Supporting Material for Modeling of Cardiac Muscular Thin Films: Pre-stretch, Passive and Active Behavior
Jongmin Shim1 , Anna Grosberg2 , Janna C. Nawroth3 , Kevin Kit Parker2 , and Katia Bertoldi1
2

School of Engineering and Applied Science, Harvard University, Cambridge, MA Disease Biophysics Group, Wyss Institute for Biologically Inspired Engineering, School of Engineering and Applied Science, Harvard University, Cambridge, MA 3 Division of Biology, California Institute of Technology, Pasadena, CA

1

S1 EXPERIMENTAL METHODS

S1

S1

Experimental Methods

This section provides a detailed description of the preparation of the silicone thin ﬁlm and cell harvesting procedure.

S1.1

Manufacturing the Silicone Thin Films

The silicone thin ﬁlms were prepared via a multi-step spin coating process, extending a previously published method (Feinberg et al., 2007) in order to allow for higher experimental throughput. The samples prepared this way have been dubbed “Heart on a chip”. To prepare the chips, a section of glass (7.5cm × 11cm) was sonicated and covered with a protective ﬁlm (Static Cling Film, McMaster-Carr, Robbinsville, NJ) on both sides. Rectangles (1cm × 0.6cm) were cut out of the top protective ﬁlm, and poly(N-isopropylacrylamide) (PIPAAm, Polysciences, Inc., Warrington, PA) in 1-butanol (10% w/v) was spin-coated onto the uncovered portions of the glass, and then the top protective ﬁlm was removed. The silicone polymer polydimethylsiloxane (PDMS, Sylgard 184-Dow Corning, Midland, MI) was pre-cured for 1 to 6 hours, spin coated onto the glass, and cured for 8 hours at 65◦ C. The bottom protective ﬁlm was removed, and the glass section was cut into cover slips of 1.4cm × 1.2cm. From each glass section, two cover slips were used to measure the thickness of the PDMS layer using a proﬁlometer (Dektak 6M, Veeco Instruments Inc., Plainview, NY). The PDMS surface of the cover slip was functionalized and sterilized for 8 minutes in UV ozone (Model No. 342, Jetlight Company, Inc., Phoenix, AZ). Fibronectin (FN, Sigma, St. Louis, MO or BD Biosciences, Sparks, MD) was micropatterned onto the surface via microcontact printing with a brick pattered PDMS stamp (each brick was 20µm wide and 100µm long, and each short edge terminated with 5µm long saw-tooth, Bray et al., 2008), such that the brick pattern would be perpendicular or at some tilted angle to the long edge of the cover slip. The gaps in the pattern were blocked by incubating the cover slip in Pluronics F127 (BASF Group, Parsippany, NJ) for 5 minutes, and then the chips were stored at 4◦ C until cell seeding. Three diﬀerent PDMS thickness were made though this procedure: 14.5µm, 18.0µm, 23.0µm.

S1.2

Cell harvesting, Seeding, and Culture

Cell harvesting was conducted in accordance with the guidelines of Institutional Animal Care and Use Committee of Harvard University. Cardiomyocytes were harvested and isolated based on published protocols (Adams et al., 2007). Brieﬂy, ventricles were extracted from two-day-old neonatal Sprague-Dawley rats (Charles River Laboratories, Wilmington, MA) and homogenized by washing in Hanks balanced salt solution, then incubated with trypsin overnight at 4◦ C. To release the myocytes into solution, the ventricular tissue was digested with collagenase at 37◦ C. Cells were re-suspended in M199 culture medium supplemented with 10% (v/v) heat-inactivated fetal bovine serum (FBS), 10mM HEPES, 3.5g/L glucose, 2mM L-glutamine, 2mg/L vitamin B-12, and 50U/ml penicillin. The myocytes were seeded on the substrates at a density of 400,000 cells per cover slip (round, diameter of 18mm), and cultured for a period of four days prior to running the assay. During the initial 48 hours, the cells were kept at 37◦ C and 5%CO2 with 10% FBS media in the culture; thereafter, the maintenance media (2% FBS) was used. The measured cell thickness was approximately 4.0µm.

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS

S2

S1.3

Contractility Experiments

To run the contractility experiments, the chip was moved to a Petri dish with 37◦ C normal Tyrode’s solution (1.192g of HEPES, 0.901g of glucose, 0.265g of CaCl2 , 0.203g of M gCl2 , 0.403g of KCl, 7.889g of N aCl, and 0.040g of N aH2 P O4 per liter of deionized water; reagents from Sigma, St. Louis, MO). The PDMS elastic layer was cut such as to leave the PDMS along the border of the cover slip intact, while in the center there were two rows of three to four rectangles (approximately 2mm × 3.5mm) separated by approximately 1mm, and connected to the PDMS border by one edge (see Figs. ??). The unwanted ﬁlm was peeled away, exposing the temperature-sensitive PIPAAm beneath the rectangular strips and causing it to dissolve as the solution was allowed to cool down towards room temperature. Subsequently, the rectangular ﬁlms were free to move vertically up from the plane of the cell culture, without completely separating from the chip. The chip with six or eight rectangular ﬁlms was moved to a heated bath (34◦ C − 37◦ C) under a stereomicroscope (Model MZ6 with darkﬁeld base, Leica Microsystems, Inc., Wetzlar, Germany), and the dynamics of the ﬁlms was recorded from above at 120 f rames/sec ( Basler A602f camera, Exton, PA). Several snapshots from the recorded movies are shown in Figs. ?? and ??. While the ﬁlm laid ﬂat on the glass, both its length and width were measured. As the ﬁlms contracted away from the surface, the horizontal projection was recorded. Unimpeded ﬁlms would ﬁrst isometrically deform to a curvature corresponding to the diastolic tension in the cells, and then continue to deform until peak systole. For substrates with cells aligned with the length of the ﬁlm, the observed deformation was a shortening of the projection, described in Fig. ?? (top). On the other hand, the substrates with cells at an angle to the length of the ﬁlm showed the inclined free edge of the ﬁlm (see Fig. ?? (bottom)), so an appropriate projection length needed to be deﬁned in order to obtain the curvature of the MTFs. To ensure all ﬁlms on the chip were moving at once, the myocytes were ﬁeld-stimulated at 1.5Hz with 5 to 20V using an external ﬁeld stimulator (Myopacer, IonOptix Corp., Milton, MA) which applied a 10msec square wave pulse.

S2

Constitutive Modeling of Bio-hybrid Films

In this section, we ﬁrst summarize the equations governing the nonlinear deformation of hyperelastic materials, following the formulation previously introduced by ?. Then, the equations are specialized to capture the behavior of the elastomeric substrate and the cardiac myocytes. As mentioned in Section 1, cardiac myocytes comprise highly aligned myoﬁbrils, so their behavior is captured extending the formulation previously developed to model ﬁber reinforced elastomers (e.g., Spencer, 1984).

S2.1

General Formulation

Let F = ∂x/∂X be the deformation gradient mapping a material point from the reference position X to its current position x, and J = detF be its determinant. The behavior of nearly incompressible materials is eﬀectively described by splitting the deformation locally into volumetric (denoted by superscript v ) and isochoric (denoted by superscript i ) components as F = Fv · Fi , where Fv = J 1/3 1, Fi = J −1/3 F. (S1)

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS

S3

For an isotropic hyperelastic material, the strain energy density ψ is a function of the three invariants of C = FT ·F, ψ = ψ(I1 , I2 , I3 ). Based on the kinematic assumption (S1), a decoupled form for ψ is introduced ¯ ¯ ψ = ψ v (J) + ψ i (I1 , I2 ), (S2) with 1 ¯ ¯ I1 = trCi , I2 = 2 trCi
2

− tr Ci

2

.

(S3)

When an isotropic material is reinforced by a family of ﬁbers with direction a0 in the reference conﬁguration, the isochoric part of the strain energy can be expressed as a function of not only ¯ ¯ ¯ ¯ I1 and I2 , but also two additional invariants (I4 , I5 ) depending on a0 (Spencer, 1984), ¯ ¯ ¯ ¯ ψ i = ψ i (I1 , I2 , I4 , I5 ), with ¯ ¯ I4 = a0 · Ci · a0 = λ2 ,
2 ¯ I5 = a0 · Ci · a0 .

(S4)

(S5)

¯ ¯ However, the dependence of the material response on I2 or I5 is generally weak, so that the isochoric part of the strain energy can be simpliﬁed as
i i ¯ ¯ ψ i = ψiso I1 + ψani I4 ,

(S6)

i i where ψ i has been further decoupled into an isotropic, ψiso , and anisotropic, ψani , contributions. Finally, the Cauchy stress T is found by diﬀerentiating ψ with respect to C as

T=

∂ψ v 2 2 ∂ψ T F F = 1+ J ∂C ∂J J

i i ¯ ¯ ∂ψiso I1 I4 ¯ ∂ψ dev Bi + I4 ani dev (¯ ⊗ a) , a ¯ ¯ ¯ ∂ I1 ∂ I4 T

(S7)

where “dev” stands for deviatoric part of 2nd order tensors, Bi = Fi ·Fi is the left Cauchy-Green ¯ ¯ tensor, and a = Fi · a0 / I4 is the unit directional vector of ﬁber in the deformed conﬁguration. In the following two sections, the above general formulation is specialized to capture the behavior of the elastomeric substrate and the cardiac myocytes.

S2.2

Elastomeric Substrate Behavior

The elastomeric substrate is fabricated using PDMS, and its behavior is well captured using a neo-Hookean model, so that the strain energy density is given by ˜ κ ˜ E ¯ ¯ ψ = ψ v (J) + ψ i I1 = (J − 1)2 + I1 − 3 , 2 6 (S8)

˜ where κ and E denote the bulk modulus and the initial elastic modulus of the elastomer, ˜ respectively. Based on (S7), the Cauchy stress is given by: T = κ(J − 1)1 + ˜ ˜ E dev Bi . 3J (S9)

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS

S4

S2.3

Cardiac Muscle Cell Behavior

The characteristic energy dissipative process of the cardiomyocite active contribution can be modeled by the physiological approach, which considers its true mechanism due to the actin potential under calcium ﬂux (e.g., Sainte-Marie et al., 2006; Chapelle et al., 2010). However, this approach is mathematically complex, and model parameter identiﬁcation is challenging. In this article, instead, we take a simple, but eﬀective phenomenological approach, where the active behavior is assumed to be elastic. While recently B¨l et al. (2009) developed a phenomenological o model of cardiac muscle cells, their model neglects the important eﬀect of the pre-stretch that the muscle cells develop as they mature. This paper presents a 3-D phenomenological model that captures the two major features of cardiac muscles: the passive behavior inducing pre-stretched deformation and their active behavior including systole and diastole. S2.3.1 Kinematics including Pre-stretch

Even when resting, muscle cells in vivo experience a state of stress due to their pre-stretched conditions. This can be clearly observed from the experimental data reported in Figs. ?? and Fig. ??, showing a non-negligible MTF curvature during diastole. In order to account for such pre-stretched conditions, previous constitutive models introduced an ad hoc strain-shift in the stress-strain curve (Blemker et al., 2005; B¨l and Reese, 2008; Calvo et al. (2010)). o However, this strain-shift leads to a stress-strain curve that depends on the level of pre-stretch. In this paper, a diﬀerent approach is adopted; inspired by the multiplicative decomposition introduced by Kroner-Lee (Kroner, 1960; Lee, 1969), the isochoric deformation gradient Fi in (S1) is decomposed into load-induced, FiL , and pre-stretched, FiS , contributions Fi = FiL · FiS . (S10)

Here, for the sake of simplicity, the pre-stretch is assumed to be fully developed during cell diﬀerentiation and maturation prior to the experiment, and not to be aﬀected by the cell active response. If we also assume that the pre-stretch deformation is incompressible and the cells deform aﬃnely, the pre-stretched contribution to the deformation gradient FiS is given by FiS = QΛQT , where ˆ ˆ Λ = λS e1 ⊗ e1 + λS
−1 2

(S11)

(e2 ⊗ e2 + e3 ⊗ e3 )

(S12) (S13)

ˆ ˆ Q = cos θ (e1 ⊗ e1 + e2 ⊗ e2 ) + sin θ (−e1 ⊗ e2 + e2 ⊗ e1 ) + e3 ⊗ e3

ˆ where λS is the pre-stretch induced into the cardiac myocytes lying in the x1 -x2 plane during ˆ maturation (i.e., myogenesis and tissue development) and θ is the angle identifying the cell alignment in the undeformed conﬁguration (see Fig. ??). The diastole curvatures of the experimental data reported in Figs. ?? and Fig. ?? clearly show a deviation in the pre-stretched response of the cells as a result of the diversity in cell conditions. Therefore, we expect the ˆ parameter λS not to be uniquely determined, but to be inﬂuenced by experimental factors such as ambient temperature/humidity, and age/health of the cells.

S2 CONSTITUTIVE MODELING OF BIO-HYBRID FILMS S2.3.2 Constitutive Equations for Passive and Active Behavior

S5

In order to describe both passive (representing resting status) and active (including systole and diastole) behavior of cardiomyocytes, we specify their free energy as
ip ia i ¯ ¯ ¯ ψ = ψ v (J) + ψiso I1 + ψani I4 + ψani I4 , q ,

(S14)

i where q is the activation level of cardiac muscle cells. In the present model, while ψ v and ψiso ip ia reﬂects the isotropic contribution of the intercellular part, ψani and ψani represent the passive and the active contributions of anisotropic eﬀect of the myoﬁbril, repectively. Here, as for the elastomeric substrate, the volumetric free energy term is chosen as

ψ v (J) =

κ ˆ (J − 1)2 , 2

(S15)

and the isotropic contribution of the intercellular part is captured using a neo-Hookean formulation
i ψiso =

ˆ Ec ¯ I1 − 3 , 6

(S16)

ˆ where κ denotes the bulk modulus of the cells and Ec is the initial elastic modulus of intercellular ˆ part. Several experimental studies (e.g., Le Guennec et al., 1990; Granzier and Irving, 1995; Palmer et al., 1996; Cazorla et al., 2000; Weiwad et al., 2000; Boron and Boulpaep, 2009) reveal that the passive response of cardiac cells is characterized by an exponential relation between stress and deformation (see Fig. ??). In order to capture such behavior of cardiac cells, we propose the following free energy ˆ Ep ˆ ¯ ip ¯ ψani λ = 2 eα(λ−1) − α(λ − 1) − 1 , ˆ ¯ α ˆ (S17)

¯ ¯ ˆ where λ = I4 , Ep is the initial elastic modulus, and α characterizes the monotonically inˆ creasing slope of the stress-strain relation of the passive response of the ﬁbers. For the active response (see Fig. ??), a bell-type relation between stress and deformation has been shown by several studies (e.g., Zajac, 1989; Weiwad et al., 2000; Boron and Boulpaep, 2009). Here, we simplify the active stress-stretch relation into a quadratic form, which is captured introducing the following free energy  ˆ  Pq ˆ ¯ ˆ ¯ ¯ ¯ ˆ 2(1 − 2λo ) ln λ + (4λo − λ − 1)(λ − 1) , if 1 < λ < (2λo − 1) ˆ ia ¯ ψani λ, q = 2(1−λo )2 (S18) 0, otherwise ˆ where P is the isometric twitch stress (i.e., the maximum contraction force per unit crossˆ sectional area) and λo is the optimal stretch (i.e., the stretch at which the maximum active stress is attained). Since Fig. ?? and Fig. ?? clearly show that diversity in cell conditions ˆ ˆ causes large variations of the peak amplitudes, as the pre-stretch λS , the parameter P is also treated as a non-uniquely determined property of cultured cardiac tissue. Finally, the activation level q is a function of time and is deﬁned as q(t) = t ˆ T
2

exp 1 −

t ˆ T

2

,

(S19)

S3 PARAMETER IDENTIFICATION FOR THE CONSTITUTIVE MODEL ˆ where T is the characteristic contraction time. Based on (S7), the proposed free energy results in the following Cauchy stress T = Tv + Ti + Tip + Tia , iso ani ani where Tv = κ(J − 1)1 ˆ ˆ Ec dev Bi Ti = iso 3J ˆ¯ Ep λ α(λ−1) ¯ Tip = eˆ − 1 dev(¯ ⊗ a), a ¯ ani αJ ˆ  ˆ ¯ ˆ  P q 1 − λ−λo 2 dev(¯ ⊗ a), a ¯ ˆ ia J 1−λo Tani =  0,

S6

(S20)

(S21) (S22) (S23) ¯ ˆ if 1 < λ < (2λo − 1) otherwise (S24)

S3

Parameter Identiﬁcation for the Constitutive Model

This section presents the procedure to identify the 11 material parameters entering into the proposed 3-D constitutive model. While the material parameters for the elastomeric substrate are provided by the manufacturers, the parameters needed for the cardiac muscle cells are obtained from both literature and contraction assays performed with neonatal rat ventricular myocytes micropatterned on PDMS thin ﬁlms.

S3.1

Summary of Model Parameters

The proposed constitutive model requires 11 material constants • Elastomeric substrate ˜ E Initial elastic modulus κ ˜ Initial bulk modulus • Pre-stretched deformation of cardiac muscle cells ˆ θ Angle deﬁning the initial alignment of the cells within the x1 − x2 plane ˆ S Pre-stretch developed in the cardiomyocytes during cell maturation λ • Volumetric and isotropic behavior of cardiac muscle cells ˆ Ec Initial elastic modulus of intercellular part κ ˆ Initial bulk modulus • Passive ﬁber behavior of cardiac muscle cells ˆ Ep Initial elastic modulus of ﬁber α ˆ Constant related to the slope of stress-strain relation of passive ﬁber • Active ﬁber behavior of cardiac muscle cells ˆ P Isometric twitch stress ˆ o Optimal stretch for the maximum active stress λ ˆ T Characteristic time scale for contraction

S3 PARAMETER IDENTIFICATION FOR THE CONSTITUTIVE MODEL

S7

S3.2

˜ ˜ Elastomeric Substrate (E, κ)

˜ ˜ Only two material constants (E, κ) are required to model the elastomeric substrates. The ˜ initial elastic modulus is directly speciﬁed by the manufacturer as E = 1.5 MPa, while its incompressible behavior (˜ = 0.49) yields an initial bulk modulus κ = 25 MPa. ν ˜

S3.3

Cardiac Myocytes

The proposed model requires seven material-speciﬁc parameters (ˆ , Ec , Ep , α, P , λo , T ) and two κ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ geometric-speciﬁc parameters (θ, λS ). Most of model parameters are available in the literature. The paper by Weiwad et al. (2000) provides in a single article nearly all the required test results to identify the material paramors entering into the proposed material model. Moreover, all their data are in a good agreement with similar test results for the heart muscle of adult rats (i.e., Granzier and Irving, 1995, Palmer et al., 1996 for passive behavior and Nishimura et al., 2000, Palmer et al., 1996 for active behavior). Thus, the test results presented in Weiwad et al. (2000) are used to identify ﬁve material-speciﬁc parameters entering into the proposed constitutive model. Experiments with straight cell alignment (Section 2) are used to identify the remaining three ˆ ˆ parameters; two of which (i.e., P , λS ) are experimental-conditions-speciﬁc, and one of which ˆ (i.e., T ) is the characteristic time scale of contractile behavior under the stimulating electric ˆ pulse. Here, we assume that the cell conditions aﬀect only the twitch stress P and the preˆS . stretch λ S3.3.1 ˆ Isochoric Behavior of Intercellular Part (Ec ) and Isochoric Passive Behavior ˆp , α) of Fiber Part (E ˆ

Several researchers have reported experimentally measured values for Young’s modulus of nonactivated rat cardiac myocytes by using atomic force microscopy (e.g., Lieber et al., 2004) or custom-designed force transducer (e.g., Weiwad et al., 2000). The measured values varies from 20kP a to 45kP a depending on the experimental protocols and the ages of rats. Following ˆ Weiwad et al. (2000), we use a Young’s modulus, Epassive = 23kP a (see Fig. ??). From (S20), recall that the measured Young’s modulus of the non-activated cell can be decomposed into the contribution of intercellular part and passive ﬁber part. Here, we assume that the passive ﬁber ˆ ˆ part is mainly responsible for the passive elastic behavior, so that Ep = 21kP a and Ec = 2.3kP a. passive Note that the passive stress (T11 ) can be expressed as:
passive i T11 = Tiso 11 ip + Tani 11

,

(S25)

where using incompressibility each stress contribution can be determined from the reduced form of (S22) and (S23):
i Tiso ip Tani

11

11

ˆ Ec ¯ 2 1 λ −¯ , 3 λ ˆ¯ Ep λ α(λ−1) ¯ = eˆ −1 . α ˆ =

(S26) (S27)

By curve-ﬁtting the experimental results with the above equations (see Fig. ??), we obtain α = 5.5. ˆ

S4 PARAMETRIC STUDIES S3.3.2 ˆ ˆ ˆ Isochoric Active Behavior of Fiber Part (P , λo , T )

S8

Weiwad et al. (2000) have reported the evaluation of the isometric twitch stress as a function of calcium concentration (see Fig. ??). Note that a constant calcium concentrations of pCa = 2.74 was employed throughout the essay tests presented in Section 2. In order to get its overall dependence on the stretch, all the active tension-stretch relations are normalized by its maximum value, and then curve-ﬁtted by the proposed quadratic form of active stress-stretch relation for uniaxial loading case:  ¯ ˆ P q 1 − λ−λo 2 , ¯ ˆ ˆ if 1 < λ < (2λo − 1) ˆ ia 1−λo Tani 11 = (S28)  0, otherwise ˆ As a result, we obtain λo = 1.24, as shown in Fig. ??. As described in Section 4, the range of ˆ the isometric twitch stress of P ∈ [2.8, 21.6]kP a is determined from experiments with straight cell alignment presented in Section 2. The characteristic contraction time can be determined by measuring the time when the signal reaches its maximum value after the activation starts. Fig. ˆ ?? shows a typical proﬁle of the active amplitude history, and T = 0.21sec was determined. S3.3.3 Volumetric Behavior (ˆ ) κ

For simplicity, we assume that the cardiac muscle is nearly incompressible and its volumetric ˆ behavior is determined by the passive response. Thus, using Epassive = 23kP a and ν = 0.49, we ˆ obtain a bulk modulus of κ = ˆ S3.3.4
ˆ Epassive 3(1−2ν)

= 380kP a.

ˆ ˆ Isochoric Pre-stretch Behavior (θ, λS )

For the model parameter identiﬁcation, the controlled test set (i.e., the case with the straight cell ˆ alignment) with θ = 0◦ is used. It is known that the level of pre-stretch (or pre-stress/pre-load) developed during cell maturation depends on the properties of substrate. ˆ From the assay experiments in Section 2, we obtain λS ∈ [1.11, 1.18]. This range of values for pre-stretch have been also reported under similar experimental conditions using a silicon substrate (e.g., Mansour et al., 2004).

S4

Parametric Studies

In this Section, we explore the eﬀect of important parameters within MTF such as the PDMS thickness, the isometric twitch stress of cells, the pre-stretch of cells, and the frequency of ﬁeldstimulation.

S4.1

Eﬀect of PDMS Thickness

˜ Here, the eﬀect of PDMS thickness on the response of the MTF is explored by varying t from 13.0µm to 28.0µm. Meanwhile, the average value of the two cell-condition-dependent parameters ˆ ˆ are used (i.e., λS = 1.14 and P = 9.0kP a), and all other model parameters are also unchanged throughout the simulations. Fig. S1a shows the eﬀect of PDMS thickness on the total curvature (K) of MTF. As expected, an increase in PDMS thickness leads to a decrease in curvature. To quantify this behavior, the

S4 PARAMETRIC STUDIES total curvature is decomposed into pre-stretch (Kp ), and active (Ka ), contributions, so that K = Kp + Ka .

S9

(S29)

Fig. S1b reports the evolution of maximum passive and active curvature contributions (i.e., max(Kp ) and max(Ka )) as a function of the PDMS thickness. The pre-stretch contribution of the curvature can be also estimated by applying a recent modiﬁcation of the Stoney’s equation (i.e., ?). Assuming that the layer of cells is much thinner than that of substrate, the Stoney curvature equation predicts the curvature of the bilayer ﬁlm as Kstoney = ˆ ˆ ˆ 6(Ec + Ep ) lnλS ˜˜ Et ˆ t ˜ t 1+ ˆ t . ˜ t (S30)

The pre-stretch induced curvature predicted by (S30) is also reported in Fig. S1b, and a good agreement with the pre-stretch contribution from the FEM calculations is observed.

S4.2

Eﬀect of Isometric Twitch Stress of Cells

ˆ ˜ We vary P from 0 to 30.0kP a while keeping all other parameters unchanged, including t = ˆ S = 1.14. 18.0µm and λ Fig. S1c shows the eﬀect of isometric twitch stress on the curvature of MTFs. While the ˆ passive contribution of the curvature (max(Kp )) is unaﬀected by P , the active contribution ˆ (Fig. S1d). (max(Ka )) is observed to vary almost linearly with P

S4.3

Eﬀect of Cell Pre-stretch

ˆ The eﬀect of pre-stretch on the cardiac muscle cell response is investigated by varying λS from ˜ 1.00 (i.e., no pre-stretch) to 1.39 while keeping all other parameters unchanged and using t = ˆ 18.0µm and P = 9.0kP a. Figure S1e reports the time history of ﬁlm curvature for various level of cells pre-stretch. The results clearly show that the level of cell pre-stretch aﬀects the curvature both during cell maturation and activation. Interestingly, for the active response, we observe the quadratic relation between the pre-stretch and the active curvature contribution. Thus, this study suggests that the maximum level of active response (i.e., the maximum active mobility) for the MTFs can be achieved by introducing an optimal pre-stretch value. Moreover, Fig. S1f clearly show that the Stoney equation (S30) largely underestimates the MTF curvature for large pre-stretch.

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S10

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(a)

(b)

(c)

(d)

(e)

(f)

Figure S1: The left column reports the time-curvature plots from FE analysis while the right column the stress contribution of pre-stretch and active response (max(Sp ) and max(Sa )). (a, b) eﬀect of PDMS thickness, (c, d) Eﬀect of isometric twitch stress of cells, (e, f) eﬀect of cell pre-stretch.