Optimal matrix rigidity for stress-fibre polarization in stem cells

Journal name:
Nature Physics
Year published:
Published online


The shape and differentiated state of many cell types are highly sensitive to the rigidity of the microenvironment. The physical mechanisms involved, however, are unknown. Here, we present a theoretical model and experiments demonstrating that the alignment of stress fibres within stem cells is a non-monotonic function of matrix rigidity. We treat the cell as an active elastic inclusion in a surrounding matrix, allowing the actomyosin forces to polarize in response to elastic stresses developed in the cell. The theory correctly predicts the monotonic increase of the cellular forces with the matrix rigidity and the alignment of stress fibres parallel to the long axis of cells. We show that the anisotropy of this alignment depends non-monotonically on matrix rigidity and demonstrate it experimentally by quantifying the orientational distribution of stress fibres in stem cells. These findings offer physical insight into the sensitivity of stem-cell differentiation to tissue elasticity and, more generally, introduce a cell-type-specific parameter for actomyosin polarizability.

At a glance


  1. Actomyosin stress-fibre alignment in hMSCs sparsely plated on 2D substrates of different elasticities.
    Figure 1: Actomyosin stress-fibre alignment in hMSCs sparsely plated on 2D substrates of different elasticities.

    ac, hMSCs immunostained for myosin NMMIIa 24h after plating on elastic substrates with Young’s moduli Em of 1 (a), 11 (b) and 34kPa (c). Images are the most representative cells of the mean values obtained for cell area A, aspect ratio of long to short axis r and stress-fibre order parameter S=left fencecos2θright fence, where θ is the angle between each stress fibre in the cell and the long axis of the fitted ellipse. df, The respective orientational plots, where the different orientations of myosin filaments are depicted with different colours. The dark grey dashed ellipses are calculated from the moments up to the second order and represent the cell shape in terms of area and long and short axes, and the red line indicates the mean orientation of the stress fibres as determined by the anisotropic filter algorithm. χ is the angle between the mean stress-fibre orientation and the principal axis of the ellipse. From symmetry considerations, we need consider only the absolute value of χ between 0 and π/2; thus, a completely random distribution has an average χ=π/4. Values given for r and S are the mean values of at least 60 cells per condition. All scale bars represent 50μm.

  2. Cell adhesion and polarization represented by a 1D spring model.
    Figure 2: Cell adhesion and polarization represented by a 1D spring model.

    Springs with constants kc and km represent the elasticity of the cell and matrix respectively. Elastic morphological changes on cell adhesion (ab) are represented here by a change in the cellular spring length Δlc=[kc/(kc+km)]Δlc0 This triggers an internal feedback mechanism (bc) that results in an enhancement of the active forces (see equation (2)), and to a further change in cell length as given by equation (1).

  3. Cell polarization as a function of the ratio of Young/'s modulus of the matrix, Em, and the cell, Ec, in both our 2D and 3D models.
    Figure 3: Cell polarization as a function of the ratio of Young’s modulus of the matrix, Em, and the cell, Ec, in both our 2D and 3D models.

    The plots are shown for different values of the cellular aspect ratio, r. a,b, The normalized average dipole elements left fencepzzaright fence (solid lines) and left fencepxxaright fence (dashed lines) corresponding to the forces in the directions that are respectively parallel ( ) and perpendicular ( ) to the long axis of the cell (dark grey: r=5, light grey: r=2) for our 3D (a) and 2D (b) models. c,d, The calculated orientational order parameter of the stress fibres that is given by the normalized difference (left fencepzzaright fenceleft fencepxxaright fence)/p for our 3D (c) and 2D (d) models. The colour coding indicates the aspect ratio. In this plot, the Poisson ratio of the matrix and the cellular domain are taken to be νm=0.45 and νc=0.3 and the magnitude of the polarizability is α=3.

  4. The effect of axial cell elongation on stress-fibre polarization and experimental values of the order parameter S for different elastic substrates.
    Figure 4: The effect of axial cell elongation on stress-fibre polarization and experimental values of the order parameter S for different elastic substrates.

    a, A calculation of the 2D order parameter as a function of the matrix rigidity, for two cases: the cell spreads isotropically on the substrate, η=0 (black curve); the cell spreads anisotropically on the substrate, η=1 (grey curve), see the text. The two illustrations left of the curves show top views over the cell, before (shown as blank) and after (shown as shaded) cell spreading. In the asymmetric spreading case, r corresponds to the cell shape in an infinitely rigid matrix. For both curves we used r=2,α=2 and Poisson ratios as in Fig. 3. b, The experimental values of the stress-fibre order parameter, S=left fencecos2θright fence, 24h after plating the cells, for the three groups of cells (of aspect ratios r=1.5,2.5,3.5) as a function of Young’s modulus of the matrix, Em;θ is the angle between each stress fibre in the cell and the long axis of the fitted ellipse. Within each of the different groups, S is maximal for Em=11kPa and generally increases with aspect ratio r, in agreement with our theoretical predictions. The error bars denote the standard error of the mean and theory curves (dotted lines) calculated from the simplified expansion of S (Supplementary Information) are shown to guide the eye.


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Author information

  1. These authors contributed equally to this work

    • A. Zemel &
    • F. Rehfeldt


  1. Institute of Dental Sciences, Faculty of Dental Medicine, and the Fritz Haber Center for Molecular Dynamics, the Hebrew University-Hadassah Medical Center, Jerusalem 91120, Israel

    • A. Zemel
  2. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • F. Rehfeldt &
    • A. E. X. Brown
  3. III. Physikalisches Institut, Georg-August-Universität, 37077 Göttingen, Germany

    • F. Rehfeldt
  4. Graduate Group of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • D. E. Discher
  5. Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel

    • S. A. Safran


A.Z. and S.A.S. developed the theory. F.R., A.E.X.B. and D.E.D. designed the experiments; F.R. carried out the experiments; A.E.X.B. wrote the image analysis algorithm. All authors analysed the data and wrote the paper.

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The authors declare no competing financial interests.

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