RESUMO
The Hertz-Sneddon elastic indentation model is widely adopted in the biomechanical investigation of living cells and other soft materials using atomic force microscopy despite the explicit viscoelastic nature of these materials. In this work, we demonstrate that an exact analytical viscoelastic force model for power-law materials, can be interpreted as a time-dependent Hertz-Sneddon-like model. Characterizing fibroblasts (L929) and osteoblasts (OFCOLII) demonstrates the model's accuracy. Our results show that the difference between Young's modulus EY obtained by fitting force curves with the Hertz-Sneddon model and the effective Young's modulus derived from the viscoelastic force model is less than 3%, even when cells are probed at large forces where nonlinear deformation effects become significant. We also propose a measurement protocol that involves probing samples at different indentation speeds and forces, enabling the construction of the average viscoelastic relaxation function of samples by conveniently fitting the force curves with the Hertz-Sneddon model.
RESUMO
Stressed soft materials commonly present viscoelastic signatures in the form of power-law or exponential decay. Although exponential responses are the most common, power-law time dependencies arise peculiarly in complex soft materials such as living cells. Understanding the microscale mechanisms that drive rheologic behaviors at the macroscale shall be transformative in fields such as material design and bioengineering. Using an elastic network model of macromolecules immersed in a viscous fluid, we numerically reproduce those characteristic viscoelastic relaxations and show how the microscopic interactions determine the rheologic response. The macromolecules, represented by particles in the network, interact with neighbors through a spring constant k and with fluid through a non-linear drag regime. The dissipative force is given by γvα, where v is the particle's velocity, and γ and α are mesoscopic parameters. Physically, the sublinear regime of the drag forces is related to micro-deformations of the macromolecules, while α ≥ 1 represents rigid cases. We obtain exponential or power-law relaxations or a transitional behavior between them by changing k, γ, and α. We find that exponential decays are indeed the most common behavior. However, power laws may arise when forces between the macromolecules and the fluid are sublinear. Our findings show that in materials not too soft not too elastic, the rheological responses are entirely controlled by α in the sublinear regime. More specifically, power-law responses arise for 0.3 ⪠α ⪠0.45, while exponential responses for small and large values of α, namely, 0.0 ⪠α ⪠0.2 and 0.55 ⪠α ⪠1.0.
Assuntos
Engenharia Biomédica , Viscosidade , ReologiaRESUMO
Living cells are constantly exchanging momentum with their surroundings. So far, there is no consensus regarding how cells respond to such external stimuli, although it reveals much about their internal structures, motility as well as the emergence of disorders. Here, we report that twelve cell lines, ranging from healthy fibroblasts to cancer cells, hold a ubiquitous double power-law viscoelastic relaxation compatible with the fractional Kelvin-Voigt viscoelastic model. Atomic Force Microscopy measurements in time domain were employed to determine the mechanical parameters, namely, the fast and slow relaxation exponents, the crossover timescale between power law regimes, and the cell stiffness. These cell-dependent quantities show strong correlation with their collective migration and invasiveness properties. Beyond that, the crossover timescale sets the fastest timescale for cells to perform their biological functions.
Assuntos
Fenômenos Fisiológicos Celulares/fisiologia , Elasticidade , Viscosidade , Linhagem Celular , Linhagem Celular Tumoral , Movimento Celular , Fibroblastos/fisiologia , Humanos , Microscopia de Força Atômica , Modelos Biológicos , Imagem Molecular , Invasividade Neoplásica/patologiaRESUMO
We propose a magnetic force modulation method to measure the stiffness and viscosity of living cells using a modified AFM apparatus. An oscillating magnetic field makes a magnetic cantilever oscillate in contact with the sample, producing a small AC indentation. By comparing the amplitude of the free cantilever motion (A0) with the motion of the cantilever in contact with the sample (A1), we determine the sample stiffness and viscosity. To test the method, the frequency-dependent stiffness of 3T3 fibroblasts was determined as a power law k(s)(f) = α + ß(f/f¯)(γ) (α = 7.6 × 10(-4) N m(-1), ß = 1.0 × 10(-4) N m(-1), f¯ = 1 Hz, γ = 0.6), where the coefficient γ = 0.6 is in good agreement with rheological data of actin solutions with concentrations similar to those in cells. The method also allows estimation of the internal friction of the cells. In particular we found an average damping coefficient of 75.1 µN s m(-1) for indentation depths ranging between 1.0 µm and 2.0 µm.
RESUMO
The viscoelastic properties of human kidney cell lines from different tumor types (carcinoma (A-498) and adenocarcinoma (ACHN)) are compared to a non-tumorigenic cell line (RC-124). Our methodology is based on the mapping of viscoelastic properties (elasticity modulus E and apparent viscosity η) over the surface of tens of individual cells with atomic force microscopy (AFM). The viscoelastic properties are averaged over datasets as large as 15000 data points per cell line. We also propose a model to estimate the apparent viscosity of soft materials using the hysteresis observed in conventional AFM deflection-displacement curves, without any modification to the standard AFM apparatus. The comparison of the three cell lines show that the non-tumorigenic cells are less deformable and more viscous than cancerous cells, and that cancer cell lines have distinctive viscoelastic properties. In particular, we obtained that E(RC-124) > E(A-498) > E(ACHN) and η(RC-124) > η(A-498) > η(ACHN).