Dynamic Contact Guidance of Myoblasts by Feature Size and

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Biological and Environmental Phenomena at the Interface

Dynamic Contact Guidance of Myoblasts by Feature Size and Reversible Switching of Substrate Topography: Orchestration of Cell Shape, Orientation and Nematic Ordering of Actin Cytoskeletons Philipp Linke, Ryo Suzuki, Akihisa Yamamoto, Masaki Nakahata, Mineko Kengaku, Takahiro Fujiwara, Takuya Ohzono, and Motomu Tanaka Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02972 • Publication Date (Web): 30 Oct 2018 Downloaded from http://pubs.acs.org on October 31, 2018

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Dynamic Contact Guidance of Myoblasts by Feature Size and Reversible Switching of Substrate Topography: Orchestration of Cell Shape, Orientation and Nematic Ordering of Actin Cytoskeletons

Philipp Linke1,2, Ryo Suzuki1, Akihisa Yamamoto1, Masaki Nakahata3, Mineko Kengaku4, Takahiro Fujiwara4, Takuya Ohzono5*, and Motomu Tanaka1,2*

1

Center for Integrative Medicine and Physics, Institute for Advanced Study, Kyoto University, 606-

8501 Kyoto, Japan

2

Physical Chemistry of Biosystems, Institute of Physical Chemistry, Heidelberg University, D69120

Heidelberg, Germany

3

Department of Material Engineering Science, Graduate School of Engineering Science, Osaka

University, 560-8531 Osaka, Japan

4

Institute for Integrated Cell-Material Sciences, Kyoto University, 606-8501 Kyoto, Japan

5

Electronics and Photonics Research Institute, National Institute for Advanced Industrial Science and

Technology, 305-8505 Tsukuba, Japan

Corresponding authors: [email protected], [email protected]

1 Environment ACS Paragon Plus

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Abstract Biological cells in tissues alter their shapes, positions, and orientations in response to dynamic changes in their physical microenvironments. Here, we investigated the dynamic response of myoblast cells by fabricating substrates displaying microwrinkles that can reversibly change the direction within 60 s by axial compression and relaxation. To quantitatively assess the collective order of cells, we introduced the nematic order parameter of cells that takes not only the distribution of cellwrinkle angles but also the degree of cell elongation into account. On the sub-cellular level, we also calculated the nematic order parameter of actin cytoskeletons that takes the rearrangement of actin filaments into consideration. The results obtained on substrates with different wrinkle wavelengths implied the presence of a characteristic wavelength beyond which the order parameters of both cells and actin cytoskeletons level off. Immunofluorescence labeling of vinculin showed that the focal adhesions were all concentrated on the peaks of wrinkles when the wavelength is below the characteristic value. On the other hand, we found the focal adhesions on both the peaks and the troughs of wrinkles when the wavelength exceeds the characteristic level. The emergence of collective ordering of cytoskeletons as well as the adaptation of cell shapes and orientations were monitored by live cell imaging after the seeding of cells from suspensions. After the cells had reached


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the steady state, the orientation of wrinkles was abruptly changed by 90 °. The dynamic response of myoblasts to the drastic change in surface topography was monitored, demonstrating the coordination of the shape and orientation of cells and the nematic ordering of actin cytoskeletons. The “dynamic” substrates established in this study can be used as a powerful tool in mechanobiology that helps us understand how cytoskeletons, cells, and cell ensembles respond to the dynamic contact guidance cues.

Introduction

Mounting evidence suggests that biological cells do not only transduce chemical signals from the surrounding environments but also adapt their shapes and functions to the physical microenvironments, such as mechanical properties of extracellular matrices (ECMs).1-3 For example, the mechanosensing of stem cells has been drawing increasing attentions due to their crucial roles in the maintenance of stem cell functions and the regulation of lineage-specific differentiation.4-8 ECMs often show highly anisotropic topography that influences the morphology, directional order, and migration behavior of cells, which is called “contact guidance”.9-11 For instance, fibrous ECMs made out of collagen type I play major roles in guiding the alignment and migration of various cells.12-13


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One of the straightforward ways to model contact guidance in vitro is to fabricate two-dimensional substrates displaying anisotropic patterns possessing different biochemical functions14-17 or mechanical properties.18-20 As an alternative strategy, substrates displaying parallelly aligned wrinkles, ridges and grooves have been drawing increasing attentions as artificial contact guidance cues.

To date, a number of studies have utilized substrates displaying ridges and grooves with sub-µm to several µm features prepared by lithographic patterning, and demonstrated the contact guidance of various types of cells, such as fibroblasts, myoblasts, neuroblasts, dendrocytes, and endothelial cells.21-25 As an alternative strategy to lithographic patterning, substrates with periodic wrinkles can be fabricated either by the hardening of polydimethylsiloxane (PDMS) surfaces by plasma treatment26-27 or by the deposition of stiff layers, such as metal,28-29 hard plastic30 and graphene oxide on PDMS.31 Compared to the lithographic patterning, the preparation of these substrates is less laborious, but the amplitude and wavelength of wrinkles are limited in order to avoid the delamination of stiff layers.32

The homeostasis of biological tissues is sustained by the establishment of collective order of cells via various mechanical interactions between cells and ECMs. Cells in the tissues alter their shapes, positions, and orientations in response to dynamic changes in their mechanical microenvironments.33


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One of the prominent examples manifesting the instrumental role of cell alignment is muscles, where the optimal function is achieved by the alignment of actin filaments. Yang et al. reported that substrates displaying 0.8 µm-wide ridges could facilitate a significantly better cell alignment compared to flat substrates, which resulted in enhanced dystrophin expression and myogenesis of cardiac cell patches taken from the model mice of muscle dystrophy.34 Since the migration of smooth muscle cells is supported by the enzymatic digestion of collagenous ECM by MMP-9,35 the overexpression of MMP-9 inhibitor suppressed the smooth muscle cell migration in vitro and the formation of hyperplasia after injury of rat arteries in vivo.36-37

Thus, substrates that are capable of modulating topographic features are ideally suited to gain further insights into the dynamic adaptation of cells and cell ensembles to the abrupt change in contact guidance cues. The dynamic formation of wrinkles (λ = 3 – 6 µm) by releasing the stress exerted on shape memory polymers coated with thin Au layers led to a distinctly improved alignment of stem cells derived from human adipose compared to the initial, flat surface.38 Although several studies have shown that the generation of wrinkle patterns by thermal39-40 or optical cues41 induced the alignment of cells, all these substrates can undergo topographic changes only once. The reversible formation and removal of sinusoidal wrinkles (λ = 6 µm) can be achieved by compression and


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relaxation of PDMS capped by brittle oxide layers, which enabled the switching of C2C12 myoblasts between aligned and random orientations in 24 h with no loss of myogenic capability.42 To further extend the strategy from the simple formation and removal of wrinkles, the next challenge is to fabricate a dynamic substrate that can change the direction of pre-formed wrinkles. Ohzono et al. reported the reversible switching of the wrinkle direction by the axial compression of PDMS coated with thin metal layers.29, 43 The angle between the original and newly generated wrinkles can be controlled by the axis of compression with respect to the orientation of the original wrinkles.43

In this study, we deposited a thin layer of polyimide on PDMS30 and fabricated the wrinkle patterns with the wavelength of λ = 1.7, 2.5, 3.7 and 6.3 µm. Following the surface coating with fibronectin, we seedeed C2C12 mouse myoblasts, a stable cell line undergoing myotube differentiation. Different from previous studies recording the angles between cells and the wrinkle direction, we assessed the directional ordering of actin cytoskeletons as well as cells by calculating the nematic order parameters.44-45 The nematic order parameter of cells was weighed by the aspect ratio (the ratio of major and minor axes of a cell) and that of actin cytoskeletons by the length and thickness of actin filaments in order to take the elongation of cells and polymerization/rearrangement of actin filaments into account. Intriguingly, we found a characteristic wavelength λ* beyond which both order


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parameters reached the saturation levels. We found that the focal adhesion contacts are confined only on the peaks of wrinkles at λ < λ*, while the focal adhesions can be found not only on the peaks but also on the troughs of wrinkles at λ > λ*. The dynamic response of cells to the abrupt change in the wrinkle direction by 90 ° was monitored by the live cell imaging of C2C12 transfected by LifeActGFP46. We found that the cells initially elongated along the original wrinkles first became round, and then elongated in the direction of new wrinkles, which is perpendicular to the original ones. The influence of wrinkle wavelength on the dynamics of cellular response was discussed in terms of the locations of focal adhesions. The details of the obtained results are presented in the following sections.

Experimental

Materials. Polydimethylsiloxane (PDMS, Sylgard 184) was purchased from Dow Corning (Midland, USA), fetal bovine serum (FBS), goat anti-mouse Alexa Fluor® 488 conjugate and rhodaminephalloidin from Thermo Fisher Scientific (Tokyo, Japan). Penicillin, streptomycin and phosphate buffered saline (PBS) were purchased from Nacalai Tesque (Kyoto, Japan), HiLyte 488-labeled fibronectin from Cytoskeleton Inc. (Denver, USA), and LifeAct–TagGFP2 and Torpedo® lipofection


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reagent were obtained from ibidi (Munich, Germany). Unless stated otherwise, other chemicals were purchased from Sigma-Aldrich (St. Louis, USA) and used without further purification.

Fabrication of substrates with periodic wrinkles. The substrates displaying periodic wrinkles were fabricated by slightly modifying the protocols reported previously30 (Scheme 1). In brief, PDMS was polymerized on a silicon wafer (Furuuchi Chemicals, Tokyo, Japan) and cut into blocks of 12 × 12 × 5 mm³ size. The PDMS blocks were treated with ambient air plasma for 20 s (PDC-001, Harrick Plasma, USA), followed by spin-coating (MS-A150, Mikasa Corporation, Hokkaido, Japan) with a solution

of

polypyromellitic

dianhydride-co-4,4'-oxydianiline

(polyamic

acid)

in

N-

methylpyrrolidone (NMP) at 5000 rpm for 60 s. The solvent was evaporated by preheating at 90 °C for 5 min, and the samples were axially compressed by 3 % strain and cured at 180 °C for 3 h during the formation of polyimide. The wrinkles were formed upon release of the external strain. 3D structures of wrinkles were characterized using an atomic force microscope (NanoWizard, JPK Instruments, Berlin, Germany) with a pyramidal cantilever (MLCT, Bruker, Billerica, USA).

Modulation of wrinkle orientation. To dynamically modulate the wrinkle orientation an external strain was exerted on the substrate in the direction of wrinkle orientation.43 A strain above 10 % induced reorganization of the wrinkle structure perpendicular to the original wrinkle orientation. To


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control the strain accurately we used a custom-built motorized pusher including a cell incubation chamber, which can be mounted on a microscope stage (Strex, Osaka, Japan).

Cell culture, immunostaining, and transfection. C2C12 mouse myoblast cells (< 20 passages) purchased from RIKEN BRC Cell Bank (Tsukuba, Japan) were cultured in RPMI-1640 medium supplemented with 10 % (v/v) FBS, 100 U/mL penicillin and 100 µg/µL streptomycin. Before seeding the cells, wrinkled substrates were incubated with 30 µg/mL human fibronectin for 2 h at room temperature. The cells were detached from the culture flasks using 0.25% trypsin-EDTA solution, and 2 × 103 cells/cm² were seeded on the wrinkled substrate. For immunostaining, the cells were fixed with 0.02 g/mL paraformaldehyde, permeabilized with 0.1% Tween®-20 and blocked with 1% BSA in PBS. Then the cells were incubated with mouse anti-vinculin, goat anti-mouse Alexa Fluor® 488 conjugate, rhodamine-phalloidin and 4′,6-diamidino-2-phenylindole (DAPI), each for 1 h at room temperature. To visualize actin fibers in living cells, C2C12 were transfected with LifeAct–TagGFP2 using Torpedo® lipofection reagent following the manufacturer’s protocol.

Image acquisition and analysis. Cells on wrinkled substrates were imaged by an upright laser scanning confocal microscope (FV1000, Olympus, Tokyo, Japan) equipped with a water dipping objective (40 ×, N.A. 0.8). Morphometric parameters, such as aspect ratio and circularity, and the


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orientation of cells with respect to the wrinkle direction were extracted by using Fiji software.47 Each cell was fitted as an ellipse from the calculated first and second order moments of the binary image48, then the aspect ratio was defined as the fraction of the major and minor axis of the fitted ellipse. The order parameters of cells and actin filaments were calculated using a self-written routine in MatLab (MathWorks, Natick, USA) as described before.45-46 The original image was convoluted with a series of elongated Laplace of Gaussian kernels, which were rotated n times between 0 and π - π/n. The application of an intensity threshold yields the orientation of fibers at each pixel. Finally, the order parameter can be calculated from the histogram of fiber orientations.

Scheme 1: Fabrication of substrates displaying periodic wrinkles. Polydimethylsiloxane (PDMS) block was first treated with ambient air plasma, and then polyamic acid dissolved in NMP was spincoated. A thin polyimide layer was formed by curing the sample at 180 °C under axial compression. The release of axial strain resulted in the periodic wrinkle structures.

Results and Discussion


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Characterization of Wrinkled Substrates. Figure 1a represents the AFM topographic image of a wrinkled substrate. The wrinkles are aligned in the direction perpendicular to the strain applied during the polymerization of the polyimide layer. The height profile along the solid yellow line (Figure 1a) measured by AFM reveals a wavelength (peak-to-peak distance) of λ = 2.5 ± 0.1 µm and a peak-totrough height difference of Δh = 0.32 ± 0.01 µm. Figure 1b shows the wavelength λ and amplitude Δh of wrinkles, plotted as a function of the thickness of polyimide layers (Supporting Information S1). The wrinkle wavelength λ is controlled by the thickness of the stiff polyimide layer dPI:28

𝜆 ∝ 𝑑PI

1 3

( ), 𝐸PI

𝐸PDMS

(1)

The bulk elastic modulus of PDMS EPDMS  10 MPa was determined by AFM indentation, while that of polyimide layer EPI  2.5 GPa was from the manufacturer. Note that the ratio between λ and Δh is almost constant, Δh/λ  0.13, implying that the wrinkles are not steep “hills” and “valleys” but rather shallow.

To establish a stable polyimide film, PDMS surface with an initial contact angle of φ = 110 ° was treated with ambient plasma for 20 s, resulting in a hydrophilic surface with a contact angle of φ < 10 ° (Figure 1c). After the incubation with 30 µg/mL human fibronectin for 2 h at room temperature, the substrates exhibited homogeneous fluorescence signals from fibronectin labeled with HiLyte 488,


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independent from the presence and absence of wrinkles. We also confirmed that the untreated control sample showed no background fluorescence (Figure 1d).

Figure 1. (a) AFM topographic image of a substrate displaying periodic wrinkles. The height profile analysis along the yellow line yields a wrinkle wavelength of λ = 2.5 ± 0.1 µm and an amplitude of Δh = 0.32 ± 0.01 µm. (b) Wavelength λ (open circles) and amplitude Δh of wrinkles (solid circles) plotted as a function of polyimide thickness dPI. The broken line corresponds to the linear fit, 𝜆 ∝ 𝑑PI. Note that the ratio Δh/λ is only 0.13 for all λ. (c) A water droplet on a flat PDMS substrate before and after the plasma treatment. (d) The polyimide surface allows for the physisorption of fluorescently labeled fibronectin.


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Directional Ordering of Cells and Cytoskeletons. To determine how the wrinkles influence the directional ordering of cells and actin cytoskeletons in a quantitative manner, we first assessed the coarse scale ordering of cell orientation. Figure 2a shows a fluorescence image of C2C12 myoblasts 4 h after seeding onto a substrate displaying wrinkles with λ = 1.7 µm. Actin filaments were labeled with rhodamine-phalloidin (green) and cell nuclei with DAPI (blue). The cells were fitted as ellipses, and the angle of the major axis with respect to the wrinkle αcell was recorded for each cell. The distribution functions of cell-wrinkle angle (Figure 2b) were fitted with Gaussian functions (broken lines). At λ = 1.7 µm (red bars), the full width at the half maximum (FWHM) of the orientation distribution was FWHMcell,1.7µm = 68 ± 15 °. The increase in wavelength to 2.5 µm (green bars), 3.7 µm (blue bars), and 6.3 µm (yellow bars) led to a decrease in the width of the distribution. To evaluate the collective order of cell orientation, we calculated the two-dimensional nematic order parameter of cells:

〈𝑆cell〉 =

∑ 𝐴𝑅𝑖cos (2𝛼𝑖) 𝑖

∑ 𝐴𝑅𝑖

.

(2)

𝑖

Note that is weighed by the aspect ratio ARi, which is the ratio between the major and minor axes of the i-th cell. Figure 2c represents the FWHMcell (open circle) and (closed circle) plotted


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as a function of wavelength λ. The nematic order parameter on the substrate with λ = 1.7 µm is already high,  0.55. It reaches to  0.91 at λ = 3.7 µm, and remains constant at λ = 6.3 µm. On the other hand, the width of orientation distribution exhibits a distinct decrease from λ = 3.7 µm (FWHMcell,3.7µm = 20 ± 3 °) to 6.3 µm (FWHMcell,6.3µm = 13 ± 1 °). These results indicate that the orientation order of cells increases with the increase in λ, and cells are almost perfectly oriented parallel to wrinkles at λ = 3.7 µm. A further increase in the wrinkle wavelength does not change the averaged order parameter but sharpens the width of orientation distribution.

Figure 2. (a) Overlay of DIC image and fluorescence image of C2C12 on a substrate displaying wrinkles with λ = 1.7 µm. The wrinkles were imaged by DIC, while fluorescently labeled actin cytoskeletons (green) and cell nuclei (blue) by fluorescence microscopy. Cells were fitted as ellipses,


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and the orientation orders were assessed by recording the angle between the major axis of each cell and wrinkles (inset). (b) Distribution functions of cell-wrinkle angle αcell fitted with Gaussian functions (broken lines). (c) The width of distribution function (FWHM, open circle) and nematic order parameter of cells (solid circle) plotted as a function of wrinkle wavelength λ. (d) Ordering of actin cytoskeletons on the same cell as in panel (a) was evaluated by recording the angle between the individual actin filaments and wrinkles, αactin (inset). (e) Distribution functions of actinwrinkle angle αactin fitted with Gaussian functions (broken lines). (f) FWHM (open circle) and nematic order parameter of actin (solid circle) plotted as a function of wrinkle wavelength λ.

Ample evidence suggests that the cell morphology is determined by actin cytoskeletons inside individual cells. Therefore, in the next step, we evaluated the ordering of cytoskeletons by staining actin filaments with rhodamine-phalloidin. As presented in Figure 2d, we indicated the directional information of individual actin filaments using a color code. Figure 2e shows the distribution functions of actin-wrinkle angle αactin recorded on substrates with different λ. The width of orientation distribution of individual filaments exhibited a monotonic decrease from FWHMactin,1.7µm = 82 ± 5 °


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to FWHMactin,6.3µm = 25 ± 1 ° concomitantly with the increase in the wavelength. The two-dimensional nematic order parameter of actin filaments was calculated using:

〈𝑆actin〉 =

∑ 𝐴𝑖cos (2𝛼𝑖) 𝑖

∑ 𝐴𝑖

.

(3)

𝑖

Ai is the pixel numbers multiplied by the corresponding fluorescence intensity of i-th filament, which takes the length and thickness of actin filaments into account.46 In Figure 2f, the FWHM (open circle) and (solid circle) are plotted as a function of λ. It is notable that the order parameter of actin at λ = 1.7 µm,  0.34, is much lower than the order parameter of cells on the same substrate,  0.55. The increase in the wavelength from λ = 1.7 to 3.7 µm led to a monotonic increase in the order parameter, resulting in = 0.73 ± 0.20 which can be explained by the rearrangement of actin filaments in the direction parallel to the wrinkle orientation. Note that the order parameter is “weighed” by the length and thickness of actin filaments (Eq. 3). On the other hand, the further increase in the wavelength to λ = 6.3 µm caused no distinct increase in the actin ordering ≈ .

Localization of Focal Adhesion. The fact that both order parameters and leveled off at λ ≥ 3.7 µm suggests that there is a characteristic wavelength λ* between 2.5 and 3.7 µm enforcing


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the alignment of both cells and actin cytoskeletons. To gain further mechano-biological insights into this characteristic length scale obtained from the phenomenological observation, we examined the effect of wrinkle wavelength λ on the location of focal adhesion contacts. The left panels of Figures 3a, 3b, 3d, and 3e represent the overlay of DIC images and immunofluorescence images, while the anti-vinculin signals integrated along the wrinkle direction inside the boxes are shown in the right panels. The blue arrows indicate the intensity maxima found on the peaks of wrinkles, which are mostly separated by a distance λ.

At λ = 1.7 µm (Figure 3a) and 2.5 µm (Figure 3b), all the intensity maxima were found at the peaks of the wrinkles, indicating that focal adhesion contacts were concentrated on the peaks (Figure 3c). On the other hand, the focal adhesion contacts of cells to the substrate with λ = 3.7 µm (Figure 3d) and 6.3 µm (Figure 3e) were concentrated not only on the peaks but also on the troughs of wrinkles, as indicated by red arrows. This suggests that the cells first establish the focal contacts on the peaks of wrinkles, but they cannot keep the membrane flat when the separation between the neighboring focal contacts exceeds λ*. In fact, the intensity from the troughs seems less than that from peaks at with λ = 3.7 µm (Figure 3d), while the intensities from trough sand peaks become comparable at λ =


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6.3 µm (Figure 3e). Consequently, the cell membrane bends and establishes the contacts in the middle, i.e., the troughs of wrinkles (Figure 3f).

Figure 3. Dependence of focal adhesions on wrinkle wavelength. (a, b, d, e) Left panels: Overlay of immunofluorescence (vinculin) images and DIC images. Right panels: vinculin signals integrated in the region of interest (yellow boxes) along the wrinkle direction. Blue arrows indicate the intensity maxima located on wrinkle peaks. Vinculin signals are localized on the wrinkle peaks at (a) λ = 1.7 µm and (b) λ = 2.5 µm, suggesting that the cell membrane is almost flat at λ < λ*, as schematically shown in (c). On the other hand, focal contacts can also be detected in the troughs of wrinkles at (d) λ = 3.7 µm and (e) 6.3 µm (red arrows), suggesting that the cell can bend and follow the undulation


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at λ > λ*, as schematically shown in (f). Red arrows indicate the intensity maxima located on wrinkle troughs.

Dynamic Cell Spreading on Wrinkles. In the next step, we performed the live cell imaging of C2C12 transfected by LifeAct-GFP to shed light on the dynamics of morphological adaptation on the cellular level and the remodeling of cytoskeletons in the sub-cellular level during the initial phase of cell-substrate contact.46 Figure 4a represents several snapshots of a cell spreading on a substrate with λ = 3.7 µm. When the cell gets in contact with the surface, the cell is round and the orientation of actin cytoskeletons seems still isotropic. After the first contact, the cell was axially elongated in the direction parallel to the wrinkle direction and then reached to a stable shape after 3 h. The elongation of cell shape was accompanied by the rearrangement of actin filaments in the wrinkle direction. Figure 4b represents the change in the aspect ratio AR (red triangles) and the nematic order parameter of actin filaments (black circles) recorded over time. The change in AR seems almost linear, while the change in can be fitted well with an exponential function. The characteristic relaxation time obtained by an exponential fit is τ ≈ 37 ± 4 min. Figure 4c shows the relationship between AR and throughout the duration of the experiment. Until t = 1.5 h, both AR and


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increased in a correlated fashion (purple). Once reached the saturation level, AR continued to increase (t = 1.5 ‒ 3 h, green). Afterwards both parameters showed no remarkable changes (t = 3 ‒ 5 h, orange). On a substrate with λ = 1.7 µm, we observed AR and to be positively correlated throughout the entire observation until the cell reached to the steady state after t = 3 h (Supporting Information S2). On the other hand, C2C12 seeded on a flat polyimide substrate underwent an isotropic spreading, AR ≈ 1 and ≈ 0 for more than 4 h (Supporting Information S3).

Figure 4. Dynamic spreading of cells on wrinkled substrates. (a) The orientation of actin cytoskeletons αactin extracted from live cell imaging of C2C12 cell after seeding on the substrate with


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λ = 3.7 µm. (b) Change in the aspect ratio AR and nematic order parameter of actin filaments recorded over time. (c) In the initial phase of cell spreading, AR and showed a positive correlation (t = 0 ‒ 1.5 h, purple). AR increased furthermore even after reached the saturation (t = 1.5 ‒ 3 h, green), which finally achieved a steady state after t = 3 h (orange).

Cellular Response to Dynamic Change in Wrinkle Orientation. The next question we wanted to address was how the cell morphology and the nematic order of actin cytoskeletons respond to a dynamic change in the surface topography, such as a drastic change in the wrinkle direction. As previously reported,43 the direction of wrinkles of our substrates can be altered by 90 ° by applying an axial stretch parallel to the wrinkle direction in air. Figure 5a shows a photograph of the custombuilt pusher mounted on top of an upright confocal microscope and Figure 5b the schematic illustration of the sample environment. Figure 5c represents the DIC images of wrinkled substrates in the cell culture medium captured at different strain levels while pushing the substrate at a constant speed of 25 µm/s. The insets are the Fourier transforms of the images presented in the main panels. In the absence of axial strain (0 %), the Fourier transformed image indicates the parallel alignment of wrinkles with a characteristic distance of 3.9 µm in real space. At the strain level of 9.5 %, the


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wrinkles perpendicular to the original direction started appearing locally. The emergence of the perpendicular pattern can also be seen in the Fourier transformed image,43 yielding the characteristic distance of 4.4 µm. The complete transition of the wrinkle orientation could be achieved at a strain level of 12 % within 60 s, resulting in wrinkles aligned perpendicular to the original direction with a characteristic distance of 4.0 µm. As soon as the strain was released by relaxation at a constant speed of 25 µm/s, we observed the recovery of the original wrinkle pattern with the wavelength of 4.1 µm. Note that the systematic optimization of bonding of polyimide and PDMS, such as plasma treatment of PDMS, was essential to avoid the delamination of polyimide in aqueous solutions. The reversible switching of wrinkle directions in cell culture medium further enables us to examine the dynamic adaptation of cell morphology and actin ordering.


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Figure 5. (a) Photograph of the custom-built pusher for live cell imaging. The axial strain can be applied by pushing the substrates at a speed of v = 25 µm/s, while performing live cell imaging using an upright confocal microscope equipped with a water dipping objective. (b) Schematic illustration of the sample environments. (c) DIC images of wrinkle patterns captured at different strain levels. The direction of axial compression is indicated by arrows, and Fourier transforms of DIC images are presented in the insets.

After confirming that the cell reached to the steady state (Figure 4), we applied an axial strain in the direction parallel to the wrinkles. As presented in Figure 5, the rearrangement of the wrinkle direction


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was completed in 60 s when the substrate was compressed at v = 25 µm/s. Figure 6a represents snapshot images of C2C12 on a substrate with λ = 3.7 µm before and after changing the wrinkle direction by 90 °. The wrinkle direction corresponding to each image is shown in the inset. From the live cell imaging data, we first plotted the changes in the global morphometric parameters, aspect ratio AR (red triangles) and cell orientation θcell (blue circles), as a function of time (Figure 6b). In contrast to α, which is used in Figure 2, we use a different angular index θ to describe the dynamic cellular response. θ = 0 ° is defined as the “initial” wrinkle direction and thus the new wrinkle direction is θ = 90 °.

Upon the change of the wrinkle direction, AR showed a sharp decrease from the initial value AR0 = 2.4 and reached the minimum level (ARmin = 1.1), suggesting that the cell took a round shape (Figure 6b). Then AR started increasing again, which coincides with the elongation of the cell in the direction perpendicular to the original one. Finally, after t ≈ 4.5 h AR reached AR = 2.7, which is comparable to AR0. The transition time point for the morphological change was defined as the time point corresponding to ARmin, tAR,3.7µm* = 2.8 h. θcell exhibited a transition from  0 ° to  75 ° in t ≈ 3 h which is comparable to tAR,3.7µm* (Figure 6b). The agreement between tAR,3.7µm* and tθ,3.7µm* was confirmed by monitoring the dynamic response of other cells using differential interference contrast


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microscopy (Supporting Information S4). It should be noted that the morphological change is not governed by the strain of the underlying PDMS substrates. Indeed, Steward et al. monitored the change in cell morphology and orientation on flat PDMS substrates under a constant strain of 20 % and reported that the characteristic response time is about 24 h.49

In the next step, we examined whether the transition of shape and orientation on the cellular level is correlated with the ordering of cytoskeletons inside the cell. Figure 6c shows the change in actin order parameter (black open circles), showing a monotonic decrease in . Note that is calculated from the angle between each actin filament and the original wrinkle direction. Namely, = 1 means that all actin filaments are aligned with respect to the “original” wrinkle direction. On the other hand, = − 1 means that all actin filaments are aligned with respect to the “new” wrinkle direction. The decrease in from 0.9 to – 0.7 suggests that the actin cytoskeletons changed the orientation almost by 90 °. The data can be fitted with an error function (black solid line), and the time corresponding to = 0 was defined as the transition time point tactin,3.7µm* = 2.8 h. Intriguingly, the transition time point obtained from the aspect ratio and the one calculated from the transition of actin order parameter showed good agreement tAR,3.7µm* = tactin,3.7µm* = 2.8 h, suggesting the coordination of sub-cellular level ordering of the actin cytoskeletons and global


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response on the cellular level. Finally, we released the strain and recovered the original wrinkle direction (Figure 6d), confirming the reversible control of cell shape and actin ordering using the dynamic change in wrinkle orientation.

Figure 6. (a) Response of transfected C2C12 to a dynamic change in the wrinkle orientation tracked by live cell imaging (λ = 3.7 µm). After confirming that the cell reached the steady state in the absence of strain, the wrinkle orientation was changed by axially compressing the substrate in the direction parallel to wrinkles (t = 0 h). As shown in Figure 5, the switching of wrinkle direction to the direction perpendicular to the initial one takes only 60 s. The wrinkle orientation corresponding to each image


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is shown in the inset. After t = 6 h, the cell did not exhibit any further change in the orientation. (b) Changes in aspect ratio AR (red triangles) and cell orientation θcell (blue circles) plotted as a function of time, yielding the transition time point tAR,3.7µm* = 2.8 h. Note that the “initial” wrinkle direction before applying the axial strain is defined as θcell = 0 ° in order to highlight the dynamic response of cells. (c) Changes in actin order parameter (black open circles) plotted as a function of time. The error function fit of (solid line) yields the transition time point tactin,3.7µm* = 2.8 h at = 0. (d) Recovery of the initial cell morphology after releasing the axial strain, confirming the reversibility.

Impact of Wrinkle Wavelength. As demonstrated in Figure 2, we found a characteristic wavelength λ* between 2.5 and 3.7 µm enforcing the alignment of both cells and actin cytoskeletons. Interestingly, the focal adhesion contacts were concentrated on the peaks at λ < λ* (Figure 3a and Figure 3b), suggesting that the cells are not able to bend the cell membranes to follow the substrate undulation (Figure 3c). On the other hand, the focal adhesion contacts can be found not only on the peaks but also on the troughs of wrinkles at λ > λ* (Figure 3d and Figure 3e), indicating that the cells bend and establish the contacts in the middle of the troughs of wrinkles, too (Figure 3f).


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Since the results presented in Figure 6 were obtained on λ = 3.7 µm corresponding to λ > λ*, the next question we addressed was how cells adapt the shape and the ordering of actin cytoskeletons in response to an abrupt change in the wrinkle direction. For this purpose, we examined the dynamic response of C2C12 to a change in wrinkle direction of a substrate with λ = 1.7 µm (Figure 7). Figure 7a shows the orientation of actin cytoskeletons extracted from live cell images, responding to an abrupt change in the wrinkle direction. After confirming that the cell reached the steady state (defined as t = 0 h), the wrinkle direction was switched by 90 ° in 60 s. As presented in Figure 7b, the change of wrinkle direction led to a rapid decrease in AR from the initial value (AR0 = 3.3), reaching the minimum level at tAR,1.7µm* = 2.3 h. It is notable that the cell on the substrate with λ =1.7 µm near the transition (t ≈ 2 h) took a spikier configuration than that on the substrate with λ = 3.7 µm (Figure 6a, t ≈ 2 h). θcell exhibited an abrupt increase from 0 ° to 90 ° at tθ,1.7µm* ≈ 2.3 h, showing excellent agreement with the transition time point of AR. We also confirmed the agreement between tAR,1.7µm* and tθ,1.7µm* by differential interference contrast microscopy observation (Supporting Information S5). Figure 7c shows the change in over time, exhibiting a monotonic decrease from 0.6 to – 0.6. The error function fit of against t (solid line) implied that the transition of actin ordering occurs at tactin,1.7µm* = 2.2 h. This is in good agreement with the transition time points of cell shape, tAR,1.7µm* ≈ tθ,1.7µm* = 2.3 h.


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Figure 7. (a) Response of transfected C2C12 to a dynamic change in the wrinkle orientation (λ = 1.7 µm). (b) AR (red triangles) and θcell (blue circles) plotted as a function of time. (c) (black open circles) plotted versus time. Solid line: the error function fit of .

Discussion. As presented in Figure 2, C2C12 seeded on substrates with wrinkles exhibited distinctly different orderings both on the cellular level as well as on the sub-cellular level, depending on the wrinkle wavelength. The distribution of angles between the major axes of cells and the underlying wrinkles, αcell, showed a broad distribution on a substrate with λ = 1.7 µm. The FWHM of distribution


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decreased according to the increase in λ (Figure 2b), denoting that the cells are more aligned parallel to the wrinkle direction on substrates with larger λ.

Using PDMS substrates displaying waves, ridges and grooves, it has been shown that the cell-wave angle θ for bovine capillary endothelial cells showed a monotonic decrease with the increase in λ: θ = 20 ± 3 ° at λ = 5 µm and θ = 14 ± 4 ° at λ = 10 µm.27 Here, θ is the average of absolute values of cell-wrinkle angles: θ = 0 ° is defined as the “perfect alignment” and θ = 45 ° as the “random orientation”. A similar definition called “cell orientation index (COI)” was used in other studies on substrates with sub-µm scale features,50 which is defined as COI = 1 ‒ θ/45. However, the quantitative comparison of the degree of directional order was not possible, due to the lack of distribution functions. Other studies utilized the percentage of aligned cells and showed that the alignment of cells and actin cytoskeletons was more pronounced on substrates displaying 1 - 5 µm-wide ridges compared to substrates with larger lateral dimensions,51-53 but the threshold between “aligned” and “not aligned” cells was not clearly stated.

Like cells in anisotropic biological microenvironments, the cells on wrinkled substrates are not only aligned but also elongated axially along the wrinkles. For example, the aspect ratio of bovine capillary endothelial cells increased from 2.9 to 3.2 following the increase in λ from 5 to 10 µm, while the


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corresponding value on flat substrates was about 2.5.27 However, the elongation of cells is not always accompanied by the increase in lateral dimensions. The aspect ratio and the percentage of aligned human mesenchymal stem cells decreased according to the increase in the lateral dimensions of ridges and grooves.54

To take both the directional order and the degree of elongation into account, we calculated the nematic order parameter of cells by weighing the distribution of cos(2αcell) by the aspect ratio AR (Eq. 2). We found that the order parameter reached to the saturation level  0.9 by increasing the wavelength from λ = 2.5 to 3.7 µm (Figure 2c), suggesting the existence of a characteristic wavelength λ* that determines the directional ordering of cells. On the other hand, the macromolecular ordering of actin cytoskeletons on the sub-cellular level was distinctly anisotropic (Figure 2d), which is clearly different from isotropic actin filaments in the cells spreading on flat substrates (Supporting Information S3). The distributions of angles between individual actin filaments and the underlying wrinkles, αactin, were broader than those of cell-wrinkle angles αcell (Figure 2e). However, when the nematic order parameter of actin was calculated by weighing the distribution of cos(2αactin) by the amount of actin filaments, reached to the saturation level when the wavelength was increased from λ = 2.5 to 3.7 µm,  0.7 (Figure 2f). These findings indicate that there is a


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characteristic wavelength λ* to change the nematic order of cells and actin filaments, despite of the difference in hierarchical levels.

Since the mechano-sensory signals transduced from focal adhesions induce the formation of actin stress fibers,55 we determined the positions of focal adhesion contacts by the immunofluorescence labeling of vinculin (Figure 3). First, we found that the focal adhesion contacts were clearly anisotropic and aligned along the wrinkles, independent of the wrinkle wavelength. More remarkably, the characteristic wavelength λ* determining the directional ordering of cells and actin filaments significantly influenced the locations of focal adhesion contacts. At λ < λ* (Figures 3a and 3b) the focal adhesions were found only on the peaks of wrinkles, suggesting that the bottom surface of cells remains flat and does not follow the wrinkles. On the other hand, at λ > λ*, the focal adhesion contacts were found not only on the peaks but also on the troughs of wrinkles (Figures 3d and 3e).

The clear difference in the positions of focal adhesions can be understood in terms of the counteracting two energy contributors, the adhesion free energy (Fadh) and the bending energy (Fbend) per unit area. Let us define the height profile of wrinkles as ℎ(𝑥, 𝑦) =

( ),

∆ℎ 2𝜋 2 sin 𝜆 𝑥

where x-axis is

perpendicular and y-axis is parallel to wrinkles. When the cell membrane fully follows the wrinkle surface (full adhesion, Figure 3f), the total free energy is written as 𝐹full = ― 𝐹adh + 𝐹bend. On the


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other hand, the free energy of the cell membrane, which remains flat and adheres only to the wrinkle peaks (partial adhesion, Figure 3c), should read, 𝐹partial = ― 𝜒𝐹adh. Here, χ is the areal fraction of cell-substrate contact, 0 < χ < 1. Note that the first scenario corresponds to the adhesion on the substrates with λ > λ*, while the latter to those with λ < λ*. The free energy difference between “full adhesion” and “partial adhesion” is given by ∆𝐹 = ― (1 ― 𝜒)𝐹adh + 𝐹bend. The bending energy of a membrane is generally represented as a function of the bending modulus of cell membrane κ and the contact curvature C:56-57

1

𝐹bend ∙ 𝐴 = ∫2𝜅(𝐶𝑥 + 𝐶𝑦 ― 2𝐶0)2d𝐴.

(4)

In our system, Cy and the spontaneous curvature C0 are zero. If one takes the contour length corresponding to one wavelength along x-axis L into account, the bending energy normalized by the

(Δℎ𝜆)𝜆𝐿𝜅 . As L can also be expressed as 𝐿 = 𝑓2(Δℎ𝜆)𝜆, one can predict the

area A = L × 1 is 𝐹bend = 𝑓1

presence of characteristic wavelength λ* corresponding to the balance of two counter-acting energies:

∗

𝜆 =

(Δℎ𝜆)𝜅 . Δℎ (1 ― 𝜒)𝑓2( 𝜆 )𝐹adh 𝑓1

(5)

(Δℎ𝜆) and 𝑓2(Δℎ𝜆) are given in the Appendix. Note that Δℎ𝜆 is almost constant

The actual forms of 𝑓1

in our experimental system,

Δℎ 𝜆

≈ 0.13 (Figure 1). If we take the values from the previous studies,


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such as Fadh ~ 10−7 to 10−5 Jm−2 58-59 (Supporting Information S6) and κ ~ 10−19 J,58, 60 the characteristic wavelength calculated from our experiments (2.5 µm < λ* < 3.7 µm) suggests 0.77 < χ < 1. Apparently large χ values could be attributed to the fact that the bending energy is neglected in the case of “partial adhesion”. From the physical viewpoint, the transition of adhesion across λ* seems to share some common features with the wetting transition across the “healing” length.61 The details of the numerical derivation are presented in the Appendix.

Some previous studies also suggested the existence of a characteristic length scale for the localizations of focal adhesions. For example, human fibroblasts and vascular endothelial cells seeded on substrates with ridges and grooves established almost all the focal adhesions on 2 µm-wide ridges but not on 2 µm-wide grooves. On the other hand, focal adhesions can be found on both 10 µm-wide ridges and 10 µm-wide grooves,25 suggesting the characteristic repeat distance lies between 4 – 20 µm. Recently, Ray et al. seeded breast cancer cells on substrates displaying ridges with 0.4 ‒ 1.2 µm width, and reported that focal adhesions are localized only on 0.4 µm-wide ridges separated by 0.4 µm-wide grooves. In contrast, the focal contacts can be found both on 0.8 µm-wide ridges and on 0.8 µm-wide grooves,50 suggesting the characteristic repeat distance of 0.8 − 1.6 µm. The characteristic wavelength suggested by our data, λ* = 2.5 − 3.7 µm, seems to be in good agreement with these reports as the


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order of the magnitude. A slightly lower characteristic repeat distance found for the breast cancer cells could be interpreted in terms of the difference in the deformability of cells. As cancer cells are generally softer than normal somatic cells,62 it is plausible that the breast cancer cells are more deformable than normal myoblasts, fibroblasts, and vascular endothelial cells. Actually, a previous report by Kim et al showed that the cardiac cells can get into 400 nm-wide ridges and grooves,63 which can qualitatively be understood in terms of the mechanical properties of cells. A typical Young’s modulus of cardiac cells is less than 0.5 kPa,64 which is much lower than that of C2C12, 12 – 15 kPa.65

To date, the morphology and alignment of cells on flat substrates vs. substrates with wrinkles and ridges/grooves have mostly been discussed from the static images of cells after the cell shape reached the steady state, using immunofluorescence staining.27, 66-67 Previously the only way to capture the dynamics of cytoskeleton remodeling was the fixation and staining of actin68 and microtuble69 at different time points ex situ. In this study, we performed the live cell imaging of C2C12 transfected by LifeAct-GFP, and monitored the actin ordering during cell spreading in situ (Figure 4a). The cell adhering to the substrate initially took a round shape, followed by an axial elongation in the direction parallel to wrinkles. Actin filaments in the cytoplasm rearranged themselves in the wrinkle direction


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over time. As presented in Figure 4b, both the cell shape (AR) and the nematic order parameter of actin () reached the saturation level after 3 h. AR and exhibited a clear positive correlation in the initial phase of spreading, while AR kept on increasing even after reached the saturation level (Figure 4c), suggesting that the shape adaptation is guided by actin cytoskeletons. It should be noted that the increase in AR and was positively correlated on a substrate with λ = 1.7 µm until the cell reached the steady state (Supporting Information S2). On the other hand, the spreading of the same cells on a flat polyimide substrate was completely isotropic throughout the entire observation period (Supporting Information S3).

One of the unique characteristics of our system is the capability to reversibly change the wrinkle direction by axial compression and relaxation (Figure 5). So far, several studies have suggested strategies to dynamically modulate the surface topography by external cues. For example, the treatment of a PDMS block by oxygen plasma led to a hardening of the surface. The axial compression enabled the formation of wrinkles with λ = 6 – 7 µm, while the relaxation of the axial strain recovered a flat surface.42 It has been reported that C2C12 cells seeded on these substrates exhibited the switching between aligned and random orientations in 24 h, following the formation and removal of wrinkles. As an alternative strategy, Kiang et al. embedded magnetic microwires in polyacrylamide


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gels, and demonstrated the capability to reversibly induce random roughness on the surface.70 However, despite of the capability of reversibly switching the surface topography, this material did not possess regular lateral features. Our system can offer a unique advantage over the abovementioned strategies, owing to the capability to reversibly change the orientation of wrinkles by 90 ° simply by axial compression and relaxation in the direction parallel to the original wrinkles (Figure 5). When an axial strain was applied in the direction parallel to the wrinkles (λ = 3.7 µm), AR decreased towards the minimum level (ARmin ≈ 1) then increased again (Figure 6b). This suggests that the cell first became round and then elongated in the new wrinkle direction. The transition time point was defined as the time point corresponding to ARmin, tAR,3.7µm* = 2.8 h. The angle between cells and the original wrinkle direction θcell exhibited a transition from  0 ° to  75 ° within a similar time window (3 h). In a previous account, the directional ordering of adipose derived stem cells before and after the removal of wrinkle patterns of a thin Au layer on shape memory polymer was monitored with the time interval of  20 h.38 The reversible switching between “aligned” and “random” states of C2C12 cells in response to the formation and removal of PDMS wrinkles was observed with a time interval of 15 ‒ 30 h.42 Our data suggest that the kinetics of dynamic response of C2C12 cells to the abrupt change in surface topography was significantly faster, t*  1 – 4 h.


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The nematic order parameter of actin showed a monotonic decrease from 0.9 to – 0.7 (Figure 6c), denoting that the actin cytoskeletons changed the orientation almost by 90 °. The transition time point obtained by the error function fit was in good agreement with that of cell shape adaptation tAR,3.7µm* = tactin* = 2.8 h, suggesting the orchestration of sub-cellular level ordering of actin cytoskeletons and the global response on the cellular level, such as cell shape and orientation. Moreover, the reversible switching of cell shape and orientation after the recovery of the original wrinkle patterns was confirmed (Figure 6d).

In the last step, we examined if the dynamic response of cells is different across the characteristic wavelength λ*, which was found in Figures 2 and 3. As λ = 3.7 µm is beyond λ* (Figure 6), we performed the experiments on the substrate with λ = 1.7 µm which is clearly below λ* (Figure 7). The cell showed the same tendency: AR first decreased then increased, and the cell orientation showed a clear transition across tAR,1.7µm* ≈ 2.3 h. Moreover, the transition time point for the ordering of actin filaments calculated from the error function fit showed good agreement, tactin,1.7µm* ≈ 2.2 h. On the substrate with λ < λ*, the cell extended spiky filopodia (Figure 7a), and the focal adhesions were confined only on the peaks of wrinkles (Figure 3a-3b). Therefore, the remodeling of focal adhesions occurred in the same plane. On the other hand, the focal adhesions on the substrate with λ = 3.7 µm


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(λ > λ*) were found both on the peaks and on the troughs of wrinkles (Figure 3c-3d). This leads to the reorganization of focal contacts and hence the remodeling of actin in three-dimensional space.

To our knowledge, this is the first systematic report of the dynamic cellular response to the reversible switching of wrinkle directions by 90 °, demonstrating the influence of characteristic length scale on the kinetics of contact guidance. Last, but not least, as shown in a previous account,43 the “dynamic” substrate established in this study enables one to reversibly change the wrinkle directions not only by 90 ° but in any direction. Further studies on the influence of wrinkle wavelength, direction of wrinkle modulation, and genetic perturbation of mechano-sensing proteins71 on the differentiation of C2C12 into myotubes would help us further understand the interplay of extrinsic topographical cues and intrinsic genetic cues.

Conclusions

To form functional tissues, it is highly important to place different cells in the right positions and right orientations. Contact guidance has been reported in various morphogenetic processes, but most of the studies on artificial substrates have been focused on contact guidance on pre-formed and thus


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static topographical features. As a consequence, little is understood how cells react when the direction of such anisotropic structures is abruptly switched.

In this study, we fabricated microwrinkles with various wavelengths, whose orientation could be switched by 90 ° within 60 s by axial compression and relaxation. As the target cells, we selected C2C12 myoblasts, as the alignment of actin filaments is known to be highly important during the formation of myotubes. In contrast to previous studies which simply recorded the average cellwrinkle angles or the percentages of aligned cells, we assessed the directional ordering of cells by calculating the nematic order parameters. Moreover, we weighed the order parameter of cells by the aspect ratio AR, taking the elongation of cells into account. On the sub-cellular level, the order parameter of actin cytoskeletons was also weighed by the length and thickness of actin filaments. Both order parameters increased concomitantly with the increase in wrinkle wavelength and then leveled off at λ ≥ 3.7 µm, indicating the presence of a characteristic wavelength 2.5 < λ* < 3.7 µm. To relate this finding to the adhesion function of cells, we performed the immunofluorescence labeling of focal adhesion protein vinculin. We found that the focal adhesions on substrates with λ = 1.7 and 2.5 µm were concentrated only on the wrinkle peaks, while the focal adhesions on substrates with λ = 3.7 and 6.3 µm were detected both on the peaks and the troughs. Such a transition could be


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understood by the competition between the gain of adhesion energy and the energy penalty by bending cell membranes.

We monitored the adaptation of cell shapes/orientation and the collective ordering of cytoskeletons during the spreading by live cell imaging. After confirming that the cells had reached the steady state, we switched the orientation of wrinkles by 90 °. On the cellular level, the changes in AR and θcell showed that the cell first became round, and then elongated in the new wrinkle direction. On the subcellular level, exhibited the transition from 0.9 to – 0.7, denoting that the direction of actin filaments switched by 90 °. Interestingly, the transition time point of cell shape corresponding to the minimum AR was in very good agreement with that of actin ordering corresponding to = 0, indicating the orchestration of the shape adaptation of the whole cell and the collective ordering of sub-cellular actin filaments. This dynamic response was perfectly reversible, when the original wrinkle direction was recovered by releasing the axial strain. Such a dynamic substrate that can reversibly switch the direction of wrinkles will serve as a new tool in mechanobiology, shedding light on the unexplored dynamics of contact guidance emerging on different levels of complexities, ranging from cytoskeletons, cells, to cell ensembles.


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Supporting Information

Characterization of polyimide layer, dynamic cell spreading on a substrate with λ = 1.7 µm, dynamic spreading on a flat substrate, estimation of adhesion free energy from cell shape, dynamic cell response to abrupt change in wrinkle direction.

Acknowledgement

M.T. thanks T. Yamaguchi for inspiring discussions, and P.L. thanks M. Hörning for technical advice. M.T. thanks the German Science Foundation (SFB873 B07), JSPS (17H00855, 16KT0070), and Nakatani Foundation for supports. P.L. thanks the fellowship from Heidelberg University within the framework of HeKKSaGOn Alliance.

Appendix

Free energy of cell membranes on wrinkled substrates. When the cell membrane perfectly follows the height profile of wrinkle substrates (full adhesion, Figure 3f), ℎ(𝑥, 𝑦) = energy exerted on the area A = L × 1 is given from Eq. 4 as:


( ), the bending

∆ℎ 2𝜋 2 sin 𝜆 𝑥

Page 43 of 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1

1

𝜆 1 𝐹bend = 𝐿∫0d𝑦∫0d𝑥2𝜅𝐶𝑥2 𝑔,

where 𝑔 = 1 +

(∂ℎ∂𝑥)

2

By substituting 𝐶𝑥 =

(6)

is the surface metric. ∂2ℎ ∂𝑥2 ∂ℎ 2 ∂𝑥

(1 + ( ) )

, one yields

32

2

𝐹bend =

3

(Δℎ2 ) (2𝜋𝜆) sin2 (2𝜋𝜆𝑥)

𝜅 𝜆 2𝐿∫0

(

1+

2

2

(Δℎ2 ) (2𝜋𝜆) cos2 (2𝜋𝜆𝑥))

d𝑥.

(7)

52

Due to the periodicity of the integrand,

2𝜅 𝐹bend = 𝐿

∫

𝜋2

2

2𝜋 3 2 2𝜋 sin 𝑥 𝜆 𝜆

( )( ) ( ) Δℎ 2

d𝑥

52

( ( ) ( ) ( ))

0

1+

Δℎ 2

2

2

2𝜋 2𝜋 cos2 𝑥 𝜆 𝜆

( )( ) 2

2

( ) ( ) Δℎ (2𝜋 ( 𝜆 ) + 1) ∙ 𝐸 𝜋 Δℎ + 1 ― 𝐾 𝜋 Δℎ + 1 (𝜆) (𝜆) 2

2

𝜋2

Δℎ 𝜆

𝜋2

2

4𝜋 3

(Δℎ𝜆)𝜆𝐿𝜅 .

≡ 𝑓1

2

2

=

Δℎ 𝜆

2

2

( ) +1

Δℎ 𝜋 𝜆

𝜅 𝜆𝐿

2

(8) 𝜋2

Here, 𝐸(𝑘) = ∫0

𝜋

2 1 ― 𝑘2sin2 𝜃d𝜃 and 𝐾(𝑘) = ∫0 1 1 ― 𝑘2sin2 𝜃d𝜃 are the complete elliptic

integrals of the second kind and the first kind, respectively.


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𝜆

Similarly, 𝐿 = ∫0 1 +

𝐿=

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2

(∂ℎ∂𝑥) d𝑥 can be represented as: 2 𝜋

1+

( )

Δℎ 2 𝜋2 𝜆 𝐸

( )

2

(Δℎ𝜆) Δℎ 2 1 + 𝜋2( 𝜆 ) 𝜋2

(Δℎ𝜆)𝜆.

𝜆 ≡ 𝑓2

(9)

Therefore, the bending energy per unit area can be written as:

(Δℎ𝜆) 𝜅 𝐹bend = Δℎ 𝜆2. 𝑓2( 𝜆 ) 𝑓1

Since

Δℎ 𝜆

(10)

≈ 0.13 in our experimental systems, 𝑓1(0.13) ≈ 1.50 and 𝑓2(0.13) ≈ 1.04. Hence, the

bending energy per unit area scales with κλ−2.

When the cell membrane remains flat while adhering only to the peaks of wrinkles (partial adhesion, Figure 3c), the adhesion energy per unit area must be corrected by the areal fraction of cell-substrate contact, 0 < 𝜒 < 1, as described in the main text. Since each adhesion area at peaks of wrinkled substrates is geometrically infinitesimal under this assumption, the adhesion energy per unit area is naturally smaller compared to the case of full adhesion. It should be noted that the bending energy is neglected in the case of partial adhesion, which causes an underestimation of the total free energy per unit area.


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For Table of Contents


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Scheme 1 178x26mm (300 x 300 DPI)

ACS Paragon Plus Environment

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Figure 1 178x109mm (300 x 300 DPI)


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Figure 2 178x90mm (300 x 300 DPI)


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Figure 3 178x86mm (300 x 300 DPI)


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Figure 4 178x106mm (300 x 300 DPI)


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Figure 5 172x97mm (300 x 300 DPI)


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Figure 6 178x116mm (300 x 300 DPI)


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Figure 7 178x81mm (300 x 300 DPI)


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TOC 82x44mm (300 x 300 DPI)


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Dynamic Contact Guidance of Myoblasts by Feature Size and

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