cable element with constant \({k}_{c,M}\). As the myosin linkers, if one of the sides of one spectrin triangle detaches, the corresponding extreme of the myosin rod randomly selects a free spectrin triangle within a radius \(r\), \({d}_{\min } < \,r < \,{d}_{\max }\), to connect. If there are no free spectrin triangles within that radius, then the myosin rod is removed from the network. Furthermore, myosin rods are randomly created and removed from the spectrin mesh at a rate \({\varphi }_{a}\) and \({\varphi }_{r}\), respectively. This was implemented in the simulation by adding (removing) a myosin rod at each time-step if a randomly drawn number from a uniform distribution \(u \sim U\left({{{{\mathrm{0,1}}}}}\right)\) is less than the probability \(\varDelta t{\varphi }_{a\left(r\right)}\). These rates and the initial number of myosin rods were informed by experimental observations.The simulation of the full cytoskeleton network composed of edges representing spectrin bundles, stress fibers, myosin linkers or myosin rods, and nodes representing short actin filaments, focal adhesions, or stress fiber connectors, is described below:At the start of the simulation:
Initialize the network with the spectrin bundles, short actin filaments, stress fibers, focal adhesions, and myosin linkers.
Add the myosin rods to the spectrin mesh to randomly selected locations, verifying that their length is within \({d}_{\min }\) and \({d}_{\max }\).
At each time-step:
For each \(j\) edge in the network spanning from \({\bar{r}}_{l}\) to \({\bar{r}}_{{l}^{{\prime} }}\):
Calculate the force generated by the potential energy of their spring and/or cable elements \(\bar{F}\left(\bar{r}\right)=-\frac{\partial U}{\partial \bar{r}}\) at \({\bar{r}}_{l}\) and \({\bar{r}}_{{l}^{{\prime} }}\).
Remove from the network an edge corresponding spectrin bundles if it generates a force larger than \({F}_{{th}}\).
Update the position of the nodes, by adding the force generated by their edges, \(\bar{r}=\bar{r}+\frac{\bar{F}\left(\bar{r}\right)}{\xi }\Delta t\).
For the focal adhesions: If time \( < {t}_{s}\), move the nodes corresponding to the focal adhesions vertically at velocity \({v}_{A}\), hence \(\bar{r}=\bar{r}+{v}_{A}\Delta t\). Otherwise, set the velocity to zero.
For the edges corresponding to myosin (rods or linkers) attached to a spectrin triangle that lost one of its sides: find a new free spectrin triangle to attach within a radius \({d}_{\min } < \,r < \,{d}_{\max }\). If there are no free spectrin triangles within that radius, remove the spectrin edge from the network.
If \(u \,\sim \,U\left({{{{\mathrm{0,1}}}}}\right) \, < \,\varDelta t{\varphi }_{a}\), add a myosin rod to a random location in the spectrin mesh.
If \(u \,\sim \,U\left({{{{\mathrm{0,1}}}}}\right) \, < \,\varDelta t{\varphi }_{r}\), randomly select a myosin rod and remove it from the network.
In the model, the short F-actin position is dictated by the balance of forces generated by the connected spectrin bundles. Hence, there is feedback from spectrin to short F-actin by design: as long as spectrin bundles are connected to short F-actin, they will influence its position. Moreover, there must be feedback from the spectrin network to the stress fibers. As shown in Supplementary Fig. 10, the interaction between spectrin and stress fibers is important for network clustering. When the focal adhesions are fixed throughout the simulations, fewer spectrin bundles are detached, which hinders the transition to the four neighbor configuration (Supplementary Fig. 10I, M). Even changing the velocity at which the focal adhesions are pulled together can have large effects. For example, if the focal adhesions are faster, the spectrin network fails to compress at that rate, and it extends beyond the stress fibers, contrary to what is observed in experiments (Supplementary Fig. 10L). Therefore, there must be feedback between the spectrin network and the stress fibers to achieve an efficient spectrin network compression. Since spectrin network compression is achieved by the action of myosin rods and stress fiber dynamics are linked to the focal adhesion movement, we conclude that the morphological transitions are the result of this complex interaction.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.