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Stress fiber growth and remodeling determines cellular
morphomechanics under uniaxial cyclic stretch
Aritra Chatterjee1, Paturu Kondaiah2, Namrata Gundiah*1,3
1Centre for Biosystems Science and Engineering,
2Department of Molecular Reproduction, Development and Genetics,
3Department of Mechanical Engineering,
Indian Institute of Science, Bangalore 560012, Karnataka
* For correspondence:
Namrata Gundiah, Department of Mechanical Engineering
Indian Institute of Science, Bangalore 560012.
Email: [email protected], [email protected]
Tel (Off/ Lab): 91 80 2293 2860/ 3366
Keywords: nonlinear fiber‐reinforced continuum, effective modulus, cytoskeleton, nuclear orientation
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Abstract
Stress fibers in the cytoskeleton are essential in maintaining cellular shape, and influence their
adhesion and migration. Cyclic uniaxial stretching results in cellular reorientation orthogonal to the
applied stretch direction via a strain avoidance reaction; the mechanistic cues in cellular
mechanosensitivity to this response are currently underexplored. We show stretch induced stress
fiber lengthening, their realignment and increased cortical actin in fibroblasts stretched over varied
amplitudes and durations. Higher amounts of actin and alignment of stress fibers were
accompanied with an increase in the effective elastic modulus of cells. Microtubules did not
contribute to the measured stiffness or reorientation response but were essential to the nuclear
reorientation. We modeled stress fiber growth and reorientation dynamics using a nonlinear,
orthotropic, fiber‐reinforced continuum representation of the cell. The model predicts the
observed fibroblast morphology and increased cellular stiffness under uniaxial cyclic stretch. These
studies are important in exploring the differences underlying mechanotransduction and cellular
contractility under stretch.
Introduction
Adherent fibroblasts in tissues, such as arteries and lungs, undergo cyclic stretch and respond
to changes in their mechanical milieu through a complex interplay between the cytoskeletal
components and integrin‐mediated pathways to influence cell contractility and secretion of
extracellular matrix constituents [1,2]. Cell driven biochemical processes result in gain or loss in mass
and induce remodeling of the underlying material properties such as stiffness and anisotropy.
Continuum mechanics‐based approaches to address biological growth and remodeling suggest an
intimate relationship between these processes which results in non‐uniform changes to overall form
and function of the underlying structures [3]. How do mechanosensing processes influence cellular
growth and remodeling under stretch? Stretch induced reorientation of cells involves both passive
mechanical response to cyclic substrate deformation and dynamic changes to the cytoskeleton [4].
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Contractile stress fibers (SF), comprised primarily of actin and myosin II, are essential in the
development of intracellular stresses which are transmitted through focal adhesion (FA) complexes
and depend on the activity of RhoA, active Rac1, or Cdc42 [5–7].
Cyclic uniaxial stretching of fibroblasts seeded on isotropic elastomeric substrates leads to cell
reorientation from random to a near perpendicular angle to the loading direction [8,9]. Vascular
smooth muscle cells under stretch show an initial rapid growth and reinforcement followed by
disassembly of SF aligned in the direction of stretch; the SF subsequently rearrange in an orthogonal
direction [10]. The associated actin reorientation under uniaxial stretch is accompanied with an
increase in FAK signaling, MAP kinases, and a corresponding dynamic increase in the size of FA
complexes [11]. Recent experiments show reorganization and remodeling of FA due to stretching of
cells [11–13]. Higher strain rates caused rapid disassembly of SF’s along the direction of deformation
[14,15]. The application of cyclic uniaxial strain also influence cell spreading on soft substrates; SF
formation persisted four hours after stretching and was related to temporal changes in the movement
of MRTF‐A (myocardin‐related transcription factor‐A) and YAP (Yes‐associated protein) from the
cytoplasm to the nucleus [13,16].
Stress fibers are exquisitely sensitive to the dynamically changing mechanical environment
and are essential in the regulation of cell polarization and transmission of tensional force from the FA
to the cell [17]. Cell spreading, dependent on the stiffness of the underlying substrate, resulted in a
bi‐phasic cellular orientation caused by de novo formation of ventrally oriented actin in the cell. Cell
alignment under cyclic uniaxial stretch is hypothesized to be an avoidance reaction to stretch which is
facilitated via cell‐substrate interactions and their links to the cytoskeleton [18]. The mean cell
orientation after stretch contributes to the tensional homeostasis of the cell and helps maintain an
optimal internal stress [19]. More recently, Chen et al showed that traction boundary conditions are
crucial in the determination of cellular orientation under stretch [20] The precise mechanistic cues in
cellular mechanosensitivity to dynamical environments and links to stress fiber growth and cell
stiffness changes however remains a fundamental open question.
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The goals of this study are to investigate SF dynamics in fibroblasts under uniaxial cyclic stretch
and relate their growth and remodeling to the cellular reorientation response using a morphoelasticity
framework. We used a custom fabricated stretcher to subject fibroblasts to uniaxial cyclic stretch for
different amplitudes over varied time periods. We show that the lengthening and realignment of SF
primarily contributes to the cellular reorientation response under cyclic stretch. We model these
morphomechanical responses using nonlinear, hyperelastic, fiber‐reinforced material model for the
cell comprising of two families of SF that grow and reorient to under uniaxial cyclic stretch. The
measured changes in cell stiffness in response to the stretch amplitude and time duration due to SF
reorientation are well captured using the model. These results show the importance of SF growth in
the resulting elongation and orientation response of cells under cyclic uniaxial stretch that are
important in the overall mechanotransduction and cellular contractility.
Results
Cyclic stretch induced changes in cell morphology and stiffness depends on SF
reorientation and cortical actin thickness
Uniaxial cyclic stretch experiments were performed on NIH 3T3 fibroblasts seeded on thin
flexible membranes (20 mm length x 10 mm width x ~150 µm thickness), fabricated from poly dimethyl
siloxane (PDMS; Sylgard®184, Dow Corning) mixed in the ratio of 10:1, using a custom stretcher
(Supplementary Fig. S1). Cell seeded constructs were stretched uniaxially at 5% and 10% amplitude
at a frequency of 1 Hz for 3 and 6 hours. Confocal images of fibroblasts stretched at 10% amplitude
for 6 hours (A10T6) show distinct changes in the overall cellular morphology. Cells were significantly
elongated and elliptical along a uniform direction perpendicular to the direction of applied stretch as
compared to unstretched controls (Fig. 1). There were clear changes that were apparent in the
cytoskeletal and nuclear orientations at higher amplitude and over longer stretching durations
(A10T6) (Fig. 2a, b). These results show significant differences in the major axis length of the cell along
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the direction of reorientation and in cellular aspect ratios (p<0.01 for both) between cyclically
stretched cells (A10T6) and corresponding unstretched controls (Fig. 2d, e). In addition, the cell spread
area increased significantly after subjecting to cyclic stretch (3087±854 m2) as compared to
unstretched controls (2450 ±357 m2; p<0.05). We quantified the SF orientation distributions, based
on a wedge‐shaped orientation filter in MATLAB (MathWorks 2016b), from immunofluorescence
images of cells (n=30) stained to visualize SF (Supplementary Fig. S2) [21,22]. SF lengths were
quantified using Hough transform. A total of ~30 cells were included in each set and the data averaged
to compute an overall SF angular and length distribution (Table 1). The results show SF realignment in
a direction near perpendicular to the direction of applied cyclic stretch; nuclear orientation was also
in the direction of cytoskeletal reorientation (Fig. 2c). SF orientations changed from a bimodal
distribution in control unstretched cells to a unimodal distribution orthogonal to the stretch direction
due to uniaxial cyclic stretch. The SF distribution showed a distinct peak at higher amplitude and
stretch duration (A10T6). These data suggest that almost all cells oriented to a uniform near
perpendicular angle to the direction of applied cyclic stretch.
We used cytoskeletal inhibitors to delineate the individual contributions of SF’s and
microtubules to the re‐orientation response of the cell under cyclic uniaxial stretch. We treated
fibroblasts seeded on ECM coated substrates with cytochalasin‐D, to inhibit actin polymerization, and
nocodazole, to disrupt microtubules, and uniaxially stretched the constructs for either 3 or 6 hours
using 5% and 10% amplitude at a frequency of 1 Hz [23]. We also quantified and compared the SF and
nuclear orientations between the different groups. These experiments clearly demonstrate that
inhibition of microtubule polymerization did not abrogate the reorientation response of cells; in
contrast, inhibition of actin led to complete loss of the reorientation response of the cells (Table 2;
Supplementary Fig. S3). Further, actin and microtubule depolymerizations significantly abrogated the
nuclear reorientation under cyclic stretch. Cytochalasin‐D treated cells had ~7° change in nuclear
orientation under high stretch amplitude and duration (A10T6) as compared to nocodazole treated
cells which did not show any change in nuclear angular orientation under stretch (Fig. 2c). These
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results show conclusively the primary contribution of actin to the cellular reorientation responses as
compared to that of the microtubules which play an important role in the nuclear reorientation.
We quantified the differences in material properties of the cells before and after cyclic uniaxial
stretch using an AFM (Atomic Force Microscope). Individual cells were indented with a spherical
indenter with tip radius, 𝑅 , and the modulus (E) was calculated using a thin layer, finite thickness,
Hertz model (22,24) (Supplementary information‐1). The force (F)‐displacement (δ) relationship is
given by
𝐹 ⁄
𝛿 1 𝜒 𝜒
𝛼 𝛽 𝜒 𝛼 𝛽 𝜒 (1)
𝑅 and 𝜒 √ (2)
where h is the cell height and µ is the Poisson’s ratio of the cell (=0.5). α and β are constants that are
functions of µ. All indentation experiments were performed for a constant indentation depth of 1 μm
which corresponded to ~14% of the total cell height. Fibroblasts stretched at 10% for 6 hours (A10T6)
were indented in the presence or absence of cytoskeletal inhibitors and the corresponding force‐
displacement curves were compared with unstretched controls. Our results show that cyclically
stretched cells had significantly higher stiffness as compared to unstretched controls (p<.05; n~25 in
each group; Fig. 2f). There was no change in the effective elastic modulus for cytochalasin‐D treated
cells under cyclic stretch. In contrast, nocodozole treated cells showed a significant increase in cell
stiffness following cyclic stretch (p<.01). The cytoskeletal reorientation response in cells demonstrates
the dominant contribution of SF in cell realignment under cyclic stretch. Depolymerization of
microtubules, using nocodozole treatment, did not affect the stretch induced increased the cell
stiffness and was linked to formation of SF which drives the cellular reorientation dynamics under
cyclic stretch.
Results from cells stretched at 10% amplitude for 6 hours (A10T6) showed ~40% increase in
the total actin fluorescence intensity as compared to unstretched controls (p<0.05) (Fig. 3f). These
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results on cyclically stretched cells were accompanied with increased SF lengths and higher cell
stiffness measured using AFM. We measured intensity distributions corresponding to the F‐actin
fluorescence along the transverse sections of the cell at various points along the major axis of the cell
to explore possible contributions of the cortical actin to the overall increase in cell stiffness due to
cyclic stretching (Fig. 3a, b, c). The full width at half maximum (FWHM) values corresponding to the
intensity distribution for each cell and the peak intensity values were used to characterize the
thickness of the cortical actin for the cell (Fig. 3d, e). These results showed that the FWHM was 40%
higher for cyclically stretched (A10T6) cells as compared to un‐stretched controls (p<0.05); cyclic
stretched cells also had significantly higher peak intensities as compared to control cells (p<0.01).
Reinforcement of the cortical actin and an increase in the SF lengths in individual cells caused by cyclic
uniaxial stretching hence resulted in the increased cell stiffness.
Morphoelastic model of the cell under uniaxial cyclic stretch
Volumetric growth in the cell, associated with mass addition, is described within a finite strain
elasticity framework based on deformations caused by non‐uniform changes in the mass at the local
level at each material point of the body [25]. The SF elongation in the fibroblast under cyclic stretch is
described using the growth tensor, G, which produces a virtual stress‐free and non‐integrable
configuration in the cell to generate residual stress. SF growth kinematics is accompanied with an
elastic deformation, A, which is induced by growth and is essential in restoring compatibility of the
overall deformation gradient, F, during growth. The elastic deformation tensor plays an important role
in maintaining the cellular integrity and is related to F as:
F = A G (3)
Coupling between the elastic responses and biological growth is integral in morphoelasticity and is
consistent from kinematic and thermodynamic perspectives. Results from experiments to quantify the
growth of SF in fibroblasts under uniaxial cyclic stretch were used to determine the growth kinematics
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based on the evolving configurations during cell reorientation under stretch (Supplementary
Information ‐2).
We model the cell as a fiber‐reinforced orthotropic hyperelastic material, comprised of two
families of SF, based on the microscopy images (Fig. 1) [26,27]. A representative element of the cell is
characterized through dependence on the elastic deformation, A, and is given by the strain energy
function, W(A). We assume that the contributions to W are given in terms of an isotropic component,
Wiso, given by a neo‐Hookean term related to the cytosol and an anisotropic component due to
presence of two families of SF, Waniso which is given using a standard reinforcing solid model as
𝑾 𝑨 𝑾𝒊𝒔𝒐 𝑾𝒂𝒏𝒊𝒔𝒐𝝁
𝟐𝑰𝟏 𝟑
𝝁𝜶
𝟐𝑰𝟒 𝟏 𝟐 𝑰𝟔 𝟏 𝟐 (4)
𝜇 is the shear modulus (𝜇 0) and 𝛼 is a measurement of the additional fiber reinforcement (𝛼 0)
[28]. The elastic counterpart of the right Cauchy‐Green tensor 𝑪 𝑨𝑻𝑨. 𝐼 𝑨 ; 𝑘 1,4,6 are invariants
of the elastic deformation given by [29]
𝐼 𝑨 𝑡𝑟 𝑪 𝐼 𝑨 𝑡𝑟 𝑪 𝑡𝑟 𝑪𝑬 𝐼 𝑨 𝑑𝑒𝑡 𝑪𝑬
𝐼 𝑨 𝒎 . 𝑪𝑬𝒎 𝐼 𝑨 𝒎 . 𝑪𝑬𝒎 (5)
𝒎 ; 𝑖 1,2 represents vectors corresponding to each of the two SF directions in the natural
configuration in the absence of stretch.
We use experimental data on SF lengthening, measured at different amplitudes and for
various amount of stretch durations, to formulate the evolution equation for increase in length () of
SF under cyclic stretch which is assumed to be a function of the applied stretch, 𝜆, and time, t, and is
given by
𝛾 𝑓 𝜆 𝑒 / 𝑐 (6a)
where 𝑓 𝜆 = 143 𝜆 1 3.8 𝜆 1 (6b)
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c is a constant with dimensions of length and is defined as the SF length in unstretched (control) cells;
𝜏 is a growth time scale for the SF. The coefficients in 𝑓 𝜆 were obtained by fitting the experimental
data to equation (6). We use these evolution equations and write the growth tensor for a single SF as
𝑮 𝑰 𝛾 1 𝒆 ⨂ 𝒆 (7)
𝒆 represents the unit vector in the direction of SF growth. We next include thermodynamic
restrictions for active growth based on the Claussius–Duhem inequality using the standard Coleman–
Noll procedure (Supplementary Information ‐3,4).
We assume a form for the SF re‐orientation equations along a uniform direction perpendicular
to applied cyclic uniaxial stretch as
𝒎𝒐𝒊
𝒏 𝒏. 𝒎𝒐𝒊 𝒎𝒐
𝒊 (8)
We have assumed the direction of applied uniaxial cyclic stretch is along the X‐axis. The vector n
represents the unit vector along the direction of preferred cytoskeletal orientation orthogonal to the
applied stretch in the Y‐axis (e2) [28,30]. Equation (8) implies that the two families of SF reorient
perpendicular to the cyclic stretch direction over time and results in reinforcement.𝜏 is the
remodeling time scale which depends on the stretch amplitude. The elongation and reorientation
dynamics of SF in cells for 10% stretch amplitude over 6 hours (A10T6) is shown in Supplementary
Video 1. The model provides insights into active SF growth that is accompanied with passive SF re‐
orientational dynamics.
Model verification using uniaxial stretch data on cells
We tested the model predictions for changes in SF lengths and angles under uniaxial stretch
and related their influence on the measured cell stiffness using experimental data. The evolution
equation (6) for increase in length of SF under stretch is in agreement with the experimentally
measured observations (r2 ~0.99). (Fig. 4a). The time scale for growth of SF in the model is
described using 𝜏 which is a function of the SF properties. We explored changes in SF growth by
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parametrically varying 𝜏 in the model (Fig. 4c, d) for the 5% and 10% stretch amplitude and
compared these results with experimental data; the model predictions match with experimental
data for 𝜏 =2 hours. The growth law also satisfies the thermodynamic incompatibility condition for
active biological growth based on various possible fiber angles and experimental duration for different
stretch amplitudes used in our study (Supplementary information 4). Fig. 4b shows the corresponding
surface plot for the inequality term (𝑴: 𝑮𝑮 𝟏 ) for 5% stretch amplitude as a function of time as well
as fiber angles. 𝑴 is the Mandel stress tensor which is a thermodynamic conjugate to the velocity
gradient, 𝑮𝑮 𝟏 , satisfies frame indifference, and is used as a natural descriptor for growth processes
[31,32 ]. From Fig. 3b, we note that 𝑴: 𝑮𝑮 𝟏 is positive for all possible values of time and fiber angle
in our model which satisfies the thermodynamic constraints.
The reorientation law predicts SF orientation under various magnitudes of stretch and time
duration using inputs corresponding to the mean SF angular distributions. We compared model
predictions from fiber reorientation law with the experimentally obtained SF orientation distributions
that were quantified using confocal images for both 5% and 10% cyclic stretch data. The model shows
a good match with experimental data for increased SF orientational dynamics with higher stretch (Fig.
5a, b). Remodeling time scale for SF orientation, 𝜏 , increased significantly with the magnitude of
applied stretch. Experimental results are superposed on the model predictions; these demonstrate a
good match for 𝜏 =3 hours for 10% stretch and 𝜏 =9 hours for 5% stretch (r2>0.9 for both). That is,
cells reorient to the orthogonal direction to direction of stretch in a shorter time when subjected to
higher stretches. Supplementary Fig. S4 shows parametric variations corresponding to various initial
conditions and times for the SF reorientation law.
We calculated changes in the effective elastic modulus, Eeff, of cells as function of fiber
reorientation angle and time (equations (8). Eeff in a given direction is defined as the gradient of
uniaxial tension, N, with respect to the stretch in that direction, is given in terms of the stress‐free
configuration as 𝐸𝑵 | 𝜆 1 (31) (Supplementary Information‐6). Variations in Eeff with fiber
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angles (𝜃 ∈ 0°, 90° 𝑎𝑛𝑑 𝜃 ∈ 90°, 180° ) from the two SF families are shown in Fig. 6a, b. These plots
show an increase in 𝐸 when both families of fibers align perpendicular to the applied stretch
direction. Variations in 𝐸 with time and the stress distributions with variations in fiber angles of two
SF family (𝜃 𝑎𝑛𝑑 𝜃 ) for 5% and 10% stretch amplitude were also calculated (Fig. 6c, 6d). Stresses
obtained from experimentally obtained fiber distributions at various stretch amplitude and time
points using structure tensors based on fiber orientation distribution functions (Supplementary
Information‐7) show a similar trend as that predicted by the model for reorientation under cyclic
stretch along the minimum stress direction (Fig. 6d). These observations are in good agreement with
results from by De et al (33) who suggested that cellular reorientation under uniaxial cyclic stretch
occured along the minimal matrix stress direction.
Discussion
Growth and dynamics of SF networks are important in maintaining cell shape, control of
cellular mechanical properties and mechanotransduction processes underlying the adhesion and
migration of cells. The primary aim of this study was to quantify the morphomechanical response
of fibroblasts under uniaxial cyclic stretch. There are three main implications of this work. First,
we show that cyclic stretch induced SF growth and reorientation alters the cellular morphometrics.
SF elongate based on the amplitude of cyclic stretch; their orientations changed from an initial random
configuration in unstretched cells to a uniform direction perpendicular to the applied stretch. Cortical actin
thickness increased with applied stretch which resulted in an increase in the effective elastic modulus of the
cell measured using AFM. Second, we quantified the differential roles of actin and microtubules in
cellular and nuclear reorientations under stretch. We show that actin is essential to the cellular
orientation changes under cyclic stretch and contributes to the measured cell stiffness; in contrast,
microtubules influence nuclear reorientations. Third, we use a nonlinear, orthotropic, hyperelastic, and
fiber‐reinforced constitutive model for the cell to elucidate the combined effects of amplitude and
duration of uniaxial cyclic stretch using a growth and remodeling framework. The model uses SF
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evolution equations that depend on the stretch amplitude and time and were determined
experimentally to predict the effects of uniaxial cyclic stretch on SF dynamics.
Actin growth dynamics is exquisitely sensitive to mechanical stimuli [34,35]. Dense
branched actin networks grown on functionalized surfaces demonstrate nonlinear mechanical
responses [34]. Parekh et al [36] measured the force‐velocity relationships of growing actin networks
in‐vitro using a differential AFM and showed the dependence of actin network growth velocity on the
history of applied loading. Mechanical stimuli like torsion and fluctuations in applied loading also
influence the growth of actin networks [37]. These studies were however reported on in vitro
preparations of isolated and purified actin networks; our results show similar responses for actin
networks in fibroblast cells under stretch. The resulting changes to the material properties of the cells,
through a process of morphoelasticity, has not been demonstrated earlier at the cellular level. The
actin cortex also plays an important role in maintaining cellular shape [38]. We show that the
application of uniaxial cyclic stretch increases cortical actin thickness which reinforces the
cytoskeleton and also results in an increase in the cell stiffness (Fig. 2f; Fig 3d). Hasse and Pelling
showed the importance of actin cortex in controlling cellular deformation by applying precise nano
Newton forces [39]. Comparison of the cell modulus to the actin network orientations based on
experimental data and model predictions show that cells with higher SF dispersions have a more
compliant response as compared cells with highly aligned SFs (Fig. 2f; Fig 3d) [22,40].
Uniaxial cyclic stretch experiments in our study show that the growth realignment of SF is
dependent on the amplitude of applied external stretch; SF aligned from a random configuration to a
uniform direction perpendicular to the direction of cyclic stretch (Fig. 1; Table 1). The hyperelastic
model for the cell under external cyclic stretch is based on the assumption of the SF as an elastic
structure that undergo elongation and realignment. The persistence length for a SF, 𝜉 , given as a
ratio of the bending stiffness to statistical fluctuations, is typically greater than 50 µm [41,42].
Filaments with lengths smaller than 𝜉 are flexible and those with greater dimensions are generally
described using statistical methods to include the effects of thermal fluctuations. The longest SF length
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in the fibroblasts in our study was less than 𝜉 which justified the use of elastic rod approaches to
model SF growth dynamics.
We used a phenomenological nonlinear, stretch‐dependent evolution equation for SF
elongation that was dependent on time and cyclic stretch amplitude from the experimental results
(Fig. 3). We used two different time scales in our model to represent changes in the SF length and
orientation under cyclic uniaxial stretch. The time scale for fiber growth, 𝜏 , is constant and depends
on the intrinsic properties of the SF. The second time scale, 𝜏 , represents SF remodeling time scale
which is sensitive to applied stretch and decreases with an increase in stretching amplitude (Fig. 4).
We quantified the individual contributions of SF and microtubules in cellular and nuclear reorientation
under stretch and show that microtubule depolymerization did not significantly affect cellular
reorientation response. However, actin depolymerisation resulted in almost complete loss of the
cellular reorientation response (Table 2; Supplementary Fig. S3). These results are in close agreement
with a previous study by Goldyn et al (17) who showed dependence of force‐induced cell reorientation
on FA sliding alone and was largely independent of the contributions from microtubule networks. AFM
indentation experiments in our study also show that the depolymerization of microtubules did not
change the cellular stiffness under stretch; formation and re‐organization of SFs in response to stretch
induced cytoskeletal stiffening.
Stretch induced cytoskeletal reorganization has previously been linked to stress or strain
homeostasis in the cells [33,43]. Recent studies show that the direction of cell realignment under the
application of stretch depended on the traction boundary conditions [20]. The reorientation dynamics
of individual cells under uniaxial cyclic stretch resulted from a minimization of the elastic strain energy
in the cell [4]. We used a morphoelasticity framework to show that the cyclic stretch dependent
cellular morpho‐mechanical properties are a result of SF growth and reorientation in a direction
perpendicular to the cyclic stretch direction. These observations are in agreement with previously
published works by De et al who predicted the direction of cellular reorientation along the minimal
matrix stress direction [33]. One limitation in our model is we have not considered the effects of
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branched networks and have only included straight filaments. SF are thick and stable and generally
distinguished into four categories as ventral and dorsal SF, transverse arcs, and perinuclear actin cap [44].
We have modelled the growth and reorientation dynamics of ventral SF alone and show the importance of
Sf reinforcement in the measured changes to the mechanical properties of the cell.
To conclude, we used a novel growth and remodeling framework to study the effects of uniaxial
cyclic stretch in fibroblasts using experimental stretching methods in combination with a continuum
mechanics framework. The model uses experimentally motivated evolution equations for SF growth and
remodeling that have not been reported earlier. We show an increase in the cellular stiffness under cyclic
stretch which is linked to SF growth and reorientation in a direction orthogonal to the direction of stretch.
These studies show the importance of uniaxial stretching in mechanotransduction processes related to
changes in the cytoskeleton which are essential to model diseases like aneurysm growth and fibrosis that
currently use qualitative representation of the cell under stretch.
MATERIALS AND METHODS
1. Design and assembly of the bioreactor for cell stretching
The custom designed cell stretcher allows for both uniaxial and biaxial stretching with equal
and different strain rates on each actuator such that the central point for viewing is fixed. We designed
a novel clamping arrangement to clamp a thin stretchable membrane (width 10mm x length 20mm x
thickness 150 μm) between two opposite translating actuator arms for the dynamic stretch
experiments. The clamps were 3D printed using a biocompatible acrylic compound (Verro White;
Stratasys) (Supplementary Fig. 1a, b). The cell stretcher was operated using user‐controlled stretch
amplitude and frequency inputs over extended time periods and placed within a humidified
temperature‐controlled incubator to maintain sterility.
2. Strain quantification
Strain quantification experiments were performed to calibrate the novel clamping design for
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the specified stretch amplitude during cyclic stretch experiments. In‐plane strains were quantified by
tracking fiduciary markers in the PDMS (poly di‐methyl siloxane; Sylgard ®184, Dow Corning) sheet,
mixed in the ratio of 10:1 elastomer to curing agent, using an overhead video camera (Sony HD HDR‐
XR100E). Marker centroids were obtained using segmentation and the corresponding displacements
were quantified. Marker displacements from the referential to the deformed configuration were
measured and the deformation gradient, F, was computed as F = I + H, where 𝐻=𝜕𝑢/𝜕𝑋 is the
displacement gradient in the referential configuration. A strain interpolation algorithm, implemented
using MATLAB (v R2016a), was used to quantify in‐plane Green‐Lagrange strains in the specimen using
measured values of the deformation gradients [27, 48]. These data were used to define a central
homogeneous region, where strains were constant in the sample, and used to quantify cell
behaviours.
3. Cell culture and immunofluorescence staining
NIH 3T3 fibroblast cells were cultured in Dulbecco’s minimum essential medium (DMEM;
Sigma Aldrich) supplemented with 10% fetal bovine serum (GIBCO BRL) and 1 vol% of penicillin‐
streptomycin (Sigma Aldrich) in a humidified temperature‐controlled incubator (Panasonic Inc.) at 37
°C and 5% CO2. The same medium was also used during cyclic stretching experiments performed using
the custom bioreactor within the incubator. Following stretching experiments, cells were fixed in 4%
paraformaldehyde, permeabilized in 0.5% TritonX‐100 (Sigma) solution for 10 minutes and stained
either for actin using rhodamine phalloidin (Thermofisher 1:200) or microtubules using α‐tubulin FITC
(Invitrogen 1:200), and a nuclear DAPI counterstain (Thermofisher 1:400). Samples were imaged using
a confocal microscope (Leica Microsystems, TCS SP5 II) using 10X, and 63X oil immersion objectives to
visualize the cytoskeletal organization.
4. Uniaxial cyclic stretching of fibroblasts in the bioreactor and cytoskeletal inhibitor
treatments
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Thin flexible PDMS membranes were used as substrates to culture cells. PDMS was mixed in the ratio
of 10:1 elastomer to curing agent, degassed in a vacuum pump, and spin coated on triacetylene
treated glass slides at 300 rpm and 60 seconds to yield 150 μm thick sheets. The substrates were
plasma treated for 2 minutes, incubated with 30μg/ml of fibronectin solution at 37⁰C for 45 minutes,
and washed twice with PBS. Prepared substrates were incubated with DMEM containing 10% FBS for
30 minutes followed by seeding of cells. Cell seeded constructs were incubated overnight at 37⁰C
and 5% CO2 to allow cell attachment and were stretched cyclically (5% and 10% amplitude) at 1 Hz for
varied durations to explore the temporal dependence of stretching on cell alignment (3 hours and 6
hours).
To delineate the individual contributions of actin and microtubule in cellular response under
cyclic stretch, we treated cells with either 0.5 µM concentration of cytochalasin‐D (Sigma‐Aldrich;
USA) to depolymerize actin or 0.5 µM concentration nocodazole (Sigma‐Aldrich) to disrupt
microtubules [23]. We ensured that the same concentration of the drug was present during the
stretching experiment to prevent possible repolymerizations of actin and microtubules that may
contribute to possible artefacts in the experiments.
5. Quantification of SF orientation under cyclic stretch
Confocal images of cells stained for actin were analysed using a custom MATLAB program
(Mathworks, 2016b) [21,22]. In this procedure, a binarized image of the actin intensity in the cell was
obtained and represented using a distribution function f(x,y); (x,y) defines a point in the 2‐D image.
We calculated the Fast Fourier transformation (FFT) of this image as ℱ (𝑓 (𝑥, 𝑦)) =(𝑢,𝑣) where (u,v)
denotes a point in the Fourier space. The power spectrum of this matrix was obtained by 𝑃 (𝑢, 𝑣) =𝐹
(𝑢, 𝑣). 𝐹∗ (𝑢, 𝑣), F* is the complex conjugate of the function, F in this expression. The SF were
distinguished based on the spatial frequencies and orientations in Fourier space. A wedge‐shaped
orientation filter was used to quantify the SF orientation distributions (Supplementary Fig. S2) [21].
6. Measurement of effective modulus of the cell using Atomic Force Microscope (AFM)
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We used a Park Systems XE‐Bio AFM to quantify the effective modulus of the cell using an App
Nano Hydra 6V‐200NG‐TL silicon dioxide cantilever with an attached spherical bead of diameter 5.2
μm. The stiffness of the cantilever based on manufacturer sheet was 0.045 N/m and was determined
to be 0.041 N/m using thermal tuning method. Specific locations for indentations were selected using
the cell topography to be approximately equidistant from the nucleus and the cell edge to minimize
the influence of the nucleus. Indentation experiments were performed for each cell using cells in
Hank’s balanced salt solution and the corresponding force‐displacement curves for each location were
used to quantify the effective Young’s modulus of the cell using the material properties of the
cantilever and a modified Hertzian contact mechanics model [24]. The force curve corresponding to
the approach of the tip towards the substrate was measured and a constant indentation depth of 1
μm was maintained. For indentation experiments on cells treated with cytoskeletal depolymerizers,
we ensured that the optimum concentration of the specific drug was present throughout the
experiment.
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List of Figures
1) Figure 1. Fibroblast cells, cultured on PDMS membranes and subjected to uniaxial cyclic stretch
(1 Hz) for different durations, were fixed and imaged to visualize actin (red), microtubules
staining (green), and the nuclei (blue). The direction of cyclic uniaxial stretch is indicated using
an arrow. Stretch amplitude (X % elongation) and duration of the experiment (Y in hrs) are
represented as AXTY. Unstretched cells on PDMC membranes were used as controls in the
study. Scale bar represents 50 m.
2) Figure 2a. F‐actin (red), microtubule (green) and DAPI nucleus (blue) stained merged confocal
images of control fibroblasts on unstretched PDMS membranes. b. Fibroblasts on
unidirectionally stretched PDMS membranes at 10% amplitude for 6 hours (A10T6). c. The
angular orientation of the nucleus was quantified under unidirectional cyclic stretch (1 Hz) with/
without cytochalasin‐D and nocodazole. No significant nuclear reorientation was observed with
depolymerization of microtubules; the re‐orientational response was also reduced in cells
following actin depolymerization d. Cyclic stretch (A10T6) induced fibroblast elongation was
quantified using the major axis of the cell. e. The cell aspect ratio increased under unidirectional
cyclic stretch. f. AFM was used to measure the effective elastic modulus of cells, subjected to
cyclic stretch in the presence and absence of cytoskeletal disruptors, and are shown for control
cells (unstretched and un‐treated) and cells stretched using 10% amplitude for 6 hours (A10T6;
n=25 each group). Significant differences between the different groups are indicated for p<0.01
(**) and p<0.05 (*).
3) Figure 3 a. Confocal image of fibroblast stained with phalloidin to visualize actin. b. The
corresponding binarized image of the cell was fit to an ellipse and the intensity distributions
were calculated at three equidistant transverse sections along the major cell axis. c. A
representative image of cortical actin intensity, calculated at one of the transverse sections
along major axis, is shown d. The full width at half maxima (FWHM) was calculated to compute
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the actin cortex thickness e. The peak intensity of cortical actin in cells subjected to uniaxial
cyclic stretch increased under stretch (A10T6) f. The total actin fluorescence intensity in
cyclically stretched cells (A10T6) also increased as compared to unstretched controls. Significant
differences in comparisons between the different groups are indicated for p<0.01 (**) and
p<0.05 (*).
4) Figure 4a. Changes in SF lengths under cyclic stretch (5% amplitude and 10% amplitude) were
calculated from experimental values (mean ± SEM) and were used to obtain the growth
evolution law for SF (equation (6)). The unknown constants in the equation were obtained by
fitting the experimental data to equation (6) (r2 = 0.99). b. Thermodynamic inequality for active
growth process was assessed as a function of experiment duration and SF orientation. c. & d.
The growth time scale (𝜏 ) was parametrically varied in the SF growth model for 5% stretch
(black) and 10% stretch (red). Experimental results are also plotted on the figures using black
and solid circles for 5% and 10% amplitude.
5) Figure 5a & b. The computed SF reorientation under cyclic uniaxial stretch agreed with
experimental observations which demonstrates the importance of SF remodeling time scale (𝜏 )
as a function of the stretch amplitude. 𝜏 was 9 hours for 5% and 3 hrs for 10% stretch
respectively (r2 > 0.9 for each)
6) Figure 6a & b. Changes to the effective elastic modulus, Eeff, of the cell (in Pa), subjected to 5%
and 10% amplitude uniaxial cyclic stretch, were measured and are plotted as a function SF
alignment angle. c. Changes in Eeff based on the model are shown as a function of SF
reorientation under cyclic stretch (5% stretch in red and 10% stretch shown in blue). d. Changes
in cell tractions (T11) under uniaxial stretch from the model calculations are compared with the
experimental results.
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List of Tables
1) Table I: Distribution of the angular orientations (Mean ± SEM) and lengths (Mean ± SEM) are
shown for fibroblasts on PDMS membranes and subjected to uniaxial cyclic stretch (1 Hz). The
stretch amplitude (X % elongation) and duration of the experiment (Y in hrs) are represented as
AXTY.
2) Table 2. The effect of cytoskeletal inhibitors (cytochalasin‐D and nocodazole) on the cells under
uniaxial cyclic stretch were assessed using the SF angular orientation (Mean ± SEM) for
fibroblasts cultured on PDMS membranes.
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Figure 1
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Figure 2
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26
Figure 3
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27
Figure 4
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28
Figure 5
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29
Figure 6
PEAK AVERAGE ANGLES
(°)
Control 43.6 and 150.9
5% for 3 hours 51.8 and 148.9
5% for 6 hours 76.05 and 127.5
10% for 3 hours 72.7 and 115.2
10% for 6 hours 79.85 and 99.17
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Table I
SF angles (°)
(Mean ± SEM)
SF length (µm)
(Mean ± SEM)
Control 43.6 ± 4.9 & 150.9 ± 7.9 25.1 ± 1.6
A5T3 51.8 ± 5.3 & 148.9 ± 4.9 28.3 ±1.7
A5T6 76.2 ± 7.5 & 127.5 ± 5.5 32.5 ± 2.2
A10T3 72.7 ± 7.6 & 115.2 ± 7.3 31.1 ± 1.7
A10T6 79.9 ± 2.8 & 99.2 ± 1.4 58.5 ± 3.6
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31
Table 2
Untreated
(Mean ± SEM)
Cytochalasin‐D
(Mean ± SEM)
Nocodazole
(Mean ± SEM)
Unstretched
Control
43.6 ±4.9 & 150.9 ±7.9 37.7± 5.7 & 154.1± 12.2 46.4 ±5.9 & 175.8±13.3
A5T6 76.2 ±7.5 & 127.5±5.5 40.7± 6.3 & 150± 7.5 50.2± 5.1 & 118± 7.9
A10T6 79.85± 2.8 & 99.2± 7.6 48.1 ±5.3 & 146.7± 7.5 90± 9.9 & 123.6±1.8
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32
List of Supplementary Figures
1) Figure S1 a & b. Custom designed cell stretcher and bioreactor. A thin PDMS membrane is
held using custom designed clamps in the 3D printed bioreactor that may be subjected to
uniaxial or biaxial stretching. c Quantification of Green‐Lagrange strains under uniaxial cyclic
strain (15% amplitude) are shown for the custom designed clamps using particle tracking
algorithm using computed strain displacements based on the underlying markers on the
substrate.
2) Figure S2. a. Confocal images show an individual 3T3 fibroblast cell, stained with phalloidin to
visualize SF and DAPI to identify the nucleus. b. Corresponding binarized image of the confocal
image is shown. c. The computed Fourier power spectrum was obtained using the binary
image d. Distribution of SF in the cell was calculated using their spatial frequency distribution.
3) Figure S3. Angular orientation of the SF (Mean ± St. Dev) under unidirectional cyclic stretch (1
Hz) with and without cytoskeletal disruptors (cytochalasin‐D and nocodazole)
4) Figure S4. Parametric study of the SF reorientation model for various remodeling time scale
(𝜏 ) in hours.
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Supplementary Figure S1
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Supplementary Figure S2
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Supplementary Figure S3
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Supplementary Figure S4
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SUPPLEMENTARY INFORMATION
1. Thin layer, finite thickness Hertzian model to compute the effective elastic modulus
The force measured during indentation on the cell (F) is related to the indentation
depth of the cantilever tip, δ, as [1]
𝐹 /
(1)
is the Poisson’s ratio, E is the Young’s modulus, and Rtip is the radius of the spherical
cantilever tip. The cell was assumed to be incompressible with = 0.5. The tip‐sample
separation, Δ, is related to indentation depth, δ, by a constant, C, and is given by δ =C – ∆ [2].
Using this expression, we modify the Hertzian relation as
𝐹 / 𝐾 /
. ∆ (2)
K is another constant and depends on both the coordinates of the substrate contact point
with the cantilever tip and the geometry of the tip. E can be directly calculated as a scaled
slope of the linear relation of the F2/3‐Δ curve. The point of maximum slope change
accompanied with change in sign of the force is considered to be the point of contact of the
cantilever tip with the cell. The force data is divided into a region prior to the contact point of
the cantilever tip with the cell and a region after contact. r2 values were calculated for each
of the raw data curves to estimate the goodness of the linear fit; values with the mean r2 value
higher than 0.9 are reported in this study.
The calculated values of E were normalized using a thin layer, finite thickness Hertz
contact model to determine the elastic modulus of soft materials [3]. The force (F)‐
displacement (δ) relationship is given as
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𝐹 ⁄
𝛿 1 𝜒 𝜒
𝛼 𝛽 𝜒 𝛼 𝛽 𝜒 (3)
𝑅1
𝑅2ℎ
𝐴𝑛𝑑 𝜒√𝑅𝛿
ℎ
In this expression, h is cell height, µ is the Poisson’s ration and the constants α and β are
functions of µ.
𝛼 . . . & 𝛽 . . .
(4)
The heights of the cells were taken to be 7 μm based on Z‐stacks of confocal images. For this
analysis, δ = 1 μm (~14% of the total cell height) for all data which lies within the 4 ‐30% range
of the cell height reported in literature that are required for the equations to be used for such
studies.
2. Constitutive modelling of fibroblast in a morphoelastic framework
We use a fiber‐reinforced orthotropic hyperelastic material model to quantify the SF
reorientation dynamics of fibroblasts under cyclic stretch. Our results show distinct growth
and reorientation of the SF along a near perpendicular direction perpendicular to stretch. We
also observe remodeling, in terms of increased cell stiffness and change in effective elastic
modulus, in the fibroblasts following cyclic stretch. The strain energy density function of the
cell combines a neo‐Hookean response for the cytoplasm and a ‘standard reinforcing model’
[4,5] for the two families of SF in the cell, as obtained from our experimental observations.
The direction of fiber orientation is obtained from the angular distributions of SF from two
term Gaussian distribution functions.
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The deformation gradient, F, is written in terms of a multiplicative decomposition of the
growth tensor, G and the elastic deformation tensor, A, in the framework of morphoelasticity
[6].
𝑭 𝑨. 𝑮 where 𝑭𝒊𝒋𝝏𝒙𝒊
𝝏𝑿𝒋 (5)
The strain energy density function of the cell is expressed as a function of the elastic
deformation A, such that,
𝑊 𝐴 𝑊 𝑊 𝐼 3 𝐼 1 𝐼 1 (6)
Where 𝜇 refers to the shear modulus (𝜇 0) and 𝛼 is a measurement of the additional fiber
reinforcement (𝛼 0). The strain energy function also satisfies the condition [7]
𝑊 𝑪 , 𝑴𝒐𝒊 𝑊 𝑸 𝑪 𝑸, 𝑸 𝑴𝒐
𝒊 𝑸 (7)
The structure tensor is given as 𝑴𝟎𝒊 𝒎𝒐
𝒊 ⊗ 𝒎𝒐𝒊 , where 𝒎𝒐
𝒊 denotes the stress fiber
distributions. 𝑪 𝑨𝑻𝑨 [8] and 𝑸 is an arbitrary orthogonal second‐order tensor. The
invariants used in the strain energy function are given as: ‐
𝑰𝟏 𝒕𝒓 𝑪𝑬 𝑰𝟐𝟏
𝟐𝒕𝒓 𝑪𝑬 𝟐 𝒕𝒓 𝑪𝑬𝟐
𝑰𝟑 𝒅𝒆𝒕 𝑪𝑬 (8)
𝑰𝟒 𝒎𝒐𝟏. 𝑪𝑬𝒎𝒐
𝟏 𝑰𝟔 𝒎𝒐𝟐. 𝑪𝑬𝒎𝒐
𝟐
We consider the biological cell as an incompressible material, such that 𝑑𝑒𝑡 𝑨 1. For
uniaxial stretch, the elastic deformation tensor, 𝑨 is given by
𝑨
𝜆 0 00
√0
0 0√
(9)
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where 𝜆 is the magnitude of stretch applied along X‐axis. The Cauchy stress can be computed
as a function of the elastic deformation 𝑨, using the relation [9]: ‐
𝑻 𝑝𝑰 2𝑨 𝝏
𝑪𝑨 (10)
Where p refers to the Lagrange multiplier due to incompressibility assumption.
The condition for mechanical equilibrium is given by 𝜵. 𝑻 0. In our model, the uniaxial cyclic
stretch is applied in the X‐direction and the cell is adhered on the substrate in the X‐Y plane.
Therefore, the unit vector along the direction SF is given as
𝒎𝒐𝟏 𝑐𝑜𝑠𝜃 𝒆𝟏 𝑠𝑖𝑛𝜃 𝒆𝟐 (11)
𝒎𝒐𝟐 𝑐𝑜𝑠𝜃 𝒆𝟏 𝑠𝑖𝑛𝜃 𝒆𝟐
Using the above relations, we can compute the following identities
𝝏𝑰𝟏
𝝏𝑪= 0,
𝝏𝑰𝟒
𝝏𝑪=𝒎𝒐
𝟏 ⊗ 𝒎𝒐𝟏 and
𝝏𝑰𝟔
𝝏𝑪=𝒎𝒐
𝟐 ⊗ 𝒎𝒐𝟐 (12)
Substituting these expressions, the constitutive equation for the cell can be written as
𝑻 𝑝𝑰 𝑨 𝜇𝑰 𝜇𝛼 𝟐 𝐼 1 𝒎𝒐𝟏 ⊗ 𝒂𝒐
𝟏 2 𝐼 1 𝒎𝒐𝟐 ⊗ 𝒎𝒐
𝟐 𝑨𝑻 (13)
The corresponding tractions for the cell in the directions of 𝒆𝟏, 𝒆𝟐 and 𝒆𝟑 are computed as
𝒕𝒊 𝑻𝒊𝒊𝒆𝒊
𝑡 𝑝 𝜇 𝜆 2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑐𝑜𝑠 𝜃 2𝛼 𝜆 𝑐𝑜𝑠 𝜃
𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑐𝑜𝑠 𝜃
𝑡 𝑝 𝜇 𝜆 2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑠𝑖𝑛 𝜃 2𝛼 𝜆 𝑐𝑜𝑠 𝜃
𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑠𝑖𝑛 𝜃
𝑡 𝑝 𝜇 𝜆 where 𝜆 𝜆 ; 𝜆 𝜆√
(14)
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3. Growth Law for evolution in the SF length under cyclic uniaxial stretch
We propose a growth law for the SF under uniaxial cyclic stretch as a combined function of
both the stretch amplitude and experimental duration.
𝛾 𝑓 𝜆 𝑓 𝑡 (15)
Based on our experimental observation, we formulate an evolution law for the increase in
length of SF under cyclic stretch as
𝛾 143 𝜆 1 3.8 𝜆 1 𝑒 / 𝑐 (16)
. 𝑒 /
The parameters are obtained by curve‐fitting to the experimental data, where c= is a constant
indicative of the length of SF in unstretched cells, 𝜏 is equivalent to a growth time scale of the
cell; 𝜆 is the corresponding stretch and t is time.
Based on the growth law of the SF under stretch, we now construct the growth tensor for the
growth of a single actin stress fiber in the cylindrical co‐ordinate system:
𝑮𝒓 𝜽 𝒛 𝑰 𝛾 1 𝒆 ⨂ 𝒆 (17)
𝛾 corresponds to the growth of actin SF in the radial direction. We can then transform 𝐺
from the cylindrical co‐ordinate system to Cartesian using the transformation: ‐
𝑮𝒙 𝒚 𝒛 𝑸. 𝑮𝒓 𝜽 𝒛. 𝑸𝑻 𝒘𝒉𝒆𝒓𝒆 𝑸𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 0𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 00 0 1
(18)
4. Thermodynamic Constraints on the Growth Law
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Thermodynamic restrictions on the growth law were computed from the Claussius–
Duhem inequality using the standard the Coleman–Noll procedure. Neglecting temperature
gradients, the thermodynamic inequality for active growth processes is written as [ 10]: ‐
𝑴: 𝑮 𝟏𝑮 𝐽 ℎ 0 (19)
M is the Mandel stress tensor; 𝑀 𝐽 𝐴 𝑇𝐴 and the entropy contribution in the process
is given by ℎ. In our case 𝐽 =1 due to incompressibility. We consider the SF as a network non‐
Gaussian chain and use a statistical treatment of randomly jointed chains to quantify the
change in entropy [11] under uniaxial tension 𝜆 𝜆 ; 𝜆 𝜆√
ℎ 𝑘𝑁
(20)
𝑛 is the number of links in one chain and 𝑙 corresponds to the length of each link. The ratio
is a measure of the fractional extension of the chain; 𝑟 is the mean square length of
chains in unstrained state. 𝑘 is the Boltzmann constant and N refers to the number of chains
in the network. This term is much smaller in magnitude as compared to the 𝑀: 𝑮 𝟏𝑮 term
and is hence neglected in the remaining calculations.
5. Dynamics of SF remodeling under cyclic stretch
Uniaxial cyclic stretching of cells causes reorientation along a near‐perpendicular direction
to the direction of applied stretch. The reorientation dynamics is dependent on the stretch
amplitude as well as the duration of stretch. Based on our observations, we propose a
reorientation law for the SF (7,12)
𝒎𝒐𝒊
𝒏 𝒏. 𝒎𝒐𝒊 𝒎𝒐
𝒊 (21)
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The vector n represents the unit vector along the direction of preferred cytoskeletal
orientation under stretch, which is along Y‐axis (e2) (when the cyclic stretch is applied along
X‐axis). The term 𝑡 refers to a remodeling time scale which also depends on the amplitude
of cyclic stretch. Using this reorientation law, we can predict the remodeling dynamics of actin
stress fiber in cells under different stretch amplitude and duration. Using the equations for
𝒎𝒐𝒊 the reorientation law becomes
=
𝑐𝑜𝑠 𝜃 (22)
6. Changes in the effective Young’s modulus of the cell due to stress fiber remodeling
We quantified the effective Young’s modulus along the direction of SF reorientation (i.e 𝒆 )
as the gradient of uniaxial tension, N, with respect to the stretch in that direction (10)
𝑬𝒆𝒇𝒇|𝒆𝟐𝝏𝑵
𝝏𝝀𝟐| 𝝀𝟏 𝝀𝟐 𝟏 (23)
Where N= 𝑡 𝑎𝑛𝑑 𝑡 𝑡 0. From the constitutive equation, we can therefore write the
tractions along the 𝒆 , 𝒆𝟐 and 𝒆 direction as
𝑡 𝑝 𝜇 𝜆 2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑐𝑜𝑠 𝜃
2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑐𝑜𝑠 𝜃
𝑡 𝑝 𝜇 𝜆 2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑠𝑖𝑛 𝜃
2𝛼 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 𝜆 𝑠𝑖𝑛 𝜃
𝑡 𝑝 𝜇 𝜆 (24)
Using the expressions of 𝑡 , 𝑡 and 𝑡 , we write two additional terms as
𝐴 + 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 (𝜆 𝑐𝑜𝑠 𝜃 ‐𝜆 𝑠𝑖𝑛 𝜃 )+ 𝜆 𝑐𝑜𝑠 𝜃
𝜆 𝑠𝑖𝑛 𝜃 1 (𝜆 𝑐𝑜𝑠 𝜃 ‐𝜆 𝑠𝑖𝑛 𝜃 )
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𝐵 + + 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 (𝜆 𝑐𝑜𝑠 𝜃
𝜆 𝑠𝑖𝑛 𝜃 )+ 𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 1 (𝜆 𝑐𝑜𝑠 𝜃 𝜆 𝑠𝑖𝑛 𝜃 )
(25)
Using the expressions of 𝐴 and 𝐵 we compute the term 𝝏𝝀𝟏
𝝏𝝀𝟐 and using
equation (23) we obtain the expression of
µ 𝐸 |𝑒 = (26)
From this expression at 𝜃=90°; we obtain 𝐸 |𝒆 =𝜇 3 8𝛼 . The parameters µ and α are
obtained by equating with the experimentally obtained Young’s modulus values obtained
from the AFM indentation experiments.
7. Formulation of generalized structure tensor based on distributed fiber orientations
We quantified the angular orientation of the distributed SF in the cell using a two term
Gaussian distribution function. The orientation distribution function 𝜌 𝒎𝒐𝒊 𝜃 is based on
the distribution of fibers in their referential configuration.
𝒎𝒐𝒊 𝜃 𝑐𝑜𝑠𝜃 𝒆𝟏 𝑠𝑖𝑛𝜃 𝒆𝟐 (27)
The Gaussian distribution has a symmetry requirement 𝜌 𝒎𝒐𝒊 𝜃 ≡ 𝜌 𝒎𝒐
𝒊 𝜃 and is
also normalized such that
𝜌 𝒎𝒐𝒊 𝜃 𝑑𝜃
√𝑒 𝑑𝜃 1 (28)
Based on these distributions we formulate a generalized symmetric structure tensor using the
relation [13]:
𝑯 𝜌 𝒎𝒐𝒊 𝜃 . 𝒎𝒐
𝒊 𝜃 ⊗ 𝒎𝒐𝒊 𝜃 𝑑𝜃 (29)
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Where 𝑖 ∈ 1,2 for the two families of distributed SF. Components of the structure tensor
H are calculated as:
𝐻 cos 𝜃 1
𝜎√2𝜋𝑒 𝑑𝜃
𝐻 sin 𝜃 1
𝜎√2𝜋𝑒 𝑑𝜃
𝐻 cos 𝜃 sin 𝜃 √
𝑒 𝑑𝜃 (30)
Because H is symmetric, 𝐻 𝐻 , and the other terms are zero. Using the structure
tensor for the distributed fiber reorientations we write the constitutive equation as:
𝑻 𝑝𝑰 𝑨 𝜇𝑰 𝜇𝛼 𝟐 𝐼 1 𝐻 2 𝐼 1 𝐻 𝑨𝑻 (31)
Where 𝐼 and 𝐼 can be re‐calculated using 𝐻 as
𝐼 𝑪 : 𝑯 𝑎𝑛𝑑 𝐼 𝑪 : 𝑯 (32)
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