In this work, we present a new experimental methodology that integrates magnetic tweezers (MT) with substrate deformation tracking microscopy (DTM) and traction force microscopy (TFM). Two types of MT-DTM/TFM experiments are described: force-control mode and displacement-control mode experiments. In model bead-on-gel experiments for each mode, an MT device is used to apply a controlled force or displacement waveform to a fibronectin-coated superparamagnetic bead attached to a fibrillar type I collagen gel containing a layer of covalently attached red-fluorescent microspheres. Serial fast time-lapse differential interference contrast and epifluorescence image acquisition steps are used to capture displacements of the bead and microspheres, respectively, in response to the applied force or displacement. Due to the large number of acquired images and the dynamic nature of the experiment, new quantitative approaches are implemented to adapt TFM for the analysis of the data, including (i) a temporospatial correction algorithm for improved tracking of microsphere displacements, (ii) a method for the objective determination of L2 regularization parameters for computing incremental traction stress solutions, and (iii) an empirical means for identifying time intervals within the data that can be approximated by elastostatic conditions. We also illustrate how force and energy balances in a force-control mode bead-on-gel experiment can be used to estimate the elastic modulus of a collagen substrate. Finally, in a proof-of-concept, bead-on-cell demonstration, measurements of incremental cell–matrix traction stresses are used to observe how a force applied to a focal contact on the apical surface of a keratinocyte is transmitted to the collagen substrate below the cell.

The human epidermis—the outermost layer of skin—comprises an organized assembly of epithelial cells known as keratinocytes.1 Within the epidermis, basal keratinocytes adhere to one another via specialized cell–cell anchoring junctions known as desmosomes and adherens junctions,1,2 whereas keratinocytes within the innermost layer of the epidermis are also anchored to the underlying connective tissue of the dermis by specialized cell–extracellular matrix (cell–matrix) anchoring junctions known as hemidesmosomes and focal adhesion contacts.2,3 Within individual keratinocytes, keratin intermediate filaments associate with desmosomes and hemidesmosomes, while actin microfilaments converge on focal adhesions and adherens junctions.2–4 Collectively, these anchoring junctions and their associated cytoskeletal proteins form a transcellular filamentous network superstructure that is critical to the mechanical barrier function of skin. In humans, several different types of congenital and acquired immunobullous skin fragility disorders exist in which specific components of this cytoarchitectural complex are rendered dysfunctional.2,5–7 Affected areas of skin form blisters, erosions, and ulcerations, thereby impairing the ability of the skin to regulate temperature and water balance while permitting invasion by external pathogens.2,8

In the autoantibody-mediated immunobullous diseases, pemphigus vulgaris (PV) and bullous pemphigoid (BP), impairment of desmosome and hemidesmosome functions, respectively, are known to cause skin fragility.9–14 Specific molecular targets of PV autoantibodies include the extracellular domains of the desmosomal cadherins, desmoglein 1 and desmoglein 3, whereas an extracellular epitope within collagen XVII (BP180) of the hemidesmosome represents the most likely target of pathogenic autoantibodies in BP.9,10 The pathophysiologic roles of other autoantibodies observed in both diseases are uncertain.9–11,13,14 To date, multiple mechanisms have been proposed to explain how autoantibodies trigger the loss of keratinocyte cohesion in PV, including steric hindrance of the adhesive molecular contacts within extracellular domains of desmosomal cadherins, initiation of intracellular signaling cascades that lead to disruption in desmosome assembly, premature disassembly of desmosomes, and/or inappropriate retraction of keratin intermediate filaments.9–11,13 Blister formation in BP is currently hypothesized to result from a cascade of events, including autoantibody binding to collagen XVII, complement fixation and activation, recruitment of neutrophils and eosinophils to the basement membrane zone, and proteolytic destruction of collagen XVII and other matrix elements localized to the vicinity of the hemidesmosome.9,10 In summary, although much is known about the existence of autoantibodies in immunobullous diseases, the biophysical mechanisms and the temporal sequence of events that culminate in mechanical disintegration of the epidermis following autoantibody exposure remain incompletely understood.9–14 

In order to bridge mechanistic knowledge gaps in our understanding of immunobullous diseases, new experimental approaches are needed. Arguably, in vitro methodologies are promising,15 specifically, techniques that might permit observation of the real-time evolution of force transmission within multicellular human epidermal tissue constructs following specific autoantibody exposures. New insight into the mechanobiological functions of desmosomes and hemidesmosomes in the presence of specific autoantibodies could provide an impetus for the development of innovative therapeutics that are designed to strengthen the structural integrity of the epidermis, a major shift away from current immunosuppressive treatments that are intended to quell autoantibody production and the inflammatory response.13 Because autoantibody targets in immunobullous disorders are localized to both the keratinocyte cell membrane and the extracellular matrix, a robust methodology would be able to interrogate force transmission between cells, between cells and matrix, and within the matrix itself. Fortunately, over the past few decades, numerous biophysical techniques have been developed for the exploration of cell mechanobiology and extracellular matrix rheology, including Förster resonance energy transfer (FRET)-based tension sensors,16,17 magnetic tweezers (MT),18–22 displacement/deformation tracking microscopy (DTM),23,24 and cell traction force microscopy (TFM) and its variants.25–28 With respect to multicellular systems such as the epidermis, MT devices have been used to explore local viscosity within embryonic tissues,29 whereas DTM/TFM has been used to directly measure traction stresses associated with collective cell migration30 and to infer cell–cell force transmission within living cell clusters, including keratinocytes.23,31–33 In contrast, only limited work has been done to combine the techniques of MT and DTM/TFM for biological applications,34,35 possibly due to difficulties that stem from the integration of dynamic MT-based rheological experiments with the quasi-static analytical framework of TFM.

As a motivation for this work, we propose that experiments that integrate MT and DTM/TFM can be used to explore force transmission within the human epidermis in the context of immunobullous skin diseases. The design, operation, and calibration of our MT device are comprehensively reviewed in a companion paper.36 Herein, we establish two model methodological approaches for integrating the techniques of MT and DTM/TFM: force-control mode and displacement-control mode MT-DTM/TFM experiments. As the initial step toward implementation in future studies of keratinocyte mechanobiology, we first apply the integrated MT-DTM/TFM methodology to investigate force transmission within a more tractable experimental paradigm: application of a controlled force or displacement to a superparamagnetic bead that is attached to a collagen gel substrate. Using two model bead-on-gel experiments as our vehicle for discussion, we introduce the relevant protocols, physical approximations, and computational methods that we have developed to adapt the standard DTM/TFM workflow for analyzing the large image-based datasets generated during an integrated MT-DTM/TFM experiment. We illustrate how force and energy balances in a force-control mode bead-on-gel experiment can be used to estimate the elastic modulus of a collagen substrate. Finally, to validate the potential utility of the methodology for more advanced explorations of keratinocyte mechanobiology, a bead-on-cell force-control mode MT-DTM/TFM demonstration is presented. In this experiment, measurements of incremental cell–matrix traction stresses are used to monitor how a force applied to a focal adhesion contact on the apical surface of an isolated living keratinocyte is transmitted to the collagen substrate subjacent to the cell. Together, the proof-of-concept model experiments detailed in this work suggest that the methodology of MT-DTM/TFM will be a useful tool in future investigations that attempt to elucidate the biophysical mechanisms of epidermal failure in bullous pemphigoid and pemphigus vulgaris.

In order to optimally explore mechanical force transmission within a human epidermal tissue construct in vitro, we postulate that the experimenter should have the ability to apply a force that is (i) specific to a cell–cell or cell–matrix anchoring junction of interest and that is (ii) of sufficient magnitude to generate cellular and/or underlying substrate deformations that can be physically observed and characterized within the framework of applied mechanics.37 Based on these constraints, we chose to integrate the techniques of MT and DTM/TFM. An idealized cartoon schematic of our methodology is detailed in Fig. 1(a). Depending on culture conditions, 2D planar multicellular aggregates of individual keratinocytes, an integrated monolayer epithelial sheet of keratinocytes, or a 3D epidermal tissue construct can be cultured on a soft deformable substrate with fluorescent microspheres embedded on its surface. The substrate should mimic the physiologically relevant matrix of the tissue environment in vivo while also being compatible with keratinocyte culture in vitro, i.e., the surface mechanochemistry must be conducive to normal cell attachment and viability. As an additional constraint, the substrate must be mechanically compliant such that forces and/or displacements applied by the MT device result in substrate deformations that can be easily observed by tracking movements of the embedded fluorescent microspheres.

FIG. 1.

The above schematic conceptually demonstrates two types of integrated MT-DTM/TFM mechanobiological experiments. (a) In force-control (FC) mode experiments, the MT device exerts a prescribed magnetic actuation force or force waveform, FMT*, on a superparamagnetic bead attached to a keratinocyte present within a 2D planar multicellular aggregate. (b) In displacement-control (DC) mode experiments, the MT device is first actuated to magnetically clamp the needle tip to a superparamagnetic bead attached to a keratinocyte that is also present within a 2D planar multicellular aggregate. Subsequent translation of the needle is then used to impose a prescribed displacement or displacement waveform on the bead, ΔMT. Application of either FMT* or ΔMT will generate a deformation in the underlying culture substrate that is a function of cell–cell and cell–matrix anchoring junctions present within the multicellular aggregate. Substrate deformations are observed by tracking displacements of fluorescent microspheres embedded in its surface. For both modes of operation, experiments require the acquisition of two sequential yet coupled DIC and epifluorescence imaging sets during which an identical FMT* or ΔMT actuation sequence is applied to the superparamagnetic bead. (c) An orthogonal x,y,z-coordinate system with its origin fixed to the needle tip of the MT device is used to spatially locate the position of the superparamagnetic bead at any instance of time, t, a vector defined here as δt. Bead position is illustrated at two different time points, t0 and t1. Yellow block arrows are used to indicate the xy- and xz-projections of the magnetic actuation force vector, FMT*. Here, zt is assumed to be constant for the duration of the experiment as motion of the bead is largely restricted to the xy-plane. Note in (c) how the angle between δt and the xy-plane changes as a function of the bead position, as shown by αt, the projection of this angle in the xz-plane. Although the bead and the y-axis of the needle are grossly aligned at t0, small y-displacements of the bead (exaggerated in the above schematic) are observed during actuation of the MT device, as imaged within the xy-focal plane of the microscope. The bead-tip Euclidean separation distance, δt, and geometry of the setup are deterministic of the vectoral component of FMT* present within the xy-plane of the substrate, a variable defined as FMT, with a scalar magnitude denoted as FMT.

FIG. 1.

The above schematic conceptually demonstrates two types of integrated MT-DTM/TFM mechanobiological experiments. (a) In force-control (FC) mode experiments, the MT device exerts a prescribed magnetic actuation force or force waveform, FMT*, on a superparamagnetic bead attached to a keratinocyte present within a 2D planar multicellular aggregate. (b) In displacement-control (DC) mode experiments, the MT device is first actuated to magnetically clamp the needle tip to a superparamagnetic bead attached to a keratinocyte that is also present within a 2D planar multicellular aggregate. Subsequent translation of the needle is then used to impose a prescribed displacement or displacement waveform on the bead, ΔMT. Application of either FMT* or ΔMT will generate a deformation in the underlying culture substrate that is a function of cell–cell and cell–matrix anchoring junctions present within the multicellular aggregate. Substrate deformations are observed by tracking displacements of fluorescent microspheres embedded in its surface. For both modes of operation, experiments require the acquisition of two sequential yet coupled DIC and epifluorescence imaging sets during which an identical FMT* or ΔMT actuation sequence is applied to the superparamagnetic bead. (c) An orthogonal x,y,z-coordinate system with its origin fixed to the needle tip of the MT device is used to spatially locate the position of the superparamagnetic bead at any instance of time, t, a vector defined here as δt. Bead position is illustrated at two different time points, t0 and t1. Yellow block arrows are used to indicate the xy- and xz-projections of the magnetic actuation force vector, FMT*. Here, zt is assumed to be constant for the duration of the experiment as motion of the bead is largely restricted to the xy-plane. Note in (c) how the angle between δt and the xy-plane changes as a function of the bead position, as shown by αt, the projection of this angle in the xz-plane. Although the bead and the y-axis of the needle are grossly aligned at t0, small y-displacements of the bead (exaggerated in the above schematic) are observed during actuation of the MT device, as imaged within the xy-focal plane of the microscope. The bead-tip Euclidean separation distance, δt, and geometry of the setup are deterministic of the vectoral component of FMT* present within the xy-plane of the substrate, a variable defined as FMT, with a scalar magnitude denoted as FMT.

Close modal

In an integrated MT and DTM/TFM experiment, a superparamagnetic bead is covalently functionalized with a specific ligand of interest (e.g., fibronectin, collagen, E-cadherin, desmocollin, and desmoglein). The bead is then allowed to bind to its native ligand(s) or receptor(s) on the surface of an individual cell forming a cell–cell or cell–ECM anchoring junction mimic, i.e., a mechanobiochemical coupling between the superparamagnetic bead and the cell that can be used to experimentally model one or more critical attributes of a true biological anchoring junction. In a force-control (FC) mode experiment [see Fig. 1(a)], a defined force or force waveform, FMT*, is selectively applied to the anchoring junction mimic using the MT device during two serial imaging experiments. Note that throughout this work, variables in bold are used to represent vector-valued functions, while non-bold variables are used to represent scalar-valued functions. In the first experiment, FMT* is applied while differential interference contrast (DIC) imaging is used to monitor displacements of the superparamagnetic bead. In the second experiment, FMT* is applied again, but this time, epifluorescence imaging is used to capture displacements of the fluorescent microspheres embedded in the surface of the culture substrate. Tracked and quantified, microsphere displacements can be used to approximate a continuum representation of the substrate displacement field, u (i.e., DTM). If substrate displacements are both small and confined to within the microscopic field of view, it is possible to calculate the incremental traction stress (vector) field, T, that develops on the surface of the substrate in response to FMT* using the analytical framework of TFM. In a displacement-control (DC) mode experiment [see Fig. 1(b)], the tip of the MT device is positioned in close proximity to the attached superparamagnetic bead of interest and then actuated to magnetically clamp the bead to the needle tip. The MT needle is then translated to impose a displacement or displacement waveform, ΔMT, on the bead in the horizontal plane of the substrate [see Fig. 1(b)]. Similar to FC mode, two serial imaging experiments are also performed in the DC mode: the first in which DIC imaging is used to capture displacements of the superparamagnetic bead and the second in which epifluorescence imaging is used to track displacements of substrate microspheres. The same prescribed ΔMT is applied for both imaging studies. Again, tracking of fluorescent microsphere displacements in the underlying substrate (u) allows for the calculation of T that develops in response to ΔMT. For both modes of operation, it is important to recognize that experiments require the acquisition of sequential yet coupled DIC and epifluorescence image sets during which an identical FMT* or ΔMT actuation waveform is applied to the superparamagnetic bead. The DIC image set records the motion of the superparamagnetic bead and the cell, whereas the epifluorescence imaging set captures the kinematics of the microspheres embedded in the culture substrate. Additional phase contrast or DIC imaging can also be used to provide wide-field views of the reconstituted cell sheet.

The instrumentation and control schemes employed to operate our MT device are extensively detailed in a companion paper.36 With respect to integrated MT-DTM/TFM experiments, in brief, our setup consisted of a Nikon Eclipse Ti-E inverted microscope (Nikon, Melville, NY) controlled by an HP Z72 Workstation running Nikon Elements AR v.4.51 software on a non-deterministic 64-bit Windows® operating system. A PCO.EDGE 4.2 sCMOS camera (PCO-Tech, Wilmington, DE) was used to acquire images at a standard frame rate of 40 frames/second (fps) and a resolution of 2048 × 2044 pixels2. Frame rates of up to 100 fps for smaller defined regions of interest (ROIs) were used for some experiments. The microscope was equipped with standard DIC, phase contrast, and epifluorescence imaging modalities. Nikon CFI Plan Fluor DL 10× (0.3 NA), Plan Apo Lambda 10× (0.3 NA), or CFI Plan Apo VC 20× (0.75 NA) air objective lenses coupled with an optional 1.5× magnifier were used in the collection of all experimental image sets. Epifluorescence imaging used a Texas Red bandpass filter cube set (560/40 nm excitation filter, 595 nm dichroic mirror, and 630/60 nm emission filter). DIC or phase contrast imaging was used for visualization of keratinocytes. The MT device was controlled via a Dell Optiplex 9010 desktop computer that used a LabVIEWTM Real-Time 15.0.1 deterministic operating system (National Instruments, Austin, TX) equipped with a 16-bit NI PCIe-6341 data acquisition (DAQ) board. The DAQ board was used to acquire analog input measurements [i.e., magnetic flux density, solenoid current, and transistor–transistor logic (TTL) pulses marking exposures of the sCMOS camera] while generating analog output voltages to control actuation of the MT device in real-time. The MT device was mounted on an MHW-3 three-axis water hydraulic manual micromanipulator (Narishige, Amityville, NY) that was fitted with an aluminum adapter arm fabricated in-house to extend the reach of the needle tip over the microscope stage.

Working surfaces of 50 mm-diameter Petri dishes with 35 mm-diameter No.1.5 coverslip glass bottoms (MatTek, Ashland, MA) were treated with a BD-20AC corona discharge treater (Electro-Technic Products, Chicago, IL) to render the surface hydrophilic due to an increase in hydroxyl moieties on the glass surface.38,39 Next, the glass was silanized by covering with 97% 3-aminopropyltriethoxysilane (3-APTMS; Sigma Aldrich, St. Louis, MO) for 5 min, followed by multiple rinses with deionized (DI) water. The surface was then covered in a freshly prepared 5% glutaraldehyde solution (Sigma Aldrich) for 30 min, followed by multiple rinses of DI water.23,40 Rat tail type I collagen solutions (1.0–3.0 mg/ml) were prepared at 4 °C from a mixture of phenol-red 10× Dulbecco’s phosphate buffered saline (DPBS; Gibco, Dublin, Ireland), DI water, and rat tail type I collagen (Corning, Inc., Corning, NY), titrated to a pH of 7.4 using 1N NaOH according to the manufacturer’s protocol. The collagen solution was then dispensed into an ∼18 mm inner diameter Sylgard 184 polydimethylsiloxane (PDMS; Dow Corning, Midland, MI) circular mold fixed to the glass surface. PDMS molds were fabricated in-house with thicknesses ranging from 250 to 750 µm and outer diameters of ∼24 mm. Specimens were subsequently incubated at 37 °C/5%CO2 for 60 min to allow for collagen fibrillogenesis and hence gel formation, after which the PDMS molds were removed. A second circular PDMS mold with an inner diameter of ∼24 mm and an outer diameter of ∼28 mm was then placed around the fully formed collagen gel to create a temporary liquid reservoir. The surface of the collagen gel was then covered with a sonicated solution of 0.5 μm-diameter carboxylate-modified polystyrene red fluorescent microspheres (FluoSpheres, ThermoFisher Scientific, Grand Island, NY) and incubated at room temperature for 2 min. Loosely physisorbed microspheres were rinsed from the surface via multiple washes with 1× DPBS. To covalently attach the fluorescent microspheres to the collagen fibril network, the PDMS reservoir was filled with a 2 mM solution of 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDAC; ThermoFisher Scientific) in 50 mM 2-(N-morpholino) ethanesulfonic acid hydrate (MES; Sigma Aldrich) at a pH of 6.0–6.5 for 15 min at room temperature. The EDAC/MES solution was then aspirated, and the specimen was rinsed multiple times with 1× DPBS. The final surface density of red fluorescent microspheres was approximately 0.06 spheres/μm2. Following the covalent attachment step, PDMS rings were removed, and the entire dish was filled with 1× DPBS to prevent gel drying. Specimens were sealed with parafilm and stored at 4 °C for up to 3 weeks prior to use.

Several discussion points on our use of collagen hydrogels are worth noting. Although polyacrylamides and silicones are routinely employed in TFM experiments, we found that these materials proved incompatible with keratinocyte culture when formulated with the soft mechanical compliances needed for our integrated MT-DTM/TFM experiments. As such, we selected type I collagen gel substrates for use. Collagen hydrogels are an attractive substrate choice because their 3D fiber networks provide an in vivo-like biomimetic surface. Collagen hydrogels have also been successfully used in the development of epidermal tissue constructs.41–43 Equally importantly, both the morphology and rheology of collagen gels have been extensively studied, with properties that are known functions of reconstitution conditions.44 In general, the vast majority of studies probing collagen gel properties were conducted using reconstitution times ranging from 30 min to 2 h.45–47 More specifically, in a parallel plate rheometry study using the exact same rat tail type I collagen reconstitution protocol used in this work, collagen gels formed at 37 °C were shown to attain a stable maximum elastic storage modulus on a time scale of ∼10 min.48 Other works have also demonstrated similar kinetics of fibrillogenesis for type I collagen gels reconstituted at 37 °C.46 Therefore, the 60-min time interval for reconstitution employed in our protocol should generate a collagen substrate with fully developed rheological properties. However, in a broader context, collagen fibril formation is an entropy-driven self-assembly/disassembly process that is affected by perturbations under environmental conditions, i.e., changes in temperature, ionic strength, and pH that occur before, during, or after the initial onset of fibrillogenesis. Moreover, concomitant cell culture can also trigger de novo cell-mediated reorganization of collagen fibril networks within a gel even when environmental conditions no longer favor continued fibrillogenesis. Together, these observations imply that because the morphologic and rheological properties of a reconstituted collagen gel are inherently dynamic, there is no single optimal time point to assess gel behavior. Therefore, as done in this work, collagen substrates used in MT-DTM/TFM experiments must be carefully controlled for reconstitution conditions, secondary processing steps, storage conditions, and culture history.

As a final comment regarding the preparation of our collagen gel substrates, we highlight that the carboxylate-modified polystyrene fluorescent microspheres conjugated to the surface of the collagen gel are used as fiducial markers for visualizing substrate deformation in our MT-DTM/TFM experiments. Microsphere-free approaches to TFM that employ confocal reflectance or fluorescence imaging of collagen fibrils to quantify matrix deformations,49,50 although possible, are intrinsically slower and incompatible with the fast time-lapse image acquisition (40 fps) necessary to track the dynamic substrate motions that occur during force-control or displacement-control mode experiments (see Secs. IV A and IV B). Consequently, the presence of fluorescent microspheres in our experiments is required for accurate DTM and TFM analyses (see Secs. III E and III F). Interestingly, in several preliminary model bead-on-gel force-control mode experiments (see Sec. IV A), we observed microsphere displacement patterns that showed no gross quantitative differences between substrates in which microspheres were allowed to attach to the collagen surface through simple physisorption vs substrates in which microspheres were attached using the EDAC/MES protocol detailed above. Moreover, in adjunctive parallel plate rheometry studies, we did not observe any statistically significant differences in the elastic storage moduli of collagen gels that had been subject to our EDAC/MES conjugation protocol compared to untreated gels (data not shown). Despite no apparent objective differences between EDAC/MES-treated and physisorbed microspheres, we ultimately opted for incorporation of the EDAC/MES conjugation step in our fabrication protocol. In our estimation, the possible existence of relative rotational movements or relative translational displacements between collagen fibrils and the fluorescent microspheres in response to an applied mechanical deformation would be much less likely for covalently coupled microspheres compared to microspheres adherent to the substrate solely via van der Waals forces.

Primary adult human keratinocytes (HEKs; ATCC, Manassas, VA) were cultured in T75 flasks (tissue culture-treated polystyrene) at 37 °C/5%CO2 in keratinocyte serum-free medium (KSFM; Gibco) containing L-glutamine and supplemented with 50 mg/ml of bovine pituitary extract, 2.5 ng/ml of the recombinant human epidermal growth factor, 100 U/ml of penicillin, 100 mg/ml of streptomycin, 0.205 mg/ml of sodium deoxycholate, and 0.25 µg/ml of amphotericin. The fully supplemented KSFM contains low concentrations of calcium (<0.2 mM) and is otherwise referred to here as a low [Ca2+] KSFM. A fully supplemented KSFM with a calcium concentration of ≈1.2 mM was also prepared via addition of sterile CaCl2, referred here to as high [Ca2+] KSFM. Two-dimensional planar multicellular aggregates of keratinocytes were reconstituted on collagen gels by harvesting cells >48 h following passage 4 or 5 of standard culture in T75 flasks. Keratinocyte aggregates were only reconstituted on gels with collagen concentrations that were 2.0 mg/ml or greater. As others have observed, keratinocytes seeded on gels with collagen concentrations <1.5 mg/ml demonstrated low rates of cell survival.51 No differences in cell viability were noted in comparing keratinocytes seeded on EDAC/MES-treated vs untreated collagen substrates.

To initiate cell seeding, the 1× DPBS covering the collagen gel for storage was aspirated and the Petri dish containing the gel was sterilized by a 10–15-min exposure to 254 nm ultraviolet light in the biosafety cabinet. Next, an 18 µl droplet containing 200 cells/μl in low [Ca2+] KSFM was dispensed onto the central area of the gel, and sterile DI water was pipetted into the Petri dish around the edges of the gel to prevent dehydration while keratinocytes attached and spread over the collagen surface. The specimens were incubated at 37 °C/5%CO2 for 3 h, after which the water was aspirated, and the dish was filled with either low [Ca2+] or high [Ca2+] KSFM and incubated for an additional 24–72 h. The seeding procedure outlined here consistently generated 2D planar multicellular aggregates of keratinocytes on the central area of the collagen gel, ∼3 to 6 mm in diameter. In some instances, however, due to localized overseeding of cells, keratinocytes were artificially stratified, meaning that some keratinocytes plated out on top of other keratinocytes that had already attached and spread to cover the underlying collagen substrate. In this sense, the aggregates were not biologically stratified as observed in the culture of an epidermal tissue equivalent.41–43 

Under low [Ca2+] conditions, individual keratinocytes adhere to the underlying collagen substrate solely via focal adhesion contacts.2,52 However, under high [Ca2+] conditions, keratinocytes form a monolayer epithelial sheet that is mechanically interconnected via cell–cell adherens junctions and desmosomes.2,23,53–56 Because this so-called calcium switch model of keratinocyte culture is well-established in the scientific literature, immunohistochemical staining for specific keratins, actin, vimentin, E-cadherin, and/or desmoplakin to verify the presence or absence of anchoring junction formation in our reconstituted multicellular aggregates was not performed. To summarize, the reconstitution protocol described here was specifically designed to generate two experimentally tractable cases for probing force transmission in human epidermal keratinocytes: one in which keratinocytes exist as isolated cells mechanically independent of neighboring cells within the aggregate and the other one in which keratinocytes have been structurally integrated into a 2D monolayer epithelial sheet. To demonstrate the potential utility of our MT-DTM/TFM methodology for future investigations of keratinocyte mechanobiology, we elected to apply the technique to observe force transmission in an isolated keratinocyte at the peripheral margin of a multicellular aggregate reconstituted under low [Ca2+] conditions (see Sec. V). Although not done in the present study, MT-DTM/TFM experiments investigating force transmission in higher order epidermal tissue constructs are plausible, although our reconstitution protocol would require significant adaptations to allow for extended periods of culture at an air/liquid interface.41–43 

Superparamagnetic beads used in both bead-on-gel and bead-on-cell MT-DTM/TFM demonstrations were prepared according to the bead manufacturer’s protocol by incubating 4.5 μm-diameter Dynabeads M-450 Tosylactivated® superparamagnetic beads with human plasma fibronectin purified protein (Catalog No. FC010; MilliporeSigma, St. Louis, MO) reconstituted at a concentration of 1 mg/ml in a 0.1M sodium phosphate buffer. For model bead-on-gel experiments (see Secs. IV A and IV B), a small droplet of fibronectin-coated superparamagnetic beads diluted to 15 beads/μl in 1× DPBS was dispersed on the surface of a collagen gel substrate and incubated at 37 °C/5%CO2 for 10 min to allow for bead settling. Following this initial incubation, the dish was gently filled with additional DPBS after which the specimen was incubated overnight at 37 °C/5%CO2 to promote more robust bead attachment to the substrate via native hydrophobic adhesive interactions between collagen and fibronectin.57 For the bead-on-cell demonstration (see Sec. V), fibronectin-coated superparamagnetic beads were again diluted in 1× DPBS. After aspirating the overlying KSFM, the bead suspension was dispensed as a small droplet on the surface of the multicellular keratinocyte aggregate and incubated at 37 °C/5%CO2 for 10 min. After this brief incubation, loosely adherent beads and excess solution were removed, and the entire Petri dish was then filled with low [Ca2+] KSFM and incubated overnight at 37 °C/5%CO2. With this bead dispersion and incubation protocol, a random but finite fraction of superparamagnetic beads came into conformal contact with the apical surface of one or more individual keratinocytes within the multicellular aggregate. Other beads became attached to the underlying collagen substrate. For superparamagnetic beads that contacted cells, focal adhesion contacts (FACs) were formed via adhesive interactions between integrins within the FACs and the fibronectin coupled to the surface of the bead.58 

Integrated MT-DTM/TFM experiments based on DIC image sets were analyzed by first selecting a global ROI appropriate for data analysis. Within this global ROI, the spatial position of the centroid of the superparamagnetic bead was quantified within each frame using a custom MATLAB (Mathworks, Inc., Natick, MA) code based on imfindcircles and/or the geometric transformation function imregtform (restricted to rigid body translation without rotation) applied to a small fixed local ROI (within the global ROI) that captured the bead’s position throughout the overall image sequence. The use of the local ROI for centroid tracking allowed for markedly more efficient computation times. For model FC and DC mode MT-DTM/TFM experiments (see Secs. IV A and IV B), positions of the needle tip of the MT device were not tracked. In addition to superparamagnetic bead positions, the code tracked displacements of small corner ROIs confined to the four corners of the global ROI (each with an area ∼1% of the global ROI) using the MATLAB function imregtform. A substrate drift correction to the superparamagnetic bead position in each frame was calculated by subtracting the average rigid body translation of the four corner ROIs from the raw measured bead position.

Epifluorescence image sets of the microspheres embedded within the surface of our collagen substrates were analyzed to estimate ux*,y*,t, a two-dimensional continuum representation of the substrate displacement field on a time-invariant equally spaced rectangular x*,y*-positional grid for each image frame or imaging time, t. A concise conceptual outline of the steps required for reduction of microsphere displacement data can be found elsewhere.59 In this work, we employed a modified version of an open-source MATLAB TFM code originally developed to investigate force modulation in sub-resolution cellular adhesions.60 Identical global ROIs were used for both DIC and epifluorescence image sets. For epifluorescence analysis, individual fluorescent microspheres were first detected using a two-dimensional Gaussian fit of the intensity distribution for a point source.61 After microsphere identification, incremental microsphere positions were tracked using pixel correlation with subpixel fitting (PCSF),60 sometimes followed by an additional tracking step based on a modified version of subpixel correlation by image interpolation (SCII).62 The template size was set to be larger than the average separation distance between microspheres, and the maximum displacement was set at least equal to the maximum displacement of the superparamagnetic bead observed in the associated DIC image set.

After tracking the raw microsphere positions, a multistep temporospatial displacement correction algorithm was applied to the dataset. First, local displacement outliers within the global ROI at each time (or imaging frame), t, were automatically detected from the dataset using a universal normalized median residual test algorithm adapted from particle tracking velocimetry and particle image velocimetry methods.63,64 Corrected displacements for these spatial outliers were interpolated from the measured displacements of neighboring microspheres. Second, unique to our DTM/TFM analysis, spurious high frequency frame-to-frame noise in the individually tracked microsphere displacements was attenuated by passing the entire dataset through a low-pass digital filter with its design specifically tailored to the actuation waveform of the experiment. For our model bead-on-gel FC mode experiment (see Sec. IV A) and our bead-on-cell FC mode demonstration (see Sec. V B), individual microsphere displacements were temporally filtered using an equiripple linear-phase FIR digital filter with 41 taps, a passband of 2 Hz, and a stopband of 10 Hz. For our model bead-on-gel DC mode experiment (see Sec. IV B), displacements were temporally filtered using an equiripple linear-phase FIR digital filter with 41 taps, a passband of 10 Hz, and a stopband of 15 Hz. Temporal filtering introduced a time shift in the filtered microsphere position/displacement data, and thus, the final 0.5 s (or ∼20 imaging frames) of each FC or DC displacement experiment were omitted from analysis. As the last step in our displacement correction algorithm, spatial outlier-corrected and temporally filtered microsphere displacements present within small square ROIs localized to the four corners of the global ROI at t = 0 (each with an area ∼1% of the global ROI) were used to compute a frame-by-frame substrate drift correction. In each frame, the average displacement of microspheres contained in these corner ROIs was subtracted from the displacement measured for every microsphere contained within the global ROI. As with displacement tracking for DIC image sets, rotational drift corrections were not performed.

As the final step in data reduction, the optimal x*,y*-grid was determined as a function of x*- and y*-domains of microsphere positions within the global ROI, the total number of tracked microspheres, and the average spacing of microspheres. For each imaging time, t, we computed ux*,y*,t by applying the MATLAB function griddata (set to cubic interpolation) to the spatial outlier-corrected, temporally filtered, and drift-corrected microsphere displacement dataset. Due to the random x*,y*-distribution of the microsphere data, cubic interpolation failed to estimate displacements for some x*,y*-grid positions. To fill these “holes,” biharmonic spline interpolation was used to calculate missing values using the interpolated displacements and their corresponding x*,y*-grid positions as inputs to the griddata function, not the original microsphere displacement dataset.65 In this manner, C2 continuity was preserved throughout the interpolated displacement field ux*,y*,t.

Excellent reviews of TFM and the various analytical and numerical approaches used to calculate cell traction stress (vector) fields from the observed substrate displacement fields can be found elsewhere.59,66,67 In brief, we computed solutions for the incremental traction stress fields, Tx*,y*,t, using datasets where ux*,y*,t were small and where ux*,y*,t along the boundary of the global ROI were tending toward zero for all t of the epifluorescence image set. For model bead-on-gel experiments and the bead-on-cell demonstration, the collagen substrate was modeled as a spatially homogeneous, isotropic, linear elastic, semi-infinite half-space with Young’s elastic modulus, E, and Poisson’s ratio, ν, which, to a first-order approximation, holds true, given the relatively small deformations observed in the majority of our experiments. To calculate Tx*,y*,t, interpolated (or gridded) displacement fields were filtered in the spatial domain with a Wiener filter (3 × 3 pixel2)28 and then padded by a margin of zeros.59,68 A Hann window was applied to the filtered and padded dataset to minimize spectral leakage.59 The gridded, padded, and windowed displacement dataset was then transformed into Fourier space.59 Solutions for Tx*,y*,t were computed for each frame of a collective fast time-lapse image sequence using an open-source MATLAB code employing the methodology of Fourier Transform Traction Cytometry (FTTC) with an L2 regularization scheme.28,60 FTTC with L2 regularization was chosen over other methodologies due to its computational efficiency given the large number of imaging frames to be analyzed per experiment (>1000 frames). Algorithmic determination of the L2 regularization parameter(s) employed a Bayesian methodology,69 as detailed in Secs. SI.A.2 and SI.B.2 of the supplementary material.

Once Tx*,y*,t were determined, the total traction force present at the surface of the substrate at time, t, denoted here as Ft, and its magnitude, Ft, were calculated by numerical integration of Tx*,y*,t,
Ft=Fx*t+Fy*t=globalROITx*x*,y*,tdx*dy*+globalROITy*x*,y*,tdx*dy*,
(1)
where the subscripts x* and y* represent the vectoral components of Tx*,y*,t and Ft in the x*- and y*-directions, respectively, and integration is carried out over a domain that includes the entire x*,y* positional grid configured for the global ROI (see Sec. SI.A.2 of the supplementary material). In addition to Tx*,y*,t, Ft, and Ft, we also computed the strain energy density field, ρx*,y*,t, as
ρx*,y*,t=12Tx*,y*,tux*,y*,t,
(2)
and the total strain energy associated with deformation of the substrate, Ut, as
Ut=globalROIρx*,y*,tdx*dy*,
(3)
where, again, the limits of integration were set to span the domain of the complete x*,y* positional grid configured for the global ROI. Note that in a typical TFM analysis of a single cell adherent on a deformable substrate, Ft as derived from the integrated cell traction stress field is identically zero because the adherent cell is mechanically static at the time scale under consideration, i.e., the traction stress field that the cell produces to adhere to the underlying substrate does not generate a net force. In contrast, for the model MT-DTM/TFM experiments and demonstrations described here, Ft are non-zero. In our work, Tx*,y*,t represents the incremental traction stress field imposed on the collagen substrate as the result of force generated by actuation of the MT device. In the idealized case where a superparamagnetic bead directly attached to the surface of a linear elastic substrate is subject to a static magnetic actuation force, FMT* [see Fig. 1(c)], the above analysis should yield Ft=FMTt, where FMTt represents the vectoral component of FMT* present within the xy-plane of the substrate.

As for the mechanical properties of the type I collagen substrates used in our TFM calculations, we assumed ν = 0.4 for all collagen substrates.70 As a first pass in our TFM analyses, we assumed concentration-dependent shear moduli, Gc, of 5.0, 55.4, and 341.8 Pa for 1, 3, and 7 mg/ml collagen gels, respectively, as measured by the collagen manufacturer (Corning) using parallel plate rheometry.48 Here, the variable, c, is used to denote the collagen concentration. In this work, collagen fibrillogenesis was carried out exactly as was done in the Corningstudy.48 For other collagen concentrations within the range of 1 mg/ml to 7 mg/ml, Gc were estimated from the equation, Gc5.0Pac2.1,45 and all concentration-dependent elastic moduli, Ec, were calculated from the implied linear elastic relationship, Ec=2Gc1+ν (assuming ν = 0.4). As detailed in Sec. IV A 3, as an alternative to assuming mechanical properties of the substrate for our TFM analyses, we also devised a method for directly estimating Ec using data acquired during a force-control mode MT-DTM/TFM experiment.

Ethics approval was not required for this study.

In this section, we introduce the methodology of MT-DTM/FTFM as a technique for investigating force transmission within the extracellular matrix using two tractable model bead-on-gel experiments in which a controlled force or displacement is applied to a superparamagnetic bead that is attached to a type I collagen gel substrate.

1. Protocol

As a model bead-on-gel FC mode MT-DTM/TFM experiment, we dispersed a dilute suspension of 4.5 μm-diameter, fibronectin-coated superparamagnetic beads directly onto a ∼680 μm-thick, 1.0 mg/ml type I collagen substrate embedded with a surface layer of covalently attached red fluorescent microspheres. Preparation of the collagen substrate, functionalization of the superparamagnetic beads, and attachment of superparamagnetic beads to the substrate were carried out as detailed in Secs. III B and III D. Attached beads were spatially separated by at least one field of view at 30× magnification. With the microscope configured for 30× magnification, an FC mode experiment was performed using sequential DIC and epifluorescence fast time-lapse image capture steps, during which identical three-cycle, 3 s/7 s, ON/OFF magnetic actuation waveform sequences with maximum calibrated ON magnetic flux density setpoints of 175 Gs were applied by the MT device. As opposed to perfectly square ON/OFF waveforms, magnetic actuation in our setup included short stepwise ramps in calibrated magnetic flux density values between ON and OFF states. Specifically, the MT device was stepped up (or down) to ON (or OFF) states passing through intermediate calibrated magnetic flux density values of 25, 50, 75, 100, 125, and 150 Gs, maintaining each of these intermediate flux density setpoints for ∼70 ms before proceeding to the next setpoint (up or down). As a consequence of this magnetic actuation scheme, there exists a ∼500 ms transient interval in magnetic actuation force, FMT*, that separates the ON and OFF (0 nN) states of a given actuation cycle. The use of stepwise ON/OFF ramps during magnetic actuation was intentional because it facilitated more reliable tracking of the kinematics of the fluorescent microspheres embedded in our collagen substrates. Explicit details of the actuation of our MT device are described in Sec. SI.A.1, Sec. SI.A.2, and Fig. S1 of the supplementary material.

The initial (x, y, z)-spatial location of the superparamagnetic bead relative to the needle tip of the MT device was x0 = 20.0 µm, y0 = 0 µm, and z0 = 13.0 µm [δt0 ∼ 23.9 µm; see Fig. 1(c)]. Prior to data collection, a three-cycle magnetic actuation waveform sequence was applied to the superparamagnetic bead to precondition the mechanical response of the collagen substrate. Following data collection, all images were analyzed according to the methods presented in the supplementary material (see Sec. SI.A.2). Of note, TTL pulses marking exposures of the sCMOS camera were used to temporally correlate DIC and epifluorescence image data to the magnetic flux density signal recorded during actuation of the MT device. Complete real-time videos of the coupled DIC and epifluorescence imaging data analyzed for this experiment can be found in Vid. 1 and Vid. 2, respectively, of the supplementary material. In all DIC images, the superparamagnetic bead remains clearly in focus throughout the magnetic actuation waveform sequence, as does the shadow of the needle tip of the MT. Although the shadow of the needle tip is apparent in the epifluorescence image set, the superparamagnetic bead is not easily identified.

2. Bead kinematics, substrate displacements, and incremental traction stresses

In comparing the kinematics of the superparamagnetic bead to the mechanical deformation of the collagen substrate in response to the same magnetic actuation waveform sequence, consider the data shown in Fig. 2. In this model bead-on-gel experiment, we define the time point, tOFF, for each cycle within the overall magnetic actuation waveform sequence as the last imaging frame captured prior to initiation of the stepwise decrease in FMT* toward a null actuation force (FMT* = 0 nN). Similarly, we define the time point, tON, for each cycle as the last imaging frame captured prior to initiation of the stepwise increase in FMT* toward the maximum actuation force. Figures 2(a) and 2(b) depict the DIC and epifluorescence images captured near tOFF for the first magnetic actuation cycle, respectively. Now, consider plots of Δmaxxmax*,ymax*,t and ΔMTt, as shown in Fig. 2(e). Here, Δmaxxmax*,ymax*,t is used interchangeably with Δmaxt, representative of the magnitude of the displacement vector of the microsphere embedded on the collagen substrate subject to the largest observed displacement during the experiment, or xmax*,ymax*. ΔMTt represents the magnitude of the displacement vector of the superparamagnetic bead. As shown in Fig. 2(e), at t = tOFF for cycles 1, 2, and 3, the maximum substrate displacement was ≈2.0 µm less than that recorded for the superparamagnetic bead, a quantity on the order of the average radius of the bead (∼2.25 µm). When testing numerous superparamagnetic beads attached to collagen substrates in model FC mode MT-DTM/TFM experiments, this finding was repeatable. For superparamagnetic beads that were allowed to establish conformal contact with the substrate for >12 h, values of ΔMT(tOFF) − Δmax(tOFF) for a given cycle were less than ∼2.25 µm for experiments in which ΔMT(tOFF) < 2.25 µm. These values plateaued at ∼2.25 µm for experiments where ΔMT(tOFF) ≥ 2.25 µm. For experiments in which superparamagnetic beads were only allowed to establish conformal contact with the substrate for a short period of time (∼1 h), greater differences were observed between ΔMT(tOFF) and Δmax(tOFF) for ΔMT(tOFF) < 2.25 µm. Moreover, these beads frequently detached from the substrate for ΔMT(tOFF) ≥ 2.25 µm. Based on these physical observations, we speculate that the superparamagnetic bead might both rotate and translate in response to the magnetic actuation exerted by our MT device, dependent on the state of conformational contact between the bead and the substrate. Although previous authors have attempted to model substrate deformations that occur when an attached superparamagnetic bead is subject to an applied torque,71 contact models characterizing the response (to force) observed here represent a subject of future investigation.

FIG. 2.

Panels (a)–(f) highlight data from a model bead-on-gel FC mode MT-DTM/TFM experiment involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the surface of a ∼680 μm-thick, 1.0 mg/ml type I collagen substrate embedded with a surface layer of covalently attached red fluorescent microspheres. (a) DIC image showing the superparamagnetic bead near its peak displacement during the first ON cycle of the applied magnetic actuation waveform sequence. (b) Texas Red epifluorescence image with overlaid quivers that represent the displacement vectors of individual microspheres present within the surface of the collagen substrate at this same time point in the magnetic actuation waveform. Dashed boxes in (a) and (b) denote the ROIs used for drift correction. The displacement field, ux*,y*, and incremental traction stress field, Tx*,y*, computed for the microsphere displacement field displayed in (b), are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. A plot of the displacement magnitude of the microsphere subject to the largest overall displacement during actuation, Δmaxxmax,ymax,t, vs the magnitude of the superparamagnetic bead displacement, ΔMTt, is shown in (e). A plot of the magnitude of the component of the magnetic actuation force vector in the xy-plane of the collagen substrate, FMTt, vs the magnitude of the integrated total traction force vector, Ft, is shown in (f), where Ft has been computed assuming υ = 0.4 and two different elastic moduli, E = 14.0 Pa and E = 76.23 Pa. Black triangles in (e) and (f) denote tON and tOFF for each actuation cycle. Vertical dashed lines in (f) indicate dynamic time intervals in which the assumption of elastostatic conditions is likely invalid.

FIG. 2.

Panels (a)–(f) highlight data from a model bead-on-gel FC mode MT-DTM/TFM experiment involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the surface of a ∼680 μm-thick, 1.0 mg/ml type I collagen substrate embedded with a surface layer of covalently attached red fluorescent microspheres. (a) DIC image showing the superparamagnetic bead near its peak displacement during the first ON cycle of the applied magnetic actuation waveform sequence. (b) Texas Red epifluorescence image with overlaid quivers that represent the displacement vectors of individual microspheres present within the surface of the collagen substrate at this same time point in the magnetic actuation waveform. Dashed boxes in (a) and (b) denote the ROIs used for drift correction. The displacement field, ux*,y*, and incremental traction stress field, Tx*,y*, computed for the microsphere displacement field displayed in (b), are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. A plot of the displacement magnitude of the microsphere subject to the largest overall displacement during actuation, Δmaxxmax,ymax,t, vs the magnitude of the superparamagnetic bead displacement, ΔMTt, is shown in (e). A plot of the magnitude of the component of the magnetic actuation force vector in the xy-plane of the collagen substrate, FMTt, vs the magnitude of the integrated total traction force vector, Ft, is shown in (f), where Ft has been computed assuming υ = 0.4 and two different elastic moduli, E = 14.0 Pa and E = 76.23 Pa. Black triangles in (e) and (f) denote tON and tOFF for each actuation cycle. Vertical dashed lines in (f) indicate dynamic time intervals in which the assumption of elastostatic conditions is likely invalid.

Close modal

Real-time videos depicting continuum representations of the substrate displacement field, ux*,y*,t, the incremental traction stress field, Tx*,y*,t, and the strain energy density field, ρx*,y*,t, for this experiment, assuming E = 76.23 Pa and υ = 0.4, are presented as Vid. 3, Vid. 4, and Vid. 5, respectively, of the supplementary material. Figures 2(c) and 2(d) depict ux*,y*,t=4.851s and Tx*,y*,t=4.851s, respectively, where t = 4.851 s is very near tOFF for the first actuation cycle. Considering the inherent assumption of linear elasticity, overall, the solution computed for Tx*,y*,t=4.851s appears to be physically reasonable for the observed ux*,y*,t=4.851s, i.e., a quasi-symmetrical distribution of traction stresses around a localized circular area that has a diameter approximately equal to that of the superparamagnetic bead (∼4.5 µm). Integrating Tx*,y*,t over the x*,y* positional grid for all time, t, provides a mean for computing the total traction force, Ft, and its scalar magnitude, Ft. In Fig. 2(f), we compare Ft to the magnitude of the component of the magnetic actuation force exerted by the MT device in the xy-plane (also the x*y*-plane) of the collagen substrate, or FMTt, as calculated from DIC imaging data (see Sec. SI.A.2 of the supplementary material). With regard to force, note that FMTt approximates a square waveform with small ∼500 ms transients between ON and OFF states [see Fig. 2(f) and Fig. S1 of the supplementary material]. This finding is somewhat counterintuitive, based on the fact that for a constant nominal calibrated magnetic flux density setpoint, the magnetic actuation force, FMT*t, increases as the superparamagnetic bead more closely approaches the needle tip.36 However, based on the geometry of our MT setup, the vectoral component of FMT*t in the xy-plane of the collagen substrate remains roughly constant due to the increase in the relative angle between δt and the xy-plane as the bead approaches the needle tip [see Fig. 1(c)].

3. Quantification of substrate modulus

Neglecting viscous and inertial effects, if the collagen substrate behaved purely as an ideal linear elastic material, both Δmax and ΔMT should have exhibited displacement responses that qualitatively mirrored the approximate square waveform of FMTt. However, as shown in Fig. 2(e), both Δmax(t) and ΔMT(t) exhibit evidence of viscoelastic behavior in response to the applied magnetic actuation force.72 Allowing for a viscoelastic response but assuming that mechanical equilibrium is achieved at tOFF for each of the three actuation cycles, one should find that FMTtOFF = FtOFF upon integrating Tx*,y*,tOFF over the (x*, y*) domain of the global ROI. However, as shown in Fig. 2(f), if one assumes E = 14.0 Pa in our TFM calculations, as suggested by the rheologic data published for a 1.0 mg/ml type I collagen gel,48  FtOFFFMTtOFF. As such, we determined the apparent E for our collagen substrates by combining information from the DIC and epifluorescence image sets collected during identical time intervals of the magnetic actuation waveform sequence. Specifically, for the final ∼500 ms of the ON portion of the second magnetic actuation cycle, we iterated the assumed value of E within our Tx*,y*,t calculations until the computed values for Ft during this time interval best fit, in a least squares sense, the magnetic actuation force, FMTt, using the derivative-free, unconstrained minimum search function fminsearch in MATLAB. Note that each iteration of the search required re-computation of the L2 regularization parameters for the entire dataset, given its dependence on E. As shown in Fig. 2(f), an apparent E of 76.23 Pa provides a much closer match between FMT and F at t = tOFF for each of the three actuation cycles. However, with the increase in modulus, the mean noise floor in F during OFF segments of magnetic actuation also increases from ∼0.03 nN for E = 14.0 Pa to ∼0.2 nN for E = 76.23 Pa (also see Fig. S2 of the supplementary material).

Using an energetics approach, we also estimated an upper bound for the apparent E from the same combined DIC and epifluorescence image data. Using DIC images, we calculated the work done by the MT device in moving the superparamagnetic bead during the ON segment of a specified cycle of the overall magnetic actuation waveform sequence, a variable denoted here as WMTt, and calculated from tracked superparamagnetic bead positions as
WMTt=tONtFMTtxtxt2+yt2dxtdtFMTtytxt2+yt2dytdtdt,
(4)
where dxtdt and dytdt represent differential displacements of the magnetic bead in the x- and y-directions, respectively, and t marks any time point such that tON < ttOFF. To estimate the upper bound of the apparent E, we first computed WMTt=tOFF for the second cycle of the three-cycle magnetic actuation waveform. We then iterated the assumed value of E within our Tx*,y*,t calculations for the corresponding epifluorescence image at time, t = tOFF, of the second actuation cycle to find the minimum difference between UtOFF and WMTtOFF. Note that again, this required computation of the L2 regularization parameters for the entire dataset for each iterated value of E. Optimization solutions utilized fminsearch in MATLAB. By neglecting the energy lost to the system as viscous dissipation within the collagen gel and instead equating WMTtOFF to UtOFF at t = tOFF (for cycle 2), we approximated an upper bound to the apparent E of the collagen substrate. Figure 3 demonstrates temporal profiles of WMT(t) and Ut computed for this model bead-on-gel FC mode MT-DTM/TFM experiment. U(t) was calculated assuming that both E = 14.0 Pa and an apparent E = 149.8 Pa, the latter representing the upper bound of E estimated for this collagen specimen using an energetics approach. Note how WMT(t) for the first actuation cycle shown in Fig. 3 is relatively greater than WMT(t) for cycles 2 and 3, whereas U(t) showed much less intercycle variation. We believe this to be due to the fact that the collagen gel had not been adequately preconditioned by the three-cycle actuation sequence that was applied to the bead prior to the initial DIC fast time-lapse image acquisition step.
FIG. 3.

Plot showing WMTt, the work done by the MT device in moving the superparamagnetic bead during the ON segments for each of the three magnetic actuation cycles, vs Ut, the total strain energy stored in the collagen substrate during the course of the magnetic actuation waveform sequence. An upper bound to the apparent elastic modulus of the collagen substrate is found by iterating on the assumed value of E in computing Tx*,y*,t and Ut and then equating Ut=tOFF to WMTt=tOFF for actuation cycle 2. Black triangles denote tON and tOFF for each actuation cycle. Vertical dashed lines indicate dynamic time intervals in which the assumption of elastostatic conditions is likely invalid.

FIG. 3.

Plot showing WMTt, the work done by the MT device in moving the superparamagnetic bead during the ON segments for each of the three magnetic actuation cycles, vs Ut, the total strain energy stored in the collagen substrate during the course of the magnetic actuation waveform sequence. An upper bound to the apparent elastic modulus of the collagen substrate is found by iterating on the assumed value of E in computing Tx*,y*,t and Ut and then equating Ut=tOFF to WMTt=tOFF for actuation cycle 2. Black triangles denote tON and tOFF for each actuation cycle. Vertical dashed lines indicate dynamic time intervals in which the assumption of elastostatic conditions is likely invalid.

Close modal

One possible explanation for the observed difference between the reported value of E and our best-fit apparent E is that the elastic modulus of our collagen substrates was altered by the EDAC/MES chemical conjugation step used to confer covalent attachment of the fluorescent microspheres to the collagen substrate. Intuitively, EDAC/MES treatment should lead to the generation of new covalent crosslinks within the collagen gel, although at an uncertain molecular length scale. However, at a pH of 6.0–6.5, the EDAC/MES conjugation step could also thermodynamically favor some degree of collagen fibril disassembly. Interestingly, in adjunctive bulk parallel plate rheometry studies, we did not observe any statistically significant differences in the elastic storage moduli of collagen gels that had been subject to our EDAC/MES conjugation protocol compared to untreated gels (data not shown). Based on this physical reasoning, coupled with our experimental observations to date, we conclude that the EDAC/MES conjugation step has little, if any, net effect on the macroscopic rheological properties of our collagen substrates. More precise characterization of the effects of our EDAC/MES microsphere conjugation step on the fibril morphology, crosslink density, and the microrheologic properties of our collagen substrates remains a subject for future investigation.

Differences between the reported value of E = 14.0 Pa for a reconstituted type I collagen gel as derived from bulk parallel plate rheometry measurements48 and our best-fit apparent E = 76.23 Pa are best explained in terms of disparities in measurement techniques and length scales. Acellular reconstituted collagen is a material that has been subject to rigorous rheological testing in multiple different configurations, including extension, compression, and shear. Even on the macroscale, the mechanical properties derived from these different techniques do not always demonstrate exact quantitative agreement with each other.44,73 Moreover, a reconstituted type I collagen gel is a multi-scale material consisting of tropocollagen monomers, fibers, fibrils, and entangled fibrillar networks. Depending on the critical length scale of any given measurement technique, collagen can be cited as having elastic moduli that span from ∼10 GPa to ∼1 Pa. In our model bead-on-gel FC mode experiment, we probed mechanical force transmission within the fibrillar collagen network at a length scale on the order of ∼4.5 µm (the size of our superparamagnetic bead), whereas the rheometry measurements performed by Corning used a standard parallel plate configuration with a diameter of 40 mm and a gap thickness of 500 µm.48 In an experiment analogous to the model bead-on-gel FC mode experiment presented here, Velegol and Lanni found that elastic storage moduli for type I collagen gels measured by laser trap microrheometry with ∼2 µm polystyrene beads were also significantly higher than moduli measured via standard parallel plate rheometry.74 Moreover, in the same study, the authors also noted significant spatial heterogeneity in elastic storage moduli within collagen gels that were interrogated via laser trap microrheometry.74 In future work, we plan to quantify both the inter-specimen variability and intra-specimen spatial heterogeneity in elastic moduli among our collagen substrates and the potential impact such heterogeneities might have on the selection of a statistical average elastic modulus that can be used to best characterize our reconstituted gels.

As a final remark on gel rheology, we consider the potential effect of collagen substrate thickness on our method for the determination of the apparent E. In preliminary bead-on-gel FC mode experiments, we observed that gels with thicknesses of ∼250 µm had apparent E values that were larger than the apparent E values computed for gels that were ≥500 µm thick, despite all gels having identical reconstitution conditions. Thickness-dependent shear moduli of collagen gels have also been observed in parallel plate rheometry experiments.75 In order to safeguard the critical TFM assumption that the collagen substrate is to behave as a semi-infinite, linear elastic half-space in an FC mode experiment, we only used substrates with thicknesses >500 µm while keeping Δmaxt< 15 µm. More precise quantification of substrate thickness effects on our apparent E measurements is a subject of ongoing investigation.

4. L2 regularization parameter selection

In this work, we observed that for a given ux*,y*,t, selection of the L2 regularization parameter, λ2, can have a profound effect on the solution for Tx*,y*,t in terms of the distribution and magnitude of the resultant incremental traction stress field. In our first attempt to objectively select λ2 for the dataset, we used a Bayesian approach69 to objectively compute a raw time-specific value of λ2 for each imaging frame to base the solution of Tx*,y*,t. However, when these independent incremental traction stress fields were assembled into a collective time sequence, subtle non-physical temporal discontinuities in the frame-to-frame evolution of Tx*,y*,t could be observed. Consequently, we devised an objective algorithmic approach for computing Tx*,y*,t that utilized two global L2 regularization parameters, designated here as λ2ON and λ2OFF, where λ2ON was used as the L2 regularization parameter to calculate Tx*,y*,t during all ON portions of the magnetic actuation sequence, while λ2OFF was used as the L2 regularization parameter in solving Tx*,y*,t for all OFF portions of the sequence (see Sec. SI.A.2 of the supplementary material for details). The use of global λ2ON and λ2OFF parameters provided more physically realistic incremental traction stress fields while minimizing traction noise, especially during the OFF segments of the actuation sequence. A graphical demonstration of the computation of λ2ON and λ2OFF corresponding to the model FC mode experiment detailed in Fig. 2 can be found in Fig. S3 of the supplementary material.

5. Identification of data intervals approximated by elastostatic conditions

All of the foregoing TFM analysis applied to our epifluorescence image data is based on the solution of a classic elastostatics problem, i.e., one that assumes infinitesimal displacements of a spatially homogeneous, isotropic, linear elastic half-space in static equilibrium with one or more applied surface traction stresses.76 However, in our model bead-on-gel FC mode MT-DTM/TFM experiment, the magnetic actuation waveform sequence generated by the MT device produces dynamic motion of the collagen substrate. Although we compute Tx*,y*,t, Ft, ρx*,y*,t, and Ut corresponding to the measured ux*,y*,t for every analyzed imaging frame, solutions for T, F, ρ, and U are only approximately valid at time points during the magnetic actuation sequence when inertial contributions to the global force balance in this boundary-value problem are negligible. Clearly, at time points where inertial contributions are not negligible, the incremental traction stress field solution, as described in Sec. III F, would be invalid.

As an ad hoc experimental means for identifying time intervals within our overall dataset for which the assumptions underlying the elastostatics model are valid, we performed an additional experiment, referred here to as the FC mode null force control experiment. With the microscope configured for 30× magnification, the identical superparamagnetic bead of interest used in the preceding experiment was positioned such that x0 = 20.0 µm, y0 = 0 µm, and z0 = 13.0 µm with respect to the needle tip. The bead was then actuated with a three-cycle, 3 s/7 s, ON/OFF waveform sequence with maximum nominal ON magnetic flux density setpoints of 0 Gs using the same protocol implemented for the previous model FC mode experiment. In other words, the MT device was actuated with a three-cycle waveform sequence such that FMTt = 0 nN for all time points in the sequential DIC and epifluorescence fast time-lapse image acquisitions. With a null actuation force, the collagen substrate remains static throughout data acquisition, in contrast to the dynamic substrate motions observed during the model bead-on-gel FC mode experiment (see Fig. 2). Real-time videos of the coupled DIC and epifluorescence imaging data analyzed for this FC mode null force control experiment and videos computed for ux*,y*,t, Tx*,y*,t, and ρx*,y*,t, respectively, are presented as Vid. 6, Vid. 7, Vid. 8, Vid. 9, and Vid. 10, respectively, of the supplementary material, the latter two assuming E = 76.23 Pa and υ = 0.4. Figure S2 of the supplementary material summarizes data from the FC mode null force control experiment in a manner that is analogous to the data presented in Fig. 2. The global ROIs used to analyze both the null force control experiment and the model bead-on-gel FC mode experiment were identical, i.e., the central 1024 × 1022 pixels2 of each image.

In this FC mode null force control experiment, 3266 microspheres were tracked over 1236 imaging frames. As done for the model bead-on-gel FC mode experiment, measured microsphere displacements for the null force control experiment were spatial outlier-corrected, temporally filtered, and drift-corrected (see Sec. III E). For each imaging frame in the dataset, we computed the x*- and y*-vectoral components of the displacement tracked for each microsphere embedded within the collagen substrate, denoted here as ux*μspherei,t and uy*μspherei,t, respectively, where μspherei denotes a unique microsphere present in the collagen substrate identified by the index, i. Starting with the third imaging frame, we also computed the x*- and y*-components of microsphere accelerations, denoted here as 2t2ux*μspherei,t and 2t2uy*μspherei,t, respectively, using a backward difference methodology. For each analyzed frame, we computed averages for the x*- and y*-components of the displacements and accelerations observed for all 3266 microspheres (i = 1 through 3266) in addition to the statistical upper and lower bounds of these data defined by ±6 standard deviations (±6SD). Disregarding the data from the first and second imaging frames where accelerations were undefined, we computed the global means of the average displacements, accelerations, and 6SD upper/lower bounds observed over the remaining 1234 imaging frames (see Fig. S4 of the supplementary material). Because none of the 3266 microspheres experienced a true displacement or acceleration during the FC mode null force control experiment, the mean 6SD upper/lower bounds on these displacement and acceleration observations represent estimates of the physical detection limits (thus measurement uncertainty) intrinsic to our experimental setup and microsphere tracking algorithms.

Assuming identical microscope settings, global ROI selection, and microsphere tracking procedures as described for the model bead-on-gel FC mode experiment, we assert that any microsphere displacement or acceleration observed during a bead-on-gel FC mode experiment as described here cannot be distinguished as being true vs artifactual if it falls within the mean 6SD upper/lower bounds of the x*- and y*-displacements and accelerations computed for the null force control experiment. Accordingly, consider the model bead-on-gel FC mode experiment originally detailed in Fig. 2 and, more specifically, the microsphere embedded within the collagen substrate that was subject to the largest overall displacement during magnetic actuation, or xmax*,ymax*. The x*- and y*-components of the displacement and acceleration of this microsphere, denoted here as ux*xmax*,ymax*,t, uy*xmax*,ymax*,t, 2t2ux*xmax*,ymax*,t, and 2t2uy*xmax*,ymax*,t, respectively, are plotted in Fig. 4. The mean 6SD upper/lower bounds for x*- and y*-displacements and accelerations as derived from the FC mode null force control experiment are included with each plot. In Fig. 4, one finds that in terms of x*- and y*-accelerations, the model bead-on-gel FC mode experiment is potentially dynamic within small 275–325 ms (11–13 imaging frames) or 150–225 ms (6–9 imaging frames) time intervals correlating with transients that follow the start of each ON or OFF segment, respectively, of the overall magnetic actuation waveform sequence. Knowing a priori that the microsphere defined by uxmax*,ymax*,t would have experienced the largest accelerations during magnetic actuation, we classified each imaging frame within the original dataset of the model bead-on-gel FC mode experiment as being elastostatic if (i) 2t2ux*xmax*,ymax*,t < 31.4 µm/s2, if (ii) 2t2uy*xmax*,ymax*,t < 30.3 µm/s2, and if (iii) the frame was continuous within a sequence of more than 15 other elastostatic imaging frames. If all three conditions were not met, the frame was classified as being dynamic and therefore potentially in violation of the elastostatic assumptions inherent to our solution of Tx*,y*,t. Dynamic time intervals for our model bead-on-gel FC mode experiment are shown in Figs. 2(f) and 3. Similarly, images of diminished brightness are used to indicate dynamic imaging frames in Vid. 4 and Vid. 5 of the supplementary material. Based on the above analysis for ±6SD acceleration bounds, we argue that the vast majority of imaging frames collected for our model bead-on-gel FC mode experiment can be approximated by elastostatic conditions and therefore associated with valid solutions for Tx*,y*,t, Ft, ρx*,y*,t, and Ut. Alternatively, reduced upper and lower acceleration bounds could be employed using this same algorithm, e.g., ±3 standard deviations of the recorded acceleration data, and accordingly, this would result in a decreased number of imaging frames being classified as elastostatic.

FIG. 4.

Plots showing time-resolved vectoral components in the x*- and y*-directions of the displacement, uxmax*,ymax*,t, and acceleration, 2t2uxmax*,ymax*,t, of the microsphere embedded within the collagen substrate that was subject to the largest overall displacement during magnetic actuation in the model bead-on-gel FC mode MT-DTM/TFM experiment. Horizontal dashed lines in each plot represent the 6 standard deviation (6SD) upper- and lower-bounds representative of the physical detection limits for measured microsphere displacements and accelerations, as derived from an FC mode null force control experiment (see Sec. IV A 5). Vertical gray boxes in the acceleration plots indicate dynamic time intervals, i.e., times during which inertial contributions to the force balance of the boundary-value problem used to find solutions for the incremental traction stress field are likely to be non-negligible.

FIG. 4.

Plots showing time-resolved vectoral components in the x*- and y*-directions of the displacement, uxmax*,ymax*,t, and acceleration, 2t2uxmax*,ymax*,t, of the microsphere embedded within the collagen substrate that was subject to the largest overall displacement during magnetic actuation in the model bead-on-gel FC mode MT-DTM/TFM experiment. Horizontal dashed lines in each plot represent the 6 standard deviation (6SD) upper- and lower-bounds representative of the physical detection limits for measured microsphere displacements and accelerations, as derived from an FC mode null force control experiment (see Sec. IV A 5). Vertical gray boxes in the acceleration plots indicate dynamic time intervals, i.e., times during which inertial contributions to the force balance of the boundary-value problem used to find solutions for the incremental traction stress field are likely to be non-negligible.

Close modal

1. Protocol

In most MT-based experiments, displacements of the superparamagnetic bead are small, i.e., <5 µm, and associated with actuation forces on the order of 5 nN or less. However, while exploring the mechanobiology of an epithelial sheet, it is conceivable that selective application of large forces (>5 nN) and/or large displacements (>5 µm) might be required to interrogate force transmission via specific cell–cell or cell–matrix anchoring junctions. Such is the case in our work, and thus, the impetus for the development of the displacement-control mode MT-DTM/TFM experiment presented here. As a model bead-on-gel DC mode experiment, 4.5 μm-diameter, fibronectin-coated superparamagnetic beads were allowed to attach to a ∼700 μm-thick, 1.0 mg/ml type I collagen substrate embedded with a surface layer of fluorescent microspheres (see Secs. III B and III D). With the microscope configured for 30× magnification, two separate FC mode experiments were performed using three-cycle, 3 s/7 s, ON/OFF magnetic actuation waveform sequences, each with maximum nominal ON magnetic flux density setpoints of 175 Gs, exactly as described in Sec. SI.A.1 of the supplementary material. For both experiments, the initial xyz-spatial location of the superparamagnetic bead relative to the needle tip of the MT device was x0 = 20.0 µm, y0 = 0 µm, and z0 = 13.0 µm. DIC and epifluorescence images collected during the second FC experiment were used to determine the apparent elastic modulus for the collagen substrate (E = 52.73 Pa, assuming υ = 0.4) using a global ROI set to the central 1024 × 1022 pixels2 of each image.

A second superparamagnetic bead a few fields of view away from this original bead was then identified. With the microscope now configured for 20× magnification and DIC fast time-lapse image acquisition at 40 fps (2048 × 2044 pixels2/frame), the bead was magnetically clamped to the needle tip by actuating the MT device with a magnetic flux density setpoint of 175 Gs. The needle was then manually translated away from this initial position in 1 μm-step increments, holding each position for ≈1 s before proceeding with the next incremental step until a maximum needle translation of 10 µm was reached. After a ≈2 s hold at this maximum displacement, the needle was manually translated back to its initial position in an identical stepwise fashion. After a brief 1–5 s pause, the MT was actuated to achieve FMTt = 0 nN, releasing the superparamagnetic bead from the needle tip. The exact same sequence of events was then carried out with the microscope configured for fast time-lapse epifluorescence imaging. More specific procedural details for this model DC mode MT-DTM/TFM experiment can be found in Sec. SI.B.1 of the supplementary material. For a DC mode null displacement control experiment (with respect to collagen substrate motion), the needle was positioned near the same superparamagnetic bead of interest such that δ < 1 µm. The MT device was then actuated with FMTt = 0 nN, i.e., a null magnetic clamping force. Serial fast time-lapse image acquisitions at 40 fps (2048 × 2044 pixels2/frame) were then performed in DIC and epifluorescence imaging modes, while the needle tip was translated to and from the same maximum displacement of 10.0 µm, exactly as done in the model bead-on-gel DC mode experiment. The substrate remained static and undeformed throughout the duration of the DC mode null displacement control experiment.

Following data collection, all image data were analyzed according to the methods presented in Sec. SI.B.2 of the supplementary material. Real-time videos of the DIC image data (full field and zoom) and the epifluorescence image data (full field) and videos for ux*,y*,t, Tx*,y*,t, and ρx*,y*,t for the model bead-on-gel DC mode MT-DTM/TFM experiment are presented as Vid.11, Vid. 12, Vid. 13, Vid. 14, Vid. 15, and Vid. 16, respectively, of the supplementary material, the latter three videos computed assuming E = 52.73 Pa and υ = 0.4. The analogous set of videos for the DC mode null displacement control experiment are also presented as Vid. 17, Vid. 18, Vid. 19, Vid. 20, Vid. 21, and Vid. 22, respectively, of the supplementary material, where the latter three videos were also computed assuming E = 52.73 Pa and υ = 0.4. Quantitative determination of imaging intervals where microsphere dynamics might preclude implementation of our elastostatic TFM analysis was not done for this DC mode experiment, as was previously shown for the model bead-on-gel FC mode experiment (see Sec. IV A 5). Although our DC mode null displacement control experiment provides a dataset amenable to such analysis, identification of these time intervals does not offer any new insight into the understanding of this potential limitation of our integrated MT-DTM/TFM methodology.

2. Temporal alignment of DIC and epifluorescence image sets

In contrast to an FC mode experiment, the model bead-on-gel DC mode experiment described here does not employ a programmed and measured magnetic actuation waveform that can be used to temporally correlate data from the serially acquired DIC and epifluorescence image sets. Translation of the needle tip was manually initiated for both imaging sequences and did not necessarily start at the same raw time point during each respective fast time-lapse image acquisition. As such, we chose to correlate the independent datasets by temporally aligning the first significant microsphere displacement observed in the epifluorescence imaging sequence to the first significant superparamagnetic bead displacement observed for the DIC imaging sequence. Experimentally significant displacements were determined via analysis of our DC mode null displacement control experiments, which can be found in the supplementary material (Figs. S5 and S6). In brief, we first searched the epifluorescence dataset to find the raw time point, tEPI, at which Δmaxxmax*,ymax*,tr first becomes ≥0.15 µm, i.e., the raw time, tr, marking the first physically significant microsphere displacement measurement (see Fig. S6). We then determined tDIC, the first time point at which ΔMTt 0.28 µm in the DIC imaging data, i.e., the time marking the first physically significant superparamagnetic bead displacement (see Fig. S5). Knowing tDIC and tEPI, we temporally shifted the entire epifluorescence dataset, re-defining the temporal variable for this dataset, t, as t = tr + (tDICtEPI). With this alignment of the epifluorescence and DIC image data, the first experimentally significant microsphere and superparamagnetic bead displacements are temporally coincident. For the model bead-on-gel DC mode experiment shown here, tEPI = 2.652 78 s, while tDIC = 5.330 58 s. Time stamps shown in Vid. 13, Vid. 14, Vid. 15, and Vid. 16 of the supplementary material have all been shifted by 2.6778s.

The alignment of the datasets in this manner assumes that superparamagnetic bead and substrate microsphere displacements are temporally coincident, although this is not known to be true a priori. However, in our FC mode experiments, ΔMT and Δmax were observed to be temporally coincident despite differences in the observed magnitudes of these displacements [see Fig. 2(e)]. Thus, we believe this to be a reasonable assumption for analyses presented here. In future DC mode experiments, we plan to use a piezo nanopositioner to control the translation of the needle tip, similar to the setup described for the MT-based pick and place magnetic particle assembly.77 DIC and epifluorescence imaging data will then be temporally correlated by measurements of the actuator’s position acquired in parallel with the TTL pulses marking exposures of the sCMOS camera during each respective DIC or epifluorescence fast time-lapse imaging sequence.

3. Bead kinematics, substrate displacements, and incremental traction stresses

Figure 5 highlights the key findings from the model bead-on-gel DC mode experiment. A similar figure for the DC mode null displacement control experiment and a graphical depiction of our computation of the L2 regularization parameters, λ2ON and λ2OFF, for the model bead-on-gel DC mode experiment detailed in Fig. 5 can be found in Figs. S7 and S8, respectively, of the supplementary material. In comparing the displacement of the superparamagnetic bead, ΔMTt, and the displacement of the microsphere subject to the largest displacement during translation of the needle tip, Δmaxt, we first note that the datasets are not perfectly correlated in time [see Fig. 5(e)]. We attribute this to the fact that displacement control was achieved via manual translation of the needle tip using our three-axis micromanipulator. With manual control, the translation step and hold times were not identical for the DIC and epifluorescence image acquisition sequences. Despite this fact, several important findings are still evident from the coupled datasets. Foremost, we see a discrepancy between the maximum of ΔMTt (10.3 µm) and the maximum of Δmaxt (8.9 µm) observed during the DC experiment. Similar to our bead-on-gel FC mode experiment (see Sec. IV A 2), the difference between maxima in ΔMT and Δmax (∼1.4 µm) is less than the radius of the superparamagnetic bead. As previously hypothesized for our FC mode experiments, we also speculate that the superparamagnetic bead can both rotate and translate during the application of the magnetic clamping force exerted by the MT device. Relative amounts of rotation and translation are likely dependent on the state of conformational contact between the bead and the collagen substrate. Another possibility to consider is that fibronectin coating of the superparamagnetic bead results in the formation of short fibronectin fibrils.78 As the bead is displaced, stretching of fibronectin fibrils could potentially account for some of the discrepancy between ΔMT and Δmax, depending on the relative mechanical compliances of collagen the fibronectin fibrils.

FIG. 5.

Panels (a)–(f) highlight data from a model bead-on-gel DC mode MT-DTM/TFM experiment involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the surface of a ∼700 μm-thick, 1.0 mg/ml type I collagen substrate containing a surface layer of covalently attached red fluorescent microspheres. (a) DIC image showing the superparamagnetic bead near its peak displacement following a ≈10 µm translation of the needle tip. The inset image shows a magnified view of the region denoted by the central red dashed square box. (b) Texas Red epifluorescence image with overlaid quivers representing the displacement vectors of individual microspheres at the same time point. Small dashed boxes in the corners of (a) and (b) denote the ROIs used for drift correction. The displacement field, ux*,y*, and incremental traction stress field, Tx*,y*, computed for the microsphere displacement field displayed in (b), are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. A plot of the displacement of the microsphere subject to the largest displacement during translation of the needle tip, Δmaxt, vs superparamagnetic bead displacement, ΔMTt, is shown in (e). A plot of the magnitude of the integrated total traction force vector, Ft, is shown in (f), where Ft has been computed assuming υ = 0.4 and E = 52.73 Pa. The asterisks (*) shown in (e) and (f) denote the time points at which the magnetic clamping force was removed, releasing the bead from the needle tip.

FIG. 5.

Panels (a)–(f) highlight data from a model bead-on-gel DC mode MT-DTM/TFM experiment involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the surface of a ∼700 μm-thick, 1.0 mg/ml type I collagen substrate containing a surface layer of covalently attached red fluorescent microspheres. (a) DIC image showing the superparamagnetic bead near its peak displacement following a ≈10 µm translation of the needle tip. The inset image shows a magnified view of the region denoted by the central red dashed square box. (b) Texas Red epifluorescence image with overlaid quivers representing the displacement vectors of individual microspheres at the same time point. Small dashed boxes in the corners of (a) and (b) denote the ROIs used for drift correction. The displacement field, ux*,y*, and incremental traction stress field, Tx*,y*, computed for the microsphere displacement field displayed in (b), are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. A plot of the displacement of the microsphere subject to the largest displacement during translation of the needle tip, Δmaxt, vs superparamagnetic bead displacement, ΔMTt, is shown in (e). A plot of the magnitude of the integrated total traction force vector, Ft, is shown in (f), where Ft has been computed assuming υ = 0.4 and E = 52.73 Pa. The asterisks (*) shown in (e) and (f) denote the time points at which the magnetic clamping force was removed, releasing the bead from the needle tip.

Close modal

Although not seen in this experiment, in other trial bead-on-gel DC mode experiments, we did observe (with DIC imaging) relative sliding contact motion between the superparamagnetic bead and the needle tip that occurred during the translation of the needle tip away from its initial position following the application of the magnetic force clamp. Thus, it is possible that relative bead-tip motion contributed to the non-uniform initial displacement steps in Δmaxt that can be observed for 5 s < t < 13 s in Fig. 5(e). However, as the substrate becomes increasingly deformed, the magnitude of displacement steps in Δmaxt and ΔMTt become nearly identical, which suggests that no further relative motion occurs between the superparamagnetic bead and the needle tip. As shown in Fig. 5(e), small residual step displacements in both Δmaxt and ΔMTt are observed following demagnetization of the needle tip and removal of the clamping force at t ≈ 37 s [also, see Fig. 5(f)]. Assuming that the collagen behaves as a linear elastic substrate, this displacement could be accounted for by hysteresis in the micromanipulator used to translate the needle tip. Alternatively, assuming no hysteresis in the translation of the needle tip, residual step displacements might be attributed to the viscous and/or plastic constitutive material properties of the collagen substrate in response to the applied deformation cycle.

With regard to our continuum models of the measured displacement field and computed incremental traction stress fields, note that although three superparamagnetic beads were present within the field of view for this bead-on-gel DC mode experiment [see Fig. 5(a)], substrate deformations and incremental traction stress were localized only to the near field of the bead that was magnetically clamped to the needle tip, as shown in Figs. 5(c) and 5(d), respectively. Due to the needle’s concentrated magnetic field gradients, our MT device possesses the ability to selectively clamp a specific bead of interest without introducing forces on neighboring superparamagnetic beads positioned >385 µm away from the needle tip. Smaller neighborhoods may be possible, but the exact spatial selectivity of our MT device was not further quantified in this work.

With that understanding, compare the maximum incremental traction stress generated in our model bead-on-gel DC mode experiment, ≈6.34 Pa [as shown in Fig. 5(d)], to the maximum incremental traction stress generated in our model bead-on-gel FC mode experiment, ≈18.2 Pa [as shown in Fig. 2(d)]. Admittedly, this result is physically inconsistent with what one would expect given that the deformation in the DC mode experiment was much larger than that observed for the FC mode experiment, even considering the known differences in apparent elastic moduli of the collagen substrates used for these two independent experiments (E = 52.73 Pa for the DC mode experiment, whereas E = 76.23 Pa for the FC mode experiment). However, it is important to recognize that our solution for the incremental traction stress field was unconstrained, meaning that we did not restrict solutions for the incremental traction stress field to consist only of tractions localized to within the area of bead–substrate contact. By happenstance, the peak incremental traction stress solution for the FC mode experiment predicted a small localized incremental traction stress field confined to a 4.5 μm-diameter circular area, roughly the same size of a superparamagnetic bead. In contrast, the peak incremental traction stress solution for the DC mode experiment predicted an incremental traction stress field localized to a linear streak-like area with a length of ∼3 superparamagnetic bead diameters [see Fig. 5(d)]. In future experiments, we will consider adopting an iterative solution scheme that constrains incremental traction stresses to an area of presumed conformational contact between the collagen substrate and the superparamagnetic bead, analogous to the methodology originally described by Butler et al. for constraining cellular focal adhesion tractions to spatial locations present within the projected boundary of cell.79 

4. Bead-tip clamping forces

As a final discussion point, we note that our model bead-on-gel DC mode experiments tend to fail by one of three mechanisms: (i) mechanical fracture within the collagen substrate, (ii) detachment of the superparamagnetic bead from the collagen substrate, or (iii) the development of incremental traction stresses within the substrate that ultimately exceeded the magnetic clamping force holding the bead to the needle tip. Fracture of the collagen substrate was only observed for extremely large deformations of soft gels with collagen concentrations of ≤1 mg/ml. When fracture occurred, grossly visible amounts of collagen debris could be observed sticking to the superparamagnetic bead. For our fibronectin-coated beads (see Sec. III D), bead detachment from the collagen substrate proved to be a rare event when beads were incubated for >12 h at 37 °C to allow sufficient time to establish conformal contact via hydrophobic adhesive interactions between fibronectin and collagen. Consequently, bead-tip separation represented the most common mode of experimental failure.

As shown in Fig. 5(f), the total traction force vector, F, reached a maximum of ∼36.2 nN during this model bead-on-gel DC mode experiment, suggesting a bead-tip clamping force, at least, of the same magnitude. In other bead-on-gel DC mode experiments using 1.0 mg/ml collagen gels, our TFM analysis suggested F = 43.0 nN for peak collagen displacements of Δmax ≈ 11.0 µm. As such, we believe that our MT device operates with a bead-tip clamping force of at least 40 nN. Although maximum displacements of ΔMT = 20.0 µm were possible with 1.0 mg/ml gels, unfortunately, tracking of substrate microspheres for these experiments proved to be exceedingly difficult. Moreover, even if we were able to track substrate microspheres for these large displacement experiments, the finite deformations resultant in the collagen gel would preclude accurate modeling of incremental traction stresses using our TFM code, given that our computational solution for T is based on a theory of infinitesimal elastic deformations. In future work, we plan to explore alternative needle tip geometries and different preparations of both collagen gels and superparamagnetic and/or ferromagnetic beads in an attempt to better define the maximum range of bead-tip clamping forces that characterize the DC mode of operation for our integrated MT-DT/TFM methodology.

In this section, we present a proof-of-concept, bead-on-cell demonstration of our MT-DTM/TFM methodology. Toward this end, we first establish our ability to reconstitute 2D planar multicellular keratinocyte aggregates on type I collagen gel substrates cultured under both low and high [Ca2+] conditions, making several important observations regarding changes in the cell morphology and cell–substrate interactions that occur as keratinocytes form an epithelial sheet. Subsequently, we perform a bead-on-cell FC mode MT-DTM/TFM experiment that interrogates force transmission across focal adhesion contacts within a single keratinocyte present at the peripheral margin of a multicellular aggregate reconstituted under low [Ca2+] conditions. Interrogation of a single keratinocyte is an intentional first step toward validation of the methodology for more advanced explorations of keratinocyte mechanobiology. In limiting force transmission to a specific type of anchoring junction within an isolated cell, conceptually, it is easier to reconcile our measured data to expected experimental outcomes.

Type I collagen gel substrates, 2.0 mg/ml, ∼500 to 600 μm-thick, and embedded with a surface layer of fluorescent microspheres, were prepared as previously described (see Sec. III B). Two-dimensional planar multicellular aggregates of primary normal human epidermal keratinocytes were reconstituted on these collagen substrates and cultured with either low [Ca2+] or high [Ca2+] KSFM for ∼24 h (see Sec. III C). Phase contrast images of the keratinocyte sheets under low and high [Ca2+] conditions and epifluorescence images of their respective underlying fluorescent microsphere distributions are shown in Fig. 6. Keratinocytes adherent to 2.0 mg/ml gels were found to survive for at least 72 h in culture, the time at which our experiments were terminated. As others have observed, keratinocytes grown on gels with collagen concentrations ≤1.0 mg/ml did not exhibit normal cell viability.51 With regard to cell morphology, keratinocytes cultured on our relatively soft 2.0 mg/ml collagen cells under low [Ca2+] conditions manifest altered morphologies compared to cells cultured on tissue culture-treated polystyrene. Specifically, compared to tissue culture-treated polystyrene, keratinocytes adherent to our collagen substrates exhibited smaller spread areas and increased numbers of spindled morphologies, the latter conspicuously observed for isolated keratinocytes localized to the peripheral margin of the aggregate [see Fig. 6(a)]. Similar spindled keratinocyte morphologies have also been observed for keratinocytes cultured under low [Ca2+] conditions on soft polyacrylamide gels.23,24 Spindled morphologies were notably absent from the peripheral margins of multicellular aggregates reconstituted under high [Ca2+] conditions [see Fig. 6(c)], likely due to the fact that these once isolated cells have become incorporated into the evolving epithelial sheet. For both low and high [Ca2+] culture conditions, keratinocytes typically adopted more polygonal forms within central areas of the aggregates [see Figs. 6(a) and 6(c)].

FIG. 6.

Paired phase contrast and Texas Red epifluorescence images of select areas of reconstituted multicellular keratinocyte aggregates cultured for ∼24 h under low [Ca2+] [(a) and (b)] or high [Ca2+] [(c) and (d)] conditions. The white dashed lines in each image crudely demarcate the peripheral margin of each multicellular aggregate. Magnified views of the ROIs delineated by white solid boxes in (a)–(d) are shown directly below each main phase contrast or epifluorescence image. ROIs that are more centrally located within the aggregate are labeled “C,” whereas ROIs that are located in closer proximity to the peripheral margin of the aggregate are labeled “P.” Phase contrast images permit the observation of individual keratinocyte morphology, whereas epifluorescence images reveal mechanical cell–substrate interactions as visualized by changes in the microsphere distributions present within the underlying 2.0 mg/ml type I collagen substrate. Note the presence of spindled cells located at the peripheral margin of the aggregate reconstituted at low [Ca2+] [white asterisk in (a)] vs the absence of such cells for the aggregate cultured under high [Ca2+] conditions (c). Within central areas of aggregates cultured at both high and low [Ca2+], cells typically adopted polygonal forms. With a low [Ca2+] culture, the size and density of microsphere footprints are roughly independent of the keratinocyte location within the aggregate (b). In contrast, with a high [Ca2+] culture, an intensification of microsphere footprints develops at the peripheral margin of the evolving epithelial sheet (d).

FIG. 6.

Paired phase contrast and Texas Red epifluorescence images of select areas of reconstituted multicellular keratinocyte aggregates cultured for ∼24 h under low [Ca2+] [(a) and (b)] or high [Ca2+] [(c) and (d)] conditions. The white dashed lines in each image crudely demarcate the peripheral margin of each multicellular aggregate. Magnified views of the ROIs delineated by white solid boxes in (a)–(d) are shown directly below each main phase contrast or epifluorescence image. ROIs that are more centrally located within the aggregate are labeled “C,” whereas ROIs that are located in closer proximity to the peripheral margin of the aggregate are labeled “P.” Phase contrast images permit the observation of individual keratinocyte morphology, whereas epifluorescence images reveal mechanical cell–substrate interactions as visualized by changes in the microsphere distributions present within the underlying 2.0 mg/ml type I collagen substrate. Note the presence of spindled cells located at the peripheral margin of the aggregate reconstituted at low [Ca2+] [white asterisk in (a)] vs the absence of such cells for the aggregate cultured under high [Ca2+] conditions (c). Within central areas of aggregates cultured at both high and low [Ca2+], cells typically adopted polygonal forms. With a low [Ca2+] culture, the size and density of microsphere footprints are roughly independent of the keratinocyte location within the aggregate (b). In contrast, with a high [Ca2+] culture, an intensification of microsphere footprints develops at the peripheral margin of the evolving epithelial sheet (d).

Close modal

Under low [Ca2+] conditions, keratinocytes within a multicellular aggregate lack adherens junctions, desmosomes, and hemidesmosomes.2,23,52–56 As shown in Fig. 6(b), individual keratinocytes within the multicellular aggregates cultured at low [Ca2+] conditions—mechanically coupled to the underlying substrate via focal adhesion contacts—each forms a distinct cell-sized zone of increased microsphere density in their immediate subjacent collagen substrate, referred to here as a microsphere footprint. Whether or not this keratinocyte-organized microsphere footprint reflects true re-organization and compaction of collagen fibrils within the substrate remains an open question subject to further investigation. Regardless, under low [Ca2+] conditions, microsphere footprints of individual cells exhibited similar sizes and densities that were independent of the keratinocyte location within a multicellular aggregate, i.e., similar footprints were noted for cells located within the center or near the peripheral margin of the aggregate [see Fig. 6(b)]. In stark contrast, keratinocytes within an aggregate that formed an epithelial sheet (due to culture under high [Ca2+] conditions) exhibited a dramatic intensification of microsphere footprints within cells present at the peripheral margin of the sheet [see Fig. 6(d)]. Qualitatively, these data suggest that under low [Ca2+] conditions, keratinocytes within a multicellular aggregate generate independent cellular traction stresses to maintain adhesion to the underlying substrate. However, following the exposure to high [Ca2+] conditions—with keratinocytes mechanically coupled to one another via both adherens junctions and desmosomes2,23,52–56—cellular traction stresses become cooperatively localized to the peripheral margin of the epithelial sheet. Collectively, these observations are congruent with traction stress measurements and modeling that have previously been reported for cultures of small colonies of murine epidermal keratinocytes when transitioned from low to high [Ca2+] conditions.32,80

One multicellular keratinocyte aggregate cultured in low [Ca2+] medium was selected for further interrogation by means of a model bead-on-cell FC mode MT-DTM/TFM experiment. Fibronectin-coated superparamagnetic beads were allowed to attach, at random, to cells and substrate within this specimen using the protocol described in Sec. III D. Next, a superparamagnetic bead that had attached directly to the collagen substrate within a cell-free area distant to the multicellular aggregate was subjected to an FC mode MT-DTM/TFM experiment exactly as described in Sec. IV A 1. Using data collected from this experiment, we quantified an apparent elastic modulus of E = 138.3 Pa (assuming υ = 0.4) for this 530 μm-thick 2.0 mg/ml collagen substrate, employing the analysis described in Sec. IV A 3. Following the determination of the substrate modulus, a superparamagnetic bead that had attached to the apical surface of a mechanically isolated keratinocyte present at the peripheral margin of the multicellular aggregate was identified. In a bead-on-gel demonstration of the methodology of MT-DTM/TFM, we subjected this bead to three sequential three-cycle, 4 s/7 s, ON/OFF FC mode actuation sequences with maximum nominal ON magnetic flux density setpoints of 175 Gs, exactly as described in Sec. SI.A.1 of the supplementary material. No imaging was performed during the first three-cycle actuation sequence, whereas fast time-lapse DIC and epifluorescence imaging were used to observe the second and third actuation sequences, respectively. A tiled phase contrast image of the multicellular keratinocyte aggregate and the location of the specific keratinocyte interrogated during this experiment can be found in the supplementary material (see Fig. S9). The initial xyz-spatial location of the superparamagnetic bead relative to the needle tip of the MT device for this experiment was x0 = 11.0 µm, y0 = 0 µm, and z0 = 13.0 µm [δt0 ≈ 17.0 µm]. Image data from the second and third actuation sequences were analyzed according to the methods presented in Sec. SI.A.2 of the supplementary material. Real-time videos of the DIC and epifluorescence imaging data for this bead-on-cell experiment are presented as Vid. 23 and Vid. 24, respectively, of the supplementary material. Videos for ux*,y*,t, Tx*,y*,t, and ρx*,y*,t are presented as Vid. 25, Vid. 26, and Vid. 27, respectively, of the supplementary material, all computed assuming that E = 138.3 Pa and υ = 0.4. A graphical determination of the L2 regularization parameters used for the solution of T and ρ in this experiment can also be found in the supplementary material (see Fig. S10). Although possible, determination of imaging intervals where microsphere dynamics prevent implementation of an elastostatic TFM analysis was not done for this experiment, as was previously shown for the model bead-on-gel FC mode experiment (see Sec. IV A 5).

Important findings from this bead-on-cell demonstration investigating force transmission through a keratinocyte cultured under low [Ca2+] conditions are graphically summarized in Fig. 7. Here, Figs. 7(a)7(d) represent the DIC image, epifluorescence image, ux*,y*, and Tx*,y*, respectively, captured or calculated at tOFF for cycle 3 (t ≈ 27.14 s) of the magnetic actuation waveform sequence. Magnified views of u and T within the white dashed boxes shown in Figs. 7(c) and 7(d), respectively, are shown in Fig. 8. In the following discussion, it is extremely important to note that u measured for this bead-on-cell experiment represent substrate displacements that develop in response to the magnetic actuation force applied to the superparamagnetic bead attached to the apical surface of keratinocyte #1 [see Fig. 7(a)]. In other words, substrate displacements, u, are defined with respect to a reference state that consists of a collagen substrate that has already been mechanically deformed by the overlying cell layer, not an undeformed substrate. Consequently, in this analysis, T represents the incremental traction stress field that develops in response to the applied magnetic actuation force, not the active traction stresses generated by cells when establishing conformal adhesive contact with the underlying collagen substrate.

FIG. 7.

Panels (a)–(f) highlight data from a bead-on-cell FC mode MT-DTM/TFM demonstration involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the apical surface of normal human epidermal keratinocyte that is adherent to a ∼530 μm-thick, 2.0 mg/ml, type I collagen substrate containing a surface layer of covalently attached red fluorescent microspheres. The keratinocyte is located at the peripheral edge of a multicellular aggregate reconstituted under low [Ca2+] conditions (see Fig. S9 of the supplementary material). (a) DIC image showing the superparamagnetic bead at its peak displacement near tOFF of actuation cycle 3. (b) Texas Red epifluorescence image with quivers that represent the displacement vectors of individual microspheres within the surface of the collagen substrate at the same time point. The displacement field, ux*,y*, and the corresponding incremental traction stress field, Tx*,y*, are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. The white dashed square boxes in (c) and (d) denote the area of the collagen substrate subjacent to keratinocyte #1. Magnified views of this region are presented in Fig. 8. A plot of the displacement of the microsphere subject to the largest displacement during magnetic actuation, Δmaxt, vs superparamagnetic bead displacement during magnetic actuation, ΔMTt, is shown in (e). A plot of the magnitude of the component of the magnetic actuation force vector in the xy-plane of the collagen substrate, FMTt, vs the magnitude of the integrated total traction force vector, Ft, is shown in (f). Both Tx*,y*,t and Ft were computed assuming υ = 0.4 and E = 138.3 Pa. Black triangles in (e) and (f) denote tON and tOFF for each actuation cycle.

FIG. 7.

Panels (a)–(f) highlight data from a bead-on-cell FC mode MT-DTM/TFM demonstration involving a fibronectin-coated 4.5 μm-diameter superparamagnetic bead attached to the apical surface of normal human epidermal keratinocyte that is adherent to a ∼530 μm-thick, 2.0 mg/ml, type I collagen substrate containing a surface layer of covalently attached red fluorescent microspheres. The keratinocyte is located at the peripheral edge of a multicellular aggregate reconstituted under low [Ca2+] conditions (see Fig. S9 of the supplementary material). (a) DIC image showing the superparamagnetic bead at its peak displacement near tOFF of actuation cycle 3. (b) Texas Red epifluorescence image with quivers that represent the displacement vectors of individual microspheres within the surface of the collagen substrate at the same time point. The displacement field, ux*,y*, and the corresponding incremental traction stress field, Tx*,y*, are shown in (c) and (d), respectively, where the white dashed circle in (d) represents a ∼4.5 μm-diameter circle. The white dashed square boxes in (c) and (d) denote the area of the collagen substrate subjacent to keratinocyte #1. Magnified views of this region are presented in Fig. 8. A plot of the displacement of the microsphere subject to the largest displacement during magnetic actuation, Δmaxt, vs superparamagnetic bead displacement during magnetic actuation, ΔMTt, is shown in (e). A plot of the magnitude of the component of the magnetic actuation force vector in the xy-plane of the collagen substrate, FMTt, vs the magnitude of the integrated total traction force vector, Ft, is shown in (f). Both Tx*,y*,t and Ft were computed assuming υ = 0.4 and E = 138.3 Pa. Black triangles in (e) and (f) denote tON and tOFF for each actuation cycle.

Close modal
FIG. 8.

Images depict magnified views of (a) the displacement field, ux*,y*, and [(b) and (c)] the incremental traction stress field, Tx*,y*, for the area of collagen substrate subjacent to keratinocyte #1 near tOFF of actuation cycle 3 for the bead-on-cell FC mode MT-DTM/TFM demonstration detailed in Fig. 7 (t = 27.139 s). For ease of visualizing the local directionality of the incremental traction stress vector field, traction quivers external to the projected keratinocyte boundary have been removed in (b), whereas quivers within the area of the substrate subjacent to the cell boundary have been omitted in (c). Scale bars = 10 µm.

FIG. 8.

Images depict magnified views of (a) the displacement field, ux*,y*, and [(b) and (c)] the incremental traction stress field, Tx*,y*, for the area of collagen substrate subjacent to keratinocyte #1 near tOFF of actuation cycle 3 for the bead-on-cell FC mode MT-DTM/TFM demonstration detailed in Fig. 7 (t = 27.139 s). For ease of visualizing the local directionality of the incremental traction stress vector field, traction quivers external to the projected keratinocyte boundary have been removed in (b), whereas quivers within the area of the substrate subjacent to the cell boundary have been omitted in (c). Scale bars = 10 µm.

Close modal

As shown in Fig. 7(a), there are four keratinocytes present within the field of view for this demonstration, with keratinocyte #1 having the fibronectin-coated superparamagnetic bead attached to its apical surface. Present at the periphery of the reconstituted multicellular aggregate, all keratinocytes possess a somewhat spindled morphology. Keratinocyte #3 is in close apposition to keratinocyte #1, and although unlikely, it is possible that their cell membranes are in direct physical contact. As clearly visible in Vid. 24 of the supplementary material, one can appreciate that each of these four keratinocytes has created a microsphere footprint in the subjacent collagen substrate prior to the onset of magnetic actuation. Now, observe in both Fig. 7(b) and Vid. 24 that how only keratinocyte #1 develops obvious displacements in its subjacent collagen substrate in response to magnetic actuation of the superparamagnetic bead. Perhaps more clearly evident in ux*,y*,t, during magnetic actuation (see Vid. 25 of the supplementary material), the subjacent collagen develops several focal peaks in displacement that are only localized to substrate areas within the projected boundary of keratinocyte #1. As evident in Fig. 7(c), no significant displacements are observed in the collagen substrate beneath keratinocytes #2, #3, and #4. When comparing the displacement field that develops during this bead-on-cell demonstration to the displacement field of the model bead-on-gel experiment, note how the latter exhibits a single local displacement peak that monotonically decays out into the surrounding substrate [see Fig. 2(c)]. In contrast, the displacement field for the bead-on-cell demonstration shown in Fig. 8(a), although continuous, exhibits a very sharp gradient in ux*,y* as displacements vary from 0 µm in the substrate immediately adjacent to the cell border but then abruptly rise to ∼1 µm within the collagen subjacent to the cell.

With regard to the mechanical response of the superparamagnetic bead, ΔMTt, we see more evidence of an overall viscoelastic response of the bead in the bead-on-cell demonstration compared to the bead-on-gel experiment. With a cycle 1 peak displacement of ∼1.35 µm [Fig. 7(e)] in response to a ≈3.45 nN load [Fig. 7(f)], the data for this bead-on-cell keratinocyte demonstration are both qualitatively and quantitatively similar to the mechanical response observed for a 4.5 μm-diameter fibronectin-coated dynabead that has been attached to a murine embryonic fibroblast and actuated with an MT device.81 When comparing ΔMTt to Δmaxt, we note that the observed difference between ΔMTtOFF and ΔmaxtOFF for each of the three actuation cycles varies from ≈0.4 to 0.5 µm [Fig. 7(e)]. However, ΔMTt and Δmaxt are temporally coincident without any observable phase shift between the two displacement signals, which is indicative of the elastic mechanical coupling between the bead and the underlying collagen substrate. Collectively, our findings strongly suggest that in this bead-on-cell FC mode MT-DTM/TFM demonstration, the force applied to the superparamagnetic bead during magnetic actuation was transmitted from the apical surface of the keratinocyte through the cytoskeleton and ultimately to focal adhesion contacts linking the cell to the subjacent collagen substrate. Evidence of cell–cell force transmission is lacking. Although not done here, bead-on-cell FC mode MT-DTM/TFM experiments could be used to test specific hypotheses of mechanical force transmission in keratinocytes, aided by the use of cytoskeletal network-disrupting agents (cytochalasin D, colchicine, and acrylamide), activators or inhibitors of the actomyosin contractile machinery (isoproterenol, histamine, and Y27632), and/or antibodies targeting extracellular adhesive epitopes within the α2β1 integrin heterodimers present in the focal adhesion contacts coupling the cell to the collagen substrate. Such experiments are the subject of future investigation.

As expected from the displacement field shown in Fig. 8(a), the solution for Tx*,y*,t=27.139s exhibits focal incremental traction stresses within the collagen subjacent to keratinocyte #1, which are directionally correlated with the underlying displacement field [see Figs. 8(a) and 8(b)]. However, these tractions are also continuous with an annular zone of ≈0 Pa incremental traction stresses that are surrounded by an outer zone of incremental traction stresses with a directionality opposite to the central tractions [see Fig. 8(c)]. Sometimes referred to as “ringing” in TFM, this finding in our solution for T is a mathematical consequence of the input displacement field regardless of whether the solution was computed using FTTC or the boundary element method (see Fig. S11 of the supplementary material).60 Low pass filters in Fourier space are often used in FTTC-based approaches to TFM to remove high frequency noise from the gridded displacement data.59 However, in this instance, ringing in the computed solution for T is a consequence of an actual measured step-like gradient in u, and therefore, filtering out the high-frequency content of u would not be appropriate.

As a final point of discussion, note that in the model bead-on-gel FC mode experiment (see Sec. IV A 3), the collagen substrate was shown to exhibit mechanical properties that approximate a spatially homogeneous linear elastic solid. However, in our bead-on-cell demonstration, individual keratinocytes adherent to the gel have presumably re-organized the fibrillar structure of the collagen, as evidenced by the increased microsphere density in the substrate subjacent to the cells. As such, we postulate that the displacement gradient observed in our bead-on-cell demonstration developed as a consequence of localized incremental traction stresses applied to a fibrillar collagen substrate with constitutive mechanical properties that are spatially heterogeneous. Practically speaking, collagen substrates that have undergone significant remodeling by adherent cells might no longer behave as a spatially homogeneous linear elastic material, invalidating the assumptions inherent to our TFM-based incremental traction stress solutions. However, until higher-order TFM models are developed that permit substrates with spatially heterogeneous constitutive mechanical properties, to a first-order approximation, we propose that our solutions for T stand as semiquantitative observations that can still provide potentially useful insight into future investigations of force transmission in multicellular keratinocyte constructs.

Motivated by the need to explore unanswered questions in the field keratinocyte mechanobiology, we have introduced a new experimental methodology that integrates the techniques of magnetic tweezers (MT) with substrate displacement/deformation tracking microscopy (DTM) and traction force microscopy (TFM). The experimental protocol and methods of data reduction that serve as the basis of the methodology were presented via two conceptually tractable model bead-on-gel experiments referred to as force-control (FC) mode and displacement-control (DC) mode MT-DTM/TFM experiments. In the model bead-on-gel FC mode experiment, we showed how the integrated methodology could be used to characterize the elastic storage modulus of a type I collagen gel substrate using quantitative approaches based on TFM analysis. Because FC mode experiments are inherently dynamic, an ad hoc empirical criterion was established to verify the existence of elastostatic conditions within the acquired data to justify the use of TFM analysis. In a model bead-on-gel DC mode MT-DTM/TFM experiment, we substantiated the idea that the integrated methodology can be implemented in experiments requiring either large substrate deformations (>5 µm) and/or large forces (>5 nN). Important considerations for DC mode experiments were identified, including a classification of bead-tip clamp failure mechanisms and estimation of maximum bead-tip magnetic clamping forces. Finally, in a proof-of-concept bead-on-cell FC mode MT-DTM/TFM demonstration, we applied a defined magnetic actuation force to a fibronectin-coated superparamagnetic bead attached to a mechanically isolated keratinocyte present at the periphery of a multicellular aggregate reconstituted under low [Ca2+] conditions. Both the measured substrate displacement fields and the calculated incremental traction stress fields observed for this bead-on-cell experiment suggest that a force applied to the apical surface of the keratinocyte is directly transmitted through the cell to its underlying substrate. Collectively, the model FC mode and DC mode bead-on-gel experiments, the FC mode bead-on-cell demonstration, and the rigorous approaches to data reduction presented in this work serve to establish the potential utility of the integrated methodology for more widespread application in future explorations of matrix rheology and keratinocyte mechanobiology in the context of immunobullous skin diseases.

See the supplementary material for a comprehensive list of abbreviations and symbols used in this work, detailed narratives of the experimental procedures, methods of data reduction developed for our model bead-on-gel FC mode and DC mode MT-DTM/TFM experiments, sample magnetic actuation waveform data for FC mode experiments, analysis of our FC mode null force control and DC mode null displacement control experiments, data utilized for L2 regularization parameter determination in our FC and DC mode experiments, and a tiled and stitched phase contrast image showing the reconstituted multicellular keratinocyte aggregate used for the bead-on-cell FC mode MT-DTM/TFM demonstration detailed in Sec. V.

The authors acknowledge Dan Witte and George DeBeck (Nikon Instruments, Inc.) for their expert technical assistance with the setup and operation of our microscope. J.C.S. thanks the Dermatology Foundation for its generous physician scientist career development award. E.A.S. recognizes the National Science Foundation (Grant No. CAREER 1452728).

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