Intracellular Mechanics Research Articles

Altering Cell Junctional Tension in Spheroids through E-Cadherin Engagement Modulation.

Cadherin-mediated tension at adherens junctions (AJs) is fundamental for cell-cell adhesion and maintaining epithelial integrity. Despite the importance of manipulating AJs to dissect cell-cell interactions, existing three-dimensional (3D) multicellular models have not adequately addressed the precise manipulation of these junctions. To fill this gap, we introduce E-cadherin-modified tension gauge tethers (TGTs) at the junctions within spheroids. The system enables both quantification and modulation of junctional tension with specific DNA triggers. Using rupture-induced fluorescence, we successfully measure mechanical forces in 3D spheroids. Furthermore, mechanically strong TGTs can maintain normal E-cadherin-mediated adhesion. Employing toehold-mediated strand displacement allowed us to disrupt E-cadherin-specific cell-cell adhesion, consequently altering intracellular tension within the spheroids. Our methodology offers a robust and precise way to manipulate cell-cell adhesion and intracellular mechanics in spheroid models.

Giant organelle vesicles to uncover intracellular membrane mechanics and plasticity

Tools for accessing and studying organelles remain underdeveloped. Here, we present a method by which giant organelle vesicles (GOVs) are generated by submitting cells to a hypotonic medium followed by plasma membrane breakage. By this means, GOVs ranging from 3 to over 10 µm become available for micromanipulation. GOVs are made from organelles such as the endoplasmic reticulum, endosomes, lysosomes and mitochondria, or in contact with one another such as giant mitochondria-associated ER membrane vesicles. We measure the mechanical properties of each organelle-derived GOV and find that they have distinct properties. In GOVs procured from Cos7 cells, for example, bending rigidities tend to increase from the endoplasmic reticulum to the plasma membrane. We also found that the mechanical properties of giant endoplasmic reticulum vesicles (GERVs) vary depending on their interactions with other organelles or the metabolic state of the cell. Lastly, we demonstrate GERVs’ biochemical activity through their capacity to synthesize triglycerides and assemble lipid droplets. These findings underscore the potential of GOVs as valuable tools for studying the biophysics and biology of organelles.

The role of a nanogrooved polydimethylsiloxane substrate on mesenchymal stem cells adhesion, self-renewing, and mechanical properties

Stem cells have a variety of applications in biological, reconstructive medicine, and tissue engineering studies owing to their capacity to differentiate into specific cell lineages and their self-renewal ability. Among the factors affecting stem cells, biochemical agents such as growth and differentiation factors, along with environmental cues including physical and mechanical characteristics of the cell microenvironment, are of interest in this field. Several studies have evaluated the impact of physical factors such as mechanical forces, electric or magnetic fields, and light pulses on stem cell behavior. Also, cell culture substrates that can simulate extracellular matrix in natural conditions are considered the most critical environmental factors. Hence, in this study, we investigated the effects of a specially designed polydimethylsiloxane substrate with a surface pattern for cell contact guidance on various biological and biophysical characteristics of mesenchymal stem cells. Plasma treatment of the substrate under lateral tension from 50 to 200 s caused the increased wavelength of the formed wrinkle structures from 500 ± 56 to 1052 ± 182 nm, respectively, leading to variations of stiffness affecting the cell behaviors. Moreover, by increasing the available surface and hydrophilicity of the substrate, the cell attachment quality improved by more than 46% compared to the control. Finally, results of the mechanics of mesenchymal stem cells on these substrates obtained by bio-AFM showed an increase in the stiffness of cells despite a slight decrease in Young's modulus parameter of different substrates. These changes in the intracellular mechanics can eventually stimulate the more pivotal mechanobiological effects on stem cell fate.

Intracellular mechanics and TBX3 expression jointly dictate the spreading mode of melanoma cells in 3D environments

Cell stiffness and T-box transcription factor 3 (TBX3) expression have been identified as biomarkers of melanoma metastasis in 2D environments. This study aimed to determine how mechanical and biochemical properties of melanoma cells change during cluster formation in 3D environments. Vertical growth phase (VGP) and metastatic (MET) melanoma cells were embedded in 3D collagen matrices of 2 and 4 mg/ml collagen concentrations, representing low and high matrix stiffness. Mitochondrial fluctuation, intracellular stiffness, and TBX3 expression were quantified before and during cluster formation. In isolated cells, mitochondrial fluctuation decreased and intracellular stiffness increased with increase in disease stage from VGP to MET and increased matrix stiffness. TBX3 was highly expressed in soft matrices but diminished in stiff matrices for VGP and MET cells. Cluster formation of VGP cells was excessive in soft matrices but limited in stiff matrices, whereas for MET cells it was limited in soft and stiff matrices. In soft matrices, VGP cells did not change the intracellular properties, whereas MET cells exhibited increased mitochondrial fluctuation and decreased TBX3 expression. In stiff matrices, mitochondrial fluctuation and TBX3 expression increased in VGP and MET, and intracellular stiffness increased in VGP but decreased in MET cells. The findings suggest that soft extracellular environments are more favourable for tumour growth, and high TBX3 levels mediate collective cell migration and tumour growth in the earlier VGP disease stage but play a lesser role in the later metastatic stage of melanoma.

Multimodal microscale mechanical mapping of cancer cells in complex microenvironments

The mechanical phenotype of the cell is critical for survival following deformations due to confinement and fluid flow. One idea is that cancer cells are plastic and adopt different mechanical phenotypes under different geometries that aid in their survival. Thus, an attractive goal is to disrupt cancer cells’ ability to adopt multiple mechanical states. To begin to address this question, we aimed to quantify the diversity of these mechanical states using in vitro biomimetics to mimic in vivo two-dimensional (2D) and 3D extracellular matrix environments. Here, we used two modalities Brillouin microscopy (∼GHz) and broadband frequency (7–15 kHz) optical tweezer microrheology to measure microscale cell mechanics. We measured the response of intracellular mechanics of cancer cells cultured in 2D and 3D environments where we modified substrate stiffness, dimensionality (2D versus 3D), and presence of fibrillar topography. We determined that there was good agreement between two modalities despite the difference in timescale of the two measurements. These findings on cell mechanical phenotype in different environments confirm a correlation between modalities that employ different mechanisms at different temporal scales (Hz-kHz versus GHz). We also determined that observed heterogeneity in cell shape is more closely linked to the cells’ mechanical state. Moreover, individual cells in multicellular spheroids exhibit a lower degree of mechanical heterogeneity when compared with single cells cultured in monodisperse 3D cultures. The observed decreased heterogeneity among cells in spheroids suggested that there is mechanical cooperativity between cells that make up a single spheroid.

Intracellular Crowding by Bio-Orthogonal Hydrogel Formation Induces Reversible Molecular Stasis.

To survive extreme conditions, certain animals enter a reversible protective stasis through vitrification of the cytosol by polymeric molecules such as proteins and polysaccharides. In this work, synthetic gelation of the cytosol in living cells is used to induce reversible molecular stasis. Through the sequential lipofectamine-mediated transfection of complementary poly(ethylene glycol) macromers into mammalian cells, intracellular crosslinking occurs through bio-orthogonal strain-promoted azide-alkyne cycloaddition click reactions. This achieves efficient polymer uptake with minimal cell death (99% viable). Intracellular crosslinking decreases DNA replication and protein synthesis, and increases the quiescent population by 2.5-fold. Real-time tracking of single cells containing intracellular crosslinked polymers identifies increases in intermitotic time (15 hvs 19 h) and decreases in motility (30 µm h-1 vs 15µm h-1 ). The cytosol viscosity increases threefold after intracellular crosslinking and results in disordered cytoskeletal structure in addition to the disruption of cellular coordination in a scratch assay. By incorporating photodegradable nitrobenzyl moieties into the polymer backbone, the effects of intracellular crosslinking are reversed upon exposure to light, thereby restoring proliferation (80% phospho-Rb+ cells), protein translation, and migration. Reversible intracellular crosslinking provides a novel method for dynamic manipulation of intracellular mechanics, altering essential processes that determine cellular function.

Reciprocity of Cell Mechanics with Extracellular Stimuli: Emerging Opportunities for Translational Medicine.

Human cells encounter dynamic mechanical cues in healthy and diseased tissues, which regulate their molecular and biophysical phenotype, including intracellular mechanics as well as force generation. Recent developments in bio/nanomaterials and microfluidics permit exquisitely sensitive measurements of cell mechanics, as well as spatiotemporal control over external mechanical stimuli to regulate cell behavior. In this review, the mechanobiology of cells interacting bidirectionally with their surrounding microenvironment, and the potential relevance for translational medicine are considered. Key fundamental concepts underlying the mechanics of living cells as well as the extracelluar matrix are first introduced. Then the authors consider case studies based on 1) microfluidic measurements of nonadherent cell deformability, 2) cell migration on micro/nano-topographies, 3) traction measurements of cells in three-dimensional (3D) matrix, 4) mechanical programming of organoid morphogenesis, as well as 5) active mechanical stimuli for potential therapeutics. These examples highlight the promise of disease diagnosis using mechanical measurements, a systems-level understanding linking molecular with biophysical phenotype, as well as therapies based on mechanical perturbations. This review concludes with a critical discussion of these emerging technologies and future directions at the interface of engineering, biology, and medicine.

Effects of Visfatin on Intracellular Mechanics and Catabolism in Human Primary Chondrocytes through Glycogen Synthase Kinase 3β Inactivation.

Osteoarthritis (OA) is still a recalcitrant musculoskeletal disease on account of its complex biochemistry and mechanical stimulations. Apart from stimulation by external mechanical forces, the regulation of intracellular mechanics in chondrocytes has also been linked to OA development. Recently, visfatin has received significant attention because of the clinical finding of the positive correlation between its serum/synovial level and OA progression. However, the precise mechanism involved is still unclear. This study determined the effect of visfatin on intracellular mechanics and catabolism in human primary chondrocytes isolated from patients. The intracellular stiffness of chondrocytes was analyzed by the particle-tracking microrheology method. It was shown that visfatin damages the microtubule and microfilament networks to influence intracellular mechanics to decrease the intracellular elasticity and viscosity via glycogen synthase kinase 3β (GSK3β) inactivation induced by p38 signaling. Further, microtubule network destruction in human primary chondrocytes is predominantly responsible for the catabolic effect of visfatin on the cyclooxygenase 2 upregulation. The present study shows a more comprehensive interpretation of OA development induced by visfatin through biochemical and biophysical perspectives. Finally, the role of GSK3β inactivation, and subsequent regulation of intracellular mechanics, might be considered as theranostic targets for future drug development for OA.

Principles and Applications of Single Particle Tracking in Cell Research.

It is a tough challenge for many decades to decipher the complex relationships between cell behaviors and cellular physical properties. Single particle tracking (SPT) with high spatial and temporal resolution has been applied extensively in cell research to understand physicochemical properties of cells and their bio-functions by tracking endogenous or exogenous probes. This review describes the fundamental principles of SPT as well as its applications in intracellular mechanics, membrane dynamics, organelles distribution, and processes of internalization and transport. Finally, challenges and future directions of SPT are also discussed.

Optical quantification of intracellular mass density and cell mechanics in 3D mechanical confinement.

Biophysical properties of cells such as intracellular mass density and cell mechanics are known to be involved in a wide range of homeostatic functions and pathological alterations. An optical readout that can be used to quantify such properties is the refractive index (RI) distribution. It has been recently reported that the nucleus, initially presumed to be the organelle with the highest dry mass density (ρ) within the cell, has in fact a lower RI and ρ than its surrounding cytoplasm. These studies have either been conducted in suspended cells, or cells adhered on 2D substrates, neither of which reflects the situation in vivo where cells are surrounded by the extracellular matrix (ECM). To better approximate the 3D situation, we encapsulated cells in 3D covalently-crosslinked alginate hydrogels with varying stiffness, and imaged the 3D RI distribution of cells, using a combined optical diffraction tomography (ODT)-epifluorescence microscope. Unexpectedly, the nuclei of cells in 3D displayed a higher ρ than the cytoplasm, in contrast to 2D cultures. Using a Brillouin-epifluorescence microscope we subsequently showed that in addition to higher ρ, the nuclei also had a higher longitudinal modulus (M) and viscosity (η) compared to the cytoplasm. Furthermore, increasing the stiffness of the hydrogel resulted in higher M for both the nuclei and cytoplasm of cells in stiff 3D alginate compared to cells in compliant 3D alginate. The ability to quantify intracellular biophysical properties with non-invasive techniques will improve our understanding of biological processes such as dormancy, apoptosis, cell growth or stem cell differentiation.

Tracking intracellular forces and mechanical property changes in mouse one-cell embryo development.

Cells comprise mechanically active matter that governs their functionality, but intracellular mechanics are difficult to study directly and are poorly understood. However, injected nanodevices open up opportunities to analyse intracellular mechanobiology. Here, we identify a programme of forces and changes to the cytoplasmic mechanical properties required for mouse embryo development from fertilization to the first cell division. Injected, fully internalized nanodevices responded to sperm decondensation and recondensation, and subsequent device behaviour suggested a model for pronuclear convergence based on a gradient of effective cytoplasmic stiffness. The nanodevices reported reduced cytoplasmic mechanical activity during chromosome alignment and indicated that cytoplasmic stiffening occurred during embryo elongation, followed by rapid cytoplasmic softening during cytokinesis (cell division). Forces greater than those inside muscle cells were detected within embryos. These results suggest that intracellular forces are part of a concerted programme that is necessary for development at the origin of a new embryonic life.

Role of a Kinesin Motor in Cancer Cell Mechanics.

Molecular motors play important roles in force generation, migration, and intracellular trafficking. Changes in specific motor activities are altered in numerous diseases. KIF20A, a motor protein of the kinesin-6 family, is overexpressed in bladder cancer, and KIF20A levels correlate negatively with clinical outcomes. We report here a new role for the KIF20A kinesin motor protein in intracellular mechanics. Using optical tweezers to probe intracellular mechanics and surface AFM to probe cortical mechanics, we first confirm that bladder urothelial cells soften with an increasing cancer grade. We then show that inhibiting KIF20A makes the intracellular environment softer for both high- and low-grade bladder cancer cells. Upon inhibition of KIF20A, cortical stiffness also decreases in lower grade cells, while it surprisingly increases in higher grade malignant cells. Changes in cortical stiffness correlate with the interaction of KIF20A with myosin IIA. Moreover, KIF20A inhibition negatively regulates bladder cancer cell motility irrespective of the underlying substrate stiffness. Our results reveal a central role for a microtubule motor in cell mechanics and migration in the context of bladder cancer.

A Biophysical Model for Curvature-Guided Cell Migration

The latest experiments have shown that adherent cells can migrate according to cell-scale curvature variations via a process called curvotaxis. Despite identification of key cellular factors, a clear understanding of the mechanism is lacking. We employ a mechanical model featuring a detailed description of the cytoskeleton filament networks, the viscous cytosol, the cell adhesion dynamics, and the nucleus. We simulate cell adhesion and migration on sinusoidal substrates. We show that cell adhesion on three-dimensional curvatures induces a gradient of pressure inside the cell that triggers the internal motion of the nucleus. We propose that the resulting out-of-equilibrium position of the nucleus alters cell migration directionality, leading to cell motility toward concave regions of the substrate, resulting in lower potential energy states. Altogether, we propose a simple mechanism explaining how intracellular mechanics enable the cells to react to substratum curvature, induce a deterministic cell polarization, and break down cells basic persistent random walk, which correlates with latest experimental evidences.

Deformation Microscopy for Dynamic Intracellular and Intranuclear Mapping of Mechanics with High Spatiotemporal Resolution

SUMMARYStructural heterogeneity is a hallmark of living cells that drives local mechanical properties and dynamic cellular responses. However, the robust quantification of intracellular mechanics is lacking from conventional methods. Here, we describe the development of deformation microscopy, which leverages conventional imaging and an automated hyperelastic warping algorithm to investigate strain history, deformation dynamics, and changes in structural heterogeneity within the interior of cells and cell nuclei. Using deformation microscopy, we found that partial or complete disruption of LINC complexes in cardiomyocytes in vitro and lamin A/C deficiency in myocytes in vivo abrogate dominant tensile loading in the nuclear interior. We also found that cells cultured on stiff substrates or in hyperosmotic conditions displayed abnormal strain burden and asymmetries at interchromatin regions, which are associated with active transcription. Deformation microscopy represents a foundational approach toward intracellular elastography, with the potential utility to provide mechanistic and quantitative insights in diverse mechanobiological applications.

Dynamic Crosslinking of the Actin Cytoskeleton Governs Intracellular Mechanics

Dynamic Crosslinking of the Actin Cytoskeleton Governs Intracellular Mechanics

Src- and confinement-dependent FAK activation causes E-cadherin relaxation and β-catenin activity.

In epithelia, E-cadherin cytoplasmic tail is under cytoskeleton-generated tension via a link that contains β-catenin. A cotranscription factor, β-catenin, is also active in morphogenetic processes associated with epithelial-to-mesenchymal transition. β-Catenin signaling appears mechanically inducible and was proposed to follow phosphorylation-induced β-catenin release from E-cadherin. Evidence for this mechanism is lacking, and whether E-cadherin tension is involved is unknown. To test this, we combined quantitative fluorescence microscopies with genetic and pharmacological perturbations of epithelial-to-mesenchymal transition-induced cells in culture. We showed that β-catenin nuclear activity follows a substantial release from the membrane specific to migrating cells and requires multicellular deconfinement and Src activity. Selective nuclear translocation occurs downstream of focal adhesion kinase activation, which targets E-cadherin tension relaxation through actomyosin remodeling. In contrast, phosphorylations of the cadherin/catenin complex are not substantially required. These data demonstrate that E-cadherin acts as a sensor of intracellular mechanics in a crosstalk with cell-substrate adhesions that target β-catenin signaling.

Intracellular Fluid Mechanics: Coupling Cytoplasmic Flow with Active Cytoskeletal Gel

The cell is a mechanical machine, and continuum mechanics of the fluid cytoplasm and the viscoelastic deforming cytoskeleton play key roles in cell physiology. We review mathematical models of intracellular fluid mechanics, from cytoplasmic fluid flows, to the flow of a viscous active cytoskeletal gel, to models of two-phase poroviscous flows, to poroelastic models. We discuss application of these models to cell biological phenomena, such as organelle positioning, blebbing, and cell motility. We also discuss challenges of understanding fluid mechanics on the cellular scale.

Noninvasive Imaging: Brillouin Confocal Microscopy.

In the past decades, there has been increased awareness that mechanical properties of tissues and cells are closely associated with disease physiology and pathology. Recognizing this importance, Brillouin spectroscopy instrumentation, already utilized in physics and material science, has been adopted for cell and tissue biomechanics. For biomedical applications, progress of Brillouin spectrometer technology has been crucial, mainly improvement in the acquisition speed and combination with confocal microscopy, to enable measurement of material longitudinal modulus in three dimensions with high spatial resolution. Micron spatial resolution and high sensitivity allow mapping intracellular modulus and distinguishing between nuclear and cytoplasmic mechanical properties as well as detecting changes due to perturbations of individual cellular components. In cancer, environmental mechanical factors and intracellular mechanics are expected to play an integral role in cancer progression and treatment success. Brillouin confocal microscopy is appealing for many studies in cancer mechanobiology involving both primary tumors and metastatic dissemination. Specifically, Brillouin technology is suitable for experimental scenarios where noncontact mechanical measurements are required such as 3D tumor models, interactions with the extracellular matrix (ECM), investigation of nuclear mechanical properties, or analysis of cells within microfluidic chips.

In This Issue

HomeCirculation ResearchVol. 121, No. 6In This Issue Free AccessIn BriefPDF/EPUBAboutView PDFView EPUBSections ToolsAdd to favoritesDownload citationsTrack citationsPermissions ShareShare onFacebookTwitterLinked InMendeleyReddit Jump toFree AccessIn BriefPDF/EPUBIn This Issue Ruth Williams Ruth WilliamsRuth Williams Search for more papers by this author Originally published1 Sep 2017https://doi.org/10.1161/RES.0000000000000174Circulation Research. 2017;121:585High Throughput Screen in Cardiomyocytes (p 604)Download figureDownload PowerPointMcLendon et al identify factors influencing protein aggregation within cardiomyocytes.Accumulations of misfolded proteins within cells can impair cellular functions. Cellular aggregates of misfolded proteins are cleared mainly by the autophagic and proteasomal pathways, and this clearance is particularly important in postmitotic cells, such as those in the brain and heart. Indeed, protein aggregates are a hallmark of certain neurodegenerative disorders as well as heart conditions including desmin-related myopathy (DRM)—a disease characterized by heart and skeletal muscle degeneration. Using a mouse cardiomyocyte model of DRM and a library of silencing RNAs, McLendon and colleagues have now carried out an unbiased genome-wide screen for novel genes that either prevent protein aggregates or promote their clearance. The team identified a number of novel candidates, including the tyrosine kinase, Jak1, whose inhibition reduced protein aggregation by at least 50%. Further analysis of Jak1 in the mouse cells revealed that knock down of the kinase increased proteosomal, but not autophagic, activity. This work not only indicates that Jak1 could be a target for promoting protein clearance in DRM and other protein aggregation diseases, but also provides a list of other protein clearance candidates worthy of consideration.ET-1 Stimulates Vasoconstriction Through Rab11A (p 650)Download figureDownload PowerPointZhai et al determine the intracellular mechanics of endothelin-1 induced vasoconstriction.Controlling contraction and relaxation of arteries involves the careful balancing of opposing vasoconstrictive and dilatory signals, which act upon the smooth muscle cells (SMCs) of the vessel walls. Ion channels that regulate both the influx and intracellular levels of calcium within SMCs control contractility, but how the vasoconstrictive and dilatory signals regulate these channels is largely unknown. To learn more, Zhai and colleagues focused on the BK channels of SMCs. They found that the vasoconstrictor endothelin-1 prevents the trafficking of an essential BK subunit (β1) to the plasma membrane of arterial SMCs, and thus diminishes BK activity. Reduced trafficking of β1 to the cell surface involved endothelin-1 activation of protein kinase C followed by phosphorylation (and inhibition) of the intracellular trafficking factor Rab11a. In contrast, nitric oxide, a potent vasodilator, activated Rab11a and promoted trafficking of β1 to the plasma membrane. The study provides insight into the mechanisms controlling arterial constriction and dilation and reveals potential targets, such as Rab11a, that might be manipulated pharmaceutically to regulate vascular contractility.Ambient Temperature Controls Monocyte Egress From Marrow (p 662)Download figureDownload PowerPointWarm ambient temperatures reduce atherosclerosis in mice, report Williams et al.Epidemiological data shows an association between cold temperatures and adverse cardiovascular events. Yet, cold temperatures also stimulate thermogenesis within brown and beige fat, which is thought to improve weight loss and insulin sensitivity. To gain a better understanding of the effects of ambient temperature on cardiovascular disease, Williams and colleagues housed mice that are prone to atherosclerosis in rooms where the ambient temperatures were 4°C, 22°C, or 30°C. Although total plasma cholesterol levels did not differ between the 3 groups, animals housed at 4°C had the greatest plaque burden, while those housed at 30°C had the lowest. The ambient temperature also inversely correlated with the abundance of monocytes emerging from bone marrow and infiltrating plaques, with animals housed at 30°C exhibiting the lowest levels of both and the lowest numbers of circulating monocytes overall. The team went on to show that monocyte counts in humans were also inversely correlated with seasonal temperature changes. Williams and colleagues argue that while activation of brown and beige fat may instigate beneficial metabolic changes for combating obesity, promoting exposure to cold as a means to achieve this end, could have adverse effects. Previous Back to top Next FiguresReferencesRelatedDetails September 1, 2017Vol 121, Issue 6 Advertisement Article InformationMetrics © 2017 American Heart Association, Inc.https://doi.org/10.1161/RES.0000000000000174 Originally publishedSeptember 1, 2017 PDF download Advertisement

Epithelial Folding Driven by Apical or Basal-Lateral Modulation: Geometric Features, Mechanical Inference, and Boundary Effects

During embryonic development, epithelial sheets fold into complex structures required for tissue and organ functions. Although substantial efforts have been devoted to identifying molecular mechanisms underlying epithelial folding, far less is understood about how forces deform individual cells to sculpt the overall sheet morphology. Here we describe a simple and general theoretical model for the autonomous folding of monolayered epithelial sheets. We show that active modulation of intracellular mechanics along the basal-lateral as well as the apical surfaces is capable of inducing fold formation in the absence of buckling instability. Apical modulation sculpts epithelia into shallow and V-shaped folds, whereas basal-lateral modulation generates deep and U-shaped folds. These characteristic tissue shapes remain unchanged when subject to mechanical perturbations from the surroundings, illustrating that the autonomous folding is robust against environmental variabilities. At the cellular scale, how cells change shape depends on their initial aspect ratios and the modulation mechanisms. Such cell deformation characteristics are verified via experimental measurements for a canonical folding process driven by apical modulation, indicating that our theory could be used to infer the underlying folding mechanisms based on experimental data. The mechanical principles revealed in our model could potentially guide future studies on epithelial folding in diverse systems.