Stress distribution on CCMV shell surface.

<p>Map of the Cauchy stress tensor projections along the direction of out-of-plane bending deformation (left) and tangential in-plane stretching (right) for different deformation <i>X</i> of the CCMV shell and corresponding indentation force <i>F</i> (indentation along the 2-fold symmetry axis with <i>R</i>_<i>tip</i> = 20 nm and <i>ν</i>_<i>f</i> = 1.0 <i>μ</i>m/s). For each amino acid residue (<i>C</i>_<i>α</i>-particle), the stress components are averaged over amino acids within a sphere of radius <i>R</i>_<i>C</i> = 15 Å (color denotation is presented in the graph). Also shown are formation and subsequent evolution of microscopic cracks in the side portion (particle barrel) of CCMV structure (shown in red circle/ellipse).</p

Dynamic evolution of mechanical degrees of freedom and survival probability for CCMV shell.

<p>Panel (a) exemplifies the dynamics of Hertzian deformation <i>x</i>_<i>H</i> and beam-bending deformation <i>x</i>_<i>b</i> vs. <i>X</i> in the Hertzian regime I and in the transition regime II. Model calculations are performed using parameter values obtained from the fit of theoretical <i>FX</i>-curves to the simulated average <i>FX</i>-spectra for CCMV nanoindentation along the 2-fold symmetry axis (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.t001" target="_blank">Table 1</a>). The solid curves correspond to the exact method of parameter estimation; the dashed and dashed-dotted curves are for the (piece-wise) approximate method of estimation. Snapshots exemplify the local flattening of CCMV structure under the tip for <i>X</i> = 1 nm and 5 nm deformation. Panel (b) displays the results of overlap function <i>χ</i>-based estimation of the survival probability <i>s</i>(<i>X</i>) from simulations of CCMV nanoindentation (<i>ν</i>_<i>f</i> = 1.0 <i>μ</i>m/s, <i>R</i>_<i>tip</i> = 20 nm, and <i>κ</i> = 0.05 N/m; <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.s007" target="_blank">S3a Fig</a>) along the 2-fold (red), quasi-3-fold (blue), and quasi-2-fold symmetry axes (green). The theoretical profiles of <i>s</i>(<i>X</i>) (solid curves; see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.e055" target="_blank">Eq (16)</a>) are compared with the simulated profiles of <i>χ</i>(<i>X</i>) (data points; see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.e053" target="_blank">Eq (15)</a>). The model parameters are summarized in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.t001" target="_blank">Table 1</a>. The values of are obtained using Lagrange multipliers and the approximate method of parameter estimation (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#sec010" target="_blank">Discussion</a>).</p

Deformation and collapse of biological particles—CCMV, TrV, and AdV.

<p>Accumulated are the Young’s moduli for Hertzian <i>E</i>_<i>H</i> and bending <i>E</i>_<i>b</i> deformations, the beam strength and the cooperativity parameter <i>m</i>. The first (second) entries correspond to the exact (approximate) methods of parameter estimation. The model predictions for <i>F</i>^<i>col</i> are compared with the peak forces (in parenthesis) from the spectra (Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.g004" target="_blank">4</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.g005" target="_blank">5</a>). For TrV and AdV particles, the shell thickness was estimated as described in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004729#pcbi.1004729.s003" target="_blank">S3 Text</a>.</p

Types of mechanical excitations exemplified using the CCMV shell.

<p>(a)-(c) Hertzian deformation <i>x</i>_<i>H</i> with normal displacements <i>u</i>_<i>tip</i> and <i>u</i>_<i>par</i> (scheme on (a)) under the influence of force (vertical arrow). Dashed contour lines show the tip and particle in their undeformed states. Structures in (b)—the native (left) and partially deformed (right) states show an amplitude of <i>x</i>_<i>H</i> ≈ 3 nm. (c) CCMV shell profile showing parts of the structure with high potential energy (>3 kcal/mol per residue; red) and low potential energy (blue). (d)-(f) Bending deformation. The side portion of the structure (barrel) is partitioned into curved beams (top-side view on (d)). Structures in (e)—the partially deformed (left) and pre-collapse (middle and right) states reveal the amplitude of <i>x</i>_<i>b</i> ≈ 4.3 nm. (f) CCMV shell profile under Hertzian and bending deformations showing the potential energy distribution.</p

Tubulin Bond Energies and Microtubule Biomechanics Determined from Nanoindentation <i>in Silico</i>

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations <i>in silico</i> of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the reversible dissociation of lateral bonds followed by irreversible dissociation of the longitudinal bonds. We have determined the free energies of dissociation of the lateral (6.9 ± 0.4 kcal/mol) and longitudinal (14.9 ± 1.5 kcal/mol) tubulin–tubulin bonds. These values in conjunction with the large flexural rigidity of tubulin protofilaments obtained (18,000–26,000 pN·nm²) support the idea that the disassembling microtubule is capable of generating a large mechanical force to move chromosomes during cell division. Our computational modeling offers a comprehensive quantitative platform to link molecular tubulin characteristics with the physiological behavior of microtubules. The developed <i>in silico</i> nanoindentation method provides a powerful tool for the exploration of biomechanical properties of other cytoskeletal and multiprotein assemblies

Tubulin Bond Energies and Microtubule Biomechanics Determined from Nanoindentation <i>in Silico</i>

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations <i>in silico</i> of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the reversible dissociation of lateral bonds followed by irreversible dissociation of the longitudinal bonds. We have determined the free energies of dissociation of the lateral (6.9 ± 0.4 kcal/mol) and longitudinal (14.9 ± 1.5 kcal/mol) tubulin–tubulin bonds. These values in conjunction with the large flexural rigidity of tubulin protofilaments obtained (18,000–26,000 pN·nm²) support the idea that the disassembling microtubule is capable of generating a large mechanical force to move chromosomes during cell division. Our computational modeling offers a comprehensive quantitative platform to link molecular tubulin characteristics with the physiological behavior of microtubules. The developed <i>in silico</i> nanoindentation method provides a powerful tool for the exploration of biomechanical properties of other cytoskeletal and multiprotein assemblies