stress fiber growth and remodeling determines cellular ...aritra chatterjee1, paturu kondaiah2,...

1

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

not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which wasthis version posted April 29, 2019. ; https://doi.org/10.1101/622092doi: bioRxiv preprint

https://doi.org/10.1101/622092

2

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].


https://doi.org/10.1101/622092

3

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.


https://doi.org/10.1101/622092

4

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


https://doi.org/10.1101/622092

5

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


https://doi.org/10.1101/622092

6

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


https://doi.org/10.1101/622092

7

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


https://doi.org/10.1101/622092

8

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)


https://doi.org/10.1101/622092

9

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


https://doi.org/10.1101/622092

10

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


https://doi.org/10.1101/622092

11

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


https://doi.org/10.1101/622092

12

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


https://doi.org/10.1101/622092

13

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


https://doi.org/10.1101/622092

14

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


https://doi.org/10.1101/622092

15

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


https://doi.org/10.1101/622092

16

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)


https://doi.org/10.1101/622092

17

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.

References

1. Robertson, A. M. & Watton, P. N. Mechanobiology of the arterial wall. Transp. Biol. Media 275–347 (2013). Doi:10.1016/B978‐0‐12‐415824‐5.00008‐4

2. Hsu, H.‐J., Lee, C.‐F., Locke, A., Vanderzyl, S. Q. & Kaunas, R. Stretch‐induced stress fiber remodeling and the activations of jnk and erk depend on mechanical strain rate, but not fak. Plos one 5, e12470 (2010).

3. Humphrey, J. D. & Rajagopal, K. R. A constrained mixture model for growth and remodeling of soft tissues. Math. Model. Methods Appl. Sci. 12, 407–430 (2002).

4. Livne, A., Bouchbinder, E. & Geiger, B. Cell reorientation under cyclic stretching. Nat. Commun. 5, 1–8 (2014).

5. Balaban, N. Q. et al. Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nat. Cell Biol. 3, 466–472 (2001).

6. Cirka, H. et al. Active Traction Force Response to Long‐Term Cyclic Stretch Is Dependent on Cell Pre‐stress. Biophys. J. 110, 1845–1857 (2016).

7. Perrault, C. M. et al. Traction Forces of Endothelial Cells under Slow Shear Flow. Biophys. J. 109, 1533–1536 (2015).

8. Buck, R. C. Reorientation response of cells to repeated stretch and recoil of the substratum. Exp. Cell Res. 127, 470–474 (1980).

9. Wang, J. H.‐C., Goldschmidt‐Clermont, P., Wille, J. & Yin, F. C.‐P. Specificity of endothelial cell


https://doi.org/10.1101/622092

18

reorientation in response to cyclic mechanical stretching. J. Biomech. 34, 1563–1572 (2001).

10. Hayakawa, K., Sato, N. & Obinata, T. Dynamic reorientation of cultured cells and stress fibers under mechanical stress from periodic stretching. Exp. Cell Res. 268, 104–114 (2001).

11. Chen, Y., Pasapera, A. M., Koretsky, A. P. & Waterman, C. M. Orientation‐specific responses to sustained uniaxial stretching in focal adhesion growth and turnover. Proc. Natl. Acad. Sci. U. S. A. 110, E2352‐61 (2013).

12. Geiger, B., Spatz, J. P. & Bershadsky, A. D. Environmental sensing through focal adhesions. Nat. Rev. Mol. Cell Biol. 10, 21–33 (2009).

13. Greiner, A. M., Chen, H., Spatz, J. P. & Kemkemer, R. Cyclic tensile strain controls cell shape and directs actin stress fiber formation and focal adhesion alignment in spreading cells. PLoS One 8, (2013).

14. Krishnan, R. et al. Substrate stiffening promotes endothelial monolayer disruption through enhanced physical forces. Am. J. Physiol. Physiol. 300, C146–C154 (2011).

15. Nekouzadeh, A., Pryse, K. M., Elson, E. L. & Genin, G. M. Stretch‐activated force shedding, force recovery, and cytoskeletal remodeling in contractile fibroblasts. J. Biomech. 41, 2964–71 (2008).

16. Cui, Y. et al. Cyclic stretching of soft substrates induces spreading and growth. Nat. Commun. 6, 6333 (2015).

17. Goldyn, A. M., Rioja, B. A., Spatz, J. P., Ballestrem, C. & Kemkemer, R. Force‐induced cell polarisation is linked to RhoA‐driven microtubule‐independent focal‐adhesion sliding. J. Cell Sci. 122, 3644–3651 (2009).

18. Jungbauer, S., Gao, H., Spatz, J. P. & Kemkemer, R. Two characteristic regimes in frequency‐dependent dynamic reorientation of fibroblasts on cyclically stretched substrates. Biophys. J. 95, 3470–3478 (2008).

19. Kaunas, R., Nguyen, P., Usami, S. & Chien, S. Cooperative effects of Rho and mechanical stretch on stress fiber organization. Proc. Natl. Acad. Sci. U. S. A. 102, 15895–900 (2005).

20. Chen, K. et al. Role of boundary conditions in determining cell alignment in response to stretch. Proc. Natl. Acad. Sci. 115, 986 LP‐991 (2018).

21. Schriefl, A. J., Reinisch, A. J., Sankaran, S., Pierce, D. M. & Holzapfel, G. A. Quantitative assessment of collagen fibre orientations from two‐dimensional images of soft biological tissues. J. R. Soc. Interface 9, 3081–3093 (2012).

22. Kulkarni, A. H., Chatterjee, A., Kondaiah, P. & Gundiah, N. TGF‐ β induces changes in breast cancer cell deformability. Phys. Biol. 15, 065005 (2018).

23. Barreto, S., Clausen, C. H., Perrault, C. M., Fletcher, D. A. & Lacroix, D. A multi‐structural single cell model of force‐induced interactions of cytoskeletal components. Biomaterials 34, 6119–6126 (2013).

24. Darling, E. M., Zauscher, S., Block, J. A. & Guilak, F. A thin‐layer model for viscoelastic, stress‐relaxation testing of cells using atomic force microscopy: do cell properties reflect metastatic potential? Biophys. J. 92, 1784–1791 (2007).

25. Rodriguez, E. K., Hoger, A. & McCulloch, A. D. Stress‐dependent finite growth in soft elastic tissues. J. Biomech. 27, 455–467 (1994).

26. Merodio, J. & Ogden, R. W. Material instabilities in fiber‐reinforced nonlinearly elastic solids under plane deformation. Arch. Mech. 54, 525–552 (2002).


https://doi.org/10.1101/622092

19

27. Gundiah, N., Ratcliffe, M. B. & Pruitt, L. A. The biomechanics of arterial elastin. J. Mech. Behav. Biomed. Mater. 2, 288–296 (2009).

28. Melnik, A. V. & Goriely, A. Dynamic fiber reorientation in a fiber‐reinforced hyperelastic material. Math. Mech. Solids 18, 634–648 (2013).

29. Göktepe, S., Abilez, O. J., Parker, K. K. & Kuhl, E. A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis. J. Theor. Biol. 265, 433–442 (2010).

30. Menzel, A. Modelling of anisotropic growth in biological tissues. Biomech. Model. Mechanobiol. 3, 147–171 (2005).

31. Goriely, A. The Mathematics and Mechanics of Biological Growth. 45, (Springer New York, 2017).

32. Imatani, S. & Maugin, G. . A constitutive model for material growth and its application to three‐dimensional finite element analysis. Mech. Res. Commun. 29, 477–483 (2002).

33. De, R., Zemel, A. & Safran, S. A. Dynamics of cell orientation. Nat. Phys. 3, 655–659 (2007).

34. Bauër, P. et al. A new method to measure mechanics and dynamic assembly of branched actin networks. Sci. Rep. 7, 15688 (2017).

35. Bernheim‐Groswasser, A., Wiesner, S., Golsteyn, R. M., Carlier, M.‐F. & Sykes, C. The dynamics of actin‐based motility depend on surface parameters. Nature 417, 308–311 (2002).

36. Parekh, S. H., Chaudhuri, O., Theriot, J. A. & Fletcher, D. A. Loading history determines the velocity of actin‐network growth. Nat. Cell Biol. 7, 1119–1123 (2005).

37. Shaevitz, J. W. & Fletcher, D. A. Load fluctuations drive actin network growth. Proc. Natl. Acad. Sci. 104, 15688–15692 (2007).

38. Salbreux, G., Charras, G. & Paluch, E. Actin cortex mechanics and cellular morphogenesis. Trends Cell Biol. 22, 536–545 (2012).

39. Haase, K. & Pelling, A. E. The role of the actin cortex in maintaining cell shape. Commun. Integr. Biol. 6, e26714 (2013).

40. Gavara, N. & Chadwick, R. S. Relationship between cell stiffness and stress fiber amount, assessed by simultaneous atomic force microscopy and live‐cell fluorescence imaging. Biomech. Model. Mechanobiol. 15, 511–523 (2016).

41. Boal, D. Mechanics of the Cell. (Cambridge University Press, 2012). doi:10.1017/CBO9781139022217

42. Hwang, Y. & Barakat, A. I. Dynamics of mechanical signal transmission through prestressed stress fibers. PLoS One 7, e35343 (2012).

43. Kaunas, R., Hsu, H.‐J. & Deguchi, S. Sarcomeric model of stretch‐induced stress fiber reorganization. Cell Health Cytoskelet. 3, 13 (2010).

44. Tojkander, S. et al. A molecular pathway for myosin ii recruitment to stress fibers. Curr. Biol. 21, 539–550 (2011).

45. Livne, A. & Geiger, B. The inner workings of stress fibers ‐ from contractile machinery to focal adhesions and back. J. Cell Sci. 129, 1293–304 (2016).

46. Hotulainen, P. & Lappalainen, P. Stress fibers are generated by two distinct actin assembly mechanisms in motile cells. J. Cell Biol. 173, 383–94 (2006).

47. Luxton, G. W. G., Gomes, E. R., Folker, E. S., Vintinner, E. & Gundersen, G. G. Linear arrays of nuclear envelope proteins harness retrograde actin flow for nuclear movement. Science 329, 956–9 (2010).


https://doi.org/10.1101/622092

20

48. Agrawal, V., Kollimada, S. A., Byju, A. G. & Gundiah, N. Regional variations in the nonlinearity and anisotropy of bovine aortic elastin. Biomech. Model. Mechanobiol. 12, 1181–1194 (2013).


https://doi.org/10.1101/622092

21

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


https://doi.org/10.1101/622092

22

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.


https://doi.org/10.1101/622092

23

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.


https://doi.org/10.1101/622092

24

Figure 1


https://doi.org/10.1101/622092

25

Figure 2


https://doi.org/10.1101/622092

26

Figure 3


https://doi.org/10.1101/622092

27

Figure 4


https://doi.org/10.1101/622092

28

Figure 5


https://doi.org/10.1101/622092

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


https://doi.org/10.1101/622092

30

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


https://doi.org/10.1101/622092

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


https://doi.org/10.1101/622092

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.


https://doi.org/10.1101/622092

33

Supplementary Figure S1


https://doi.org/10.1101/622092

34



https://doi.org/10.1101/622092

35



https://doi.org/10.1101/622092

36



https://doi.org/10.1101/622092

37

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


https://doi.org/10.1101/622092

38

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.


https://doi.org/10.1101/622092

39

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)


https://doi.org/10.1101/622092

40

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)


https://doi.org/10.1101/622092

41

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


https://doi.org/10.1101/622092

42

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)


https://doi.org/10.1101/622092

43

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 )


https://doi.org/10.1101/622092

44

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)


https://doi.org/10.1101/622092

45

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)


https://doi.org/10.1101/622092

46

References

1. Johnson, K. L. Contact Mechanics. (Cambridge University Press, 1985).

doi:10.1017/CBO9781139171731.

2. Guo, S. & Akhremitchev, B. B. Packing density and structural heterogeneity of insulin

amyloid fibrils measured by afm nanoindentation. (2006). doi:10.1021/bm0600724

3. Dimitriadis, E. K., Horkay, F., Maresca, J., Kachar, B. & Chadwick, R. S. Determination of

elastic moduli of thin layers of soft material using the atomic force microscope.

Biophys. J. 82, 2798–2810 (2002).

4. Qiu, G. Y. & Pence, T. J. Loss of ellipticity in plane deformation of a simple directionally

reinforced incompressible nonlinearly elastic solid. J. Elast. 49, 31–63 (1997).

5. Merodio, J. & Ogden, R. W. Material instabilities in fiber‐reinforced nonlinearly elastic

solids under plane deformation. Arch. Mech. 54, 525–552 (2002).

6. Rodriguez, E. K., Hoger, A. & McCulloch, A. D. Stress‐dependent finite growth in soft

elastic tissues. J. Biomech. 27, 455–467 (1994).

7. Melnik, A. V. & Goriely, A. Dynamic fiber reorientation in a fiber‐reinforced hyperelastic

material. Math. Mech. Solids 18, 634–648 (2013).

8. Göktepe, S., Abilez, O. J., Parker, K. K. & Kuhl, E. A multiscale model for eccentric and

concentric cardiac growth through sarcomerogenesis. J. Theor. Biol. 265, 433–442

(2010).

9. Vandiver, R. M. Morphoelasticity: the mechanics and mathematics of elastic growth.

144 (2009). doi:10.1109/ICTAI.2005.64

10. Goriely, A. The mathematics and mechanics of biological growth. (Springer, 2017).

11. Treloar, L. R. G. The physics of rubber elasticity. (Clarendon Press, 2005).

12. Menzel, A. Modelling of anisotropic growth in biological tissues. Biomech. Model.

Mechanobiol. 3, 147–171 (2005).

13. Gasser, T. C., Ogden, R. W. & Holzapfel, G. A. Hyperelastic modelling of arterial layers

with distributed collagen fibre orientations. J. R. Soc. Interface 3, 15–35 (2006).


https://doi.org/10.1101/622092

stress fiber growth and remodeling determines cellular ...aritra chatterjee1, paturu kondaiah2,...

Documents