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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.

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.

Lemon- or spindle-shaped viruses are common in the archaeal domain of life, but structural studies have been limited by the intrinsic heterogeneity of these uniquely shaped virions. Using a combination of cryo-electron microscopy and X-ray crystallography to study Acidianus tailed spindle virus, a large tailed spindle virus, we have shed light on the architectural principles that underlie assembly of a spindle-shaped virus. The architecture suggests a metastable multistart helical assembly of variable radius that, through a remarkable transition to a more stable cylindrical assembly, could be used to drive genome ejection. The structural architecture is clearly relevant to spindle-shaped viruses in general and other members of the large tailed spindle virus superfamily in particular.

We review the nonlinear mechanics of cross-linked networks of stiff filaments with a quenched anisotropic (nematic) alignment. A combination of numerical simulations and analytic calculations shows that the broken rotational symmetry of the filament orientational distribution leads to a dramatic nonlinear softening of the network at very small strain (on the order of 0.1%). We argue that one can understand this softening in terms of Euler buckling, i.e. the loss of further load-carrying capacity in compression within the network. With increasing shear strain, this source of geometric nonlinearity appears as heterogeneous nucleation (originating in particularly fragile regions, which may be identified by a linear stability analysis) and subsequently grows into 'buckling scars' that eventually spread throughout the system. We develop a simple mean-field model for the nonlinear mechanics of such networks and suggest applications of these ideas to a variety of fiber networks and biopolymer systems.

We review the application of statistical mechanics, elasticity theory, and condensed matter physics to the assembly and maturation of viral capsids.

We present a generalized Landau-Brazovskii free energy for the solidification of chiral molecules on a spherical surface in the context of the assembly of viral shells. We encounter two types of icosahedral solidification transitions. The first type is a conventional first-order phase transition from the uniform to the icosahedral state. It can be described by a single icosahedral spherical harmonic of even $l$. The chiral pseudo-scalar term in the free energy creates secondary terms with chiral character but it does not affect the thermodynamics of the transition. The second type, associated with icosahedral spherical harmonics with odd $l$, is anomalous. Pure odd $l$ icosahedral states are unstable but stability is recovered if admixture with the neighboring $l+1$ icosahedral spherical harmonic is included, generated by the non-linear terms. This is in conflict with the principle of Landau theory that symmetry-breaking transitions are characterized by only a \textit{single} irreducible representation of the symmetry group of the uniform phase and we argue that this principle should be removed from Landau theory. The chiral term now directly affects the transition because it lifts the degeneracy between two isomeric mixed-$l$ icosahedral states. A direct transition is possible only over a limited range of parameters. Outside this range, non-icosahedral states intervene. For the important case of capsid assembly dominated by $l=15$, the intervening states are found to be based on octahedral symmetry.

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.

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.

Membrane proteins deform the surrounding lipid bilayer, which can lead to membrane-mediated interactions between neighboring proteins. Using the mechanosensitive channel of large conductance (MscL) as a model system, we demonstrate how the observed differences in protein structure can affect membrane-mediated interactions and cooperativity among membrane proteins. We find that distinct oligomeric states of MscL lead to distinct gateway states for the clustering of MscL, and predict signatures of MscL structure and spatial organization in the cooperative gating of MscL. Our modeling approach establishes a quantitative relation between the observed shapes and cooperative function of membrane~proteins.

We propose a physical model for the capsids of tailed archaeal viruses as viscoelastic membranes under tension. The fluidity is generated by thermal motion of scarlike structures that are an intrinsic feature of the ground state of large particle arrays covering surfaces with nonzero Gauss curvature. The tension is generated by a combination of the osmotic pressure of the enclosed genome and an extension force generated by filamentous structure formation that drives the formation of the tails. In continuum theory, the capsid has the shape of a surface of constant mean curvature: an unduloid. Particle arrays covering unduloids are shown to exhibit pronounced subdiffusive and diffusive single-particle transport at temperatures that are well below the melting temperature of defect-free particle arrays on a surface with zero Gauss curvature.

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., Author Summary Ventricular fibrillation (VF) is one of the leading causes of sudden death. During VF, the electrical wave of activation in the heart breaks up chaotically. Consequently, the heart is unable to contract synchronously and pump blood to the rest of the body. In our work we formulate and validate a model of heart failure (HF) that allows us to evaluate the arrhythmogenic potential of individual and combined electrophysiological changes. In diagnostic cardiology, the electrocardiogram (ECG) is one of the most commonly used tools for detecting abnormalities in the heart electrophysiology. One of our goals is to use our numerical model to link changes at the cellular and tissue level in a failing heart to a numerically computed ECG. This allows us to characterize the precursor to and the risk of VF. In order to understand the mechanisms underlying VF in HF, we design a test that simulates a HF patient performing physical exercise. We show that under fast heart rates with changes in pacing, HF patients are more prone to VF due to a new conduction blocking mechanism. In the long term, our mathematical model is suitable for investigating the effect of drug therapies in HF.

Quantitative measurement of the material properties (e.g., 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 — i.e., 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, e.g., 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.

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.

We present a generalized Landau-Brazovskii free energy for the solidification of chiral molecules on a spherical surface in the context of the assembly of viral shells. We encounter two types of solidification transitions. The first type is a conventional first-order phase transition from a uniform to an icosahedral state, described by a single icosahedral spherical harmonic of even l . The chiral pseudo-scalar term in the free energy does not affect the transition. The second type is anomalous: icosahedral spherical harmonics with odd l are unstable. Stability is recovered when admixture with the neighboring l + 1 icosahedral spherical harmonic is included. This is in apparent conflict with the principle of Landau theory that symmetry-breaking transitions are characterized by a single irreducible representation of the symmetry group of the uniform phase. The chiral term selects one of two isomeric mixed-l icosahedral states. A direct transition is possible only over a limited range of parameters. Outside this range, a non-icosahedral state with the symmetry of an isotropy subgroup of the icosahedral group interposes between the uniform and icosahedral states.This paper is dedicated to the memory of our friends and colleagues William Klug and Vladimir Lorman .

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.

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.

Semiflexible gels are composed of a crosslinked network of filaments that can support both bending and extensional forces. We study numerically the mechanical effect of adding a low density of highly incompliant semiflexible filaments to a random network of softer semiflexible filaments. Such heterogeneous networks form simple models of the mechanics of cytoskeletal networks composed primarily of F-actin but containing a low density of significantly stiffer microtubules. Networks composed solely of these two filament types were recently studied in the in vitro experiments of Lin et al. Here we determine the effect of the stiffer impurity filaments generally on the collective mechanics of the heterogeneous filament network and, more specifically, on the affine to non-affine (A/NA) crossover in the softer filament matrix, which occurs in semiflexible networks as a function of their network density. We show that the addition of a small fraction of longer and stiffer filaments to a nonaffine network leads to a significant increase in its collective elastic moduli, even though the stiff filaments do not themselves form a stress bearing network. We also determine the relationship between the density of the stiff filaments and the geometric measure of nonaffinity for the network. Here the effect of the stiffer impurity filaments is complex: their addition makes affine networks slightly more affine, but highly nonaffine networks even more nonaffine. Moreover, there is a strong negative spatial correlation between density of the stiff filaments and local geometric measure of nonaffinity. Taken together, these two observations show that the stiffer filaments serve to locally suppress nonaffine deformation but redistribute it to regions of the network where the stiffer filaments are sparse.

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.

As revealed by techniques of structural biology and single-molecule experimentation, the protein shells of viruses (capsids) are some of nature’s best examples of highly symmetric multiscale self-assembled structures, with impressive mechanical properties of strength and elasticity. Mechanical models of viral capsids built “from the bottom up,” i.e., from all-atom models in the context of molecular dynamics and normal mode analysis, have chiefly focused on unforced vibrational capsid dynamics. Due to the size of viral capsids, which can contain several hundred thousand atoms, the computer power needed for these types of methods has only recently reached the level required for all-atom simulations of entire viral capsids. Coarse-grained normal mode analysis has provided a simplified means of studying the unforced vibrational dynamics of viral capsids. Recent focus on “top-down” mechanical models of viral capsids based on two- and three-dimensional continuum elasticity have provided a theoretical complement to single molecule experiments such as atomic force microscopy, and have advanced the fundamental understanding of the forced mechanics. This review serves to assess the current state of modeling techniques for the study of the mechanics of viral capsids, and to highlight some of the key insights gained from such modeling. In particular, a theme is established of a link between shape—or geometry—and the global mechanical properties of these hierarchical multiscale biological structures.

We present a model to understand quantitatively the role of symmetry breaking in assembly of macromolecular aggregates in general, and the protein shells of viruses in particular. A simple dodecahedral lattice model with a quadrupolar order parameter allows us to demonstrate how symmetry breaking may reduce the probability of assembly errors and, consequently, enhance assembly efficiency. We show that the ground state is characterized by large-scale cooperative zero-energy modes. In analogy with other models, this suggests a general physical principle: the tendency of biological molecules to generate symmetric structures competes with the tendency to break symmetry in order to achieve specific functional goals.

Many viral capsids undergo a series of significant structural changes following assembly, a process known as maturation. The driving mechanisms for maturation usually are chemical reactions taking place inside the proteins that constitute the capsid (“subunits”) that produce structural changes of the subunits. The resulting alterations of the subunits may be directly visible from the capsid structures, as observed by electron microscopy, in the form of a shear shape change and/or a rotation of groups of subunits. The existing thin shell elasticity theory for viral shells does not take account of the internal structure of the subunits and hence cannot describe displacement patterns of the capsid during maturation. Recently, it was proposed for the case of a particular virus (HK97) that thin shell elasticity theory could in fact be generalized to include transformations of the constituent proteins by including such a transformations as a change of the stress-free reference state for the deformation free energy. In this study, we adopt that approach and illustrate its validity in more generality by describing shape changes occurring during maturation across different T-numbers in terms of subunit shearing. Using phase diagrams, we determine the shear directions of the subunits that are most effective to produce capsid shape changes, such as transitions from spherical to facetted capsid shape. We further propose an equivalent stretching mechanism offering a unifying view under which capsid symmetry can be analyzed. We conclude by showing that hexamer shearing not only drives the shape change of the viral capsid during maturation but also is capable of lowering the capsid elastic energy in particular for chiral capsids (e.g., T = 7 ) and give rise to pre-shear patterns. These additional mechanisms may provide a driving force and an organizational principle for virus assembly.

A numerical simulation framework is presented for the study of biological membranes composed of lipid bilayers based on the finite element method. The classic model for these membranes employs a two-dimensional-fluid-like elastic constitutive law which is sensitive to curvature, and subjects vesicles to physically imposed constraints on surface area and volume. This model is implemented numerically via the use of C^1-conforming triangular Loop subdivision finite elements. The validity of the framework is tested by computing equilibrium shapes from previously-determined axisymmetric shape-phase diagram of lipid bilayer vesicles with homogeneous material properties. Some of the benefits and challenges of finite element modeling of lipid bilayer systems are discussed, and it is indicated how this framework is natural for future investigation of biologically realistic bilayer structures involving nonaxisymmetric geometries, binding and adhesive interactions, heterogeneous mechanical properties, cytoskeletal interactions, and complex loading arrangements. These biologically relevant features have important consequences for the shape mechanics of nonidealized vesicles and cells, and their study requires not simply advances in theory, but also advances in numerical simulation techniques, such as those presented here.

Silicon micro cantilevers are used as transducers for a wide range of physical, chemical and biochemical stimuli, where they exhibit exquisite sensitivity (10?18?g, 10?15?J, 10?9?M, etc) over a wide range of temperatures (100?mK?1300?K). This is accomplished by inducing static bending in the cantilever structure or by changing the cantilever's resonant behaviour, both easily measurable responses. There is increasing interest in using higher-order resonant modes to achieve extra sensitivity; however, this raises the question of exactly which modes are excited in the cantilever. Using strobed interferometric microscopy we are able to probe the dynamic behaviour of individual (100 ? 500 ? 1??m3) cantilevers in an eight cantilever array over frequencies from 0 to 1?MHz. We present data that enable spatial visualization of 16 cantilever modes with nanometre-scale amplitudes; combined with finite element analysis calculations we directly assign mode indices. We discuss the reversal of specific modes between experiment and theory, the uniformity of individual cantilevers in the array and the clear relationship between boundary conditions and resonant behaviour. Our conclusion is that the assignment of a resonant frequency spectrum is fairly complex and does not necessarily follow simple intuition.

The elastic properties of capsids of the cowpea chlorotic mottle virus have been examined at pH 4.8 by nanoindentation measurements with an atomic force microscope. Studies have been carried out on WT capsids, both empty and containing the RNA genome, and on full capsids of a salt-stable mutant and empty capsids of the subE mutant. Full capsids resisted indentation more than empty capsids, but all of the capsids were highly elastic. There was an initial reversible linear regime that persisted up to indentations varying between 20% and 30% of the diameter and applied forces of 0.6-1.0 nN; it was followed by a steep drop in force that is associated with irreversible deformation. A single point mutation in the capsid protein increased the capsid stiffness. The experiments are compared with calculations by finite element analysis of the deformation of a homogeneous elastic thick shell. These calculations capture the features of the reversible indentation region and allow Young's moduli and relative strengths to be estimated for the empty capsids. © 2006 by The National Academy of Sciences of the USA.

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., Shankarjee Krishnamoorthi et. al

In this work we develop a discrete director-field model for coarse-grained description of packing arrangements of DNA within bacteriophage virus heads. This computational lattice model allows us to explore the complex energy landscape of fully three-dimensional configurations of packaged DNA. By minimizing the system’s free energy by means of the simulated annealing and the conjugate gradient methods, we make predictions about favorable packing conformations. In particular we show that the planar-wrapped inverse spool conformation is stable everywhere inside a virus except in a central core region, where the DNA tends to buckle out of the spooling plane.

The present work is concerned with the formulation of a continuum theory of viral DNA packaging based on a director field representation of the encapsidated DNA. The point values of the director field give the local direction and density of the DNA. The continuity of the DNA strand requires that the director field be divergence-free and tangent to the capsid wall. The energy of the DNA is defined as a functional of the director field which accounts for bending, torsion, and for electrostatic interactions through a density-dependent cohesive energy. The operative principle which determines the encapsidated DNA conformation is assumed to be energy minimization. We show that torsionless toroidal solenoids, consisting of planar coils contained within meridional planes and wrapped around a spool core, and fine mixtures of the solenoid and spool phase, beat the inverse spool construction.

An advanced physics-based simulation of firearms injury to the human cranium is presented, modeling by finite elements the collision of a firearm projectile into a human parietal bone. The space-discretized equations of motion are explicitly integrated in time with Newmark's time-stepping algorithm. The impact of the projectile on the skull, as well as the collisions between flying fragments, are controlled through a nonsmooth contact algorithm. Cohesive theories of fracture, in conjunction with adaptive remeshing, control the nucleation and the propagation of fractures. The progressive opening of fracture surfaces is governed by a thermodynamically irreversible cohesive law embedded into cohesive-interface elements. Numerical results compare well with forensic data of actual firearm wounds to human crania.

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.

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.

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.

The mechanosensitive channel of large conductance (MscL) provides a paradigm for mechanosensation and mechanotransduction. It also provides a model system for how bilayer material properties regulate membrane protein function. The basic phenomenology of MscL gating as a function of membrane tension, and the dependence of MscL gating on bilayer material properties, can be understood by considering the difference in membrane deformation energy between open and closed states of MscL. However, even basic issues such as the physiologically relevant oligomeric states of MscL remain a matter of intense debate. Is MscL a tetramer or a pentamer under physiological conditions, or do both structures occur in cell membranes? Moreover, MscL proteins have recently been observed to form clusters, and the gating behavior of MscL in clusters was found to be different from the gating behavior of single MscL. What are the physical mechanisms behind the clustering of MscL, and what is the biological significance of MscL clusters? The complexity of the observed MscL structures and the large size of MscL clusters have so far prohibited a theoretical analysis of the relation between the oligomeric states of MscL and the properties of MscL clusters. Here we develop a mathematical approach capable of predicting the coupling between MscL shape and function. Combining this approach with the theory of regular lattices of polygons, we study the interplay between the oligomeric state of MscL, membrane-mediated interactions between MscL proteins, the structure of MscL clusters, and cooperative gating of MscL. We predict that the architecture of MscL clusters may provide a signature of MscL shape, with distinct molecular structures of MscL yielding distinct spatial and orientational organization patterns. Our work establishes a bridge between the oligomeric state of MscL and the self-assembly of MscL into clusters.

The two-dimensional structure of graphene is known to impart high strength, but can be hard to synthesize without grain boundaries. Here, the authors find that strength increases with grain boundary mismatch, which results from low atomic-scale strain in the carbon–carbon bonds at the boundary.

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.

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.

We present a version of continuum elasticity theory applicable to aggregates of functional biomolecules at length scales comparable to that of the component molecules. Unlike classical elasticity theory, the stress and strain fields have mathematical discontinuities along the interfaces of the macromolecules, due to conformational incompatibility and large scale conformational transitions. The method is applied to the P-II to EI shape transition of the protein shell of the virus HK97. We show that protein residual stresses generated by incompatibility drive a ``reverse buckling'' transition from an icosahedral to a dodecahedral shape via a ``critical'' spherical shape, which can be identified as the P-II state.

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.

Circular domains in phase separated lipid vesicles with symmetric leaflet composition commonly exhibit three stable morphologies: flat, dimpled, and budded. However, stable dimples (i.e., partially budded domains) present a puzzle since simple elastic theories of domain shape predict that only flat and spherical budded domains are mechanically stable in the absence of spontaneous curvature. We argue that this inconsistency arises from the failure of the constant surface tension ensemble to properly account for the effect of entropic bending fluctuations. Formulating membrane elasticity within an entropic tension ensemble wherein tension represents the free energy cost of extracting membrane area from thermal bending undulations of the membrane, we calculate a morphological phase diagram that contains regions of mechanical stability for each of the flat, dimpled, and budded domain morphologies.

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.

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.