William S. Klug

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.

Finite element modeling of lipid bilayer membranes

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.

Nonlinear finite-element analysis of nanoindentation of viral capsids

Recent atomic force microscope (AFM) nanoindentation experiments measuring mechanical response of the protein shells of viruses have provided a quantitative description of their strength and elasticity. To better understand and interpret these measurements, and to elucidate the underlying mechanisms, this paper adopts a course-grained modeling approach within the framework of three-dimensional nonlinear continuum elasticity. Homogeneous, isotropic, elastic, thick-shell models are proposed for two capsids: the spherical cowpea chlorotic mottle virus (CCMV), and the ellipsocylindrical bacteriophage phi29 . As analyzed by the finite-element method, these models enable parametric characterization of the effects of AFM tip geometry, capsid dimensions, and capsid constitutive descriptions. The generally nonlinear force response of capsids to indentation is shown to be insensitive to constitutive particulars, and greatly influenced by geometric and kinematic details. Nonlinear stiffening and softening of the force response is dependent on the AFM tip dimensions and shell thickness. Fits of the models capture the roughly linear behavior observed in experimental measurements and result in estimates of Youngs moduli of approximately 280-360 MPa for CCMV and approximately 4.5 GPa for phi29 .

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.

Squeezing Protein Shells: How Continuum Elastic Models, Molecular Dynamics Simulations, and Experiments Coalesce at the Nanoscale

The current rapid growth in the use of nanosized particles is fueled in part by our increased understanding of their physical properties and ability to manipulate them, which is essential for achieving optimal functionality. Here we report detailed quantitative measurements of the mechanical response of nanosized protein shells (viral capsids) to large-scale physical deformations and compare them with theoretical descriptions from continuum elastic modeling and molecular dynamics (MD). Specifically, we used nanoindentation by atomic force microscopy to investigate the complex elastic behavior of Hepatitis B virus capsids. These capsids are hollow, approximately 30 nm in diameter, and conform to icosahedral (5-3-2) symmetry. First we show that their indentation behavior, which is symmetry-axis-dependent, cannot be reproduced by a simple model based on Föppl-von Kármán thin-shell elasticity with the fivefold vertices acting as prestressed disclinations. However, we can properly describe the measured nonlinear elastic and orientation-dependent force response with a three-dimensional, topographically detailed, finite-element model. Next, we show that coarse-grained MD simulations also yield good agreement with our nanoindentation measurements, even without any fitting of force-field parameters in the MD model. This study demonstrates that the material properties of viral nanoparticles can be correctly described by both modeling approaches. At the same time, we show that even for large deformations, it suffices to approximate the mechanical behavior of nanosized viral shells with a continuum approach, and ignore specific molecular interactions. This experimental validation of continuum elastic theory provides an example of a situation in which rules of macroscopic physics can apply to nanoscale molecular assemblies.

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.

Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

As two-dimensional fluid shells, lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses. This property gives membranes the ability to adopt dramatic shape changes. In this paper, a finite element model is developed to study static equilibrium mechanics of membranes. In particular, a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers. The augmented Lagrangian method is used to enforce global constraints on area and volume during membrane deformations. As a validation of the method, equilibrium shapes for a shape-phase diagram of lipid bilayer vesicle are calculated. These numerical techniques are also shown to be useful for simulations of three-dimensional large deformation problems: the formation of tethers (long tube-like extensions); and Ginzburg-Landau phase separation of a two lipid-component vesicle. To deal with the large mesh distortions of the two-phase model, modification of viscous regularization is explored to achieve r-adaptive mesh optimization.

Observation of nanoscale dynamics in cantilever sensor arrays

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 cantilevers 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.

A director-field model of DNA packaging in viral capsids

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.

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.