Programming the mechanics of cohesive fiber networks by compression.

Fibrous networks are ideal functional materials since they provide mechanical rigidity at low weight. Here, we demonstrate that fibrous networks of the blood clotting protein fibrin undergo a strong and irreversible increase in their mechanical rigidity in response to uniaxial compression. This rigidification can be precisely controlled by the level of applied compressive strain, providing a means to program the network rigidity without having to change its composition. To identify the underlying mechanism we measure single fiber-fiber interactions using optical tweezers. We further develop a minimal computational model of cohesive fiber networks that shows that stiffening arises due to the formation of new bonds in the compressed state, which develop tensile stress when the network is re-expanded. The model predicts that the network stiffness after a compression cycle obeys a power-law dependence on tensile stress, which we confirm experimentally. This finding provides new insights into how biological tissues can adapt themselves independently of any cellular processes, offering new perspectives to inspire the design of reprogrammable materials.

Porosity Governs Normal Stresses in Polymer Gels.

When sheared, most elastic solids including metals, rubbers, and polymer gels dilate perpendicularly to the shear plane. This behavior, known as the Poynting effect, is characterized by a positive normal stress. Surprisingly, fibrous biopolymer gels exhibit a negative normal stress under shear. Here we show that this anomalous behavior originates from the open-network structure of biopolymer gels. Using fibrin networks with a controllable pore size as a model system, we show that the normal-stress response to an applied shear is positive at short times, but decreases to negative values with a characteristic time scale set by pore size. Using a two-fluid model, we develop a quantitative theory that unifies the opposite behaviors encountered in synthetic and biopolymer gels.

Elasticity of fibrous networks under uniaxial prestress.

We present theoretical and experimental studies of the elastic response of fibrous networks subjected to uniaxial strain. Uniaxial compression or extension is applied to extracellular networks of fibrin and collagen using a shear rheometer with free water in/outflow. Both uniaxial stress and the network shear modulus are measured. Prior work [van Oosten, et al., Sci. Rep., 2015, 6, 19270] has shown softening/stiffening of these networks under compression/extension, together with a nonlinear response to shear, but the origin of such behaviour remains poorly understood. Here, we study how uniaxial strain influences the nonlinear mechanics of fibrous networks. Using a computational network model with bendable and stretchable fibres, we show that the softening/stiffening behaviour can be understood for fixed lateral boundaries in 2D and 3D networks with comparable average connectivities to the experimental extracellular networks. Moreover, we show that the onset of stiffening depends strongly on the imposed uniaxial strain. Our study highlights the importance of both uniaxial strain and boundary conditions in determining the mechanical response of hydrogels.

Uncoupling shear and uniaxial elastic moduli of semiflexible biopolymer networks: compression-softening and stretch-stiffening.

Gels formed by semiflexible filaments such as most biopolymers exhibit non-linear behavior in their response to shear deformation, e.g., with a pronounced strain stiffening and negative normal stress. These negative normal stresses suggest that networks would collapse axially when subject to shear stress. This coupling of axial and shear deformations can have particularly important consequences for extracellular matrices and collagenous tissues. Although measurements of uniaxial moduli have been made on biopolymer gels, these have not directly been related to the shear response. Here, we report measurements and simulations of axial and shear stresses exerted by a range of hydrogels subjected to simultaneous uniaxial and shear strains. These studies show that, in contrast to volume-conserving linearly elastic hydrogels, the Young's moduli of networks formed by the biopolymers are not proportional to their shear moduli and both shear and uniaxial moduli are strongly affected by even modest degrees of uniaxial strain.

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: subdiffusion of heavy tracers.

We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.

Area coverage of radial Lévy flights with periodic boundary conditions.

We consider the area coverage of radial Lévy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Lévy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Lévy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Lévy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Lévy flights in a square area. Our findings are of interest for a general understanding of Lévy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.

Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium.

We investigate the diffusive motion of micron-sized spherical tracers in a viscoelastic actin filament network over the time span of 8 orders of magnitude using optical-tweezers single-particle tracking. The hydrodynamic interactions of a tracer with the surrounding fluid are shown to dominate at microsecond time scales, while subdiffusive scaling due to viscoelastic properties of the medium emerges at millisecond time scales. The transition between these two regimes is analyzed in the frame of a minimal phenomenological model which combines the Basset force and the generalized Stokes force. The resulting Langevin equation accounts for various dynamical features of the thermal motion of endogenous or exogenous tracers in viscoelastic media such as inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. Simple analytical formulas for the mean-square displacement and velocity autocorrelation function are derived.