Alpha-helical protein networks are self-protective and flaw-tolerant

Abstract

Alpha-helix based protein networks as they appear in intermediate filaments in the cell's cytoskeleton and the nuclear membrane robustly withstand large deformation of up to several hundred percent strain, despite the presence of structural imperfections or flaws. This performance is not achieved by most synthetic materials, which typically fail at much smaller deformation and show a great sensitivity to the existence of structural flaws. Here we report a series of molecular dynamics simulations with a simple coarse-grained multi-scale model of alpha-helical protein domains, explaining the structural and mechanistic basis for this observed behavior. We find that the characteristic properties of alpha-helix based protein networks are due to the particular nanomechanical properties of their protein constituents, enabling the formation of large dissipative yield regions around structural flaws, effectively protecting the protein network against catastrophic failure. We show that the key for these self protecting properties is a geometric transformation of the crack shape that significantly reduces the stress concentration at corners. Specifically, our analysis demonstrates that the failure strain of alpha-helix based protein networks is insensitive to the presence of structural flaws in the protein network, only marginally affecting their overall strength. Our findings may help to explain the ability of cells to undergo large deformation without catastrophic failure while providing significant mechanical resistance.

Figure 1. Model formulation, geometry and setup.

Subplot A shows a schematic of the coarse-graining procedure, replacing a full atomistic representation of an alpha helical protein domain by a mesoscale bead model with bead distance r ₀. Subplot B shows a snapshot of a quasi-regular lamin meshwork (scale bar 1 µm) as observed in experimental imaging of oocytes; where structural imperfections are highlighted in white. Image of lamin meshwork reprinted with permission from Macmillan Publishers Ltd., from Nature , copyright © 1986. Subplot C depicts a schematic of the coarse grained protein network geometry used in this study, with the applied mode I tensile boundary conditions. The size of the network equals to 24 nm×24 nm, where each filament is represented by one alpha helix, as shown in the blow-up. A constant strain rate is applied in y-direction to apply mode I tensile loading through displacing the outermost rows of beads. The crack represents a geometrical flaw or inhomogeneity as they appear in vivo. Subplot D depicts characteristic force-strain curves for pulling individual alpha-helices as used in our mesoscale bead model. As explained in Materials and Methods, this force-strain behavior is derived from full-atomistic simulations and theoretical analysis, and has been validated against experimental studies. The labels α, β and γ identify the three major regimes of deformation.

Figure 2. Effect of large uniaxial stretch on the intermediate filament network in MDCK cells, illustrating the ability of intermediate filament network to undergo very large deformation without catastrophic failure.

The cells were grown on collagen-coated silastic membranes and stretched using a custom cell stretcher that was mounted on a confocal microscope. Cells were fixed and stained for immunofluorescence (red = keratin IFs, blue = DNA). Subplot A: Control cells were processed on a relaxed silastic membrane. Subplot B: Stretched cells were fixed, stained and imaged on membranes that were held in the stretched state. The approximate uniaxial strain in stretched cells is 75%. Scale bar is approximately 25 µm. Images reprinted with permission of John Wiley & Sons, Inc. from reference , Biomechanical properties of intermediate filaments: from tissues to single filaments and back, Vol. 29, No. 1, 2007, pp. 26–35, copyright © 2007 John Wiley & Sons, Inc.

Figure 3. Hierarchical structure of the alpha-helical protein network considered here.

The plot shows a schematic of five levels of hierarchies (H0..H4). Intrabackbone H-bonds provide the basic structural building block (H0). A cluster of 3–4 H-bonds stabilize the basic building block of alpha-helices, a alpha-helical convolution (H1). The linear arrangement of many convolutions leads to an alpha-helix filament (H2). The squared arrangement of several alpha-helix filament (H3) provides the basic structure of the network level (H4). The structure at the network level (H4) may also contain structural defects, as illustrated in Figure 1C.

Figure 4. Mechanical response of the alpha-helical protein network.

The graph shows stress-strain curves of a protein network, with and without a crack, as well as for two different crack sizes. The relative crack size is given as ratio of crack length ξ divided by the system size L, defined as χ  = ξ / L. We observe two major regimes, (I–III) a very flat increase in stress until approximately 100 MPa, followed (III–IV) by a very steep increase in stress due to strain hardening of the protein backbone up to strains of close to 140..150%. Eventually, strong bonds between different alpha-helical protein chains break, and the entire system fails catastrophically. Interestingly, there exists only little difference in terms of the failure strain between all three systems, indicating the fault tolerance of the studied structure. The perfect system (without a crack) has a strength of ≈600 MPa.

Figure 5. Failure strain and failure stress as a function of crack size and comparison with theoretical model.

Panel A: Systematic analysis of the failure strain of the system, showing the failure strain over the relative crack size χ. The simulation results show that the failure strain is largely insensitive to the presence and size of cracks. Further, the plot includes the prediction based on eq. (2), corresponding to a scaling as . This behavior reflects that the scaling parameters are much different (−0.0362 vs. −0.5), and that linear elastic fracture mechanics (LEFM) fails to describe the fracture behavior of this material. Panel B: Analysis of the failure stress of the system as function of χ. The analysis also shows a deviation from the prediction of LEFM. The blunted crack-tip model is also shown for comparison (dashed line), providing an overall better fit than LEFM through the scaling law σ _0,f ∼ 1/(1+Cχ). Note that for relative crack sizes <5% the maximum strain and stress in panels A and B, respectively, does not change as the material has reached a complete insensitivity with respect to imperfections (data not shown in graph).

Figure 6. Snapshots of the protein network deformation.

Panel A shows a schematic of the characteristic force-extension curve of a single alpha helix (consisting of three regimes) to provide the color code for the snapshots shown below. Panel B shows snapshots of the network with crack at different laterally applied strains (snapshot numbering refers to points shown in Figure 4). The deformation mechanism of the network is characterized by molecular unfolding of the alpha-helical protein domains, leading to the formation of very large plastic yield regions. These plastic yield regions represent an energy dissipation mechanism to resist catastrophic failure of the system. Once the entire structure reaches the rupture strain the crack propagates, leading to catastrophic failure, characterized by breaking of backbone atomic bonds as shown in the circled areas I and II. The white ellipsoids in the first and the last snapshot highlight the crack shape transformation that occurs during deformation (they show the surface geometry of the crack). The blowups show the nanoscale structural arrangements of the alpha-helical protein filaments under different levels of strain. The α structure is an intact helix, with 3–4 H-bonds per turn (yellowish thick lines). The β structure is a partially unfolded alpha-helix, with some of the H-bonds broken along the filament axis whereas others are still intact. The γ structure shows a completely unfolded alpha-helix, where the protein's backbone is being stretched. These three structures correspond to the color codes blue, yellow and red, respectively. Panel C shows the change of the crack geometry under macroscale deformation (crack shapes correspond to the white ellipses in panel B).

Figure 7. Detailed view into structure at crack tip for two distinct strain levels, and illustration of microscopic deformation mechanism.

Panel A shows results associated with Figure 4, snapshot III. Panel B shows results associated with Figure 6, snapshot IV. The same color code as shown in Figure 6A applies here. The results depicted in panel A reveal that the filaments are relaxed in the x-direction (orthogonal to loading), and are highly stretched in the y-direction (direction of loading). There is a slight stress concentration at the tip of the crack, as can be seen by the red color indicating stretching of the protein filament's covalent backbone. In panel B, alpha-helices of the entire domain to the right (and left) of the crack are unfolded and the alpha helix protein backbones are stretched, whereas only the center part of the system has unfolded. This indicates that stress localization does not appear; instead, the entire network carries the load. This could explain the different behavior compared with conventional LEFM (see Figures 5). Panel C illustrates a possible microscopic deformation mechanism (as seen similarly in the circled area in panel B). The particular geometry of the square-lattice structure provides the structural basis for filaments to independently stretch without affecting neighboring bonds, since there are no immediate interactions between individual filaments in the network that prevent microscopic rotations and shear. This facilitates extremely large strain gradients at low energy cost (≈2x10¹¹%/Å).

Figure 8. Change of the microscopic crack shape as the protein network undergoes macroscopic mode I tensile deformation.

Panel A shows shape of the initial crack (an elliptical geometry where the length in the x-direction is much greater than the extension in the y-direction). Panel B shows shape of the final crack before onset of failure, representing an elliptical geometry where the length in the y-direction is much greater than the extension in the x-direction. The plots also indicate the distribution of stresses for both cases (the solution for the stress field is symmetric, but shown here only for the right half). The crack shapes reflect those measured in the simulations shown in Figure 6 (there highlighted in white color). The initial geometry and crack shape is shown in panel B (left part) in dashed lines to illustrate the significant transformation.