Elastic deformation and failure in protein filament bundles: Atomistic simulations and coarse-grained modeling

Abstract

The synthetic peptide RAD16-II has shown promise in tissue engineering and drug delivery. It has been studied as a vehicle for cell delivery and controlled release of IGF-1 to repair infarcted cardiac tissue, and as a scaffold to promote capillary formation for an in vitro model of angiogenesis. The structure of RAD16-II is hierarchical, with monomers forming long beta-sheets that pair together to form filaments; filaments form bundles approximately 30-60 nm in diameter; branching networks of filament bundles form macroscopic gels. We investigate the mechanics of shearing between the two beta-sheets constituting one filament, and between cohered filaments of RAD16-II. This shear loading is found in filament bundle bending or in tensile loading of fibers composed of partial-length filaments. Molecular dynamics simulations show that time to failure is a stochastic function of applied shear stress, and that for a given loading time behavior is elastic for sufficiently small shear loads. We propose a coarse-grained model based on Langevin dynamics that matches molecular dynamics results and facilities extending simulations in space and time. The model treats a filament as an elastic string of particles, each having potential energy that is a periodic function of its position relative to the neighboring filament. With insight from these simulations, we discuss strategies for strengthening RAD16-II and similar materials.

Figure 1

A filament of RAD16-II comprises two β-sheets, each with a hydrophobic surface of alanine side chains oriented toward the filament center, and an outer hydrophilic surface of arginine and aspartic acid side chains. A filament has a roughly rectangular cross-section. Here the upper β-sheet is cut away to show the hydrophobic core.

Figure 2

Shear loading schemes for one (A) or two (B) double-layered filaments are shown. (A) Shear stress is applied over the hydrophobic surface at the core of a single filament. (B) Shear stress is applied over the hydrophobic core of each filament and the charged surface between the filaments. In each case a constant force is applied to the α-carbons in one β-sheet, and an opposite force to the α-carbons of another β-sheet. Forces are directed along the filament axis.

Figure 3

Under small shear loads the shear deformation of RAD16-II filaments (loaded as in Figure 2B) is elastic. Under 250 MPa of applied shear, the filaments slip irreversibly within a few ps.

Figure 4

The trajectories in Figure 3 exhibit a linear stress-strain relationship only up to about 5% strain. In this low-strain region, the shear modulus is 1.8 GPa. At larger strains, strain softening occurs.

Figure 5

The trajectory of the methyl carbon in an alanine side chain is shown as a the hydrophobic core of an RAD16-II filament is forced to slip. The discrete jumps indicate energy minima with a spacing that corresponds to the 4.8 Ǻ between β-strands.

Figure 6

The hydrophobic surface of a filament of RAD16-II is represented, with each pixel representing one alanine side chain. Failure initiates at the white-colored points, then propagates laterally and longitudinally to the rest of the surface.

Figure 7

The time from when a load is first applied to when failure occurs is plotted against the applied shear stress for both MD simulations (n=3) and coarse-grained simulations (n=100). The triangle indicates MD simulations terminated before failure occurred (n=3). At 183 MPa the coarse-grained simulations had a mean failure time of 2935 ps, standard deviation 2552 ps.

Figure 8

In coarse-grained simulations, a power law was found to relate time to initial failure to applied load and to the number of particles simulated. The lines show a global best-fit described by Equation (6), with r²=0.935.

Figure 9

(A) Five periodic energy functions are shown, with arbitrary positioning on the y-axis. Each is shown with dimensionless energy barrier Π_b set to 30, 25, and 20 for illustration, all under a dimensionless load of Π_P = 50. From top to bottom these are 3o, 4o, U, S, and V, as defined in the text. (B) Dimensionless time to failure is plotted from simulations performed with a system of 16 particles using each energy function over a range of values for Π_b Each line style matches that of the function in (A) used to generate it.

Figure 10

(A) When a load is applied to an RAD16-II gel at the bulk level, the load is thought to be borne at the microstructural level by the bending of filament bundles as they are loaded at their points of intersection. (B) This bending is idealized as a beam whose ends are fastened such that they resist both moments and shear, and whose center is loaded by force P. (C) The problem may be simplified by noting that each quarter of the beam is loaded as a cantilevered beam with a shear load but no moment at the tip.