Stochastic fluctuation-induced cell polarization on elastic substrates: A cytoskeleton-based mechanical model

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Stochastic fluctuation-induced cell polarization on elastic substrates: A cytoskeleton-based mechanical model Yuan Qin , Yuhui Li , Li-Yuan Zhang , Guang-Kui Xu PII: DOI: Reference:

S0022-5096(19)30925-1 https://doi.org/10.1016/j.jmps.2020.103872 MPS 103872

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Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

30 September 2019 5 December 2019 8 January 2020

Please cite this article as: Yuan Qin , Yuhui Li , Li-Yuan Zhang , Guang-Kui Xu , Stochastic fluctuation-induced cell polarization on elastic substrates: A cytoskeleton-based mechanical model, Journal of the Mechanics and Physics of Solids (2020), doi: https://doi.org/10.1016/j.jmps.2020.103872

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Stochastic fluctuation-induced cell polarization on elastic substrates: A cytoskeleton-based mechanical model

Yuan Qin,1# Yuhui Li, 2# Li-Yuan Zhang,3 and Guang-Kui Xu1

1

International Center for Applied Mechanics, SVL, School of Aerospace Engineering, Xi’an Jiaotong University, Xi'an 710049, China 2

3

# 

Bioinspored Engineering and Biomechanics Center, Xi’an Jiaotong University, Xi’an 710049, China

School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

These authors contributed equally to this work. Corresponding author. E-mail: [email protected] 1

ABSTRACT Mechanical cues from the microenvironments play an important role in many physiological and pathological processes, e.g., stem cell differentiation and cancer cell metastasis. Recent experiments showed that the spreading and polarization of cells are highly associated with the substrate rigidity. However, the underlying mechanisms of these cell behaviors are still unknown. Here, we develop a cytoskeleton-based mechanical model to study the cell morphology on substrates of various rigidities, and carry out the experiments of mouse embryo fibroblasts on hydrogels of different rigidities. Our theoretical model involves both biomechanical and biochemical mechanisms, including actin polymerization, myosin motors contractility, integrin binding dynamics, membrane deformation and substrate stiffness. Using this model, we can simulate the spatiotemporal evolution of cell morphology on substrates of various rigidities. Interestingly, we find that the stochastic fluctuation in the initial cell shape can lead to the spontaneous generation of cell polarization on elastic substrates. Moreover, a cell can exhibit a more anisotropic geometry on stiff substrates than on soft ones. Our theoretical predictions are in good agreement with experimental results. The proposed model is capable of exploring the cell morphology regulated by substrate rigidities, and sheds lights on the functioning of cellular mechanosensing systems.

Keywords: Substrate stiffness, cell polarization, cell spreading, mechanical model

2

1 INTRODUCTION Cells sense and respond to both biomechanical and biochemical cues of their residing microenvironments (Boerckel et al., 2011; Cheng et al., 2017). It becomes increasingly apparent that the rigidity of underlying substrates serves as a regulator for a variety of cellular behaviors and functions (Engler et al., 2006; He et al., 2014; Wang et al., 2017). For example, recent experiments evidenced that the dynamics of cell morphology on elastic substrates strongly correlates with the substrate compliance (Nisenholz et al., 2014; Yeung et al., 2005). Cells spread well on stiff substrates, achieving a broader and flatter morphology, but fail to spread on very soft substrates. Moreover, cells cultured on stiff substrates are prone to polarize and possess high migration speed and motility (Prager-Khoutorsky et al., 2011; Solon et al., 2007). The spreading dynamics of cells lies at the heart of orchestrating internal cytoskeleton organization, sculpting cell shape, and regulating cell fate (McBeath et al., 2004; Zhao et al., 2017). The spreading process depends on a rich range of biomechanical and biochemical mechanisms, including actin polymerization (Döbereiner et al., 2004), focal adhesions (FAs) dynamics (Nicolas et al., 2004; Kalappurakkal et al., 2019), cytoskeleton contractility (Giannone et al., 2004; Wilson et al., 2010) and membrane deformation (Gauthier et al., 2011; Schweitzer et al., 2014). After the initial contact of a cell with its underlying substrate, continuous cell spreading is driven by the protrusion of the dense lamellipodium network at the leading edge (Ponti et al., 2004). The protrusion of lamellipodium leads to membrane deformation and thus increases the membrane tension that resists the extension of a cell (Gauthier et al., 2011). Due to the contractive activities of myosin motors within cytoskeleton, the polymerizing actin network flows continuously in the nucleus direction, known as retrograde flow (Giannone et al., 2004; Wilson et al., 2010). At the cell-substrate interface, a collective of adhesion bonds are formed within FAs, which provide traction forces for actin polymerization and in turn cell spreading 3

(Ponti et al., 2004; Shao et al., 2012). Cell polarization plays a fundamental role in many cellular functions, such as cell division, cell formulation, and directional migration (Huttenlocher, 2005; Kapsenberg, 2003). Cell polarization results from either the external environmental cue, including both biochemical and biomechanical stimuli, or the intrinsic ability (Théry et al., 2006; Thompson, 2013). A number of studies tried to reveal the mechanisms of the generation of cell polarization, by combining experimental observations with mathematical/computational models (Freisinger et al., 2013; Slaughter et al., 2013). Most of these studies owed cell polarization to signaling reaction-diffusion processes on the basis of Turing’s theory (Mogilner et al., 2012; Turing, 1990). Based on the local-excitation-global-inhibition mechanism in which a slowly diffusing activator can self-amplify locally while a fast diffusing inhibitor spreads across the cell (Slaughter et al., 2013), Goryachev and Leda (2017) provided a good explanation for yeast cell polarization formation. However, in Turing’s theory, the effects of stress, motion and material properties (e.g., elasticity and viscosity) are neglected (Howard et al., 2011; Goehring and Grill, 2013). Recent studies have shown that the mechanical properties of cytoskeleton are able to trigger cell polarization, direct the motility of migrating cells, and polarize actin tails formed by intercellular pathogens (Tondon and Kaunas 2014; Zhang et al., 2014; Wang et al., 2017). Furthermore, the stochastic fluctuations within or without cells have been found to play a key role in cell polarization formation (Bressloff and Xu, 2015; Lawson et al., 2013). These findings suggest that the signaling diffusing, mechanical properties of the cell and the substrate, as well as the environmental or cellular internal stochastic fluctuations, should be coupled to contribute to cell polarization generation. In this study, we develop a cytoskeleton-based mechanical model to qualitatively and quantitatively investigate the influence of the substrate rigidity on cell spreading and polarization behaviors. The present model involves both biomechanical and biochemical mechanisms during cell morphology evolution, including actin polymerization, myosin motors contractility, integrin-mediated adhesion dynamics, 4

membrane deformation and substrate stiffness. If the stochastic fluctuation is neglected, our model can capture the effects of substrate rigidity on cell spreading area but fails to simulate cell polarization generation. When the fluctuation is introduced in the initial configuration of a cell, we can successfully simulate both cell spreading and polarization behaviors on elastic substrates. We show that a cell can possess higher polarization degree on stiff substrate than on soft one, because the larger traction force on stiff substrate can amplify the initial length difference of stress fibers (SFs) in different directions. Our theoretical results are in good agreement with both our experiment results and experiment data from other groups.

2 MATERIALS AND METHODS Figure 1 illustrates the morphologies of adhesive cells attached to soft and stiff substrates that lead to weak and strong cell polarization, respectively (Solon et al., 2007; Prager-Khoutorsky et al., 2011). In addition, cells spreading on stiff substrates achieve larger steady area than those on soft substrates (Yeung et al., 2005; Nisenholz et al., 2014). In this section, we first introduce our experiments of mouse embryo fibroblasts

on

hydrogels

with

various

rigidities,

and

then

develop

a

cytoskeleton-based mechanical model to simulate cell morphology on elastic substrates.

Fig. 1. Schematics of (a) a cell adhering on a soft substrate and showing weak polarization, and (b) a cell adhering on a stiff substrate and achieving strong polarization.

5

2.1 Experiments 2.1.1 PA gel preparation and modification Cover glasses (22 x 22 mm2) were clean by being immersed in 0.1M NaOH water solution for 5 mins and subsequently washed thoroughly by ddH2O on a rotator for 20 mins before being dried and treated with 3-(Trimethoxysilyl) propyl acrylate for 1 h. After being washed by ddH2O, hydrophilic cover glasses were dried by clean air flow. To prepare hydrogels with different rigidities, 400 μl Acrylamide solution (40% in HEPES) and corresponding amounts of bis-acrylamide solution (2% in HEPES) were pipetted into an Eppendorf tube, followed by 100 μl of N-Hydroxyethyl acrylamide solution. After vortex, the air in the mixture was removed by vacuum pump. To promote the crosslinking process, 25 μl water solution of ammonium persulfate (10%, APS) and 2.5 μl N,N,N,N′-Tetramethylethylenediamine (TEMED) was added into the mixture. After gently mixing, 20 μl mixture was dropped on a hydrophilic cover glass. A plasma treated cover glass (18 x 18 mm2) was immediately place upon it. After 15 mins, the upper cover glass is removed in water, followed by three times of PBS washing and swelling in PBS overnight. The Young’s moduli of the hydrogels for experiments are 1.58±0.28 kPa, 9.16±2.43 kPa and 34.05±9.8 kPa measured by an atomic force microscopy. The hydrogel was then incubated in Fibronectin solution (50 mg/ml in PBS) for 1 h to improve cell adhesion on the hydrogel surface. After being washed in PBS for three times, the hydrogels were well-prepared for cell culture. 2.1.2 Cell polarization measurement Mouse embryo fibroblasts (MEFs) were cultured in Dulbecco's modified Eagle's medium (DMEM) with 10% fetal bovine serum (FBS) and 1% penicillin streptomycin. Cells were fostered at 37 ℃ with 5% CO2 and subculture for every three days. Before being fixed by 4% paraformaldehyde for 10 mins, the MEFs were seeded at a concentration of 50,000 cells/ml on the fibronectin-coated hydrogels and incubated for 24 hrs. To stain the cell nuclei and cytoskeleton, fixed MEFs were 6

firstly permeabilized by Triton X-100 for 5 mins and blocked by 10% goat serum in PBS for 1h. After that, MEFs were incubated with Alexa Fluor 594 labeled anti-alpha tubulin at 4 ℃ overnight before staining F-actin with Alexa Fluor 488 Phalloidin for 1 h. MEFs were then treated with DAPI for 10 mins to label cell nuclei. A confocal laser scanning microscope was employed on capturing fluorescent images of MEFs. 2.2 Cytoskeleton-based mechanical model Figure 2 illustrates a cell spreading on a compliant substrate. As shown in Fig. 2(a), the SFs, mainly consisting of actins, myosin motors, and adapter proteins, are treated as m rods which bridge the nucleus and the membrane. In the periphery of a cell (Fig. 2(b)), each SF connects to the substrate via FAs formed by integrin-ligand bonds, through which the cell can sense mechanical properties of the substrate. The growth of each SF derives cell spreading, and the forces acting on a single SF are illustrated in Fig. 2(c), which will be described below. Using this structural configuration, the spatiotemporal evolution of cell morphology can be characterized by SF-membrane joints.

Fig. 2 (a) Schematic of a cell. The cell is mainly supported by SFs which connect the nucleus and membrane; (b) A cell adhering on an elastic substrate, which is connected via FAs; (c) The forces acting on each SF are illustrated. The binding of integrins and ligands provides the traction force ftrac . The myosin motors within SF generate contractive force f myo . f mc and f mr represent 7

the forces from the circumferential and radial deformations of cell membrane, respectively. f SF is the viscous force.

By assembling a branched network formed by a rapid polymerization of cross-linked filamentous actin, spreading cells can extend a thin, broad, and flat leading edge (Ponti et al., 2004). At the leading edge, the addition of actin monomers onto the barbed ends of filaments generates pushing forces that drive the leading edge forward (Ponti et al., 2004; Shao et al., 2012). In addition, the actin network undergoes retrograde flow toward the cell center due to myosin motors-generated contraction (Giannone et al., 2004; Wilson et al., 2010). As a result, the extension rate of actin network is controlled by two simultaneous but oppositely directed mechanisms: actin polymerization with a speed, Vactin , and actin retrograde flow with a speed, Vmyo . Thus, the elongation velocity of a single SF can be expressed as V  Vactin  Vmyo . Recent experiments showed that at the deceleration of cell spreading

velocity, the retrograde flow velocity increases, whereas the sum Vmyo  V is constant (Wilson et al., 2010). Therefore, the actin polymerization rate Vactin remains constant along the SF growing direction in our model. Cells adhere to substrates mainly through FAs which are composed of a collective of molecular bonds (Nicolas et al., 2004). FAs serve as mechanical connections between the cytoskeleton and the substrate, and transmit myosin motors-generated contractile forces to the substrate (Plotnikov et al., 2012; Elosegui-Artola et al., 2014). Here, we describe the FAs in terms of a cluster of bonds formed by integrins in the membrane

and

their

ligands

in

the

deformable

substrate.

The

myosin

motors-generated contractile forces in the cytoskeleton can give rise to local elastic displacements of these bonds, which will induce a tangential force, known as traction force, at the cell-substrate interface (Xu et al., 2018). Let x denotes the sum of the tangential displacement of the bond and the displacement of the elastic substrate. The integrins within a FA may either be bound to the ligands on substrates or be 8

disconnected, and their binding kinetics can be significantly affected by the thermal fluctuation. The fluctuations can disrupt the bond-mediated adhesion (Qian et al., 2017), and lead to the patterning and clustering of integrin-based binding (Yu et al., 2017). Here, using a stochastic model for binding kinetics of integrins, the probability density distribution n(x, t) of integrin-ligand bonds is expressed as (Nisenholz et al., 2014; Xu et al., 2018)





 n n  Vmyo  b 1   n dx  ub n , 0 t x

(1)

where  b and ub represent the integrin binding and unbinding rate, respectively. The solution of Eq. (1) at steady state is

n( x ) 

Vmyo

b

 1 x  exp   ub ( x) d x   Vmyo 0  .  1 x     exp    ( x) d x dx  0 0 ub V  myo 

(2)

Let N denotes the total number of integrin molecules within a FA. The total number 

of bound integrins is Nb  N0  n( x) d x  N0b b  ub  . The average of the 0

tangential deformation x is written as

x

 



0

N 0 xn( x) d x Nb



Vmyo

ub

,

(3)

and then, the traction force between the FA and substrate is given by

f trac  kFA x  kFA

Vmyo

ub

,

(4)

where kFA represents the spring constant of a FA. To calculate kFA , we treat the bond and the substrate as Hookean springs with elastic constant kint and ksub , respectively, where ksub  Esub with Esub being the Young’s modulus of the substrate. Then, the effective elastic constant of integrin-substrate system can be 9

described by kb  kint ksub  kint  ksub  . Due to the parallel connection between adhesive

bonds,

the

spring

constant

of

a

FA

can

be

expressed

as

kFA  Nb kb  Nb kint ksub  kint  ksub  .

According to the Hill’s law (Hill, 1938), the contractive force originating from myosin motors in each SF is  V  f myo  nmyolSF f 0 1  myo  , V0  

(5)

where nmyo denotes the density of myosin motors in a SF, lSF is the length of a SF,

f 0 is the stall force of myosin motors, and V0 is the velocity of myosin motors without loads. Cell spreading can induce large deformation of the cell membrane, and thus needs to overcome the resistance from membrane deformation (Gauthier et al., 2011; Schweitzer et al., 2014). The membrane force comprises two components corresponding to the radial and circumferential deformations, respectively. Let s + and s  represent the distances between a vertex and its two adjacent vertices, respectively. The membrane tension fmr in the radial direction can be expressed by the linear Hooke’s law (Sokabe et al., 1991)

f mr  sˆ

A , A0

(6)

where sˆ   s   s   2 ,  denotes the initial tension of the cell membrane, A and

A0 are the current and initial areas, respectively. For the initial distances between a vertex and its two neighbors, s0+ and s0 , the membrane force fmc along the circumferential direction is calculated by

10

f mc  km  s   s0  ,

(7)

where km represents the cell membrane stiffness. It should be noted that the spatial distributions of molecular components within the cell membranes are inhomogeneous and dynamic. Interestingly, Różycki and coworkers found that phase separation can occur within the adhering membranes if the cell membrane adhesion is mediated by several types of receptor-ligand bonds (Asfaw et al., 2006; Różycki et al., 2010). For multicomponent membranes, the initial tension γ of the cell membrane and the membrane rigidity km in Eqs. (6) and (7) may vary with the membrane components, which deserves further study. In addition, a spreading cell suffers from the viscous force arising from its composite shell envelope and intracellular cytoskeleton consisting of actin filaments, various organelles, nucleus, and associated proteins (Bausch et al., 1998). The viscous force exerting on the SF-membrane joint can be expressed as f SF  sˆcV ,

(8)

where  c is the viscous coefficient. For simplicity, the nucleus is modeled as a rigid sphere with the radius r , and the drag force acting on it is given by Stoke’s law (Reinsch and Gonczy, 1998) f -nuc  6πrmVnuc ,

(9)

where  m denotes the cytoplasm viscosity and Vnuc is the velocity of the nucleus. In the light of above analysis, the equilibrium condition of the end of a SF is expressed as ftrac  fmyo  fmr  fmc  f SF  0 ,

and that of the nucleus is given by

11

(10)

f

myo

where

f

myo

 f -nuc  0 ,

(11)

represents the sum of forces of all SFs exerting on the nucleus.

In fact, there exist a large number of stochastic fluctuations within the cell itself or in its residing environments. For instance, there could be a small asymmetry in the initial shape of an adhering cell. To account for this effect, we assume that the lengths initial of SFs at the initial configuration follow a Gaussian distribution lSF

P  ,   ,

with  and  denoting the mean and the variance values of the distribution, respectively. As shown in experiments (Parsons et al., 2010; Grashoff et al., 2010), a longer SF can provide a larger myosin motors-generated force, which in turn promotes the growth of FAs . Hence, there exists a positive feedback between the min max length of SF and the size of FA at the end of SF. Let lSF and lSF represent the d max min minimal and maximal lengths among all SFs, and lSF be their difference.  lSF  lSF

To consider the positive feedback relation between the FA size and the SF length, we use a step function

 la   lFA = lb   lc 

1 d min min lSF  lSF  lSF  lSF 3 1 d 2 d min min lSF  lSF  lSF  lSF  lSF 3 3 2 d min max lSF  lSF  lSF  lSF 3

(12)

where la , lb and lc ( la  lb  lc ) are three characteristic lengths of FAs. In this section, we have established a cytoskeleton-based mechanical model to investigate the spatiotemporal evolution of the cell morphology on compliant substrates. This model takes both biochemical and biomechanical factors into account, including

actin

polymerization,

myosin

motors

contractility,

integrin

binding/unbinding, membrane elasticity, cell viscosity, and substrate stiffness. Next, we will employ this cytoskeleton-based model to simulate the dynamical evolution of 12

cell morphology on elastic substrates with and without considering the stochastic fluctuations.

3 RESULTS AND DISCUSSION 3.1 Dynamical evolution of cell morphology on elastic substrates without fluctuations We firstly consider the case that a cell spreads on elastic substrates with the same initial length of SFs in each direction. In other words, the cell spreads isotropically on compliant substrates. In the initial configuration, 30 SFs emanate uniformly from the nucleus with a length 3 μm . The radius of nucleus is 1 μm , and the length of each FA is 0.1 μm . The values of other parameters are listed in Table 1. Table 1 Key Parameters Used in Our Model Physical meaning

Symbol

Value

Reference

Myosin motor stall force

f0

1.7 pN

Unloaded myosin motor velocity

V0

~200 nm/s

Actin polymerization velocity

Vactin

~500 nm/s

Integrin spring constant

kint

1-10 pN/nm

Integrin-ligand binding rate

b

0.1-5 s-1

Integrin-ligand unbinding rate

 ub

0.01-0.1 s-1

Cell membrane stiffness

km

0.01-1 pn/nm

(Zhao et al., 2017)

Initial tension of membrane



0.1-2 pn/nm

(Zhao et al., 2017)

Cortex viscosity

c

100 Pa·s

(Bausch et al., 1998)

Cytoplasm viscosity

n

0.01 Pa·s

(Reinsch and Gonczy, 1998)

(Chan and Odde, 2008; Cao et al., 2015; Bangasser et al., 2017) (Chan and Odde, 2008; Cao et al., 2015; Bangasser et al., 2017) (Döbereiner et al., 2004) (Chan and Odde, 2008; Cao et al., 2015; Bangasser et al., 2017) (Chan and Odde, 2008; Cao et al., 2015; Bangasser et al., 2017) (Chan and Odde, 2008; Cao et al., 2015; Bangasser et al., 2017)

Figure 3(a) and 3(b) show the spatiotemporal evolutions of a cell spreading on 13

soft (E=5 kPa) and stiff (E=100 kPa) substrates, respectively. It can be clearly seen that for two cases, the adhesion areas remarkably increase at the beginning, while finally remain constant. The steady adhesion area on stiff substrate is much larger than that on soft one, in agreement with experimental measurements (Nisenholz et al., 2014). This is because a stiffer substrate, i.e., larger ksub , leads to a larger effective elastic coefficient, kFA  Nb kint ksub  kint  ksub  , and thus can provide a larger traction force, f trac  kFA x . In the initial cell spreading phase, the spreading velocities of cells on soft and stiff substrates are 4.97 μm2/s and 7.05 μm2/s, respectively, which means that the cell spreads faster on stiff substrate than on soft one. This is because the traction force on stiffer substrates is larger (see Eq. (4)), which can induce a larger cell spreading velocity. It should be noted that the steady area of cells on stiff substrates is three times larger than that on soft ones, as shown in Fig. 3. As a result, the cells on soft substrate approach the plateau values faster than those on stiff ones, although they spread slower on soft substrates. In addition, for both soft and stiff substrates, the cell keeps an isotropic geometry during the spreading process (see Fig. 3), which is also shown in previous studies (Nisenholz et al., 2014; Zhao et al., 2017).

(a)

(b)

Soft substrate (5 kPa)

Stiff substrate (100 kPa)

4000

Cell area (m2)

Cell area (m2)

1000 800 600 400 200 0

3000 2000 1000 0

0

200

400

600

800

1000

0

Time (s)

500

1000

1500

Time (s)

Fig. 3. Dynamics of cell morphology on the substrates with rigidities of (a) 5 kPa and (b) 100 kPa. Here, the results are caculated by the cytoskeleton-based model without fluctuations.

We then study the quantitative influence of substrate rigidity on the steady adhesion area of a cell, as shown in Fig. 4. It can be clearly seen that our theoretical 14

predictions are in good agreement with experimental data (Nisenholz et al., 2014). It should be noted that a biophysical model was also proposed by Nisenholz et al. (2014), who does not directly consider the effects of membrane deformation. In fact, a cell does not exhibit isotropic geometry due to the complicacy of its biological process, and always exhibits a polarized morphology, especially on stiff substrates, as observed in our experiments. We will focus on investigating this polarization phenomenon in the sequel. 3000

Cell area (m2)

Cell area (m2)

4000 3000 2000 1000 0

Experiment Our model

0

20

40

60

80

Substrate rigidity (kPa)

100

2500

Experiment Our model

2000 1500 1000 500 0

1.56 kPa

9.16 kPa

Substrate rigidity

Fig. 4. Steady adherion areas of cells spreading on substrates of various rigidities. The experimental data are obtained from Nisenholz et al. (2014).

3.2 Dynamical evolution of cell morphology on elastic substrates with fluctuations In the case of considering stochastic fluctuations, the initial lengths of SFs are generated by a Gaussian distribution with mean 3 μm and variance 0.3 μm . The values of la , lb and lc in Eq.(12) are taken as 0.03 μm , 0.1 μm and 0.3 μm , respectively. In this case, the adhesion areas and spatiotemporal configurations of cell morphology on soft (E=5 kPa) and stiff (E=100 kPa) substrates are illustrated in Figs. 5(a) and 5(b), respectively. It is clear that the morphological evolutions are quite different with the isotropic spreading case in Fig. 3. When stochastic fluctuations are taken into account, the cell morphologies change obviously, and finally exhibit evident polarized configurations. Remarkably, the polarization degree of cell morphology on stiff substrate is significantly larger than on soft substrate. This finding shows that the substrate rigidity plays a regulating role in the formation of cell polarization, and our cytoskeleton-based model can capture this effect. In addition, the 15

34.05 kPa

spreading velocities of cells on soft and stiff substrates are respectively 4.28 μm2/s and 6.86 μm2/s in the initial spreading phase. The steady area of cells on stiff substrates is almost three times larger than that on soft ones. Therefore, similar to the case in Fig. 3, the cells on soft substrate approach the plateau values faster than those on stiff ones.

(a)

(b)

Soft substrate (5 kPa)

Stiff substrate (100 kPa)

Cell area (m2)

Cell area (m2)

1200

800

400

0

3000 2000 1000 0

0

400

800

1200

1600

Time (s)

0

500

1000

1500

2000

Time (s)

Fig. 5. Dynamical evolution of cell morphology on the substrates with rigidities of (a) 5 kPa and (b) 100 kPa. Here, the results are caculated by the cytoskeleton-based model with fluctuations.

We also carried out several groups of experiments where cells were cultured on the substrates with rigidities of 1.56 kPa, 9.16 kPa and 34.05 kPa, respectively. As shown in Fig. 6, the steady adhesion areas obtained from our model agree well with our experimental results. The representative cell morphologies on substrates of different rigidities are plotted in Fig. 7. As observed in our experiments, the cell exhibits a more anisotropic geometry on stiffer substrates. For soft substrate of 1.54 kPa, the cell polarization is weak, and the cell keeps almost circular (see Fig. 7(a)). For stiff substrate of 34.05 kPa, the cell polarization is strong, and the cell is approximately needle-like (see Fig. 7(c)). The geometric configurations predicted by our model are in good agreement with our experimental observations. Both our experiment and simulation results evidence that the degree of cell polarization is strongly related to the rigidity of substrates.

16

00

Cell area (m2)

3000

00

00

00 0

Experiment Our model

0

20

40

60

80

Substrate rigidity (kPa)

100

2500

Experiment Our model

2000 1500 1000 500 0

1.56 kPa

9.16 kPa

34.05 kPa

Substrate rigidity

Fig. 6. Steady adhesion areas of cells on substrates of different rigidities. The blue and red data are from our model and experiment, respectively.

Fig. 7. Steady cell morphologies from both our model and experiment on the substrates of various rigidities: (a) experimental observation and (d) theoretical result on the substrate of 1.56kPa; (b) experimental observation and (e) theoretical result on the substrate of 9.16kPa; (c) experimental observation and (f) theoretical result on the substrate of 34.05kPa.

3.3 Quantitative study of the degree of cell polarization on elastic substrates To quantitatively describe the degree of cell polarization, we introduce a parameter, known as aspect ratio (Prager-Khoutorsky et al., 2011). As shown in Fig. 8(a) or 8(b), the steady shape of a cell in our experiments or simulations can be approximately fitted by an ellipse. Then, the aspect ratio is defined as the ratio of the 17

length of long axis to the length of short axis of the ellipse. One can see that the larger the aspect ratio, the stronger the cell polarization. We have plotted the aspect ratios obtained from our experiments and models in Fig. 8(c). It is shown that the stiffer substrate can induce a larger aspect ratio, in which case the cell possesses a more anisotropic geometry. Moreover, the theoretical predictions are in excellent agreement with our experimental results, indicating that the proposed cytoskeleton-based model can elucidate the quantitative relation between cell polarization and substrate rigidity.

(a)

(b)

l

l

s

s

Aspect ratio

(c) 6 5

Experinment Our model

4 3 2 1

1.56 kPa

9.16 kPa

34.05 kPa

Substrate rigidity Fig. 8. Cell outlines in our (a) experimental and (b) theoretical results can be fitted by an ellipse. The aspect ratio of the length of long axis to the length of short axis is used to quantitatively describe the degree of cell polarization. (c) Comparison of theoretical results with our experimental measurements on substrates of rigidities: 1.56 kPa, 9.16 kPa and 34.05 kPa.

To further verify this cytoskeleton-based mechanical model, we compare our simulation results with more experimental data from Prager-Khoutorsky et al. (2011). In their work, Prager-Khoutorsky et al. cultured human foreskin fibroblasts (HFFs) on two elastomeric substrates: PDMS with rigidities varying from 4 kPa to 2 MPa and PAA gels with rigidities varying from 4 kPa to 150 kPa. We compare our theoretical predictions with these two cases (see Fig. 9), and find that they agree very well. These 18

findings further evidence the ability of the present model to quantitatively capture the influence of substrate rigidity on cell polarization. (a)

PDMS 2.8

Experiment Our model

Aspect ratio

3.0

Aspect ratio

(b)

PAA

3.5

2.5 2.0 1.5

Experiment Our model

2.4 2.0 1.6 1.2

1.0 4 kPa

30 kPa

150 kPa

Glass

2 kPa 20 kPa 700 kPa 2 MPa Glass

Substrate rigidity

Substrate rigidity

Fig. 9. Comparison of cell polarization from theoretical results and experimental measurements on substrates of different rigidities. The experimental data are obtained from Prager-Khoutorsky et al. (2011). HFF cells are plated on (a) PAA and (b) PDMS substrates of different rigidities.

To understand these results, we explore the potential mechanism underlying the effect of substrate rigidity on cell polarization. Due to the stochastic fluctuations in a cell, it will exhibit a small asymmetry in its initial shape, which is described by SFs with slightly different lengths. This length difference affects the sizes of FAs associated with the SFs by a positive feedback loop, and in turn affects the bound numbers Nb of integrins within FAs. Then, the traction force f trac  Nb kb Vmyo ub for each SF is different. As a result, the cell will achieve an anisotropic morphology. Moreover, the effective elasticity kb  kint ksub  kint  ksub  depends on the substrate rigidity. Since a stiffer substrate induces a larger value of kb , the difference of traction forces on each SF will be further amplified on stiffer substrates, giving rise to a higher cell polarization. Therefore, the coupling of the stochastic difference in initial shape and the substrate rigidity could be responsible for cell polarization that is stronger on stiff substrates but weaker on soft substrates.

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3.4 Parametric study of cell polarization In the above subsection, we show that the generation of cell polarization on elastic substrates is related to the intrinsic fluctuations within cells and the positive feedback relation between SFs and FAs. The important parameters describing these two factors are the extents of fluctuations and the feedback between the SF length and the FA size. The values of these parameters could be distinctly different for different cell types and thus, we investigate the influence of these factors on cell polarization in this subsection. To quantitatively explore the effect of stochastic fluctuations on cell polarization, we vary the variance (  ) of the Gaussian distribution from 0.0001 μm to 1 μm. A larger value of  means that the distribution of initial SF lengths is broader. Figure 10 plots the influences of fluctuation extent (  ) on cell polarization in the cases of soft and stiff substrates. It can be seen that the extent of cell polarization increases with  for both soft and stiff substrates. This can be understood that a larger asymmetry in the initial shape can induce a more anisotropic morphology, i.e., a higher degree of cell polarization. It should be mentioned that the fluctuation plays a more important role in the case of stiff substrate (Fig. 10(b)) than soft one (Fig. 10(a)), because of the amplification effect of substrate stiffness on the traction force. These results suggest that the larger difference among initial SF lengths could lead to a higher cell polarization on stiffer substrates.

(a)

Soft substrate (2 kPa)

4.0

2.0

3.5

1.9

Aspect ratio

Aspect ratio

Stiff substrate (1 MPa)

(b)

2.1

1.8 1.7 1.6 1.5

3.0 2.5 2.0 1.5

1.4

1E-4 0.001

0.01

m

0.1

1.0

1

1E-4 0.001

0.01

m

20

0.1

1

Fig. 10. Effects of fluctuations on cell polarization for (a) soft substrate (2 kPa) and (b) stiff substrate (1 MPa). The variance, representing the fluctuation extent, of the Gaussian distribution of initial SF lengths varies from 0.0001 μm to 1 μm. For clarity, the figures are plotted by logarithmic coordinates.

To quantitatively study the influence of the feedback relation between SFs and FAs, we vary the characteristic lengths in Eq.(12). For convenience, let lb/la=lc/lb. If the ratio lb/la or lc/lb is larger, the SF length can play a more important impact on the FA size. Figure 11 shows the aspect ratio with respect to the value of lb/la on soft and stiff substrates. For both substrates, we can find that the extent of cell polarization is highly associated with the extent of the feedback relation between SFs and FAs. The results show that a stronger positive feedback between SFs and FAs can lead to higher cell polarization on both stiff and soft substrates. Furthermore, the positive feedback between SFs and FAs has more significant impact on cell polarization for stiff substrate than for soft one, because stiffer substrates can provide larger traction forces.

(b)

Soft substrate (2 kPa)

1.8

2.2

1.6

2.0

1.5 1.4 1.3 1.2

1.8 1.6 1.4 1.2

1.1 1.0

Stiff substrate (1 MPa)

2.4

1.7

Aspect ratio

Aspect ratio

(a)

1.0 1

2

3

4

1

5

2

3

4

5

lb/la

lb/la

Fig. 11. Effects of positive feedback relation on cell polarization for (a) soft substrate (2 kPa) and (b) stiff substrate (1 MPa). lb /la, representing the extent of the feedback between SFs and FAs, varies for both soft and stiff substrates.

4 CONCLUSIONS By considering the mechanisms of actin polymerization, integrin-mediated FA dynamics, myosin motors contractility, membrane deformation and substrate stiffness, 21

we propose a cytoskeleton-based mechanical model to qualitatively and quantitatively study the influence of substrate rigidity on cell spreading and polarization behaviors. If the stochastic fluctuation is not taken into account, this model can only capture the effects of substrate rigidity on cell spreading but fails to simulate the polarized morphology of a cell. When the fluctuation is considered, the proposed model can simultaneously simulate cell spreading and polarization behaviors on elastic substrates. It is found that a cell can exhibit a stronger polarization on stiffer substrates, because the stiff substrate can amplify the initial length difference of SFs induced by the fluctuation. To verify this model, we carry out the experiments of mouse embryo fibroblasts on hydrogels with different rigidities. Our theoretical predictions are in line with our experiment results and experimental data from other research groups. These demonstrate that the proposed model is applicable to explore the effects of substrate rigidity on cell polarization and cell morphology, and deepen our understanding of biophysical mechanisms underpinning the cell mechanosensing.

Acknowledgements Supports from the National Natural Science Foundation of China (Grant Nos. 11672227 and 11872106), the Natural Science Basic Research Plan in Shaanxi Province of China (2019JC-02), and the State Key Laboratory of Structural Analysis for Industrial Equipment (GZ18101) are acknowledged.

Author Statement Guang-Kui Xu designed the research. Yuan Qin and Guang-Kui Xu proposed the theoretical model. Yuhui Li performed the experiments. Yuan Qin, Li-Yuan Zhang, and Guang-Kui Xu analyzed the data. Yuan Qin, Yuhui Li, Li-Yuan Zhang, and Guang-Kui Xu wrote the article.

Conflict of interest The authors declare that there are no conflicts of interest associated with the present study.

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