Figure 4. Modulation of substrate mechanics by micro-post arrays.

(a) Schematic of the beam theory, where the flexure displacement is proportional to the lateral force via a factor called spring constant, K. The spring constant is determined by the material and geometry of the beam and thus enables geometrical modulation of the effective stiffness of the substrate. (b) Finite element (FE) simulation results of beams of different heights subject to the same amount of lateral forces (15 nN) applied over the tips, illustrating that the higher the beam the easier it bends. (c) SEM images of hMSCs cultured on micropost arrays with different spring constants, demonstrating cell shape was sensitive to the structural rigidity of the substrate.^[107] Reproduced with permission from [107]. Copyright 2010, Nature Publishing Group. (d) Theoretical prediction of the anisotropic spring constant of a beam with oval cross-section, where θ is the angle between the force and the longitudinal axis of the cross-section.^[106] The aspect ratio of the ellipse, a/b=3, was used for the plot. Inset: SEM image of the top surfaces of microposts with oval cross-sections. Adapted with permission from [106]. Copyright 2007, United States National Academy of Sciences. (e) FE simulation results of the effect of local force on the overall effective spring constant of a beam under lateral force applied to its top. When the same amount of lateral force was applied just within a nano-scale local area of the post top (D/d=1/10, d=1.8 μm, E=2.5 MPa), rather than over the entire post tip, the effect of local rigidity sensing by local force generally decreased the effective spring constant of the beam. Such effect of local sensing is expected to be greater when the material stiffness of the beam becomes smaller.