text:
## Introduction

From the point of view of energy metabolism, cell migration is costly. Cytoskeletal dynamics generate and transmit the forces that drive cellular migration and, in doing so, can consume an estimated 50% of the cell’s ATP budget1,2. As such, a relationship between cellular metabolism and cellular migration is to be expected but the nature of such a relationship has remained unclear. On this topic, much of what is known focuses on cancer biology, the roles of the extracellular matrix (ECM; Supplement 1), and the epithelial-to-mesenchymal transition (EMT). There is consensus that the migrating mesenchymal phenotype exhibits increased aerobic glycolysis3,4,5,6,7,8 and that the Warburg Effect, which is typical of many cancer cells9,10, may promote migratory or invasive behavior11,12. For example, mesenchymal relative to epithelial prostate cancer cells exhibit increased rates of aerobic glycolysis as well as increased migration speeds and traction forces exerted by each cell on its substrate4. Even non-cancerous migratory processes, such as the neural crest cell undergoing EMT, exhibit increased glycolytic flux relative to the non-migratory cell3. Similarly, compared to the uninjured epithelial layer, the wounded migrating epithelial cell layer preferentially utilizes glycolysis in much the same manner as migrating mesenchymal cells13. Recent spatial imaging of the metabolic state of the wounded epithelium shows large variations in fluorescence lifetimes of NADH14,15, an electron carrier molecule ubiquitously involved in energy metabolism. From a mechanistic perspective, cells can modulate their metabolic state in response to extracellular forces through mechanotransduction that is associated with the adhesome and the cytoskeleton16,17,18, but it remains unclear how metabolic events are connected to collective epithelial cell migration more generally and the cellular unjamming transition (UJT) in particular.

UJT is known to occur during embryonic development19,20, injury repair21, tumor metastasis22,23, and in the asthmatic airways19,24. Both the UJT and the EMT potentiate epithelial plasticity and migration, but underlying mechanisms of UJT versus EMT are distinct19,20,25,26. Unlike the case of the EMT, for example, in the case of the UJT the cell–cell junction and barrier function remain fully intact25. The cardinal feature of the UJT, however, is the transition of the confluent epithelial collective from a solid-like, non-migratory, jammed phase to a fluid-like, migratory, unjammed phase27. In the jammed epithelium, each cell remains in a fixed spatial relationship to each of its neighbors, as if the collective were frozen. Even though cell–cell junctions remain intact, the UJT enables these cells to unfreeze and thereby more readily swap positions with their neighbors and migrate in fluid-like swirling multicellular packs and eddies. This physical solid-to-fluid transformation is typified by dynamical, morphological, and mechanical alterations19,24,25,27. This mechanical plasticity allows for rapid response to tissue perturbations while maintaining uncompromised barrier function25. It also suggests the possibility of hijacking or disruption of these processes in epithelial diseases. But little is known about the energetic events that fuel these changes. Data presented here demonstrates that the unjamming cell layer undergoes a dramatic shift in energy metabolism toward glycolysis.

## Results

We utilized multiple modes of metabolic imaging simultaneously with measurements of cell mechanics, dynamics, and morphology. These measurements, taken together, reveal the complex spatiotemporal mechano-bioenergetic changes that occur in the confluent epithelial layer both while confined by a barrier and while undergoing collective migration into free space. We used a removable polydimethyl-siloxane (PDMS) mask on polyacrylamide (PA) gel to grow an epithelial cell monolayer under geometric confinement (Fig. 1a,b). The PA gel was 100 μm thick, had a shear modulus of 9.6 kPa, and was coated with collagen-I (Methods). Just below its apical surface this gel also contained many fiducial markers comprised of fluorescent labeled beads (0.2 µm diameter) whose displacements in response to cellular contractile forces were used for traction microscopy28. This gel was then seeded with Madin-Darby Canine Kidney (MDCKII) cells which were grown to confluence until they tiled the entire available surface (Fig. 1c). After the cell layer reached confluence, the PDMS barrier was removed and, in response, the cell layer migrated into the newly created free space (Fig. 1d,e). As has been characterized elsewhere, this simplified model recapitulates cardinal features of epithelial wound healing but allows for measurement of cell mechanics, dynamics, and morphology29. In order to measure cell metabolic properties simultaneously with mechanics, these MDCKII cells were stably transfected with mCherry-Peredox-NLS (Methods) which enabled assessment of the cytoplasmic redox potential (NAD + /NADH) across the cell layer with single cell resolution30 (Fig. 1f–j, Supplementary Fig. 1). Additionally, we used fluorescence lifetime imaging (FLIM) to isolate contributions from free and enzyme bound NADH across the cell layer. To further map metabolic heterogeneities developing during collective migration, we quantified the total glucose uptake per cell using the 2-NBDG glucose uptake assay and measured the mitochondrial membrane potential using a membrane potential dependent dye, TMRE (Methods).

### Redox ratios decrease as cells unjam and become migratory

Tendencies noted above, as well systematic relationships between morphological, migratory, mechanical and metabolic indices, are analyzed statistically and quantified on a cell-by-cell basis in Supplement 3. These analyses indicate that among all variables measured, the strongest statistical predictor of local migration speed was local cell perimeter, but with an important contribution from the NAD + /NADH ratio. Similarly, the strongest statistical predictor of the local NAD + /NADH ratio was also local cell perimeter.

### NADH lifetimes decrease at the migrating edge and in the non-migratory bulk

Prior to barrier lifting and after barrier lifting at 4 h and 24 h, non-migrating cells in the confined jammed layer exhibited a uniform and elevated NADH mean lifetime (Fig. 3a,b) indicative of a larger fraction of NADH in the bound-state (Fig. 3c,d). Relative to before barrier lift, at 4 h the mean NADH lifetime was slightly reduced across the layer (Fig. 3a,b) and the free-to-bound ratio increased (Fig. 3c,d). At 24 h after lifting the barrier, a spatial gradient emerged whereby cells at the migrating unjammed front showed a sharp decrease in the mean NADH lifetime (Fig. 3a,b) and an increase in the free-to-bound ratio (Fig. 3c,d) with respect to the jammed bulk. Surprisingly, soon after barrier lifting, cells in the non-migratory and jammed center of the layer also exhibited a marked decrease in mean NADH lifetimes. Together with the results from the Peredox bioprobe, these FLIM results are consistent with an overall shift in the unjamming epithelial layer towards glycolysis40,42,43,44.

### Glucose uptake increases in the migrating edge and in the non-migratory bulk

To examine more specifically cell glycolytic activity, we then mapped glucose uptake in wild-type MDCKII cells using 2-NBDG, a fluorescent, non-catabolizable glucose analog (Methods)45,46,47. In the non-migratory and confined layer prior to barrier lifting, cells throughout the layer showed uniform and small glucose uptake per cell, i.e., after normalization by cell area (Fig. 4a,b). At timepoints after lifting the barrier, glucose uptake per cell area progressively increased everywhere throughout the layer, with the largest increase occurring in cells near the leading edge (Fig. 4a,b). This glucose uptake profile across the unjamming cell layer is consistent with the trends in the NADH lifetimes measured by FLIM. Further, it confirms that a metabolic shift toward glycolysis occurred in regions deep within the layer, beyond those regions where cells had unjammed, mobilized, and generated increased tractions.

### Mitochondria membrane potential increases with cell migration, but only near the leading edge

Mitochondrial membrane potential (MMP) drives ATP production by powering ATP synthase. As such, MMP is a key bioenergetic quantity related to ATP flux during cell respiration. To capture spatiotemporal changes in MMP across the cell layer, we used the lipophilic cationic dye, Tetramethylrhodamine ethyl ester perchlorate (TMRE) (Methods)48. TMRE fluorescence intensity across the confined, immotile, and jammed epithelial layer was small and uniform (Fig. 4c,d). At 4 h and 24 h after the barrier was lifted, only the cells within a few cell-lengths of the migrating edge displayed increased TMRE fluorescence signal (Fig. 4c,d). When we short-circuited the mitochondrial membrane by treating the cells with the mitochondrial membrane potential decoupler, FCCP, TMRE fluorescence signals were substantially reduced (Supplementary Fig. 4). To test if the variation of TMRE fluorescence was related to mitochondrial mass, we stained mitochondria with MitoView-green, a membrane potential independent dye. With the exception of first row leader cells, the MitoView signal was essentially uniform throughout the layer (Supplementary Fig. 4), thereby suggesting that MMP, as indicated by TMRE, is moderately increased at the advancing edge but remains low everywhere else in the layer.

## Discussion

The epithelial layer is ordinarily non-migratory, jammed and quiescent but all the time retains the capacity to unjam and, in doing so, transition to a migratory phase in response to the demand for tissue development, growth, remodeling or wound repair19,29,49. It has been suggested that these salutary unjamming behaviors can sometimes become hijacked in disease, however, and thereby initiate unwelcomed cellular migration and tissue remodeling as in the cases of tumor invasion or asthmatic remodeling of the airway wall24,50,51,52,53. In the advancing confluent MDCKII monolayer, here we have integrated multiple modes of metabolic imaging in order to characterize with single cell precision the metabolic state simultaneously with metrics of cell morphology, migration, and mechanics. The jammed non-migratory epithelial layer confined by the PDMS barrier exhibits small and uniform cell speeds, traction forces, cell areas, and aspect ratios. The confined cell layer also exhibits a homogenous metabolic state as evidenced by the uniform cytoplasmic NAD + /NADH ratio, NADH lifetime, glucose uptake, and MMP.

What drives the metabolic gradient in the jammed non-migratory bulk? In cells grown on PA gels, long distance substrate-deformation fields caused by cell–cell interactions during collective migration can drive large-scale and long-distance correlations in cell motion, particularly in regions of high cell density as is found far from the advancing edge58. These deformation fields have been shown to induce mechanical cooperativity throughout the cell layer. As such, we hypothesize that the long-distance transmission of extracellular forces through the elastic substrate, generated by collective cell migration near the advancing front, has the potential to induce glycolysis, even far back from the migrating edge, through the stimulation of cytoskeletal-based mechanotransductive pathways such as PI3-K mediated RAC activation found in MCF10A cells18, or LKB1 mediated AMPK activation as found in MCF10A and MDCK cells16. While mechanical force transmission through the cell layer is likely to affect the metabolic state of those cells, the degree to which biochemical signaling and mechanical perturbations respectively induce a metabolic shift in the unjamming cell layer remains an open question.

As regards molecular mechanisms of unjamming, our laboratory has recently reported results using a different epithelial system26,61. In the case of human bronchial epithelial cells, genome-wide RNA-Seq, protein–protein interaction analysis, and gene ontology analysis of temporal networks show that epithelial cell unjamming involves the interplay of downstream signaling pathways involving integrins, ERK and JNK. In that system, moreover, gene ontology analyses on differentially expressed genes at early time points identify enrichment of the lipid metabolic process, the cellular ketone metabolic process, regulation of small molecule metabolic processes, and regulation of cellular carbohydrate metabolic process. These processes involve oxidization of fatty acids and are subsequently metabolized in mitochondria for ATP/ energy generation. Additionally, the expression of the mitochondrial carrier protein SLC25A25 involved in uptake and efflux of adenine nucleotides into or from the mitochondria, was significantly induced and hint toward a high energy need and production in the initial phase of UJT that is covered by fatty acid oxidization. In unjammed and migrating cells, the use of unsaturated fatty acid metabolic process to generate energy and cell communication by electrical coupling were significantly enriched. In addition, these unjammed epithelial cells induced the expression of the glucose transporter (SLC2A6) to allow higher glucose uptake from the environment for energy production. To meet the energy need of unjammed migrating cells in that system, these observations, taken together, suggest that the unjamming response initially employs fatty acid oxidation and later enhanced glucose uptake.

Based upon cell-by-cell statistical analysis as shown in Supplement 3, an empirical relationship emerged linking local redox ratio mainly to cell perimeter. This relationship calls to mind how integrins and cadherins interact in a force-dependent manner with a master regulator of energy homeostasis, namely, AMP-activated protein kinase (AMPK) and its upstream activator, Liver Kinase B1 (LKB1)62. Accordingly, a plausible mechanistic explanation of this empirical relationship between cell morphology and cell energy metabolism is that local integrin-dependent traction forces generate greater cadherin-dependent intercellular forces28,63. These forces lead to more cell stretch and thus additional cell perimeter. Through this mechanism, it is therefore possible that a larger cell perimeter leads to a greater extent of glycolytic activation. As the cytoplasmic redox ratio is sensitive to both glycolytic and mitochondrial activity, there exists an alternative notion linking the redox ratio to cell perimeter. Repeated stretching has been shown to regulate ATP synthase and mitochondrial ATP production17. As such, it is possible that cells with excessive perimeter, indicative of cell-stretch, could exhibit an altered cellular redox ratio through this mechanism.

### Epithelial unjamming is energetically expensive whereas jamming is economical

The epithelial layer is understood to be a highly dynamic tissue. Here we show this statement to be true but incomplete. As a result of the interconversion between jammed versus unjammed phases, cellular dynamics in the very same tissue can differ dramatically in space and time. The jammed phase of the confluent epithelial collective is solid-like and non-migratory19,21,24,28,64. The unjammed phase of the confluent epithelial collective is fluid-like, migratory, and shifted toward glycolytic energy metabolism. This metabolic shift suggests that the unjammed layer possesses an increased metabolic demand that is fulfilled by the fast but inefficient glycolytic energy metabolism.

It has been suggested previously that the UJT may have evolved in the early multi-cellular organism as a primitive adaptation in response to a specific evolutionary pressure, namely, the need to accommodate epithelial migration, plasticity, or development under the physiological constraint of preserving epithelial continuity, integrity and barrier function25. Observations reported here suggest that the jammed phase might be seen as a further adaptation of the confluent epithelial collective that favors metabolic economy while operating under those same physiological constraints. The unjammed phase would then be seen as an adaptation that accommodates the confluent epithelial collective to episodic dynamic events—including pattern formation, growth, remodeling, plasticity, and wound healing—but comes at elevated metabolic expense. Adaptation of energy metabolism to physiological constraints, as hypothesized here, is reminiscent of the behavior of smooth muscle, wherein the latch state is seen as the adaptation of the contractile machinery of the cell to the evolutionary pressure of minimizing metabolic energy expense under the constraint of maintaining physiological levels of isometric tone in hollow organs65,66,67,68 (Supplement 2). So too, this hypothesis concerning epithelial jamming and unjamming proposes that the epithelial collective has become well adapted to—and specialized for—metabolic economy.

## Methods

### MDCKII peredox cell line

A stable cell line was made from MDCKII wild-type cells (ECACC 85011435 purchased from Sigma MTOX1300-1VL) using Lipofectamine 3000 transfection of the Peredox30,69 construct (pcDNA3.1-Peredox-mCherry-NLS, AddGene 32384). As it is widely considered that no diffusion barriers exist between the cytoplasmic and nuclear compartments70, we regard the redox ratio as measured by the Peredox NLS construct to be indicative of the redox ratio in the cytoplasm. Additionally, the NLS tag allows us to distinguish individual cells during multi-channel fluorescence microscopy by facilitating cell segmentation. Transfection is followed by a period of G418 (Geneticin) selection, FACS sorting, and monoclonal expansion. MDCKII cells were expanded, passed, and cultured in DMEM (Gibco) supplemented with 5.6 mM glucose, 4 mM glutamine, 110 mg/L sodium pyruvate and phenol red, 5% Fetal bovine serum (FBS) (Atlanta Biologicals), 100 U/mL Penicillin, and 100 μg/mL Streptomycin (Gibco). Cells were cultured in incubators maintained at 37 °C and at 5% CO2. Cells are cultured in T75 flasks in 13 mL of culture media. Medium was exchanged once 24 h after each passage and then every second day thereafter. Cells were passed once they reached 70–80% confluency.

### Polyacrylamide gels

We use polyacrylamide (PA) gels with a stiffness of 9.6 kPa71. Gels are cast in 12-well glass bottom plates. Each well is treated with 1 mL of bind silane solution (80 μL acetic acid glacial, 50 μL 3-(trimethoxysilyl)propyl methacrylate, and 200 mL deionized water) for 1 h at room temperature. Glass bottom plates were then rinsed thoroughly with distilled water and dried with an air gun. The polyacrylamide gel consisted of 188 μL of a 40% acrylamide solution, 59 μL of a 2% bisacrylamide solution, 0.5 μL TEMED, 100 μL of 5% ammonium persulfate (APS) solution, and 650 μL ultrapure water. A volume of 24 μL of the polyacrylamide mixture was pipetted into each well of the glass bottom plate and subsequently covered with an 18 mm glass coverslip to uniformly spread the liquid across the bottom of the well resulting in a 100 µm thick gel. After the gel was completely polymerized, DI water was added to each well to prevent gels from drying out. The glass coverslips were then removed.

Polymerized polyacrylamide gels were first treated with 440 μL of a 1:50 (v/v) mixture of Sulfosuccinimidyl 6-(4′-azido-2′-nitrophenylamino)hexanoate (Sulfo-SANPAH) (ProteoChem) and 50 mM HEPES solution under a UV light for 10 min. Afterwards, gels were rinsed with HEPES buffer followed by 3 rinses with PBS. 150 μL of fluorescent carboxyl polystyrene microspheres (Glacial Blue, 0.2 μm diameter, Bangs Labs FCGB003) diluted at 1:6000 in DI water was applied to each gel for 30 min and then subsequently washed 5 × with PBS. Bovine Collagen-1 (Advanced BioMatrix) stock solution was diluted in HEPES buffer solution to make a 0.2 mg/mL collagen solution. This solution was filtered with a 0.2 μm Acrodisc syringe filter (Pall). Immediately after filtering, 1 mL of collagen diluted solution was pipetted onto each gel. The plate was covered and placed in a 4 °C fridge overnight. The following day, collagen was aspirated and rinsed twice with HEPES buffer solution.

### PDMS barrier preparation

Large, thin Polydimethylsiloxane (PDMS) layers were made by mixing 8 g Silicone Elastomer Base with 0.8 mg Silicone Elastomer curing agent (Sylgard 184 silicone elastomer kit). Mixture was spread in a 150 × 15 petri dish (VWR) and degassed for 30 min. Petri dishes were placed in an oven to cure overnight at 65 °C. PDMS blanks were cut from the resulting film using an 18 mm diameter hollow punch. In the center of each circular PDMS disk, a 3 × 12 mm rectangle was cut by hand using a scalpel. Cut PDMS masks were put in 70% ethanol overnight. Following the ethanol treatment, PDMS masks were treated with a 1% BSA (Sigma) and 1% Pluronic (Sigma) mixture overnight in a plate shaker set to low speed, and subsequently rinsed 5 times with PBS and air dried before use72. Immediately prior to cell seeding, gels and PDMS stencils were sterilized by exposure to UV light for 10 min before placing them in the biosafety hood. PDMS stencils were briefly air dried in the biosafety hood. The PBS in the wells was aspirated and PDMS stencils were placed on the gels.

### Cell redox experiments

80,000 cells suspended in 50 μL of culture media were seeded on the exposed collagen-1 coated polyacrylamide gel surface within each 3 × 12 mm PDMS mask region. After 2 h, wells were filled with 3 mL of culture media. Culture Media was exchanged every subsequent day before measurement. Cells were grown for two days after seeding until dense, confluent layers were formed before lifting PDMS barriers. PDMS barriers were removed with tweezers at timepoints 24 h and 4 h prior to imaging. The staggard removal of the barriers in time allowed for all conditions to be imaged simultaneously and ensured that all cells were in culture for identical amounts of time before data acquisition. Two hours prior to the Peredox bioprobe measurements, cell culture medium was replaced with media containing no sodium pyruvate in order to reduce effects of exogenous pyruvate or lactate excreted by the cells on the redox state. This imaging media contained no phenol red to minimize background during fluorescence microscopy imaging. Following this media exchange, cells were moved onto a fluorescent microscope stage housed within an incubator set to 37 °C and 5% CO2 to stabilize fluctuations in redox prior to imaging. Cells were imaged on a Leica (DMi8) using a 20 × objective at 2048 × 2048 pixels and with a resolution of 0.325 µm/pixel. We used a Texas Red fluorescence cube to image the Peredox mCherry fluorophore and a custom filter (Chroma filter set: ET405/20x, T425lpxr, ET525/50 m) to image the Peredox t-Sapphire fluorophore. Cell boundaries are also imaged with phase contrast microscopy and glacial blue beads were imaged with a DAPI channel. Images in all four channels were acquired across the narrow (3 mm) axis of the cell layer and at the center of the long (12 mm) axis of the cell layer from edge-to-edge with 10% imaging overlap, thus capturing roughly 20,000 cells per layer. After each experiment, the Peredox bioprobe was calibrated as outlined in Hung et al.69. Medium containing varying amounts of lactate and pyruvate (Supplementary Table 2) were added to each well, incubated for 8 min, and imaged in the Texas Red and t-Sapphire channels, resulting in approximately 15 min between successive calibration images. The calibration media contains no glucose and no phenol red. After the final calibration step, wells were rinsed with PBS. Then, 350 μL of Trypsin (Corning) with 10% Triton X-100 (Sigma) was pipetted onto the surface of each gel and allowed to sit for 5 min in order to delaminate cells from the collagen-1 coated PA gel surface. Final bead positions were acquired using the same fluorescence imaging channels as the initial data measurement. Redox experiments were repeated on three separate occasions with each repeat containing four biological replicates per condition. Here, biological replicates are considered individual wells that contain an epithelial cell layer prepared in a manner as described above. Each replicate was only measured once as described above. We subsequently pooled data of single-cell measurements from all repeats and replicates to generate graphical data. The analysis of the pre-barrier lift condition contained a total of n = 53,826 cells, the 4-h condition contained a total of n = 129,455 cells, and the 24-h condition contained a total of n = 156,866 cells.

### Glucose uptake experiment

MDCKII wild-type cells were seeded and cultured in collagen-1 coated polyacrylamide gels in 12-well glass bottom plates and PDMS barriers were removed as described above. Glucose uptake was measured by the addition of 400 µM 2-deoxy-2-[(7-nitro-2,1,3-benzoxadiazol-4-yl)amino]-D-glucose (2-NBDG, Cayman Chemical Company 11046) diluted in 2 mL of glucose-free DMEM to each well. After 1 h of incubation, cells were rinsed with 1 ml of warm DPBS and fixed with 4% PFA for 20 min at room temperature. Cells were rinsed 3 × with PBS and refrigerated for 24 h. The following day, cells were imaged on a Leica (DMi8) fluorescent microscope using a GFP channel and 20 × objective. Glucose uptake experiments were repeated twice, with two biological replicates for each condition. In one preparation, each biological replicate was imaged at two different regions per cell layer, and the other replicate was imaged at four different regions per cell layer for a combined total of six measurements per condition.

### Mitochondrial membrane potential experiments

MDCKII wild-type cells were seeded and cultured in collagen-1 coated polyacrylamide gels in 12-well glass bottom plates and PDMS barriers were removed as described above for the Peredox redox experiment. Cells were treated with a spike of 25 nM Tetramethylrhodamine ethyl ester perchlorate (TMRE, Sigma 87917) for 30 min before imaging. A negative control was performed on a subset of the wells by treating cells with a spike of 10 µM Carbonyl cyanide 4-(trifluoromethoxy)phenylhydrazone (FCCP, Abcam ab120081) 10 min prior to the application of TMRE. Cells were imaged on a Leica (DMi8) fluorescent microscope using a Texas Red channel and 20 × objective. Immediately following TMRE imaging, cells were rinsed with 1 ml of warm DPBS and fixed with 4% PFA for 20 min at room temperature. Cells were then rinsed 3 × with PBS. After fixation, 200 nM MitoView-Green (Biotium 70054) diluted in PBS was added to each well and left to incubate for 2 h. MitoView-Green is a membrane potential independent dye and is used here to assess whether mitochondrial mass varies across the cell layer. After incubation, MitoView-Green is imaged as before using a GFP filter cube. Mitochondrial membrane potential experiments were completed once with three biological replicates. Each replicate was imaged at four different regions per cell layer.

MDCKII wild-type cells were seeded and cultured in collagen-1 coated polyacrylamide gels inside 35 mm glass bottom petri dishes and PDMS barriers were removed as described above. FLIM on expanding monolayers was carried out using a multiphoton microscope (Ultima Investigator, Bruker) equipped with a plate heater (TC-324B, Warner Instruments) and an objective heater (TC-HLS-05, Bioscience Tools) to maintain the sample at 37 °C for the duration of the imaging study. A titanium:sapphire laser (InSight X3, Spectra-Physics) was tuned to 76o nm for two-photon excitation of NADH. The laser beam was focused onto the sample through a 16 × water-immersion objective (Nikon, 0.8 N.A., 3 mm working distance). The fluorescence emission from NADH was transmitted through a dichroic mirror (700 nm, short-pass) that separates the excitation and emission paths, passed through a clean-up filter (720 nm, short-pass), and was collected through a bandpass filter (440/40 nm, band-pass) to a photon counting photomultiplier tube (10770 PB-50, Hamamatsu). Images were collected using 1024 × 1024 pixels at a resolution of 0.792 µm/pixel and a pixel dwell time of 4 µs. Tiled z-stacks were acquired to capture a 811 µm wide strip of cells from edge to edge, with the number of tiles (4 to 7, using a 2% overlap) and slices (4 to 9, in 5 µm steps) that were determined depending on the width and on the inclination of each monolayer. Time-correlated single photon counting electronics (SPC-150, Becker&Hickl) was set to 60 s collection time and 256 time bins were used to acquire the temporal decay of NADH fluorescence. Second-harmonic generated signals from self-assembled collagen fibers73 (760 nm excitation, 375/30 nm emission) were used to measure the instrument response function (IRF), which had a full width at half maximum of 311 ps. FLIM experiments contain three biological replicates. Each replicate was imaged at two different regions per cell layer. As such, each imaging region provides data corresponding to two migrating edges (one moving leftward, and one moving rightward). Graphical data is therefore generated by pooling all 12 images for each condition.

### Peredox cell analysis

All images are first flat field corrected using the ImageJ BaSiC plugin74. Using the ImageJ BaSiC plugin, fluorescence images are additionally dark-field corrected. Images were registered using custom MATLAB (R2017b Mathworks, Natick, MA) software to account for microscope stage positioning error when computing cell nuclei displacements. All analysis steps were completed on individual imaging fields-of-view and resulting data was reconstructed into a contiguous ‘stitched’ dataset after image analysis was completed. Quantities measured on a per cell basis (cell speed, tractions, shape, area, and cytoplasmic redox potential) are linearly interpolated onto a grid with grid spacing chosen to be consistent with typical cell spacing in the dense layer (35 pixels or ~ 11 µm). Composite colormaps are then generated by computing the average x-position of cells (i.e., the center of mass of the layer) across the entire imaging region, and aligning all layers by their center x-position. After alignment, the gridded quantities are then averaged in a pixel-by-pixel fashion. Traces are generated by binning data along the entire y-direction as well as in bins along the x-direction of approximately 260 µm in width. Error bars represent layer-to-layer standard deviation of the means.

For dynamical measurements, we use the sequence of images collected during the Peredox cell calibration in which images are recorded once every 15 min for a total of 8 frames. During this image sequence, nuclei positions, trajectories, and segmentations are measured from the mCherry fluorescence channel of the Peredox sensor using custom MATLAB analysis software which we call tracking with dynamic segmentation (TWDS). This algorithm uses optical flow to estimate the motion of nuclei from one frame to the next, which is combined with segmentation of the nuclei to determine the new position of the nucleus. Accordingly, if in subsequent frames, overlapping nuclei are not properly segmented, the two overlapping nuclei can be distinguished by TWDS based upon their previous position history and their interpolated position as given by optical flow. Finally, care was taken in the implementation of TWDS not to rely on intensity thresholding to identify nuclei. From the tracked nuclei data, cell speeds are computed from inter-frame displacements representing the magnitude of the velocity vector only. Cell morphology is derived using built-in MATLAB region-property functions of a Voronoi tessellation generated from the nuclei centroid positions.

Cell tractions are measured using images of the Glacial Blue labeled microbeads. Bead image pairs consist of one image taken during the Peredox redox measurement and one image taken after the Peredox calibration and cell trypsinization are completed. Image pairs are registered with custom MATLAB code to correct for microscope stage positioning error which occurs as a result of repeated removal of the multi-well plate for media exchanges. Bead displacements are measured using optical flow. From bead displacement measurements, cell tractions are calculated as previously described28. Using the cell segmentation information outlined above, we extract the traction forces beneath each individual cell footprint and calculate the average magnitude of the tractions exerted by each cell.

Redox ratio: The Peredox biosensor was analyzed on a per-cell basis (Supplementary Fig. 1). Using the nuclei tracking and segmentation outlined above, the average intensity of pixels associated with each cell nucleus was calculated separately for both the mCherry (red) and T-Sapphire (green) fluorescence channels for every image over the 8 image sequence (1 data measurement plus 7 calibration steps outlined above). The ratio of the red-to-green fluorescence signal (proportional to NAD + /NADH) was then computed for each cell in each image in the sequence. To compute a fluorescence response and calibration curve, use the cells imaged in a single field-of-view at the leading edge of a 24 h timepoint, and we follow steps and fitting procedures as outlined by Hung et al.69. Briefly, we fit the fluorescence response obtained during the lactate-pyruvate titration to a logistic function:

$$F = 1 + \frac{A}{{1 + \left( \frac{B}{X} \right)^{1.7} }}$$
(1)

where F is the normalized fluorescence ratio of the Peredox red-to-green channels for each cell, X is the lactate-to-pyruvate ratio used in the titration, and A and B are fit parameters. We use a Hill Coefficient of 1.7 as characterized elsewhere69. After fitting the normalized fluorescence response of each cell within a single image at the leading edge of the advancing cell layer at the 24 h timepoint to the logistic function using MATLAB, we find average fit parameters of A = 1.04 (± 0.24) and B = 44.13 (± 26.08). Next to generate the calibration curve, which we use to convert the fluorescence measurement of the Peredox bioprobe to the NAD + /NADH ratio, we apply the method which uses the LDH equilibrium31.

$$K = \frac{{\left[ {Pyr} \right] \times \left[ {NADH} \right] \times \left[ {H^{ + } } \right]}}{{\left[ {Lac} \right] \times \left[ {NAD + } \right]}} = 1.1 \times 10^{ - 11}$$
(2)

In Eq. (2), K is the equilibrium constant for LDH. We assume a physiological pH of 7.4 and thus [H+] = 10−7.4. Using the logistic fit parameters from the leading edge of the 24 h timepoint, we solve Eqs. (1) and (2) for the NAD + /NADH ratio resulting in a fluorescence-ratio-to-redox-ratio calibration.

All Peredox datasets were combined while in the fluorescence (red/green) ratio format into average composite images or x-profile traces of the fluorescence ratio. After the composites were generated, we converted the fluorescence ratio to NAD + /NADH using the calibration as outlined above. To compute error bars, we measured the standard deviation of the mean fluorescence ratio (red/green) of each sample. The standard deviation was then added or subtracted to the mean ratio, and then this resulting fluorescence ratio was transformed to NAD + /NADH ratio using the calibration method outlined above.

### Analysis of 2-NBDG and TMRE images

Prior to analysis of glucose uptake (2-NBDG) and mitochondrial membrane potential (TMRE) images, all images are flat field corrected using the ImageJ BaSiC plugin74. Fluorescence images are additionally dark-field corrected to reduce background. Images were stitched together by concatenating adjacent fields-of-view into long strips which span the epithelial cell layer from edge-to-edge. For the 2-NBDG and TMRE images, binary masks which distinguish the free-space where no cells exist from epithelial cells within the layer were generated by hand using ImageJ. The center of each layer was then found by computing the center-of-mass of the cell layer based on the masked area. Each imaged strip across the cell layer thus results in two advancing fronts which were treated as two separate layers. Composite colormaps are then generated by aligning all layers by their center x-position with the advancing layer moving in the positive X-direction. After alignment, images are then averaged in a pixel-by-pixel fashion. Traces are generated by binning data along the entire y-direction as well as in bins along the x-direction of approximately 260 µm in width. Error bars represent layer-to-layer standard deviation of the means.

The 2-NBDG data was normalized by cell area in order to capture the amount of 2-NBDG, and hence glucose uptake, in each cell as opposed to each pixel as shown in the fluorescence image. The cell area normalization curve comes from the average cell areas measured from the Peredox cell experiments which is the same data as shown in the composite colormaps of Fig. 2.

MitoView analysis followed the same procedure as the 2-NBDG and TMRE data with the exception of the masking step. Images from the MitoView tended to have extensive large and saturating bright puncta throughout the layer indicative of aggregated dye or non-specific binding to particulate debris left over from the PDMS barrier. To mask these regions and omit them from analysis, we used Ilastik, the interactive learning and segmentation toolkit75. After this masking step, analysis followed as before.

### FLIM image analysis

$$I\left( t \right) = \alpha_{1} \exp \left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} {\tau_{1} }}} \right. \kern-\nulldelimiterspace} {\tau_{1} }}} \right) + \alpha_{2} \exp \left( {{{ - t} \mathord{\left/ {\vphantom {{ - t} {\tau_{2} }}} \right. \kern-\nulldelimiterspace} {\tau_{2} }}} \right) + C,$$


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# Stanley's Saga

The protagonist of this story, Stanley, is standing peacefully at coordinates $$(0, \space 0)$$ without any concern in his life whatsoever.

Now, Stanley will move at coordinates $$(3, \space 4)$$.

If Stanley only goes either up, or right, or up-right (corresponding vectors in their respective order: $$\vec{u}(0, \space 1)$$, $$\vec{r}(1, \space 0)$$ , $$\vec{ur}(1, \space 1)$$), on how many ways can Stanley accomplish this tremendous stunt?

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# NCERT Class 9 Statistics Exercise 02 Maths Solutions

## Class 9 Statistics Exercise 02 Maths NCERT Solutions

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### NCERT Class 9 | STATISTICS | Exercise 02 | Question No. 06

Three coins were tossed 30 times simultaneously. Each time the number of heads occurring was noted down as follows:   0 1 2 2 1 2 3 1 3 0 1 3 1 1 2 2 0 1 2 1 3 0 0 1 1 2 3 2 2 0   Prepare a frequency distribution table for the data given above.

### NCERT Class 9 | STATISTICS | Exercise 02 | Question No. 07

The value of pi upto 50 decimal places is given below: 3.14159265358979323846264338327950288419716939937510 (i) Make a frequency distribution of the digits from 0 to 9 after the decimal point. (ii) What are the most and the least frequently occuring digits?

### NCERT Class 9 | STATISTICS | Exercise 02 | Question No. 04

The heights of 50 students, measured to the nearest centimetres, have  been found to be as follows:   161 150 154 165 168 161 154 162 150 151 162 164 171 165 158 154 156 172 160 170 153 159 161 170 162 165 166 168 165 164 154 152 153 156 158 162 160 161 173 166 161 159 162 167 168 159 158 153 154 159   (i) Represent the data given above by a grouped frequency distribution table, taking the class intervals as 160 - 165, 165 - 170, etc. (ii) What can you conclude about their heights from the table?

### NCERT Class 9 | STATISTICS | Exercise 02 | Question No. 05

A study was conducted to find out the concentration of sulphur dioxide in the air in parts per million (ppm) of a certain city. The data obtained for 30 days is as follows:   0.03 0.08 0.08 0.09 0.04 0.17 0.16 0.05 0.02 0.06 0.18 0.20 0.11 0.08 0.12 0.13 0.22 0.07 0.08 0.01 0.10 0.06 0.09 0.18 0.11 0.07 0.05 0.07 0.01 0.04   (i) Make a grouped frequency distribution table for this data with class intervals as 0.00 - 0.04, 0.04 - 0.08, and so on. (ii) For how many days, was the concentration of sulphur dioxide more than 0.11 parts per million?

### NCERT Class 9 | STATISTICS | Exercise 02 | Question No. 02

The distance (in km) of 40 engineers from their residence to their place of work were found as follows: 5 3 10 20 25 11 13 7 12 31 19 10 12 17 18 11 32 17 16 2 7 9 7 8 3 5 12 15 18 3 12 14 2 9 6 15 15 7 6 12 Construct a grouped frequency dist


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Notes on an analogue of the Fontaine-Mazur conjecture.(English)Zbl 1077.11080

Let $$\ell$$ be a prime number. According to a function field analogue of the Fontaine-Mazur conjecture [J.-M. Fontaine, B. Mazur, “Geometric Galois representations”. International Press Ser. Number Theory 1, 41–78 (1995; Zbl 0839.14011)], it should be rare to find a function field over a finite field with $$\ell\not=0$$ that has an infinite unramified pro-$$\ell$$ extension virtually with uniformly powerful geometric Galois group. Exceptional counter examples are found in [Y. Ihara, Adv. Stud. Pure Math. 2, 89–97 (1983; Zbl 0542.14011)] and [G. Frey, E. Kani, H. Völklein, Lond. Math. Soc. Lect. Note Ser. 256, 85–118 (1999; Zbl 0978.14021)]. In [J. Number Theory 81, No. 1, 16–47 (2000; Zbl 0997.11096)], the second author gave two sorts of criteria for the non-existence of such an infinite extension in terms of the Frobenius property of the Jacobian $$\ell$$-torsion. Applying Katz’ equidistribution theorem to these criteria, the authors show that most of the (under some mild assumption, approximately $$1-\frac{1}{\ell^2}$$ of all) function fields do not have such infinite extensions. By a similar argument, it is shown that approximately $$\frac{\ell}{\ell^2-1}$$ of abelian varieties over a finite field have non-trivial rational $$\ell$$-torsion points.

MSC:

 11R58 Arithmetic theory of algebraic function fields 11R32 Galois theory
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References:

 [1] Boston, N., Some cases of the Fontaine-Mazur conjecture, II. J. Number Theory75 (1999), 161-169. · Zbl 0928.11050 [2] Chavdarov, N., The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Math. J.87 (1997), 151-180. · Zbl 0941.14006 [3] De Jong, A.J., A conjecture on arithmetic fundamental groups. Israel J. Math.121 (2001), 61-84. · Zbl 1054.11032 [4] Deligne, P., La conjecture de Weil. II. Hautes Études Sci. Publ. Math.52 (1980), 137-252. · Zbl 0456.14014 [5] Deligne, P., Mumford, D., The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math.36 (1969), 75-109. · Zbl 0181.48803 [6] Dixon, J., Du Sautoy, M.P.F., Mann, A., Segal, D., Analytic Pro-p Groups. 157 (1991). · Zbl 0744.20002 [7] Ekedahl, T., The action of monodromy on torsion points of Jacobians. Arithmetic algebraic geometry (Texel, 1989). BirkhäuserBoston (1991), 41-49. · Zbl 0728.14028 [8] Fontaine, J.-M., Mazur, B., Geometric Galois representations. J. Coates and S.-T. Yau, editors, Elliptic Curves, Modular Forms, & Fermat’s Last Theorem, Series in Number Theory1 (1995), 41-78. · Zbl 0839.14011 [9] Frey, G., Kani, E., Völklein, H., Curves with infinite K-rational geometric fundamental group. H. Völklein, D. Harbater, P. Müller, and J. G. Thompson, editors, Aspects of Galois theory (Gainesville, FL, 1996), 256, 85-118. · Zbl 0978.14021 [10] Holden, J., On the Fontaine-Mazur Conjecture for number fields and an analogue for function fields. J. Number Theory81 (2000), 16-47. · Zbl 0997.11096 [11] Ihara, Y., On unramified extensions of function fields over finite fields. Y. Ihara, editor, Galois Groups and Their Representations, Adv. Studies in Pure Math.2 (1983), 89-97. · Zbl 0542.14011 [12] Katz, N.M., Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society, 1999. · Zbl 0958.11004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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## College Algebra (11th Edition)

By calculating the values of $f(x)=-\sqrt{x}$ and $g(x)=\sqrt{x}$ we can see that the value of $f(x)$ is exactly the opposite of the value of $g(x)$ for each corresponding x. In order to graph the exact parent graph, here is the table of values: $g(0)=\sqrt0=0$ $g(1)=\sqrt1=1$ $g(2)=\sqrt2=\sqrt2$ $g(3)=\sqrt3=\sqrt3$ $g(4)=\sqrt4=2$ (The square root function is not defined for negative numbers.) Therefore the graph of $f(x)$ is exactly the same as the graph of $g(x)=\sqrt{x}$ but is reflected across the x-axis.


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# Practice Questions on AlgebraAptitude Questions and Answers

## Moderate Algebra Question - 21

 Q21. A number of three digits in base 7, when expressed in base 9, has its digits reversed in the order. What is the sum of the digits of the number?
 A. 5 B. 6 C. 7 D. 8

## Moderate Algebra Question - 22

 Q22. When the index of an exponential expression with a positive base is doubled, then the expression increases by $700%$. If one of the values that the base cannot have is $X$ which of the following is not a possible value of $P$?
 A. 4 B. 8 C. 5 D. 1

## Moderate Algebra Question - 23

 Q23. (x^3-6x^2+px+q) is exactly divisible by (x^2-3x+2) then,
 A. $p+q > 0$ and $pq > 0$ B. $p+q > 0$ and $pq < 0$ C. $p+q < 0$ and $pq < 0$ D. $p+q < 0$ and $pq > 0$

## Moderate Algebra Question - 24

 Q24. The mean salary in ICM ltd. was Rs 1500, and the standard deviation was Rs 400. A year later each employee got a Rs 100 raise. After another year each employee's salary (including the above mentioned raise) was increased by $20%$. The standard deviation of the current salary is:
 A. 460 B. 480 C. 560 D. none of these

## Moderate Algebra Question - 25

 Q25. Three persons Suresh, Devesh and Prashant were born on different days in the same year. If the date and month of birth of Suresh, Devesh and Prashant are numerically equal, then what could be the minimum difference in the ages of youngest and oldest in days?
 A. 56 B. 60 C. 61 D. 62
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