Biopolymer network geometries: Characterization, regeneration, and elastic properties
Started by Lauren Hartle, Fall 2011.
"Biopolymer network geometries: Characterization, regeneration, and elastic properties" S. B. Lindström, D. A. Vader, A. Kulachenko, and D. A. Weitz. Phys. Rev. E, 82, 051905 (2010).
- IN PROGRESS********
Researchers in George Whitesides and David Weitz's groups designed a model system of a polymer network with active constituents to explore the way that cells control their mechanical properties. Cells use networks composed of active filaments controlled by molecular motors. Cells use networks composed of active filaments controlled by molecular motors. The system in question is a filamentous actin (F-actin) network, made active via myosin molecular motors anchored by the cross-linking agent filamin A (FLNa). By inducing internal stress in the network, researchers were able to manipulate the elastic properties of the gel over 3 orders of magnitude. It has been demonstrated that systems containing only a few percent protein by volume can experience a transition from liquid-like to solid-like mechanical behavior. This fact, coupled with the tuning of active components opens up the possibility of active biomaterials that adapt mechanically to their environment.
A number of properties of the system were explored. Some key findings are listed below:
Effects of System Components on Stiffness
Researchers isolated the effects of each component of the system. It was found that FLNa crosslinking only increased the networks stiffness slightly. According to the researchers, the relative softness, or "floppiness" of these crosslinks caused them to dominate the mechanical behavior in the linear regime. Once active myosin II thick filaments (molecular motors, which cause anti-parallel movement of network filaments, hence "contracting" the network and inducing internal stress) were included in the network, the stiffness and the ratio of loss to storage decreased, implying more solid-like behavior. When blebbistatin was used to de-activate the myosin motors, all effect on stiffness disappeared. To create the aforementioned stiffening effects, all three components (filaments, cross-linkers, and molecular motors) must be present. A threshold effect was noted: A slight increase in loss with increasing concentration was noted. Figure 1 shows the associated data.
Role of long-range structure and network alignment
Local alignment is present without or with FLNa cross-linkers, but the long-range structure showed no overall alignment. When myosin molecular motors formed permanent bonds (due to an ATP-free environment), significant filament "bundling" and alignment was noted. Both FLNa and myosin are required for local "bundling", and network "contraction". Figure 2 shows the bundling effect.
Comparison of active networks under internal stress to passive networks under external stress
By comparing active networks under internal stress to passive networks under external stress, researchers were able to approximate the internal stress induced by myosin motor activity. A schematic of the model shown in Figure 3A, with a passive network under shear in the left-most diagram, and an active network under internal stress in the right-most diagram. For the highest stress values, the ratio of the loss modulus to the storage modulus increased slightly for the active networks for frequencies below 0.2 rad/s, while passive networks showed frequency-independence of the loss modulus in high external stress states. Figure 3 B shows the aforementioned data. These results imply that for active networks, stress relaxation is mediated by a process with a characteristic time scale of larger than 5 seconds--a time scale consistent with measured myosin motor release rates. In other words, the passive model (which includes no creep) seems suitable for describing the viscous response of the active network, provided that the testing frequency corresponds to a timescale smaller than the myosin motor release rate. Results also demonstrated that myosin motor activity can make actin networks behave more like a solid by increasing the internal stress. This is demonstrated by the decrease in the exponent of a power law fit of elastic modulus vs frequency. For solids, the elastic modulus is frequency independent, whereas a viscous liquid has a linear dependence on frequency. As shown in Figure 3 C, the change in the exponent with stress is the same for a passive network undergoing external shear, and an active network undergoing internal stress. Both have a decreasing exponent, indicating more solid-like behavior.
At higher shear stresses, the passive networks exhibit nonlinear behavior in the elastic modulus. With increasing myosin to actin ratios, the nonlinear behavior is activated at increasingly high critical stresses, as shown in Figure 4 A. The stiffening behavior of the active networks approach and merge with the behavior of the passive networks above a certain stress, but this "merging" is delayed by a higher myosin to actin ratio. This implies that the myosin/actin network modulates the nonlinear behavior of the system by applying internal stress. As shown in Figure 4C, all the curves representing elastic modulus vs external stress collapse once the zero-stress stiffness and critical stress are used to normalize the axes. If one looks at critical strain rather than critical stress, the passive networks exhibit stiffening above 30%, while active networks exhibit stiffening after strain of a fraction of a percent. Figure 4 D explains this phenomenon in terms of pre-stresses: the critical strain has an inverse relationship with the internal stress. The pull of the molecular motors "straightens out" the cross-linkers, hence after only a small strain, the elongation of the far stiffer actin filaments must be the source of subsequent strain increases. As Figure 4 A and B demonstrate, the rupture strength is roughly equal for all passive and active networks of the same cross-link density (as determined by FLNa concentration), suggesting that cross-link rupture is the dominant failure mode of the network.
The researchers provide two "design principles" for the described active networks:
1. Myosin motors must be active to create stiffening
2. F-actin cross-linkers are required as anchor points, otherwise the myosin motors permit sliding, creating a weaker network.
Several other ideas emerge from the data presented:
- The stiffening response of the cross-linker critically influences the strain-stiffening of the network.
- Specific alignment or ordering of filaments might not be required to see stress stiffening.
- Cell-adhesion seem to operate in a nonlinear regime. This may enhance the cell's control of, and adaptability to its environment.
- The networks studies have an order of magnitude lower elastic moduli than measured in adherent cells. This may be due due to a discrepancy in filament length between the two systems.
Soft Matter Connection
The system studied touches upon a number of important concepts in soft matter. The accessibility and usefulness of nonlinear regimes in this biologically-derived network mirror those in other compliant systems, from muscle cell networks to spider webs. The complexity of the system, including concentration-, stress-dependent mechanical and chemical "cross-talk" among the different components highlights both the immense potential of and difficulty inherent in studying this, and any other soft matter system. Whether it's stiffening behavior (studied using the classic methods presented here), surface chemistry, electrostatics, thermodynamics, or kinetics, soft matter systems are often marked with complexity. The careful teasing out of the relevant fundamental principles in a given system, as well as the insight into one might usefully apply such an understanding, is the major pursuit and defining philosophy of soft matter scientists today.