Structural studies of Acidianus tailed spindle virus reveal a structural paradigm used in the assembly of spindle-shaped viruses.

The spindle-shaped virion morphology is common among archaeal viruses, where it is a defining characteristic of many viral families. However, structural heterogeneity intrinsic to spindle-shaped viruses has seriously hindered efforts to elucidate the molecular architecture of these lemon-shaped capsids. We have utilized a combination of cryo-electron microscopy and X-ray crystallography to study Acidianus tailed spindle virus (ATSV). These studies reveal the architectural principles that underlie assembly of a spindle-shaped virus. Cryo-electron tomography shows a smooth transition from the spindle-shaped capsid into the tubular-shaped tail and allows low-resolution structural modeling of individual virions. Remarkably, higher-dose 2D micrographs reveal a helical surface lattice in the spindle-shaped capsid. Consistent with this, crystallographic studies of the major capsid protein reveal a decorated four-helix bundle that packs within the crystal to form a four-start helical assembly with structural similarity to the tube-shaped tail structure of ATSV and other tailed, spindle-shaped viruses. Combined, this suggests that the spindle-shaped morphology of the ATSV capsid is formed by a multistart helical assembly with a smoothly varying radius and allows construction of a pseudoatomic model for the lemon-shaped capsid that extends into a tubular tail. The potential advantages that this novel architecture conveys to the life cycle of spindle-shaped viruses, including a role in DNA ejection, are discussed.

Ground state instabilities of protein shells are eliminated by buckling.

We propose a hybrid discrete-continuum model to study the ground state of protein shells. The model allows for shape transformation of the shell and buckling transitions as well as the competition between states with different symmetries that characterize discrete particle models with radial pair potentials. Our main results are as follows. For large Föppl-von Kármán (FvK) numbers the shells have stable isometric ground states. As the FvK number is reduced, shells undergo a buckling transition resembling that of thin-shell elasticity theory. When the width of the pair potential is reduced below a critical value, then buckling coincides with the onset of structural instability triggered by over-stretched pair potentials. Chiral shells are found to be more prone to structural instability than achiral shells. It is argued that the well-width appropriate for protein shells lies below the structural instability threshold. This means that the self-assembly of protein shells with a well-defined, stable structure is possible only if the bending energy of the shell is sufficiently low so that the FvK number of the assembled shell is above the buckling threshold.

Method for the unique identification of hyperelastic material properties using full-field measures. Application to the passive myocardium material response.

Quantitative measurement of the material properties (eg, stiffness) of biological tissues is poised to become a powerful diagnostic tool. There are currently several methods in the literature to estimating material stiffness, and we extend this work by formulating a framework that leads to uniquely identified material properties. We design an approach to work with full-field displacement data-ie, we assume the displacement field due to the applied forces is known both on the boundaries and also within the interior of the body of interest-and seek stiffness parameters that lead to balanced internal and external forces in a model. For in vivo applications, the displacement data can be acquired clinically using magnetic resonance imaging while the forces may be computed from pressure measurements, eg, through catheterization. We outline a set of conditions under which the least-square force error objective function is convex, yielding uniquely identified material properties. An important component of our framework is a new numerical strategy to formulate polyconvex material energy laws that are linear in the material properties and provide one optimal description of the available experimental data. An outcome of our approach is the analysis of the reliability of the identified material properties, even for material laws that do not admit unique property identification. Lastly, we evaluate our approach using passive myocardium experimental data at the material point and show its application to identifying myocardial stiffness with an in silico experiment modeling the passive filling of the left ventricle.

Electrophysiology of Heart Failure Using a Rabbit Model: From the Failing Myocyte to Ventricular Fibrillation.

Heart failure is a leading cause of death, yet its underlying electrophysiological (EP) mechanisms are not well understood. In this study, we use a multiscale approach to analyze a model of heart failure and connect its results to features of the electrocardiogram (ECG). The heart failure model is derived by modifying a previously validated electrophysiology model for a healthy rabbit heart. Specifically, in accordance with the heart failure literature, we modified the cell EP by changing both membrane currents and calcium handling. At the tissue level, we modeled the increased gap junction lateralization and lower conduction velocity due to downregulation of Connexin 43. At the biventricular level, we reduced the apex-to-base and transmural gradients of action potential duration (APD). The failing cell model was first validated by reproducing the longer action potential, slower and lower calcium transient, and earlier alternans characteristic of heart failure EP. Subsequently, we compared the electrical wave propagation in one dimensional cables of healthy and failing cells. The validated cell model was then used to simulate the EP of heart failure in an anatomically accurate biventricular rabbit model. As pacing cycle length decreases, both the normal and failing heart develop T-wave alternans, but only the failing heart shows QRS alternans (although moderate) at rapid pacing. Moreover, T-wave alternans is significantly more pronounced in the failing heart. At rapid pacing, APD maps show areas of conduction block in the failing heart. Finally, accelerated pacing initiated wave reentry and breakup in the failing heart. Further, the onset of VF was not observed with an upregulation of SERCA, a potential drug therapy, using the same protocol. The changes introduced at the cell and tissue level have increased the failing heart's susceptibility to dynamic instabilities and arrhythmias under rapid pacing. However, the observed increase in arrhythmogenic potential is not due to a steepening of the restitution curve (not present in our model), but rather to a novel blocking mechanism.

Bilayer-thickness-mediated interactions between integral membrane proteins.

Hydrophobic thickness mismatch between integral membrane proteins and the surrounding lipid bilayer can produce lipid bilayer thickness deformations. Experiment and theory have shown that protein-induced lipid bilayer thickness deformations can yield energetically favorable bilayer-mediated interactions between integral membrane proteins, and large-scale organization of integral membrane proteins into protein clusters in cell membranes. Within the continuum elasticity theory of membranes, the energy cost of protein-induced bilayer thickness deformations can be captured by considering compression and expansion of the bilayer hydrophobic core, membrane tension, and bilayer bending, resulting in biharmonic equilibrium equations describing the shape of lipid bilayers for a given set of bilayer-protein boundary conditions. Here we develop a combined analytic and numerical methodology for the solution of the equilibrium elastic equations associated with protein-induced lipid bilayer deformations. Our methodology allows accurate prediction of thickness-mediated protein interactions for arbitrary protein symmetries at arbitrary protein separations and relative orientations. We provide exact analytic solutions for cylindrical integral membrane proteins with constant and varying hydrophobic thickness, and develop perturbative analytic solutions for noncylindrical protein shapes. We complement these analytic solutions, and assess their accuracy, by developing both finite element and finite difference numerical solution schemes. We provide error estimates of our numerical solution schemes and systematically assess their convergence properties. Taken together, the work presented here puts into place an analytic and numerical framework which allows calculation of bilayer-mediated elastic interactions between integral membrane proteins for the complicated protein shapes suggested by structural biology and at the small protein separations most relevant for the crowded membrane environments provided by living cells.

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.

Architecture and Function of Mechanosensitive Membrane Protein Lattices.

Experiments have revealed that membrane proteins can form two-dimensional clusters with regular translational and orientational protein arrangements, which may allow cells to modulate protein function. However, the physical mechanisms yielding supramolecular organization and collective function of membrane proteins remain largely unknown. Here we show that bilayer-mediated elastic interactions between membrane proteins can yield regular and distinctive lattice architectures of protein clusters, and may provide a link between lattice architecture and lattice function. Using the mechanosensitive channel of large conductance (MscL) as a model system, we obtain relations between the shape of MscL and the supramolecular architecture of MscL lattices. We predict that the tetrameric and pentameric MscL symmetries observed in previous structural studies yield distinct lattice architectures of MscL clusters and that, in turn, these distinct MscL lattice architectures yield distinct lattice activation barriers. Our results suggest general physical mechanisms linking protein symmetry, the lattice architecture of membrane protein clusters, and the collective function of membrane protein lattices.

Computational mechanics of viral capsids.

Viral capsids undergo significant mechanical deformations during their assembly, maturation, and infective life-span. In order to characterize the mechanics of viral capsids, their response to applied external forces is analyzed in several experimental studies using, for instance, Atomic Force Microscope (AFM) indentation experiments. In recent years, a broader approach to study the mechanics of viral capsids has leveraged the theoretical tools proper of continuum mechanics. Even though the theory of continuum elasticity is most commonly used to study deformable bodies at larger macroscopic length scales, it has been shown that this very rich theoretical field can still offer useful insights into the mechanics of viral structures at the nanometer scale. Here we show the construction of viral capsid continuum mechanics models starting from different forms of experimental data. We will discuss the kinematics assumptions, the issue of the reference configuration, the material constitutive laws, and the numerical discretization necessary to construct a complete Finite Element capsid mechanical model. Some examples in the second part of the chapter will show the predictive capabilities of the constructed models and underline useful practical aspects related to efficiency and accuracy. We conclude each example by collecting several key findings discovered by simulating AFM indentation experiments using the constructed numerical models.

Simulation Methods and Validation Criteria for Modeling Cardiac Ventricular Electrophysiology.

We describe a sequence of methods to produce a partial differential equation model of the electrical activation of the ventricles. In our framework, we incorporate the anatomy and cardiac microstructure obtained from magnetic resonance imaging and diffusion tensor imaging of a New Zealand White rabbit, the Purkinje structure and the Purkinje-muscle junctions, and an electrophysiologically accurate model of the ventricular myocytes and tissue, which includes transmural and apex-to-base gradients of action potential characteristics. We solve the electrophysiology governing equations using the finite element method and compute both a 6-lead precordial electrocardiogram (ECG) and the activation wavefronts over time. We are particularly concerned with the validation of the various methods used in our model and, in this regard, propose a series of validation criteria that we consider essential. These include producing a physiologically accurate ECG, a correct ventricular activation sequence, and the inducibility of ventricular fibrillation. Among other components, we conclude that a Purkinje geometry with a high density of Purkinje muscle junctions covering the right and left ventricular endocardial surfaces as well as transmural and apex-to-base gradients in action potential characteristics are necessary to produce ECGs and time activation plots that agree with physiological observations.

Mechanical collapse of confined fluid membrane vesicles.

Compact cylindrical and spherical invaginations are common structural motifs found in cellular and developmental biology. To understand the basic physical mechanisms that produce and maintain such structures, we present here a simple model of vesicles in confinement, in which mechanical equilibrium configurations are computed by energy minimization, balancing the effects of curvature elasticity, contact of the membrane with itself and the confining geometry, and adhesion. For cylindrical confinement, the shape equations are solved both analytically and numerically by finite element analysis. For spherical confinement, axisymmetric configurations are obtained numerically. We find that the geometry of invaginations is controlled by a dimensionless ratio of the adhesion strength to the bending energy of an equal area spherical vesicle. Larger adhesion produces more concentrated curvatures, which are mainly localized to the "neck" region where the invagination breaks away from its confining container. Under spherical confinement, axisymmetric invaginations are approximately spherical. For extreme confinement, multiple invaginations may form, bifurcating along multiple equilibrium branches. The results of the model are useful for understanding the physical mechanisms controlling the structure of lipid membranes of cells and their organelles, and developing tissue membranes.

Numerical quadrature and operator splitting in finite element methods for cardiac electrophysiology.

We study the numerical accuracy and computational efficiency of alternative formulations of the finite element solution procedure for the monodomain equations of cardiac electrophysiology, focusing on the interaction of spatial quadrature implementations with operator splitting and examining both nodal and Gauss quadrature methods and implementations that mix nodal storage of state variables with Gauss quadrature. We evaluate the performance of all possible combinations of 'lumped' approximations of consistent capacitance and mass matrices. Most generally, we find that quadrature schemes and lumped approximations that produce decoupled nodal ionic equations allow for the greatest computational efficiency, this being afforded through the use of asynchronous adaptive time-stepping of the ionic state variable ODEs. We identify two lumped approximation schemes that exhibit superior accuracy, rivaling that of the most expensive variationally consistent implementations. Finally, we illustrate some of the physiological consequences of discretization error in electrophysiological simulation relevant to cardiac arrhythmia and fibrillation. These results suggest caution with the use of semi-automated free-form tetrahedral and hexahedral meshing algorithms available in most commercially available meshing software, which produce nonuniform meshes having a large distribution of element sizes.

Microrheology of highly crosslinked microtubule networks is dominated by force-induced crosslinker unbinding.

We determine the time- and force-dependent viscoelastic responses of reconstituted networks of microtubules that have been strongly crosslinked by biotin-streptavidin bonds. To measure the microscale viscoelasticity of such networks, we use a magnetic tweezers device to apply localized forces. At short time scales, the networks respond nonlinearly to applied force, with stiffening at small forces, followed by a reduction in the stiffening response at high forces, which we attribute to the force-induced unbinding of crosslinks. At long time scales, force-induced bond unbinding leads to local network rearrangement and significant bead creep. Interestingly, the network retains its elastic modulus even under conditions of significant plastic flow, suggesting that crosslinker breakage is balanced by the formation of new bonds. To better understand this effect, we developed a finite element model of such a stiff filament network with labile crosslinkers obeying force-dependent Bell model unbinding dynamics. The coexistence of dissipation, due to bond breakage, and the elastic recovery of the network is possible because each filament has many crosslinkers. Recovery can occur as long as a sufficient number of the original crosslinkers are preserved under the loading period. When these remaining original crosslinkers are broken, plastic flow results.

Linear invariant tensor interpolation applied to cardiac diffusion tensor MRI.

PURPOSE: Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape. METHODS: Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets. RESULTS: EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI. CONCLUSION: GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.

Viral capsid equilibrium dynamics reveals nonuniform elastic properties.

The long wavelength, low-frequency modes of motion are the relevant motions for understanding the continuum mechanical properties of biomolecules. By examining these low-frequency modes, in the context of a spherical harmonic basis set, we identify four elastic moduli that are required to describe the two-dimensional elastic behavior of capsids. This is in contrast to previous modeling and theoretical studies on elastic shells, which use only the two-dimensional Young's modulus (Y) and the bending modulus (κ) to describe the system. Presumably, the heterogeneity of the structure and the anisotropy of the biomolecular interactions lead to a deviation from the homogeneous, isotropic, linear elastic shell theory. We assign functional relevance of the various moduli governing different deformation modes, including a mode primarily sensed in atomic force microscopy nanoindentation experiments. We have performed our analysis on the T = 3 cowpea chlorotic mottle virus and our estimate for the nanoindentation modulus is in accord with experimental measurements.

On the role of the filament length distribution in the mechanics of semiflexible networks.

This paper explores the effects of filament length polydispersity on the mechanical properties of semiflexible crosslinked polymer networks. Extending previous studies on monodisperse networks, we compute numerically the response of crosslinked networks of elastic filaments of bimodal and exponential length distributions. These polydisperse networks are subject to the same affine to nonaffine (A/NA) transition observed previously for monodisperse networks, wherein the decreases in either crosslink density or bending stiffness lead to a shift from affine, stretching-dominated deformations to nonaffine, bending-dominated deformations. We find that the onset of this transition is generally more sensitive to changes in the density of longer filaments than shorter filaments, meaning that longer filaments have greater mechanical efficiency. Moreover, in polydisperse networks, mixtures of long and short filaments interact cooperatively to generally produce a nonaffine mechanical response closer to the affine prediction than comparable monodisperse networks of either long or short filaments. Accordingly, the mechanical affinity of polydisperse networks is dependent on the filament length composition. Overall, length polydispersity has the effect of sharpening and shifting the A/NA transition to lower network densities. We discuss the implications of these results on experimental observation of the A/NA transition, and on the design of advanced materials.

Affine-nonaffine transition in networks of nematically ordered semiflexible polymers.

We study the mechanics of nematically ordered semiflexible networks showing that they, like isotropic networks, undergo an affine to nonaffine crossover controlled by the ratio of the filament length to the nonaffinity length. Deep in the nonaffine regime, however, these anisotropic networks exhibit a much more complex mechanical response characterized by a vanishing linear-response regime for highly ordered networks and a dependence of the shear modulus on shear direction at both small (linear) and finite (nonlinear) strains that is different from the affine prediction of orthotropic continuum linear elasticity. We show that these features can be understood in terms of a generalized floppy modes analysis of the nonaffine mechanics and a type of cooperative Euler buckling.

Morphology and interaction between lipid domains.

Cellular membranes are a heterogeneous mix of lipids, proteins and small molecules. Special groupings enriched in saturated lipids and cholesterol form liquid-ordered domains, known as "lipid rafts," thought to serve as platforms for signaling, trafficking and material transport throughout the secretory pathway. Questions remain as to how the cell maintains small fluid lipid domains, through time, on a length scale consistent with the fact that no large-scale phase separation is observed. Motivated by these examples, we have utilized a combination of mechanical modeling and in vitro experiments to show that membrane morphology plays a key role in maintaining small domain sizes and organizing domains in a model membrane. We demonstrate that lipid domains can adopt a flat or dimpled morphology, where the latter facilitates a repulsive interaction that slows coalescence and helps regulate domain size and tends to laterally organize domains in the membrane.

Mechanical stress analysis of a rigid inclusion in distensible material: a model of atherosclerotic calcification and plaque vulnerability.

The role of atherosclerotic calcification in plaque rupture remains controversial. In previous analyses using finite element model analysis, circumferential stress was reduced by the inclusion of a calcium deposit in a representative human anatomical configuration. However, a recent report, also using finite element analysis, suggests that microscopic calcium deposits increase plaque stress. We used mathematical models to predict the effects of rigid and liquid inclusions (modeling a calcium deposit and a lipid necrotic core, respectively) in a distensible material (artery wall) on mechanical failure under uniaxial and biaxial loading in a range of configurations. Without inclusions, stress levels were low and uniform. In the analytical model, peak stresses were elevated at the edges of a rigid inclusion. In the finite element model, peak stresses were elevated at the edges of both inclusions, with minimal sensitivity to the wall distensibility and the size and shape of the inclusion. Presence of both a rigid and a soft inclusion enlarged the region of increased wall stress compared with either alone. In some configurations, the rigid inclusion reduced peak stress at the edge of the soft inclusion but simultaneously increased peak stress at the edge of the rigid inclusion and increased the size of the region affected. These findings suggest that the presence of a calcium deposit creates local increases in failure stress, and, depending on relative position to any neighboring lipid pools, it may increase peak stress and the plaque area at risk of mechanical failure.

Membrane shape as a reporter for applied forces.

Recent advances have enabled 3-dimensional reconstructions of biological structures in vivo, ranging in size and complexity from single proteins to multicellular structures. In particular, tomography and confocal microscopy have been exploited to capture detailed 3-dimensional conformations of membranes in cellular processes ranging from viral budding and organelle maintenance to phagocytosis. Despite the wealth of membrane structures available, there is as yet no generic, quantitative method for their interpretation. We propose that by modeling these observed biomembrane shapes as fluid lipid bilayers in mechanical equilibrium, the externally applied forces as well as the pressure, tension, and spontaneous curvature can be computed directly from the shape alone. To illustrate the potential power of this technique, we apply an axial force with optical tweezers to vesicles and explicitly demonstrate that the applied force is equal to the force computed from the membrane conformation.

Influence of nonuniform geometry on nanoindentation of viral capsids.

A series of recent nanoindentation experiments on the protein shells (capsids) of viruses has established atomic force microscopy (AFM) as a useful framework for probing the mechanics of large protein assemblies. Specifically these experiments provide an opportunity to study the coupling of the global assembly response to local conformational changes. AFM experiments on cowpea chlorotic mottle virus, known to undergo a pH-controlled swelling conformational change, have revealed a pH-dependent mechanical response. Previous theoretical studies have shown that homogeneous changes in shell geometry can play a significant role in the mechanical response. This article develops a method for accurately capturing the heterogeneous geometry of a viral capsid and explores its effect on mechanical response with a nonlinear continuum elasticity model. Models of both native and swollen cowpea chlorotic mottle virus capsids are generated from x-ray crystal structures, and are used in finite element simulations of AFM indentation along two-, three-, and fivefold icosahedral symmetry orientations. The force response of the swollen capsid model is observed to be softer by roughly a factor of two, significantly more nonlinear, and more orientation-dependent than that of a native capsid with equivalent elastic moduli, demonstrating that capsid geometric heterogeneity can have significant effects on the global structural response.