# Extracellular fluid viscosity enhances cell migration and cancer dissemination

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### Experimental methods

#### Cell culture and reagents

Human MDA-MB-231 adenocarcinoma (ATCC), MDA-MB-231 BrM218 (MDA-MB-231 cells isolated from brain metastasis were a gift from J. Massagué), HOS (ATCC), U87 (ATCC) and HEK293 (ATCC) cells and dermal fibroblasts (Coriell Institute) were grown in DMEM (Life Technologies/Gibco) supplemented with 10% heat-inactivated FBS (Gibco) and 1% penicillin–streptomycin (10,000 U ml−1; Gibco). hAOSMCs (PromoCell) were grown in smooth muscle cell growth medium (PromoCell). SUM159 cells were obtained from D. Wirtz (Johns Hopkins University) and were grown in Ham’s F-12 medium (Gibco) containing 5% FBS, 1% penicillin–streptomycin (10,000 U ml−1; Gibco), 1 μg ml−1 hydrocortisone (Sigma-Aldrich) and 5 μg ml−1 insulin (Sigma-Aldrich). Cells were maintained in an incubator at 37 °C and 5% CO2 and passaged at 60–90% confluency every 3 to 5 days. Cells were regularly checked for mycoplasma contamination by PCR using the following primer pairs: forward, 5′-GGGAGCAAACAGGATTAGATACCCT-3′; reverse, 5′-TGCACCATCTGTCACTCTGTTAACCTC-3′.

In select experiments, cells were treated with the following pharmacological agents (purchased from Sigma-Aldrich unless otherwise stated) and corresponding vehicle controls: 2-APB (Calbiochem, 100 μM), CK666 (100 µM), latrunculin A (2 µM), NSC 668394 (Calbiochem, 10 μM), Y-27632 (10 µM), blebbistatin (50 µM), BAPTA (50 µM), GSK2193874 (15 µM), GSK1016790A (1 or 15 µM), BAPTA AM (Invitrogen, 25 µM), EIPA (40 µM), HC-067047 (1 μM) and verteporfin (0.1 µM).

#### Preparation and characterization of media of varying viscosities

Methylcellulose stock solution (3%) in IMDM was purchased from R&D Systems, and media of desired viscosities were prepared by diluting the stock solution with appropriate amounts of IMDM (Life Technologies/Gibco) containing 10% heat-inactivated FBS (Gibco) and 1% penicillin–streptomycin (10,000 U ml−1; Gibco). As an alternative means of increasing viscosity, Dextran 500 kDa (Spectrum Chemical) and PVP (K-90; Sigma Aldrich; molecular mass, 360 kDa) were used as suspension in appropriate basal medium. In select experiments, Detran 6 kDa (Alfa Aesar) was used as a control to Dextran 500 kDa, as low-molecular-mass (6 kDa) dextran at the same molarity does not change viscosity appreciably2. Cannon–Fenske capillary viscometers were used to measure the viscosity of the resulting medium at 37 °C. Osmolarity was measured using freezing-point depression with a 3205 Single-Sample Osmometer (Advanced Instruments). To ensure the successful entry of the viscous medium into the microchannels, red fluorescent beads of 0.5 μm diameter (FluoSpheres) were dispersed uniformly in the medium, and their flow through the devices was imaged by epifluorescence microscopy.

During medium preparation, up to 0.6% or 92 µM of 65 kDa methylcellulose was used. Thus, osmolarity variations due to addition of methylcellulose can be only up to 100 μOsm, which is insignificant relative to the osmolarity range of basal media that is 3,000× higher (~300 mOsm).

#### 2D assays

In wound-healing assay, cells were allowed to grow to confluence on collagen-I-coated (20 µg ml−1; collagen I rat tail; Thermo Fisher Scientific) tissue culture plastic plates (Falcon). After reaching confluency, a scratch in the cell monolayer was created using a 200 µl pipette tip with a plastic ruler as a guide. In 2D single-cell motility assays, cells were plated at 5–8% confluency on collagen-I-coated (20 µg ml−1) tissue culture plastic plates (Falcon). After cell spreading or after wound creation, the culture medium was replaced with that of the appropriate viscosity, and the acquisition of time-lapse images started 30 min after medium change. In select 2D assays involving high-resolution imaging, cells were plated on collagen-I-coated (20 μg ml−1) dishes (Cellvis or IBIDI).

#### Microfluidics device fabrication and cell seeding

PDMS-based microfluidics devices containing an array of parallel microchannels of specific dimensions (200 µm in length, 10 μm in height, and 3.5 μm or 10 μm in width) were fabricated as described previously20,46. A laser profilometer was used to verify the height and width of microchannels. Microchannels were coated with collagen I (20 µg ml−1) before medium loading and cell seeding. All cell types except for hAOSMCs and human dermal fibroblasts were tested inside narrow (3.5 × 10 µm2) channels. hAOSMCs and human dermal fibroblasts were examined inside 10 × 10 µm2 channels.

Microfluidics devices were loaded with medium of appropriate viscosities by filling the inlet wells with 100 μl of medium 15 min before adding cells. Cells were detached from culture flasks or plates using 0.05% trypsin-EDTA (Gibco), followed by centrifugation at 300g for 5 min and resuspension in medium of the specified viscosity at a concentration of 5 × 106 cells per ml. After removing the medium from the inlets and outlets of the device, 20 μl of cell suspension was added into the cell-seeding inlet to create a pressure driven flow of cells into the cell-seeding channel. Cells were allowed to adhere and spread outside the microchannel entrances for at least 20 min before filling all inlet wells to 100 μl with medium at the appropriate viscosities. Devices were incubated at 37 °C and 5% CO2 before imaging. Cells were not subjected to any chemotactic stimulus or medium flow during the migration assay. In experiments with pharmacological agents, the medium in the devices was replaced with that containing the drug of consideration or its respective vehicle control once cells had fully entered the microchannels. By imaging the flow of fluorescent beads dispersed in 8 cP medium, we confirmed their successful entry into the microchannels (Supplementary Video 5).

#### Cell migration tracking and analysis

Cells were imaged every 2 to 20 min for 12 to 20 h on an inverted Nikon Eclipse Ti microscope (Nikon) with automated controls (NIS-Elements; Nikon) and a ×10/0.45 NA ph1 objective using time-lapse microscopy. In select experiments, cells were imaged using fluorescein isothiocyanate filters. For experiments with NHE1-GFP-transfected cells, a Nikon A1 confocal microscope (Nikon) with ×20 or ×60 objective was used. During the course of the experiments, cells were maintained on a stage-top incubator (Tokai Hit) at 37 °C and 5% CO2. The A1 confocal system also had a temperature-controlled cage.

Live-cell videos were exported to ImageJ (National Institutes of Health), and the MTrackJ plugin47 was used for cell motility tracking. In single-cell 2D migration assays, cells that were not in contact with any neighbours were chosen and manually tracked. Cell tracking was performed until the end of the video unless a neighbouring cell touched the tracked cell, in which case tracking was stopped. In microfluidics experiments, cell paths were recorded from the time of complete entry into the microchannel until contact was made with the end of the microchannel. A custom MATLAB (MathWorks) script was used to analyse cell migration speed and velocity from the cell paths. Cell entry time was calculated by manually evaluating the time elapsed between leading and trailing edge entry into a microchannel. Dividing or apoptotic cells were excluded from analysis in both 2D and microchannel migration. Wound-healing assays were analysed using ImageJ by drawing a line indicating the migrating fronts on two sides of the scratch wound. Wound closure was recorded when both apposing fronts touched each other.

For cell migration phenotype classification, live cells expressing Lifeact–GFP or fixed cells stained with Alexa Fluor 488 phalloidin (1:100; Invitrogen) and Hoechst (1:2,000; Invitrogen) were observed using an inverted Nikon Eclipse Ti microscope (Nikon) using a ×40 air objective 4 to 8 h after channel entry. Cell phenotype under confinement was tabulated manually, as reported previously47.

#### Spheroid assays

Spheroids with MDA-MB-231 cells were generated as described previously48. In brief, growth-factor-reduced Matrigel (Corning) was diluted 1:3 with cold DMEM supplemented with 10% heat-inactivated FBS and 1% penicillin–streptomycin. Then, 50 µl of the diluted Matrigel was transferred to each well of a 96-well plate (Falcon) and allowed to polymerize at 37 °C and 5% CO2 for 1 h. Cells were collected from culture flasks, resuspended in diluted Matrigel and were gently dispensed (2,000 cells per 50 µl) on top of the previously polymerized Matrigel in each well. After incubation at 37 °C and 5% CO2 for 1.5 h, 100 µl of DMEM with 10% heat-inactivated FBS and 1% penicillin–streptomycin was added into each well. The medium was replaced every 2 days until spheroids were collected for experiments (typically ~10–15 days later). Spheroids were collected after dissolving the Matrigel in cold DMEM and centrifuged at 2,655g for 5 min. After removing the supernatant, the spheroids were resuspended in either 0.77 or 8 cP medium and plated onto collagen-I-coated (20 μg ml−1) 24-well plates. After 60 min, the spheroids were imaged every 20 min for at least 20 h on an inverted Nikon Eclipse Ti microscope with automated controls (NIS-Elements; Nikon) with a ×10/0.45 NA ph1 objective using time-lapse microscopy. During the course of the experiments, the spheroids were maintained on a stage-top incubator at 37 °C and 5% CO2. First-cell dissociation times were acquired using the NIS Element Software (Nikon) by manually recording the time required for the first cell to fully detach from each intact spheroid.

#### Cloning, lentivirus preparation, transduction and transfection

To generate MDA-MB-231 cells with stable knockdown of ARP3 and/or ARPC4, or TRPV4, the pLKO.1 puro (plasmid 8453; Addgene; a gift from B. Weinberg) backbone was used. The following sequences were subcloned into the pLKO.1 puro backbone: Non-targeting scramble sequence: 5′-GCACTACCAGAGCTAACTCAGATAGTACT-3′; shARP3 sequence 1: 5′-GTAACACCAAACATGATTATA-3′; shARPC4 sequence 1: 5′-GCTGAAGAGTTCCTTAAGAAT-3′; shITGB1 sequence 1: 5′-TGCCTACTTCTGCACGATGT-3′; shTRPV4 sequence 1: 5′-TCTTGGTAACAAACTTGGT-3′; and shTRPV4 sequence 2: 5′-TGCTCCTATGGAGTCACATAA-3′.

Whereas shTRPV4 sequence-1 was used in Extended Data Fig. 6, in the rest of the manuscript, sequence-2 was used. For AQP5 knockdown, the following targeting sequence was subcloned into the pLVTHM (plasmid 12247; Addgene; a gift from D. Trono) backbone: shAQP5 sequence 1: 5′-ACGCGCTCAACAACAACACAA-3′.

For NHE1 depletion, the following sequences were subcloned into both the pLKO.1 puro and pLVTHM backbones and used with SC sequences subcloned into the respective backbones: shNHE1 sequence 1: 5′-GACAAGCTCAACCGGTTTAAT-3′; and shNHE1 sequence 2: 5′-CCAATCTTAGTTTCTAACCAA-3′.

Whereas the shNHE1 sequences were used individually in Extended Data Fig. 5, both sequence-1 and sequence-2 were used in combination in all of the other experiments. MDA-MB-231 cells stably transduced with shMIIA and shMIIB3 were used for migration assays. For actin retrograde flow analysis, the shMIIA sequence was cloned into the pLKO.1 puro backbone. After every cloning, sequence integrity and orientation were verified by Sanger Sequencing at the JHU Genetic Resources Core Facility.

For lentivirus production, HEK293T/17 cells were co-transfected with psPAX2, pMD2.G and the lentiviral plasmid of interest. Then, 48 h after transfection, the lentivirus was collected and concentrated using centrifugation. Wild-type MDA-MB-231 cells at 60–80% confluency were incubated for 24 h with 100× virus suspension and 8 µg ml−1 of Polybrene transfection reagent (Millipore Sigma). To maintain stable knockdown, cells transduced with the pLKO.1 puro backbone were cultured in medium containing 0.5 µg ml−1 puromycin (Gibco).

#### Live-cell reporters

The pGP-CMV-GCaMP6s (40753; a gift from D. Kim and the GENIE Project), FUGW-pHRed (65742; a gift from M. Tantama), pLentiRhoA2G (40179; a gift from O. Pertz), pLenti.PGK.LifeAct-GFP.W and pLenti.PGK.LifeAct-Ruby2.W (51010 and 51009 respectively; gifts from R. Lansford) plasmids were purchased from Addgene and used for lentiviral cell transduction. The GFP-AHD plasmid was offered by M. A. Glotzer49, and was cloned into the lentiviral backbone modified from pLV-EF1a-IRES-Puro (plasmid 85132; Addgene; a gift from T. Meyer) using BamHI and NotI. The following sequences were used to amplify GCaMP6s: 5′-CAAATGTGGTATGGCTGATTATG-3′ and 5′-GTACGCGTCACCATGGGTTCTCATCATC-3′. The PCR product was subsequently inserted into pLV-EF1a-IRES-Puro using MluI and NotI. pEYFP-C1 (Clontech)2 was cloned into pLV-EF1a-IRES-Purousing the following sequences and NotI and HpaI restriction sites: 5′-GTGCGGCCGCGTCGCCACCATGGTGAGC-3′ and 5′-CTGTTAACTCACTTGTACAGCTCGTCCATGC-3′. The aforementioned plasmids were used to prepare lentiviral particles for stable cell transduction.

Arp3-pmCherryC1 (27682; a gift from C. Merrifield) and β-actin-mRFP-PAGFP (62382; a gift from G. Charras and T. Mitchison) were purchased from Addgene. The NHE1-GFP plasmid was a gift from J. Orlowski. The above-mentioned plasmids were used for transient transfections. About 60–80% confluent MDA-MB-231 cells were transfected using Lipofectamine 3000 reagent according to the manufacturer’s recommendations. To enrich the population of cells expressing Arp3-pmCherryC1, β-actin-mRFP-PAGFP or NHE1-GFP, cells were selected with 0.5 mg ml−1 G418 (Corning) and sorted. Cells expressing Myosin IIA-GFP were generated as previously described3.

#### Lamellipodial actin staining and imaging

To stain for F-actin, a previously described protocol50 was used for cells plated on collagen-I-coated glass bottom dishes (Cellvis). In brief, when cells reached about 20% confluency, their medium was replaced with that of the desired viscosity and they were incubated for 2–6 h at 37 °C. Cells were then washed once with PBS containing Ca2+/Mg2+ (Gibco), fixed and extracted for 1 min using 0.25% glutaraldehyde and 0.5% Triton X-100 in cytoskeleton buffer (150 mM NaCl, 10 mM MES buffer, 5 mM EGTA, 5 mM glucose, 5 mM MgCl2 in water; pH 6.8) and then fixed for 5 min in 2% glutaraldehyde in cytoskeleton buffer. The samples were either processed immediately or stored with 2% glutaraldehyde in cytoskeleton buffer at 4 °C until subsequent staining. The background autofluorescence from glutaraldehyde was quenched with 1% NaBH4 for 15 min before blocking with 5% bovine serum albumin (BSA) for 1 h. Cells were stained with rhodamine or Alexa Fluor 488 phalloidin (1:100; Invitrogen) and Hoechst (1:2,000; Invitrogen). High-resolution images were acquired using the Zeiss LSM800 Confocal with AiryScan super-resolution module with an x,y resolution of 120 nm and a z-resolution of 350 nm.

#### STORM sample preparation, imaging and analysis

Samples were prepared, fixed and extracted as described in the ‘Lamellipodial actin staining and imaging’ section. Next, fixed samples were treated with 0.2% NaBH4 for 7 min, and three times with 0.5% glycine in PBS for 10 min, before blocking with 5% BSA solution containing 0.1% Triton X-100 (Sigma-Aldrich) for 1 h. The samples were subsequently stained overnight at 4 °C with Alexa Fluor Plus 647 Phalloidin (1:400; Invitrogen). On the day of imaging, the following solutions were prepared: buffer A (10 mM Tris at pH 8.0 containing 50 mM NaCl), buffer B (50 mM Tris at pH 8.0 containing 10 mM NaCl and 10% glucose) and GLOX (14 mg glucose oxidase (Millipore Sigma) and 0.85 mg catalase (Millipore Sigma) dissolved in 200 µl buffer A and 50 µl distilled H2O). The samples were washed five times with PBS containing 0.05% Triton X-100 for 5 min. Immediately before imaging, fresh STORM imaging buffer (7 µl GLOX solution, 7 µl 2-mercaptoethanol and 690 µl buffer B) was prepared and added to the dishes. A glass coverslip was gently placed on top to seal off the samples from atmospheric oxygen, and the dishes were mounted on the scope.

Images were acquired on a Nikon Eclipse Ti-E (Nikon) microscope equipped with a ×100/1.49 NA Plan-Apo objective, 647 nm fibre laser operated at 500 mW and Evolve 512 EMCCD camera (Photometrics). A total of 50,000 cropped images (128 × 128 px2) were captured in NIS-Elements (Nikon) at 120 fps. TIRF was used to limit fluorescence to a thin region of the sample and the Nikon Perfect Focus System was used to reduce z-drifts. STORM images were reconstructed and visualized using Picasso software51. Localizations were filtered by the point spread function width (sx = 2) and height (sy = 2) and localization precision in x (lpx = 2) and y (lpy = 2) in camera pixels. For analysing actin density, the reconstructed images were exported to ImageJ and the fibres were transformed into binary masks using the Auto Local Threshold function of ImageJ, using the Phansalkar algorithm. The edge of the cells was manually traced and the percentage of ON pixels over total pixels was calculated within 1.5 µm from the cell edge.

#### Cell volume and lamella area measurements

Cell volume measurements were performed as described previously20. Cells stably expressing Lifeact–GFP were imaged using a Nikon A1 confocal microscope with a ×60 oil-immersion objective within 2–6 h after the replacement of the medium with that of the desired viscosity. Images were acquired at 1,024 × 1,024 px resolution, and confocal slices were spaced at 0.5 µm apart. A 488 nm laser was used to image the cells and obtain their boundaries from the Lifeact–GFP signal. Throughout the course of the experiments, cells were maintained at 37 °C and 5% CO2 inside a stage-top incubator and cage (Tokai Hit).

Cell volume was measured from z-stacks using a custom MATLAB script described previously20. Out-of-focus z-planes were removed. Images from each analysed focal plane were processed using the binary thresholding function in MATLAB to remove noise. The cell boundary was detected in each image using the Canny Edge Detector operator. The edge was dilated and refilled as to obtain the cross-sectional area of the cell from each slice. The volume was calculated by multiplying the z-slice interval by the average area from two adjacent slices and integrating all of the values. In select experiments, cell volume was analysed using volume measurement wizards of NIS-Elements v.5.02.01 (Nikon) and Imaris v.9.7.0 (Bitplane).

In addition to measuring the projected area of the entire cell, we assessed the area of the part of the cell above a critical height from the substrate. A critical height of 2 µm was chosen after taking into consideration the typically reported lamella thickness of 1 µm. Notably, this threshold eliminated possible z-aberrations from confocal imaging. The projected lamella area was calculated by subtracting the projected area of the cell above the critical height from the overall projected cell area. To normalize the measurements with respect to the cell size, data were reported as projected area of lamella over projected area of the entire cell.

#### Lamella growth imaging and analysis

Cells stably expressing Lifeact–GFP or Lifeact–Ruby were plated on collagen-I-coated glass-bottom dishes (Cellvis) at about 20% confluency. After treating the cells with medium of the desired viscosities for 2–6 h, time-lapse images were acquired every 5 s for 110 s or 2 min with a Nikon A1 confocal microscope equipped with a ×60 oil-immersion objective. Cells were maintained at 37 °C and 5% CO2 inside a stage-top incubator and cage.

Time-lapse images were imported into ImageJ and the growing front of the cells was manually extracted. Subsequently, a custom macro was used to identify the instantaneous cell area and edge length (or perimeter). The lamella growth at every instance (i), was calculated by using the ratio of area change over the last perimeter, using the formula:

$${\rm{Growth}}(i)=\frac{{\rm{Area}}\left(i\right)-{\rm{Area}}(i-1)}{{\rm{Perimeter}}(i-1)}$$

A heat map summarizing instantaneous leading-edge lamella growth was prepared by combining all of the instances from cells at a particular viscosity.

#### Actin retrograde flow imaging and analysis

For measuring actin retrograde flow, MDA-MB-231 cells stably expressing β-actin-mRFP-PAGFP were plated on collagen-I-coated glass-bottom dishes (Cellvis). Cells were imaged using the FRAP capabilities of a Nikon A1 confocal microscope (Nikon) with a ×60 oil-immersion objective. The direction of cell motility was manually identified by visual inspection over 3 min, and then a rectangular region of interest (ROI; 60.75 × 7.25 µm2) was drawn with its length aligned perpendicular to the cell leading edge. A point in the cytoplasm was exposed to ultraviolet light (405 nm) at 20% intensity for 500 ms, and then imaged at GFP and RFP channels, every 100 ms or 400 ms for 30 s after ultraviolet illumination.

For analysis of retrograde flow, the time-lapse videos were imported into ImageJ. RFP and GFP channels were individually extracted, converted to binary and smoothed twice. Next, a kymograph was drawn perpendicular to the leading edge for the image with GFP signal, and the angle subtended by the retrograde flow of the photo-converted PAGFP molecule was used to estimate the flow rate with respect to the laboratory frame. The uninterrupted leading edge growth rate was quantified by measuring the slope generated by the cell leading edge in the kymograph before a retraction event occurred. The image from the RFP channel was used to locate the cell edge before photoactivation.

#### Measurement of epifluorescent FRET-based RHOA activity

Cells expressing the RhoA2G biosensor were plated onto collagen-I-coated glass-bottom dishes. Within 2–6 h of exposing cells to medium of the desired viscosity, images at CFP (excitation: 430/24, emission: 470/24), YFP (excitation: 500/20, emission: 535/30) and FRET (excitation: 430/24, emission: 535/30) settings were acquired. For RHOA-activity quantification, the mean pixel ratio of FRET over CFP was measured after subtraction of the background signal from the corresponding channels20,52. The image in the YFP channel was used for tracing the cell boundary. Imaging and quantification were performed using Nikon Elements. For control experiments with FBS (10%) activation, cells were serum-starved for 24 h before adding medium containing 10% FBS, and the images were acquired within 0.5–2 h.

#### FLIM analysis of membrane tension and RHOA activity

For membrane-tension measurements, cells, plated on collagen-I-coated glass-bottom dishes and incubated for 3–4 h in media of the specific viscosities in the absence or presence of the indicated drugs, were stained with the live-cell membrane tension probe Flipper-TR (Spirochrome, 0.5 mM) and imaged immediately thereafter. Confocal fluorescence lifetime imaging microscopy (FLIM) analysis of live cells stained with Flipper-TR or stably expressing the RhoA2G sensor was performed as described previously20,47 using a Zeiss LSM 780 microscope and a PicoQuant system consisting of a PicoHarp 300 time-correlated single-photon counting (TCSPC) module, two-hybrid PMA-04 detectors and a Sepia II laser control module. For Flipper-TR imaging, a 485 nm laser line was used for excitation with 600/50 band-pass detector unit. Cells were maintained at 37 °C and 5% CO2 during imaging inside a PeCon environment chamber and stage.

#### FLIM reconvolution, image segmentation and segmentation quantification

The FLIM data were processed as described previously20,47 using SymPhoTime 64 (PicoQuant) software. In brief, a customized script was used for the calculation of the internal response function from 100 data points with no smoothing. After binning the data to ensure at least 500 photons per binned pixel, a cell-specific threshold was applied to eliminate out-of-cell fluorescence. The fluorescence decays were fit into every binned pixel using the three-exponential reconvolution. For Flipper-TR quantification, the longest lifetime (τ1) with the higher fit amplitude was recorded for each cell according to the manufacturer’s recommendations. For RHOA-FLIM segmentation quantification, the intensity-weighted fluorescence lifetime averages (τAI) were measured using SymPhoTime 64 in different areas of the cells. Data from the nuclear region were excluded during FLIM segmentation using a free-form ROI because the RHOA FRET sensor was excluded from the nucleus, resulting in low and variable photon counts in the centre of the cells. The entire cell without the nucleus is defined as ‘whole’, whereas the lamella region at cell edge is identified from DIC images (Extended Data Fig. 8d). The ‘body’ of the cell comprises of the remainder of the cell without the lamella and the nucleus.

#### Calcium dynamics imaging and quantification

Cells transiently or stably expressing GCaMP6s were plated on collagen-I-coated glass coverslips with PDMS walls3 and were grown to 60–80% confluency. Within 5 min of the addition of the indicated stimulus, cells were imaged every minute for 80–120 min with a ×20 objective on a Nikon A1 confocal microscope using the 488 laser and the GFP filter settings. The GFP signal of representative cells was manually traced in ImageJ over time by measuring the mean intensity of a circular ROI inside the cell. Calcium spikes were identified as instances with greater than 2× intensity over the baseline levels.

#### Osmotic shock experiments

For application of osmotic shocks, hypotonic media were prepared and their osmolarity was measured as described previously20. Medium solutions were prepared by diluting basal IMDM medium (292 mOsm) twofold (2×, resulting in 141 mOsm) or four-fold (4×, 71 mOsm) with ultrapure water. For FLIM measurements, cells were serum-starved for 24 h before measuring RHOA activity within 5–30 min after application of the osmotic shock. Cell volume measurements were also completed within 50 min after shock.

#### Immunofluorescence and image analysis

For pMLC, Ki-67 and YAP1 staining, cells were fixed with 4% paraformaldehyde (Thermo Fisher Scientific), permeabilized in 0.1% Triton X-100 and blocked for 1 h at room temperature with 5% BSA solution containing 2% (for YAP1) or 5% normal goat serum (Cell Signaling) and 0.1% Triton X-100. Next, cells were incubated at 4 °C with an anti-pMLC (Ser19) antibody (raised in rabbit; Cell Signaling; 3671; 1:100), anti-YAP antibody (raised in mouse; clone 63.7; Santa Cruz Biotechnology; sc-101199; 1:50) or anti-Ki-67 antibody (raised in mouse; 8D5; Cell Signaling, 9449; 1:800) overnight. For NHE1 and ezrin staining, cells were fixed with 4% paraformaldehyde, permeabilized in 0.1% Triton X-100 and blocked for 1 h at room temperature with 1% BSA containing 0.1% Triton X-100. Subsequently, cells were incubated at 4 °C with an anti-NHE1 antibody (raised in mouse; 54; Santa Cruz Biotechnology; sc-136239; 1:50) or anti-ezrin antibody (raised in rabbit; Cell Signaling; 3145S; 1:200) overnight. After washing three times with PBS, samples were incubated for 1 h at room temperature with Alexa Fluor 488 goat anti-rabbit immunoglobulin G (IgG) H+L (Invitrogen; A11034; 1:200), Alexa Fluor 488 goat anti-mouse immunoglobulin G (IgG) H+L (Invitrogen; A11029; 1:200) or Alexa Fluor Plus 647 goat anti-mouse immunoglobulin G (IgG) H+L (Invitrogen; A32728; 1:100). All antibodies were prepared in the respective blocking buffer. Nuclei and actin were also stained with Hoechst 33342 (Invitrogen; 1:2,000) and rhodamine phalloidin (Invitrogen; 1:200), respectively. Samples were imaged using a ×60 oil-immersion objective on a Nikon A1 confocal system. Samples stained with only secondary antibodies were used as negative controls.

The spatial distribution of pMLC and ARP3 was evaluated by drawing a rectangle along the entire length of the cell and recording the average intensity across the rectangle’s width using the Plot Profile function in ImageJ. Raw intensity data from each cell were compiled in Excel, and the moving average was calculated after normalizing the intensity with respect to the lowest signal for the cell. For quantification of NHE1 and ezrin polarization, free-form regions at the front and rear edges of the cells were manually drawn, and their mean intensity values were recorded. The front-to-rear ratio was calculated from the mean intensity values for each cell. For evaluation of the percentage of Ki-67-positive cells, 2,396 × 1,917 µm2 images were exported to ImageJ, and the total number of nuclei identified by DAPI as well as the number of Ki-67-positive cells were enumerated using the Find Maxima function on the respective fluorescent channels. To quantify the nuclear-to-cytoplasmic ratio of YAP1, the mean fluorescence intensity of YAP1 within a manually drawn ROI inside the nucleus was divided by the mean intensity of the same ROI placed outside the nucleus.

#### qPCR analysis

Scramble control, shTRPV4 or shAQP5 cells were grown to 95% confluency and their total RNA was isolated using Direct-zol RNA isolation kit (Zymo Research) according to manufacturer’s recommendations. Reverse transcription and qPCR were performed using standard techniques described previously19 using the following primer sets: TRPV4: F, 5′-CCCGTGAGAACACCAAGTTT-3′ and R, 5′-GTGTCCTCATCCGTCACCTC-3′; AQP5: F, 5′-CAGCTGGCACTCTGCATCTT-3′ and R, 5′-TGAACCGATTCATGACCACC-3′; MIIA: F, 5′-ATCCTGGAGGACCAGAACTGCA-3′ and R, 5′-GGCGAGGCTCTTAGATTTCTCC-3′; MIIB: F, 5′-GCTGATGGCAACTCTCCGAAAC-3′) and R, 5′-CTTCCAGGACACCATTACAGCG-3′); MIIC: F, 5′-CAGCCGTCAAATGCAAACCGAG-3′ and R, 5′-TTGCCTCTGTCGTCACCTTCTC; 18S: F, 5′-CAGCCACCCGAGATTGAGCA-3′ and R, 5′-TAGTAGCGACGGGCGGTGTG-3′.

#### Western blotting

Western blots were performed as described previously19,20,47 using NuPage 4–12% gels and the following antibodies. Uncropped, original blots are provided in Supplementary Fig. 1.

The following primary antibodies were used: anti-TRPV4 antibody (raised in mouse; 1B2.6; Millipore Sigma; MABS466; 1:1,000), anti-ARP3 (mouse; FMS338; Abcam; ab49671; 1:5,000), anti-ARPC4 (rabbit; Abcam; ab217065; 1:2,000), anti-integrin β1 antibody (rabbit; Cell Signaling; 4706S; 1:1,000) and anti-NHE1 (raised in mouse; 54; Santa Cruz Biotechnology; sc-136239; 1:200). GAPDH was used as a loading control (rabbit; 14C10; Cell Signaling; 2118S; 1:5,000).

The following secondary antibodies were used: anti-mouse IgG, HRP-linked antibody (Cell Signaling; 7076S; 1:1,000) and anti-rabbit IgG, HRP-linked antibody (Cell Signaling, 7074S; 1:1,000).

#### Electrophysiological recording

Ionic currents were recorded in the whole-cell patch-clamp mode32. Patch pipettes were filled with a solution containing 140 mM N-methyl-d-glucamine chloride, 1 mM MgCl2, 5 mM EGTA, 10 mM HEPES, 4 mM ATP and 0.1 mM GTP (300 mOsm l−1, pH 7.3). The bath solution contained 125 mM NaCl, 1.5 mM MgCl2, 1 mM EGTA, 10 mM HEPES and adjusted to 305 mOsm l−1 (with d-mannitol) and pH 7.36. Cells were held at 0 mV and ramps from −100 mV to +100 mV (400 ms) were applied at a frequency of 0.2 Hz. Ramp data were acquired at 10 kHz and were low-pass filtered at 1 kHz using Axon p-Clamp software. Experiments were performed at room temperature.

#### NHE1 activity measurement

To measure NHE1 activity, two different probes and methods were used. A methodology described in ref. 53 was used, which quantifies the rate of pH recovery in cells stably expressing the ratiometric pH sensor pHRed28 after intracellular acidosis due to brief exposure to NH4Cl. The second methodology used was the quantification of intracellular pH in SC and shNHE1 cells loaded with the pHrodo red-AM (Molecular Probes; P35372) as well as the measurements of changes in pHrodo signal in cells exposed to the NHE1 inhibitor EIPA. MDA-MB-231 cells expressing pHRed were excited at 561 nm and 405 nm laser lines, and emission was collected using a 600/50 detector. Intracellular pH is inversely proportional to the ratio of intensities at 561 nm over 405 nm (I561/I405).

Cells expressing pHRed were plated on collagen-I-coated glass-bottom dishes (Cellvis) at 20% confluency. After 2–6 h exposure to media of desired viscosities, cells were incubated with DMEM supplemented with 10% heat-inactivated FBS and 1% penicillin–streptomycin for 10 min. Next, cells were imaged every 30 s for 2 min on a Nikon A1 confocal microscope with a Plan Apo ×60 objective. Thereafter, the medium was gently aspirated and replaced with DMEM containing 10% heat-inactivated FBS, 1% penicillin–streptomycin and 15 mM NH4Cl. Cells were then imaged every 30 s for 4 min before replacing NH4Cl-containing medium with DMEM supplemented with 10% heat-inactivated FBS and 1% penicillin–streptomycin. Imaging was continued every 30 s for another 5 min. For analysis, the boundary around each cell was automatically traced at every time point using the ROI detection function in NIS Element (Nikon), and the ratiometric intensity values over time were exported. GraphPad Prism (GraphPad) was used to plot the time-dependent variation in intracellular pH and linear curves were fit to the pH recovery phase of each cell. The slope from the fitted line was used to quantify the rate of proton efflux or NHE1 activity where the rate of pH recovery is -dR/dt.

Intracellular pH was measured using pHrodo Red AM according to the manufacturer’s instructions. In brief, cells grown on 13 mm coverslips were exposed to 1 ml of bathing solution containing a dilution of 1 µl of 5 mM pHrodo Red AM in 10 µl of PowerLoad 100× concentrate at 37 °C for 30 min. Cells were washed twice with isotonic solution and allowed to settle for 5 min at room temperature in a handmade perfusion chamber. Video microscopic measurements of pHrodo Red AM fluorescence were obtained using an Olympus IX70 inverted microscope with a ×20 objective (Olympus). ROIs of GFP-positive cells were selected before running the experiment using a 488 nm excitation filter. To monitor pHrodo Red signal, cells were excited at 572 nm using a xenon arc lamp coupled to computer-controlled filter wheel (Lambda Series, Sutter Instrument). Fluorescence images were collected by an ORCA Flash4plus camera (Hamamatsu Photonics) after being passed through eGFP/mCherry dual-emission filter (Chroma Technology) using the HCI software (Hamamatsu Photonics). Images were acquired every 5 s after 65 ms exposure time. Basal fluorescence levels were recorded for 2 min in isotonic solution and the medium was then exchanged to isotonic medium containing 10 µM EIPA for 10 min. Subsequent intraexperimental calibration of the probe was performed using a commercially available intracellular pH calibration buffer kit (Molecular Probes; P35379) according to the manufacturer’s instructions. In brief, after each experiment, 2–3 ml of 7.5, 6.5, 5.5 and 4.5 pH calibration standards containing 10 µM valinomycin and 10 μM of nigericin were gently perfused for 3–5 min. An average of the last ten data points of each standard measurement were plotted and fitted to a standard linear curve. A linear trend equation was used to extrapolate the experimental pH values.

#### Cell exposure to medium of different viscosities

In some experiments, cells were plated in tissue culture flasks/plates in medium of various viscosities. The medium was replaced every alternate day, and cells were passed to a fresh flask/plate after reaching 90% confluency. Cells were maintained in medium of the desired viscosity for 6 days before being tested in in vitro and in vivo assays.

#### RNA-seq and analysis

Total RNA was extracted and purified using the Zymo Research Quick-DNA/RNA MiniPrep Plus Kit (Zymogen) according to the manufacturer’s recommendations. Strand-specific mRNA libraries were generated using the NEBNext Ultra II Directional RNA library prep Kit for Illumina (New England BioLabs, E7760), and mRNA was isolated using the poly(A) mRNA magnetic isolation module (New England BioLabs, E7490). Preparation of libraries was performed according to the manufacturer’s protocol (New England BioLabs, 2.2 05/19). Input was 1 µg and the samples were fragmented for 15 min for an RNA insert size of ~200 bp. The following PCR cycling conditions were used: 98 °C for 30 s; followed by 8 cycles of 98 °C for 10 s and 65 °C for 75 s; and a final extension of 65 °C for 5 min. Stranded mRNA libraries were sequenced on an Illumina NovaSeq instrument, SP flowcell using 100 bp paired-end dual-indexed reads and 1% PhiX control. All of the samples had over 39 million reads.

We mapped the reads to the human GRCh38.p13 genome (hg38) using the HISAT2 package54, annotated each gene using Ensemble104 and totalled the number of exon reads for each pair using the HTSEQ package. The human GRCh38.p13 genome (hg38) was obtained from Ensembl. The DESeq2.r package was used (in R, v.4.0) to normalize and compare the reads for each gene between the samples of interest, with respective P values for each gene per comparison and PCA score for each sample (Extended Data Fig. 11a). We selected differentially expressed genes with P ≤ 0.05 between SC samples at 0.77 cP versus 8 cP and between SC versus shTRPV4 at 8 cP. These data were presented in volcano plots (Extended Data Fig. 11b) and used for pathway analysis using ingenuity pathway analysis (Qiagen) (Extended Data Fig. 11c). The P values for the differentially expressed genes were generated by the DESeq2.r package (assuming a negative binomial model and corrected for FDR). Fisher’s exact method was used to calculate the significance of pathway enrichment from ingenuity pathway analysis with a threshold of P ≤ 0.05. Sample preparation and RNA-seq analysis were performed in a blinded manner.

#### Zebrafish husbandry

Animal studies were conducted in K. Tanner’s laboratory under protocols approved by the National Cancer Institute and the National Institutes of Health Animal Care and Use Committee. Sexually mature zebrafish used for breeding were maintained at 28.5 °C under a 14 h–10 h light–dark cycle. Transgenic Tg(mpx:eGFP/flk:mCherry) zebrafish55,56 were obtained from natural spawning, kept at 28.5 °C until the time of cell injection, and maintained in fish water (60 mg Instant Ocean sea salt (Instant Ocean) per litre of deionized water). To inhibit melanin formation and maintain optical transparency, embryos were transferred to fish water supplemented with N-phenylthiourea (PTU; Millipore Sigma) between 18–22 h after fertilization. PTU water was prepared by dissolving 16 µl of PTU stock (7.5% w/v in DMSO) per 40 ml of fish water. Water was replaced daily. Injected fish were maintained at 33 °C after injection and during intravital imaging.

#### Zebrafish circulatory injections

Before injection, cells were washed three times with PBS and incubated for 20 min at 37 °C at a concentration of 2 × 106 cells per ml in PBS containing 1 µM CellTracker Deep Red (Thermo Fisher Scientific). Stained cells were then washed with PBS and resuspended to a concentration of 1 × 106 cells per 20 µl in PBS for injection.

For injection to the circulation, 3 days post-fertilization zebrafish were anaesthetized in 0.4% Tris-buffered tricaine and oriented in a lateral orientation on an agarose bed. Then, 2–5 nl of cell suspension was injected using a pulled-glass micropipette in the circulation and directed towards the tail through the dorsal aorta. Fish from a given clutch were randomly divided into experimental groups before injection, and experiments were blocked so that treated and control cells were injected into larvae from the same clutch for three independent clutches. Injected fish were screened between 1–4 h after injection to check for successful dissemination of cells through the circulatory system, and were then imaged as described below. Fish lacking cells in the circulation were euthanized. Gonad differentiation to determine sex had not been completed by the time points used in these experiments.

#### Intravital microscopy of cell trafficking

Zebrafish larvae were anaesthetized and immobilized in a lateral orientation in 1% (w/v) low-gelling-temperature agarose (Millipore Sigma) dissolved in fish water approximately 4 h after cell injection. To enable high-resolution confocal imaging, fish were laterally oriented in coverglass-bottom chamber slides (Nunc Lab-Tek Chambered #1.0 Borosilicate Coverglass slides, Thermo Fisher Scientific). PTU water supplemented with 0.4% buffered tricaine was then added to the imaging chamber to keep the larvae anaesthetized over the course of the experiment.

One-photon, confocal 2D images were acquired at a resolution of 512 × 512 pixels, which were stacked to acquire 3D images. Images were obtained on the Zeiss 780 or 880 confocal microscope. For each larva, 1–2 confocal z-stacks centred on the intersegmental vessels were acquired every 10 min for ~50 frames at 2 µm axial steps to image a total depth of ~100 µm. Images were acquired with a Zeiss ×20 EC Plan-Apochromat, 0.8 NA objective. The samples were simultaneously excited with 488 nm light from an argon laser, 561 nm light from a solid-state laser and 633 nm light from a HeNe633 solid-state laser. The zebrafish larva was maintained at 33 °C for the course of imaging on a stage top incubator.

#### Intravital cancer cell tracking

Time-lapse microscopy images were exported to ImageJ for analysis. Images were analysed only if blood flow was visually confirmed throughout the larvae. To adjust for fish growth during imaging, images were first registered by phase-correlation-based translational registration using the Correct 3D Drift plugin57, with the vasculature of the fish used as a topographical reference. The multi-time scale computation, subpixel drift correction and edge enhancement options were enabled during registration. Cancer cells were tracked in 3D in registered images for up to 12 h from the registered image stacks using the TrackMate plugin for ImageJ58. Tracks were visually inspected for accuracy over the entire acquisition period and were manually edited to ensure that point-to-point tracks were generated for the entire time that a cell moved through the intersegmental vessels.

Cell tracks (x,y,t,z) were exported to MATLAB for analysis. For each cell, the frame-to-frame speed was calculated by dividing the displacement of the cell by the time interval between frames. An average speed for that cell over the course of imaging was then calculated by averaging these frame-to-frame speeds. The persistence of migration was calculated by dividing the net displacement of a given cell over the course of tracking by the total point-to-point distance travelled. Speed and persistence values were calculated only for cells remaining in the intersegmental vessels for at least 4 frames, and values were calculated over a maximum of 71 frames. The average speed of cell movement through the intersegmental vessels for a given larva was calculated from mean cell speeds.

#### Ex ovo chick embryo cancer cell extravasation model

Fertilized white leghorn chicken eggs were acquired from the University of Alberta Poultry Research Centre and maintained at 38 °C. Embryos were isolated from their shells after 4 days of incubation and maintained under shell-less conditions in a covered dish placed in an air incubator at 38 °C and 60% humidity2. A protocol described previously59 was adopted to image and quantify tumour cell extravasation in chick embryos, which were randomly divided into experimental groups before injection. In brief, 25–50 × 103 YFP-tagged MDA-MB-231 cells, preconditioned for 6 days at either 3 cP or 0.77 cP (naive cells), were suspended in ice-cold PBS and injected intravenously into the CAM vasculature. Tumour cell extravasation was assessed by intravital confocal imaging 5 h after cell injection. Then, 15 min before intravital imaging, lectin-649 was injected into the CAM vasculature for visualization of blood vessels. At least 16 animals were used for each condition for 3 independent experiments. All of the procedures were approved by University of Alberta Institutional Animal Care and Use Committee (IACUC).

#### Tail-vein injection into mice

The protocol followed for intravenous injection of MDA-MB-231 cells was previously described45. In brief, female NOD-SCID Gamma (NSG) mice (aged 5 to 7 weeks) were obtained from the Johns Hopkins animal core facility, and were distributed into groups to obtain a random weight distribution across conditions. Injections were performed in a blinded manner, without knowledge of cell pretreatment conditions or identity of shRNA-mediated modifications. Before injection, the mice were warmed for 5–10 min with an overhead heat lamp to dilate the tail veins. MDA-MB-231 cells (5 × 105) suspended in 200 µl of normal saline were injected. To assess the extent of micro- and macro-metastasis in the lungs, mice were euthanized 48 h or 3 weeks after inoculation, respectively. Lungs were inflated with optimal cutting temperature (OCT) compound (Fisher Healthcare) and excised for both image and qPCR analysis. Animal studies were conducted with all relevant ethical regulations outlined in protocols approved by the Johns Hopkins University Animal Care and Use Committee. For the 3 week lung colonization experiments, mice were observed for any potential humane experimental end-point signs, including respiratory distress, hunched posture, poor grooming and loss of weight according to IACUC approved protocols. No humane end-point signs were observed during any experiment. Animal rooms were maintained at 30–70% relative humidity and a temperature of 18–26 °C with a minimum of 10 room air changes per hour. Cages were changed once a week. Mice were fed a diet containing low fibre (5%), protein (20%) and fat (5–10%).

#### Metastatic burden analysis by imaging

Excised lungs inflated with OCT were formalin-fixed, saturated in 30% sucrose at 4 °C overnight, embedded in OCT compound, flash-frozen in liquid nitrogen and sectioned (15 µm thick) using a cryotome CM1100 (Leica). The sections were mounted onto Superfrost Plus Microscope Slides (Thermo Fisher Scientific), permeabilized with 1% Triton X-100 for 10 min and blocked with 2% BSA for 30 min. The sections were stained for human vimentin (raised in mouse; O91D3; BioLegend; Alexa Fluor 647 conjugated; 677807; 1:200 dilution) for 1.5 h followed by DAPI (1:1,000) incubation for 15 min and mounted for imaging with anti-fade solution (90% glycerol, 20 mM Tris, pH 8.0, and 0.5% N-prolyl gallate). Background fluorescence was reduced with 0.1% Sudan Black treatment for 25 min.

Tissue-mounted slides were imaged using a Nikon A1 confocal microscope with a ×10/0.45 NA air objective. Five or ten areas of 1,272 × 1,272 μm2 each were randomly selected for samples collected 48 h or 3 weeks after injection, respectively. Images were exported to ImageJ and vimentin-bearing metastatic sites were annotated for manual counting. Micrometastases after 48 h were enumerated as any vimentin-positive single cell, whereas macrometastases after 3 weeks were defined as vimentin-positive clusters containing >5 nuclei.

#### Lung colonization analysis by qPCR

qPCR analysis of hLINE19,60 was performed in triplicate and the average value per mouse was reported. In brief, two small lobes of the excised lungs were cut and flash-frozen in liquid nitrogen. For DNA extraction, the lung tissue was lysed in lysis buffer (1 M Tris, pH 8.0, 5 M NaCl, 0.5 M EDTA, 10% Tween-20, 10% NP-40 and 40 µg of proteinase K), homogenized and incubated at 55 °C. After 2 h, proteinase K was deactivated at 95 °C for 2 min. DNA was then extracted by phenol–chloroform, precipitated with 2.5 mM NaCl and 100% ethanol (prechilled at −20 °C), and washed with 70% ethanol. DNA samples were resuspended in pure water after air drying. qPCR was performed using the primer pair for hLINE: forward, 5′-TCACTCAAAGCCGCTCAACTAC-3′; reverse, 5′-TCTGCCTTCATTTCGTTATGTACC-3′. A primer pair detecting both mouse and human beta actin sequences in the DNA samples (forward, 5′-ACGAGGCCCAGAGCAAGAGA-3′; reverse, 5′-GCCACACGCAGCTCATTGTAG-3′) was used as baseline to normalize the expression level of human DNA (measured by hLINE) over the total amount of DNA input (measured by mouse and human actin).

#### Statistics and reproducibility

Data represent the mean ± s.d. of ≥3 experiments (reflecting independent biological replicas) with ≥30 cells analysed per condition per experiment, and data points denote values from each cell, unless otherwise specified. Shapiro–Wilk tests were used for normality testing in cases in which the number of data points was between 3 and 8. For samples with >8 data points, the D’Agostino–Pearson omnibus normality test was used to determine whether data were normally distributed. Datasets with Gaussian distributions were compared using two-tailed Student’s t-tests and one-way ANOVA followed by Tukey’s post hoc test. For log normal distribution, statistical comparison was made after logarithmic transformation of the data followed by unpaired two-tailed Student’s t-tests or one-way ANOVA followed by Tukey’s post hoc test. For non-Gaussian distributions, nonparametricMann–Whitney U-tests were used comparing two conditions, and comparisons for more than two groups were performed using Kruskal–Wallis tests followed by Dunn’s multiple-comparison test. Select experiments were analysed using two-way ANOVA followed by Šidák’s or Tukey’s multiple-comparisons test. Analysis was performed using GraphPad Prism 7, 8 or 9 (GraphPad Software). P < 0.05 was considered to be statistically significant; *P < 0.05, **P < 0.01, ***P < 0.001, ****P < 0.0001. The exact sample size, number of replicates, P values and statistical tests performed are provided in Supplementary Information 5–19.

In Fig. 1, images are representative of 3 (Fig. 1d,h,l) or 2 (Fig. 1f,i) independent biological replicas. In Fig. 2, images are representative of 3 (Fig. 2b,g) or 2 (Fig. 2o) independent biological replicas. In Fig. 3, images are representative of ≥3 (Fig. 3a,d) or 2 (Fig. 3g) independent experiments. In Fig. 4, images are representative of 3 (Fig. 4e) or 2 (Fig. 4i) independent experiments.

### Theoretical methods

#### Stochastic model of actin network

A stochastic 2D model of actin-based lamellipodia protrusion was constructed in Python v.3.8 using frameworks and public code established in ref. 23. As actin filaments at the cell leading edge undergo elongation, branching and capping events at the barbed ends following a Poisson process, the actin network grows against the plasma membrane tension, and exerts a force that determines the protrusion velocity of the network leading edge. The model used previously determined force–velocity relationships recapitulating not only the Brownian ratchet model of polymerization but also other models such as the tethered ratchet or end-tracking motors. While the published model examined the effects of transient changes in membrane tension on leading edge network organization, we incorporated dynamic forces acting on the actin network due to extracellular fluid of a defined viscosity into the current model. We introduced viscous resistive force into the model using a linear-viscous dashpot61 where viscous loading forces on the leading edge scaled linearly with extracellular viscosity. The total force per unit width at the leading edge can be described as

$$F={F}_{0}+{F}_{\eta },$$

(1)

where, F0 and Fη are forces due to membrane tension and the viscous external fluid, respectively. The viscous force at every iteration (i) can be written as

$${F}_{\eta }=k\times \eta \times {v}_{i-1},$$

(2)

where, k(=100,000) is a dimensionless scaling constant, η is the extracellular viscosity and vi − 1 is the edge velocity from the last iteration.

Using viscosity values tested experimentally, the effects of viscous loading in actin network formation was modelled.

### Two-phase model for confined cell migration

#### Model description

A two-phase, actin–water-osmosis-coupled cell migration model26,62 was used to explain the cell behaviour observed experimentally at different extracellular viscosities. The two phases refer to the F-actin network and the cytosol. G-actin is in the cytosol phase. The model also includes ion diffusion and flux. Active ion fluxes arise from ion pumps on the cell membrane. Without loss of generality, we assume that the ions are electroneutral. The steady-state model is briefly described below.

To model cell migration in confined channels, we use a 1D coordinate system to represent the cell. For a cell of length L, we let x [0,L] represent the cell domain, which is attached to the moving cell. The cytosol phase, with pressure p and velocity vc, can be considered to be an incompressible fluid and the pressure gradient should be balanced by the friction between the cytosol and the actin network. These two conditions can be written as

$$-\frac{{\rm{d}}p}{{\rm{d}}x}-\eta {\theta }_{n}\left({v}_{c}-{v}_{n}\right)=0,\,\frac{{\rm{d}}{v}_{c}}{{\rm{d}}x}=0,$$

(3)

where η is the coefficient of interfacial friction between the two phases; θn and vn are the concentration and velocity of the actin network phase, respectively. At the cell boundary, water flows across the cell membrane due to the chemical potential difference of water between the inside and outside of the cell7. If we assume the water flux, Jwater, to be positive in the inward direction, then the boundary condition for the cytosol phase is

$${v}_{c}-{v}_{0}=-{J}_{{\rm{w}}{\rm{a}}{\rm{t}}{\rm{e}}{\rm{r}}}^{{\rm{f}}}\,{\rm{a}}{\rm{t}}\,x=L\,;\,{v}_{c}-{v}_{0}={J}_{{\rm{w}}{\rm{a}}{\rm{t}}{\rm{e}}{\rm{r}}}^{{\rm{b}}}\,{\rm{a}}{\rm{t}}\,x=0,$$

(4)

where v0 is the steady-state velocity of the cell. The superscript notes f and b indicate quantities associated with the front and rear (or back) of the cell, respectively. The water flux is calculated from

$${J}_{{\rm{water}}}^{{\rm{f}}\left({\rm{b}}\right)}=-{\alpha }^{{\rm{f}}\left({\rm{b}}\right)}[({p}^{{\rm{f}}\left({\rm{b}}\right)}-{p}_{* }^{{\rm{f}}\left({\rm{b}}\right)})-RT({c}^{{\rm{f}}\left({\rm{b}}\right)}-{c}_{0}^{{\rm{f}}\left({\rm{b}}\right)})],$$

(5)

where α is the permeability coefficient of water, c is the concentration of ion, and RT is the ideal gas constant multiplied by the absolute temperature. The subscript 0 indicates the extracellular environment. p* is the hydraulic pressure experienced by the cell, which is different from the hydraulic pressure at the end of the channel, p0, owing to hydraulic resistance developed in the channel. From the analysis of a generic pipe flow, the pressure experienced by the cell can be expressed as

$${p}_{\ast }^{{\rm{f}}}={p}_{0}^{{\rm{f}}}+{d}_{g}^{{\rm{f}}}({v}_{0}-{J}_{{\rm{w}}{\rm{a}}{\rm{t}}{\rm{e}}{\rm{r}}}^{{\rm{f}}}),\,{p}_{\ast }^{{\rm{b}}}={p}_{0}^{{\rm{b}}}-{d}_{g}^{{\rm{b}}}({v}_{0}+{J}_{{\rm{w}}{\rm{a}}{\rm{t}}{\rm{e}}{\rm{r}}}^{{\rm{b}}}),$$

(6)

where dg is the coefficient of external hydraulic resistance, which depends on the geometry of the channel and viscosity of the extracellular medium.

The actin network phase also has a pressure, σn, arising from actin swelling. Without loss of generality, we can assume that the passive swelling pressure is proportional to the concentration of the actin network, that is, $${\sigma }_{n}={k}_{{\sigma }_{n}}{\theta }_{n}$$, where $${k}_{{\sigma }_{n}}$$ is a constant. The force from focal adhesions provides an effective body force on the actin network. The force balance on the actin network is

$$-\frac{{\rm{d}}{\sigma }_{n}}{{\rm{d}}x}+\eta {\theta }_{n}\left({v}_{c}-{v}_{n}\right)-{\eta }_{{\rm{st}}}{\theta }_{n}{v}_{n}=0,$$

(7)

where ηst is the strength of focal adhesions. Actin polymerization typically occurs at the front of the cell, whereas actin depolymerization occurs within the cytoplasm. We assume that γ is a constant rate of actin depolymerization. As F-actin and G-actin convert interchangeably, the mass conservations of the two actin species are described by

$$\frac{{\rm{d}}}{{\rm{d}}x}\left({v}_{n}{\theta }_{n}\right)=-\gamma {\theta }_{n},\frac{{\rm{d}}}{{\rm{d}}x}\left({v}_{c}{\theta }_{c}\right)={D}_{{\theta }_{c}}\frac{{{\rm{d}}}^{2}{\theta }_{c}}{{\rm{d}}{x}^{2}}+\gamma {\theta }_{n},$$

(8)

where θc and $${D}_{{\theta }_{c}}$$ are the concentration and diffusion coefficient of G-actin, respectively. At the front end of the cell, the flux boundary condition for F-actin and G-actin are related to the rate of actin polymerization, Jactin, and the boundary condition for F-actin and G-actin is

$${\theta }_{n}({v}_{0}-{v}_{n})={J}_{{\rm{a}}{\rm{c}}{\rm{t}}{\rm{i}}{\rm{n}}},{\theta }_{c}({v}_{0}-{v}_{c})=-{J}_{{\rm{a}}{\rm{c}}{\rm{t}}{\rm{i}}{\rm{n}}},\text{at}\,x=L,$$

(9)

where $${J}_{{\rm{actin}}}={J}_{{\rm{actin}}}^{{\rm{f}}}{\theta }_{c}/({\theta }_{c,c}+{\theta }_{c})$$ and $${J}_{{\rm{actin}}}^{{\rm{f}}}$$ and θc,c are constants. As actin depolymerization occurs throughout the cytoplasm, the flux boundary conditions for F-actin and G-actin are zero at the rear of the cell. At the time scale we are interested in, the total amount of F-actin and G-actin remain a constant so that $${\int }_{0}^{L}\left({\theta }_{n}+{\theta }_{c}\right){\rm{d}}x=L{\theta }_{* }$$, where θ* is the average concentration of actin.

The diffusion–convection equation for ions is

$$\frac{{\rm{d}}}{{\rm{d}}x}\left({v}_{c}c\right)={D}_{c}\frac{{{\rm{d}}}^{2}c}{{\rm{d}}{x}^{2}},$$

(10)

where Dc is the diffusion coefficient of ions. The flux at the boundary is a combination of passive, Jc,p, and active, Jc,active, fluxes. The passive flux follows the concentration difference of the ion, that is, $${J}_{c,{\rm{p}}}^{{\rm{f}}\left({\rm{b}}\right)}=-{k}_{{\rm{sol}}}^{{\rm{f}}\left({\rm{b}}\right)}({c}^{{\rm{f}}\left({\rm{b}}\right)}-{c}_{0}^{{\rm{f}}\left({\rm{b}}\right)})$$, where ksol is the permeability of the passive ion flux. The negative sign arises from the model convention that all inward fluxes are considered positive. The active ion flux is prescribed in the model.

Frictional forces exist between the cell’s lateral membrane and the channel wall. The magnitude of the frictional force can be expressed as Ff = ξv0, where ξ is a friction coefficient. This force is in the opposite direction of cell migration. By considering all the external forces acting on the cell, the force balance at the cellular level is

$$-\left({p}_{0}^{{\rm{f}}}-{p}_{0}^{{\rm{b}}}\right)-\left({d}_{g}^{{\rm{f}}}+{d}_{g}^{{\rm{b}}}\right)\left({v}_{0}-{J}_{{\rm{water}}}^{{\rm{f}}}\right)-{\eta }_{{\rm{st}}}{\int }_{0}^{L}{\theta }_{n}{v}_{n}{\rm{d}}x-{F}_{f}=0.$$

(11)

The system is solved by considering all of the equations together.

#### Model specifications and parameters

From experiments, we observed a polarized distribution of focal adhesions, myosin II activity (pMLC) and NHE1 at elevated viscosities. The distributions of focal adhesions and pMLC are highly correlated (Extended Data Figs. 4i and 8j), which is consistent with the idea that myosin II activity facilitates the maturation of focal adhesions63,64. We therefore assume that the distribution of focal adhesions occurs as a result of the distribution of myosin II, which is not explicitly included in the model. The polarized distribution of NHE1 (Fig. 2c) is also a part of the model input.

The experimental data suggest that the distribution of focal adhesions is non-uniform across the cell length (Extended Data Fig. 4i). For cells at 0.77 cP medium (low viscosity (LV)), the distribution of focal adhesions is high at both ends and low in the middle of the cell. We use a quadratic function

$${\bar{\eta }}_{st,{\rm{LV}}}=4\left(1-{b}_{{\rm{LV}}}\right){\left(\bar{x}-1/2\right)}^{2}+{b}_{{\rm{LV}}}$$

(12)

to describe the normalized distribution of focal adhesions in low viscosity medium (Extended Data Fig. 5e), where $$\bar{x}$$ is the normalized coordinate of the cell length, and bLV [0,1] is the minimum value of the profile at $$\bar{x}=1/2$$. For cells in 8 cP medium (high viscosity (HV)), the distribution of focal adhesions is high at the leading edge and low at the trailing edge of the cell. Therefore, we use a different function

$${\bar{\eta }}_{st,{\rm{HV}}}=\left(1-{b}_{{\rm{HV}}}\right){\bar{x}}^{4}+{b}_{{\rm{HV}}}$$

(13)

to describe the normalized distribution of focal adhesion for high viscosity medium (Extended Data Fig. 5e), where the minimum value bHV occurs at $$\bar{x}=0$$. Experimental data suggest that bLV = 0.3 at 0.77 cP and bHV = 0.25 at 8 cP (Extended Data Fig. 4i).

The non-normalized local strength of focal adhesions is expressed as $${\eta }_{st,i}={\eta }_{st,i}^{0}{\bar{\eta }}_{st,i}$$ where i {LV,HV} and $${\eta }_{st,i}^{0}$$ is the coefficient of focal adhesions, which varies with the extracellular viscosity. To approximate the strength of focal adhesions, we combined the distribution of focal adhesion size and number for all cells in each group (Extended Data Fig. 4j) and divided them by the corresponding number of cells to obtain a normalized focal adhesion strength map along the cell length. From experimental data (Extended Data Fig. 4j), the maximum local focal adhesion area at 8 cP is about 8× larger than that at 0.77 cP. We therefore assume $${\eta }_{st,{\rm{HV}}}^{0}=8{\eta }_{st,{\rm{LV}}}^{0}$$. The value of $${\eta }_{st,{\rm{LV}}}^{0}$$ is estimated from literature62,65 with the consideration that cells under confinement experience a relatively low density of focal adhesions compared with cells in 2D66,67.

For confined cell migration in a 1D channel, the coefficient of hydraulic resistance is estimated from a pipe flow68 as dg = 12 μl0/w2, where μ is the viscosity of the extracellular medium, l0 is the effective channel length and w is the smallest dimension of the cross-sectional geometry, which is 3.5 μm in this work. While the length of cells varies with viscosity by virtue of their volume alteration, the channel length (L0 = 200 μm) and cross-sectional area (35 μm2) are constants across both viscosities. For this maths model, we assume that the cell occupies the entire cross section of the channel and therefore the cell lengths can be calculated from the measure of average cell volume in confinement (Fig. 2a): for 0.77 cP, the cell length is L = 85 μm, and for 8 cP, the cell length is L = 125 μm. The effective channel length is calculated by l0 = L0 − L. On the basis of these parameters, we can then calculate the coefficient of hydraulic resistance for each given viscosity μ. The viscosity of the medium not only increases the hydraulic resistance experienced by a cell, but also increases the friction between the cell and channel. Thus, we use a higher frictional coefficient for 8 cP medium (ξHV = 900 Pa s μm−1) than 0.77 cP (ξLV = 180 Pa s μm−1).

The polarized ion fluxes are determined by the distribution of membrane ion channels. We use $${J}_{c,{\rm{active}}}^{{\rm{b}}}$$ and $${J}_{c,{\rm{active}}}^{{\rm{f}}}$$ to represent active ion fluxes at the back (trailing edge) and front (leading edge) of a cell, respectively. For polarized cells, $$\left|\,{J}_{c,{\rm{active}}}^{{\rm{f}}}\right|{\ne {\rm{| }}J}_{c,{\rm{active}}}^{{\rm{b}}}\,{\rm{| }}$$ and we let the ratio $$\gamma ={\rm{| }}{J}_{c,{\rm{active}}}^{{\rm{f}}}{/J}_{c,{\rm{active}}}^{{\rm{b}}}{\rm{| }}$$ be the same as the polarization ratio of NHE1. Thus, for 0.77 cP medium, γLV = 1.67, and for 8 cP medium, γHV = 2.84 (Fig. 2c). In the model, we fix $${J}_{c,{\rm{active}}}^{{\rm{b}}}$$ and let $${J}_{c,{\rm{active}}}^{f}=-\gamma {J}_{c,{\rm{active}}}^{{\rm{b}}}$$. When NHE1 is inhibited, $${J}_{c,{\rm{active}}}^{f}={J}_{c,{\rm{active}}}^{{\rm{b}}}=0$$.

The remaining parameters are identical at both viscosities and are listed in Supplementary Information 4.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

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