Nonuniform elastic properties of macromolecules and effect of prestrain on their continuum nature

Many experimental and theoretical methods have been developed to calculate the coarse-grained continuum elastic properties of macromolecules. However, all of those methods assume uniform elastic properties. Following the continuum mechanics framework, we present a systematic way of calculating the nonuniform effective elastic properties from atomic thermal fluctuations obtained from molecular dynamics simulation at any coarse-grained scale using a potential of the mean-force approach. We present the results for a mutant of Sesbania mosaic virus capsid, where we calculate the elastic moduli at different scales and observe an apparent problem with the chosen reference configuration in some cases. We present a possible explanation using an elastic network model, where inducing random prestrain results in a similar behavior. This phenomenon provides a novel insight into the continuum nature of macromolecules and defines the limits on details that the elasticity theory can capture. Further investigation into prestrains could elucidate important aspects of conformational dynamics of macromolecules.

Figure 1

(a) SeMV coordinates obtained from viperdb [21] and visualized in chimera [22] and (b) an icosahedron shape showing two-, three-, and fivefold symmetry axes.

Figure 2

We start by hypothesizing that all the nodes have same elastic properties (i.e., homogeneous). This leads to probability distributions (top) that have a maximum at zero strains. The corresponding energy-density plots (bottom) show a softening response which also varies dramatically as the model is coarsened successively.

Figure 3

Top row: Probability distribution of strain invariants at a representative node. Bottom row: Energy of that node in the continuum description. Points (circles) are obtained from the probability distributions in top row by (9). Solid (blue) curves are fits to the Mooney-Rivlin constitutive law (12).

Figure 4

Top row: Probability distributions at an example node, where the most probable state is observed at large strains (${\overline{I}}_1\approx 4, {\overline{I}}_2\approx 4$, and $J\approx 0.8$) severely violating the assumption of known reference configuration. Bottom row: Corresponding energy plots of the node.

Figure 5

3D bulk elasticity moduli distribution on SeMV capsid calculated using the last 2 ns of the full capsid MD trajectory and icosahedral symmetry imposed (with a net overlay showing the symmetry of its constituent proteins).

Figure 6

Histograms of the distribution of all three elastic moduli over SeMV using a 3D continuum model and full MD show a large tail. Similar distributions are obtained for all the cases.

Figure 7

(a) Subdivision of an icosahedron used for calculating the mean surface, (b) the average mean surface, and (c) its limit surface used to calculate the curvature tensor.

Figure 8

Averaging through the surface provides a coarsening of $\approx 18$ residues per node, and the problem of wrong reference configuration disappears completely.

Figure 9

The 2D shell elasticity moduli distribution on SeMV capsid calculated using the last 2 ns of full capsid MD trajectory and icosahedral symmetry imposed (with a net overlay showing the symmetry of its constituent proteins).

Figure 10

Force response of SeMV capsid to AFM indentation along three different axes. The results are calculated using nonuniform elastic properties without any scaling parameter with both 3D elasticity (solid lines) and 2D shell models (dashed lines).

Figure 11

Two-springs model; ${\epsilon }_0=\frac{l_1-l_2}{l_1+l_2}$ is the prestrain.

Figure 12

(a) 1D spring network model for prestrain and (b) its energy-strain relationship. Combining several springs is equivalent to sufficient coarse-graining that nullifies the effect of random prestrain.

Figure 13

Energy landscapes of SeMV elastic network (a) without prestrain, (b) with prestrain in the 3D bulk elasticity model, and (c) with prestrain in the 2D shell elasticity model (after averaging through shell thickness).

Figure 14

The effective energy plot of a series of springs under spurious homogeneous assumption at two levels: When all springs are considered separately and when every two consecutive springs are combined. We observe the same softening behavior as we saw in the homogenization of SeMV.

Figure 15

Average amplitude of spherical harmonic decomposition for a spherical shell with all the displacements allowed [solid line; Eq. (B5)] and with only radial displacements allowed [dashed line; Eq. (B6)].