A method to determine Young s modulus of soft gels for cell adhesion

Transcription

1 Acta Mech Sin (2009) 25: DOI /s TECHNICAL NOTE A method to determine Young s modulus of soft gels for cell adhesion Xiaoling Peng Jianyong Huang Lei Qin Chunyang Xiong Jing Fang Received: 4 January 2009 / Accepted: 9 April 2009 / Published online: 30 May 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract A convenient technique is reported in this note for measuring elastic modulus of extremely soft material for cellular adhesion. Specimens of bending cylinder under gravity are used to avoid contact problem between testing device and sample, and a beam model is presented for evaluating the curvatures of gel beams with large elastic deformation. A self-adaptive algorithm is also proposed to search for the best estimation of gels elastic moduli by comparing the experimental bending curvatures with those computed from the beam model with preestimated moduli. Application to the measurement of the property of polyacrylamide gels indicates that the material compliance varies with the concentrations of bis-acrylamide, and the gels become softer after being immersed in a culture medium for a period of time, no matter to what extent they are polymerized. Keywords Young s modulus Bending measurement Cell adhesion Polyacrylamide gel 1 Introduction Interaction between an elastic substrate and adherent cells has received extensive attention since it plays an important role in regulating cellular functions and behaviors in contraction, migration, and invasion [1 4]. With the help of distortion of flexible substrates or elastic deformation of micro-columns induced by the adherent cells, the cellular The project supported by the National Basic Research Program (2007CB935602), the National Natural Science Foundation of China ( , ). X. Peng J. Huang L. Qin C. Xiong (B) J. Fang Laboratory of Cell Mechanics, Department of Biomedical Engineering, Peking University, Beijing, China cyxiong@pku.edu.cn mechanical responses can be investigated by quantitative analyses [5 10]. To match the mechanical properties of the cells, some extremely soft polymeric gels, such as polydimethylsiloxane (PDMS), polyacrylamide (PAA), are used as the flexible substrates to culture various cells so that their biological activities can be evaluated. It has been reported that elastic stiffness of the substrate gels exerts markedly influences on mechanical behaviors and biochemical expressions of the adherent cells [5 9]. Therefore, accurate determination of Young s moduli of these extremely soft gels becomes essential for obtaining exact responses of the cells to the substrate flexibilities. PAA gel is among the widely used materials for exploring the responses of cells to their substrates since the hardness of this polymer can be modulated by varying percentages of monomers and its varied modulus is of the order of that of cells [11]. The gels, such as PDMS and PAA, usually display large elastic deformation in resisting the applied forces. According to the tension results reported by Pelham et al. [4], when the specimens with a cross-section area of 3 mm 10 mm were loaded by a stretching force of N, the PAA gels could undergo a strain to the extent of %, showing the fact that the material behaved as linear stress strain relation in this strain range. Likewise, the indentation tests on gels with a small steel ball [7], as well as the studies using a micro-probe in atom force microscopy (AFM) [12] exhibited similar properties [13]. From the view point of experimental techniques, the method of stretching gel specimens often fails due to stress concentration at the clamping end of the specimen with low strength intensity. For the technique using AFM or nanoindentation [14] to detect the probe force and the indented depth, interfacial adhesions are usually involved in the contact surfaces between the materials, and thus the surface adhesions have strong influence on the deduction of Young s modulus of the material, and adhesive work needs

2 566 X. Peng et al. to be considered in the models in the process of mechanical property evaluation [15]. This report presents a new technique for measuring the Young s modulus of soft gels, which are useful for cell culture and adhesion. A bending specimen of gel cylinder with natural gravity load is adopted for measuring the beam deflection, since the manipulation is straightforward and the relation between the material deformation and the elastic modulus is simple. Theoretical analysis of large deformation is carried out to provide an analytical model for processing the measured displacement data, and numerical simulations are performed to validate the proposed method. The results for characterizing properties of porous PAA gels are presented to show the variation of elastic modulus with time after the specimens are immersed into a cellular culture medium. Fig. 1 An image of the bent PAA cylinder, which is produced by pushing the gel out of the pinhead of a syringe and bending under gravity 2 Specimen preparation and computational algorithm The gel material was made from polyacrylamide prepolymer prepared as described by Dembo et al. [6]. A mixture containing acrylamide (10%, Sigma) and bis-acrylamide was pumped into a syringe (1 ml) for polymerization using TEMED (N, N, N, N -tetramethylethylenediamine) and ammonium persulfate. Three sets of samples of different percentages (0.03, 0.13 and 0.26%) of bis-acrylamide were produced. In the experiments, one set was tested immediately after polymerization (50 min), while the other two sets were tested after being immersed in Dulbecco s modified Eagle s medium (DEME), containing 10% fetal bovine serum (v/v, Hyclone) and antibiotics (50 mg/ml streptomycin, 50 U/ml penicillin), and maintained in an oven at 37 Cfor 24 and 48 h. The bending specimens were prepared by simply pushing the polymerized polyacrylamide out of the pinhead of the syringe, which was placed in a horizontal bench to allow the polymer to move out and drop downwards by gravity, as shown in Fig. 1. The length of the polyacrylamide beam was measured by the moving distance of the piston as the specimen incision leaves the syringe output. Images of the bending PAA specimen were collected by a CCD camera to measure the exact diameter of the polymer cylinder, and to record the curvature of beam deflection. Taking into account the interaction between shear strain, extension strain and geometrically large bending deformation, we can derive an analytical expression for the beam deflection, as the material keeps linear stress strain relationship [16]. For a circular cross-section beam with radius R, the natural deformation under gravity, as illustrated in Fig. 2, can be described as dθ dt = s t q(s t) cos[θ(t)]dt κq cos θ + EI(θ) GA(θ), (1) Fig. 2 An illustration of the bent beam with large elastic deformation, used for the finite difference solution of the deflection where θ is the beam slope, s the arc length of the bent beam, t the length from the fixed end to the current point, q the equivalent load of gravity, E the Young s modulus, G the shear modulus and κ the shear coefficient (equal to 4/3 for the circular cross-section [16]).The firsttermin Eq. (1) represents the bending effect, and the second denotes the shear effect which should be taken into account when the geometric structure of the specimen can not satisfy the slender condition. For large deflection of the gel beam under gravity load, the bending rigidity EI(θ) and shear rigidity GA(θ)/κ are altered along the neutral axis as a result of the change in the radius of cross-section. The Poisson s ratio of PAA gels is known as 0.48 with approximately incompressible property [17,18]. Taken together, all these factors are considered in solving the deflection of the largely deformed beam by means of a finite difference approach, as given in the following.

3 A method to determine Young s modulus 567 Fig. 3 Flow chart of the iteration algorithm for data processing of deflection experiments By assuming that the beam is divided into N segments, each with length of h, the position of the nth node is determined by x n = x n 1 + h(1 + ε n ) cos θ n (2) y n = y n 1 + h(1 + ε n ) sin θ n (3) where θ n is the beam slope at the nth node, ε n the tensile strain which can be obtained by integrating Eq. (1) intwo steps. The first step is to calculate (dθ/dt) in an iterative form ( ) dθ (k) qh ( N m=n = dt n x m (k 1) EI (k 1) n ) x n (k 1), (4) where the superscript k is the current iteration number. This iteration process stops when the residual error satisfies the limited condition x (k) N x(k 1) N e (e is a beforehand given error). And the second step is to solve θ by forward difference, or θ n = θ n 1 +(dθ/dt) n h(1+ε n ). Thus the beam slope can be expressed as θ n+1 = q(s t) cos θ n EI n h 2 + 2θ n θ n 1. (5) Figure 3 displays a flow chart to describe the computation process. It is worth noting that the accumulation of computational errors in each iterative step, as implied in the flow chart of Fig. 3, is most likely to induce divergence of the differential solution and accordingly leads to unsteady computational results. Therefore, a boundary constraint must be introduced here by limiting the beam slop within a range of ( 0, π 2 ),sayθn = min [θ n 1 + ( dθ dt )n 1 h, π 2 ], to avoid divergence during the iterative computation. And then, a self-adaptive algorithm is developed to determinate the soft material Young s moduli from the beam deflections. For a real image of the bent beam captured by the CCD camera, we adopt a four-order polynomial, say y = f (x), to fit the curve of the deformed cantilever. On the other hand, by employing the above iteration program, a series of computational beam curves can be obtained by varying the value of Young s modulus within a prescribed range roughly pre-estimated. The discrepancies between the real beam shape and the specified calculated curves can be quantitatively characterized as ε 2 = r ( ) yi f (x i ) 2 / (r 1) (6) i=1 f (x i )

4 568 X. Peng et al. Fig. 4 Comparison between the bending curves solved by finite differential method (Matlab) and by FEM simulation (ANSYS) where (x i, y i ) represents the ith sampling point selected from the computational beam curves, whereas (x i, f (x i )) denotes the corresponding location of the actual deformed cantilever; r is the total number of sampling points. Differentiating Eq. (6) with respect to the Young s modulus and setting the outcome equal to zero, we can thus obtain an optimal estimation of the Young s modulus E. The self-adapted process is carried out to search for the minimum value of the error function, and the entire algorithm is programmed in Matlab (MathWorks, Natick, MA). 3 Results and discussions To validate the feasibility of the proposed method for processing experimental data, we firstly computed a soft-gel cantilever by using the finite element method (FEM) (ANSYS, Inc. Houston, PA) and the present method, respectively. The beam was chosen with 18 mm of length and 4.5 mm in diameter, and Young s modulus is 13kPa and Poisson ratio In the computation with ANSYS, the BEAM 189 element type was selected for modeling the structure; and the keys of large deformation analysis and linearly elastic material were switched on. Figure 4 gives a comparison between the beam curves from FEM (dotted line) simulation and from our finite difference computation (solid line), with the same set of geometrical and mechanical parameters. The maximum discrepancy between the two deflection curves is only mm, less than one pixel (0.05 mm) in the corresponding digital images, indicating that both results are in close agreement. Based on the curvature pictures of the bent cylinders, the Young s moduli of the PAA material were obtained via the fitting algorithm of the beam images, as showed in Fig. 5.The moduli of the non-immersed PAA gels were 13.1 ± 2.1 kpa, 34.7±4.1 kpa and 59.9 ± 4.7 kpa, respectively, as the percentages of bis-acrylamide in the soft gels were 0.03, 0.13 and 0.26%. With experimental uncertainties taken into account, these moduli are of the same order as the results obtained by using other methods, as previously reported in Refs. [4,6,17]. More specifically, the modulus of non-immersed PAA with Fig. 5 Young s moduli of PAA gels with different concentrations of cross-linker, showing their changes before and after being immersed in DMEM medium for one and two days 0.03% bis-acrylamide is about twice as large as that reported in Ref. [4], whereas the other sets show no distinct differences. More important, our results show that immersion of the specimens in DMEM brings a sharp decrease of the Young s modulus, which has not been paid attention in all previous reports. As shown in Fig. 5, the moduli of the samples immersed for 24 h are nearly one half of those without immersion. The question why the polymer moduli decline during this period is still open and needs further physicalchemistry analysis. Anyhow, there is no evident discrepancy between the immersed samples for 24h and 48 h, suggesting that for the PAA gel used to culture adherent cells, the elastic modulus of the soft substrate becomes stable after the material has been immersed in DMEM for a period of time, possibly more than 24 h. Actually, for most experiments dedicated to the study of interaction between adherent cells and polymeric substrates, the substrates are usually immersed in culture medium for at least several hours or even more than one day, in order to guarantee steady cell adhesions. During this process, the immersion might well have altered the mechanical properties of the gel substrates, especially for those hydrated materials like PAA. This feature seems neglected in previous measurements, in which the Young s moduli of the elastic substrates were normally surveyed just after gel polymerization without any immersion process [4,6,17], and thus might induce big errors in evaluating the response of adherent cells to substrate flexibilities. It is therefore recommended that the Young s moduli of the soft gels be measured under the same condition as the real one these materials will experience. Note that the above theoretical analysis and experimental data processing is based on a premise that the gel material lies within linear elasticity range even when the bent specimens

5 A method to determine Young s modulus 569 Table 1 The suggested lengths for the PAA gel-cantilever specimens to keep the material within linear stress strain relationship Duration (h) Concentration of bis-acrylamide (%) mm 35mm 45mm mm 25 mm 35 mm mm 25 mm 35 mm adhesive effect of contact in using AFM probing to measure force and deformation. We present analytical expression to describe large elastic deflection of the flexible gel beam and develop the algorithm to fit the beam curvature so as to obtain optimal estimation of the Young s modulus. The comparison of our results with those from FEM simulation validates the proposed method in data processing, and the application to property measurement of PAA material indicates that the elastic moduli of the gel vary with the percentages of the cross-linker involved. The present results show that the PAA gel becomes more compliant after immersed in DMEM for a period of time, which is important in estimating the mechanical properties of the soft matter and in evaluating the responses of the cultured cells to the stiffness change of the flexible substrates [5,10]. References Fig. 6 Comparison of the real curvature shape of a specimen containing 0.03% bis-acrylamide with the best-fitted bent beam calculated by the iterative computation with the optimal estimation of Young s modulus undergo large geometrical deformation. In actual tests, therefore, the PAA specimen bending must be constrained within a limited strain range so that such an assumption is satisfied. As mentioned above, for this kind of soft materials including PDMS and PAA, a number of tensile (or compressive) tests [4,6,19] confirmed that they responded to stress in a linearly elastic manner if the applied strain was less than 30%, and thus these requirements can be achieved in our specimen design by properly controlling the original lengths of the PAA cylinders, as given in Table 1, which are determined from FEM analysis (ANSYS) with preestimated Young s moduli. Moreover, the best-fitted bent curve derived by our processing algorithm indeed agrees well with the corresponding real cantilever shape, as shown in Fig. 6, with a mean error less than 0.05 mm, which not only further verifies that the PAA behaves as a linearly elastic material in our experiment, but also implies that it is feasible to use the current data processing to precisely estimate the elastic modulus. In conclusion, we propose an efficient way to measure the Young s modulus of soft gels with the aid of beam bending under gravity. The specimen preparation is very simple, with neither fracture of sample clamping in tensile tests, nor 1. Galbraith, C.G., Sheetz, M.P.: Forces on adhesive contacts affect cell function. Curr. Opin. Cell Biol. 10, (1998) 2. Vogel, V., Sheetz, M.P.: Local force and geometry sensing regulate cell functions. Nat. Rev. Mol. Cell Biol. 7, (2006) 3. Yeung, T., Georges, P.C., Flanagan, L.A., Marg, B., Ortiz, M., Funaki, M., Zahir, N., Ming, W.Y., Weaver, V., Janmey, P.A.: Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion. Cell Motil. Cytoskelet. 60, (2005) 4. Pelham, R.J., Wang, Y.L.: Cell locomotion and focal adhesions are regulated by substrate flexibility. Proc. Natl. Acad. Sci. USA. 94, (1997) 5. Qin, L., Huang, J.Y., Xiong, C.Y., Zhang, Y.Y., Fang, J.: Dynamical stress characterization and energy evaluation of single cardiac myocyte actuating on flexible substrate. Biochem. Biophys. Res. Commun. 360, (2007) 6. Dembo, M., Wang, Y.L.: Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys. J. 76, (1999) 7. Munevar, S., Wang, Y.L., Dembo, M.: Traction force microscopy of migrating normal and H-Ras transformed 3T3 fibroblasts. Biophys. J. 80, (2001) 8. Wang, N., Tolic-Norrelykke, I.M., Chen, J.X., Mijailovich, S.M., Butler, J.P., Fredberg, J.J., Stamenovic, D.: Cell prestress. I. stiffness and prestress are closely associated in adherent contractile cells. Am. J. Physiol. Cell Physiol. 282, C (2002) 9. Lee, J., Leonard, M., Oliver, T., Ishihara, A., Jacobson, K.: Traction force generated by locomoting keratocytes. J. Cell Biol. 127, (1994) 10. Huang, J.Y., Peng, X.L., Qin, L., Zhu, T., Xiong, C.Y., Zhang, Y.Y., Fang, J.: Determination of cellular tractions on elastic substrate based on an integral Boussinesq solution. J. Biomech. Eng.-Trans. ASME. 131, (2009). doi: / Li, Y., Wen, C., Xie, H., Ye, A., Yin, Y.: Mechanical property analysis of stored red blood cell using optical tweezers. Colloids Surf. B. 70, (2009) 12. 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, (2002) 13. Zhang, M., Zheng, Y.P., Mak, A.F.T.: Estimating the effective Young s modulus of tissue from indentation tests nonlinear finite element analysis of effects of friction and large deformation. Med. Eng. Phys. 19, (1997)

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