Nano-structural stiffness measure for soft biomaterials of heterogeneous elasticity

Measuring the structural stiffness aims to reveal the impact of nanostructured components or various physiological circumstances on the elastic response of material to an external indentation. With a pyramidal tip at a nano-scale, we employed the atomic force microscopy (AFM) to indent the surfaces of two compositions of polyacrylamide gels with different softness and seedling roots of Arabidopsis thaliana. We found that the stiffness-depth curve derived from the measured force exhibits a heterogeneous character in elasticity. According to the tendency of stiffness-depth curve, we decomposed the responding force into depth-impact (F_C), Hookean (F_H) and tip-shape (F_S) components, called trimechanic, where F_S and its gradient should be offset at the surface or subsurfaces of the indented material. Thereby, trimechnic theory allows us to observe how the three restoring nanomechanics change with varied depth. Their strengths are represented by the respective spring constants (k_C, k_H, k_S) of three parallel-connected spring (3PCS) analogs to differentiate restoring nanomechansims of indented materials. The effective Young's modulus Ê and the total stiffness k_T (= k_H + k_S) globally unambiguously distinguish the softness between the two gel categories. Data fluctuations were observed in the elasticity parameters of individual samples, reflecting nanostructural variations in the gel matrix. Similar tendencies were found in the results from growing plant roots, though the data fluctuations are expectedly much more dramatic. The zone-wise representation of stiffness by the trimechanic-3PCS framework demonstrates a stiffness measure that reflects beneath nanostructures encountered by deepened depth. The trimechanic-3PCS framework can apply any mechanical model of power-law based force-depth relationship and is compatible with thin layer corrections. It provides a new paradigm for analyzing restoring nanomechanics of soft biomaterials in response to indenting forces.