Competing hydrophobic and screened-coulomb interactions in hepatitis B virus capsid assembly

Codon usage bias of goose circovirus and its adaptation to host

Goose circovirus (GoCV), a potential immunosuppressive virus possessing a circular single-stranded DNA genome, is widely distributed in both domesticated and wild geese. This virus infection causes significant economic losses in the waterfowl industry. The codon usage patterns of viruses reflect the evolutionary history and genetic architecture, allowing them to adapt quickly to changes in the external environment, particularly to their hosts. In this study, we retrieved the coding sequences (Rep and Cap) and the genome of GoCV from GenBank, conducting comprehensive research to explore the codon usage patterns in 144 GoCV strains. The overall codon usage of the GoCV strains was relatively similar and exhibited a slight bias. The effective number of codons (ENC) indicated a low overall extent of codon usage bias (CUB) in GoCV. Combined with the base composition and relative synonymous codon usage (RSCU) analysis, the results revealed a bias toward A- and G-ending codons in the overall codon usage. Analysis of the ENC-GC3s plot and neutrality plot suggested that natural selection plays an important role in shaping the codon usage pattern of GoCV, with mutation pressure having a minor influence. Furthermore, the correlations between ENC and relative indices, as well as correspondence analysis (COA), showed that hydrophobicity and geographical distribution also contribute to codon usage variation in GoCV, suggesting the possible involvement of natural selection. In conclusion, GoCV exhibits comparatively slight CUB, with natural selection being the major factor shaping the codon usage pattern of GoCV. Our research contributes to a deeper understanding of GoCV evolution and its host adaptation, providing valuable insights for future basic studies and vaccine design related to GoCV.

Impact of DNA on interactions between core proteins of Hepatitis B virus-like particles comprising different C-terminals

Hepatitis B virus (HBV) virus-like particles (VLPs) are promising therapeutic agents derived from HBV core proteins (Cp). This study investigates the assembly dynamics of HBV VLPs, which is crucial for their potential as drug carriers or gene delivery systems. Coarse-grained molecular dynamics simulations explore the impact of C-terminal domain length (in the Cp ranging from Cp149 to wild-type Cp183) on Cp assembly and stability, particularly in the presence of DNA. Our findings reveal that the C-terminal nucleic acid binding region significantly influences Cp assembly and stability of trimers comprising Cp dimers. Shorter C-terminal domains (Cp164, Cp167) enhance stability and protein-protein interactions, while interactions between naturally occurring Cp183 are destabilized in the absence of DNA. Interestingly, DNA addition further stabilizes Cp assemblies, and this effect is influenced by the length of the nucleic acid binding region. Shorter C-terminal domains show less dependency on DNA content. This stabilization is attributed to electrostatic forces between positively charged C-terminal chains and negatively charged nucleic acids. Our study sheds light on the molecular mechanisms governing protein-protein and protein-DNA interactions in HBV VLP assembly, providing insights into Cp processability and informing the development of efficient gene therapy carriers using VLP technology.

Relaxational dynamics of the T-number conversion of virus capsids

We extend a recently proposed kinetic theory of virus capsid assembly based on Model A kinetics and study the dynamics of the interconversion of virus capsids of different sizes triggered by a quench, that is, by sudden changes in the solution conditions. The work is inspired by in vitro experiments on functionalized coat proteins of the plant virus cowpea chlorotic mottle virus, which undergo a reversible transition between two different shell sizes (T = 1 and T = 3) upon changing the acidity and salinity of the solution. We find that the relaxation dynamics are governed by two time scales that, in almost all cases, can be identified as two distinct processes. Initially, the monomers and one of the two types of capsids respond to the quench. Subsequently, the monomer concentration remains essentially constant, and the conversion between the two capsid species completes. In the intermediate stages, a long-lived metastable steady state may present itself, where the thermodynamically less stable species predominate. We conclude that a Model A based relaxational model can reasonably describe the early and intermediate stages of the conversion experiments. However, it fails to provide a good representation of the time evolution of the state of assembly of the coat proteins in the very late stages of equilibration when one of the two species disappears from the solution. It appears that explicitly incorporating the nucleation barriers to assembly and disassembly is crucial for an accurate description of the experimental findings, at least under conditions where these barriers are sufficiently large.

Codon usage bias of Venezuelan equine encephalitis virus and its host adaption

Venezuelan equine encephalitis virus (VEEV) is an emerging zoonotic virus in the alphavirus genus. It can be transmitted to humans due to spillover from equid-mosquito cycles. The symptoms caused by VEEV include fever, headache, myalgia, nausea, and vomiting. It can also cause encephalitis in severe cases. The evolutionary features of VEEV are largely unknown. In this study, we comprehensively analyzed the codon usage pattern of VEEV by computing a variety of indicators, such as effective number of codons (ENc), codon adaptation index (CAI), relative synonymous codon usage (RSCU), on 130 VEEV coding sequences retrieved from GenBank. The results showed that the codon usage bias of VEEV is relatively low. ENc-GC3s plot, neutrality plot, and CAI-ENc correlation analyses supported that translational selection plays an important role in shaping the codon usage pattern of VEEV whereas the mutation pressure has a minor influence. Analysis of RSCU values showed that most of the preferred codons in VEEV are C/G-ended. Analysis of dinucleotide composition found that all CG- and UA-containing codons are not preferentially used. Phylogenetic analysis showed that VEEV isolates can be clustered into three genera and evolutionary force affects the codon usage pattern. Furthermore, a correspondence analysis (COA) showed that aromaticity and hydrophobicity as well as geographical distribution also have certain effects on the codon usage variation of VEEV, suggesting the possible involvement of translational selection. Overall, the codon usage of VEEV is comparatively slight and translational selection might be the main factor that shapes the codon usage pattern of VEEV. This study will promote our understanding about the evolution of VEEV and its host adaption, and might provide some clues for preventing the cross-species transmission of VEEV.

Electrostatic Theory of the Acidity of the Solution in the Lumina of Viruses and Virus-Like Particles

Recently, Maassen et al. measured an appreciable pH difference between the bulk solution and the solution in the lumen of virus-like particles, self-assembled in an aqueous buffer solution containing the coat proteins of a simple plant virus and polyanions (Maassen, S. J.; et al. Small 2018, 14, 1802081). They attribute this to the Donnan effect, caused by an imbalance between the number of negative charges on the encapsulated polyelectrolyte molecules and the number of positive charges on the RNA binding domains of the coat proteins that make up the virus shell or capsid. By applying Poisson-Boltzmann theory, we confirm this conclusion and show that simple Donnan theory is accurate even for the smallest of viruses and virus-like particles. This, in part, is due to the additional screening caused by the presence of a large number of immobile charges in the cavity of the shell. The presence of a net charge on the outer surface of the capsid we find in practice to not have a large effect on the pH shift. Hence, Donnan theory can indeed be applied to connect the local pH and the amount of encapsulated material. The large shifts up to a full pH unit that we predict must have consequences for applications of virus capsids as nanocontainers in bionanotechnology and artificial cell organelles.

Engineering Metastability into a Virus-like Particle to Enable Triggered Dissociation

For a virus-like particle (VLP) to serve as a delivery platform, the VLP must be able to release its cargo in response to a trigger. Here, we use a chemical biology approach to destabilize a self-assembling capsid for a subsequent triggered disassembly. We redesigned the dimeric hepatitis B virus (HBV) capsid protein (Cp) with two differentially addressable cysteines, C150 for reversibly crosslinking the capsid and C124 to react with a destabilizing moiety. The resulting construct, Cp150-V124C, assembles into icosahedral, 120-dimer VLPs that spontaneously crosslink via the C-terminal C150, leaving C124 buried at a dimer-dimer interface. The VLP is driven into a metastable state when C124 is reacted with the bulky fluorophore, maleimidyl BoDIPY-FL. The resulting VLP is stable until exposed to modest, physiologically relevant concentrations of reducing agent. We observe dissociation with FRET relaxation of polarization, size exclusion chromatography, and resistive-pulse sensing. Dissociation is slow, minutes to hours, with a characteristic lag phase. Mathematical modeling based on the presence of a nucleation step predicts disassembly dynamics that are consistent with experimental observations. VLPs transfected into hepatoma cells show similar dissociation behavior. These results suggest a generalizable strategy for designing a VLP that can release its contents in an environmentally responsive reaction.

Heat-induced transitions of an empty minute virus of mice capsid in explicit water: all-atom MD simulation

The capsid-like structure of the virus-based protein nanoparticles (NPs) can serve as bionanomaterials, with applications in biomedicines and nanotechnology. Release of packaged material from these nanocontainers is associated with subtle conformational changes of the NP structure, which in vitro, is readily accomplished by heating. Characterizing the structural changes as a function of temperature may provide fresh insights into nanomaterial/antiviral strategies. Here, we have calculated heat induced changes in the properties of an empty minute virus of mice particle using large-scale ≈ 3.0 × 10⁶ all-atom molecular dynamics simulations. We focus on two heat induced structural changes of the NP, namely, dynamical transition (DT) and breathing transition (BT), both characterized by sudden and sharp change of measured parameters at temperatures, T_DT and T_BT, respectively. While DT is assessed by mean-square fluctuation of hydrogen atoms of the NP, BT is monitored through internal volume and permeation rate of water molecules through the NP. Both the transitions, resulting primarily from collective atomistic motion, are found to occur at temperatures widely separated from one another (T_BT>T_DT). The breathing motions, responsible for the translocation events of the packaged materials through the NP to kick off, are further probed by computing atomic resolution stresses from NVE simulations. Distribution of equilibrium atomistic stresses on the NP reveals a largely asymmetric nature and suggests structural breathing may actually represent large dynamic changes in the hotspot regions, far from the NP pores, which is in remarkable resemblance with recently conducted hydrogen-deuterium exchange coupled to mass spectrometry experiment. Communicated by Ramaswamy H. Sarma.

Self-association features of NS1 proteins from different flaviviruses

Flaviviruses comprise a large group of arboviral species that are distributed in several countries of the tropics, neotropics, and some temperate zones. Since they can produce neurological pathologies or vascular damage, there has been intense research seeking better diagnosis and treatments for their infections in the last decades. The flavivirus NS1 protein is a relevant clinical target because it is involved in viral replication, immune evasion, and virulence. Being a key factor in endothelial and tissue-specific modulation, NS1 has been largely studied to understand the molecular mechanisms exploited by the virus to reprogram host cells. A central part of the viral maturation processes is the NS1 oligomerization because many stages rely on these protein-protein assemblies. In the present study, the self-associations of NS1 proteins from Zika, Dengue, and West Nile viruses are examined through constant-pH coarse-grained biophysical simulations. Free energies of interactions were estimated for different oligomeric states and pH conditions. Our results show that these proteins can form both dimers and tetramers under conditions near physiological pH even without the presence of lipids. Moreover, pH plays an important role mainly controlling the regimes where van der Waals interactions govern their association. Finally, despite the similarity at the sequence level, we found that each flavivirus has a well-characteristic protein-protein interaction profile. These specific features can provide new hints for the development of binders both for better diagnostic tools and the formulation of new therapeutic drugs.

Electrostatics drive the molecular chaperone BiP to preferentially bind oligomerized states of a client protein

Hsp70 chaperones bind short monomeric peptides with a weak characteristic affinity in the low micromolar range, but can also bind some aggregates, fibrils, and amyloids, with low nanomolar affinity. While this differential affinity enables Hsp70 to preferentially target potentially toxic aggregates, it is unknown how a chaperone can differentiate between monomeric and aggregated states of a client protein and why preferential binding is only observed for some aggregated clients but not others. Here we examine the interaction of BiP (the Hsp70 paralog in the endoplasmic reticulum) with the client proIGF2, the pro-protein form of IGF2 that includes a long and mostly disordered E-peptide region that promotes proIGF2 oligomerization. By dissecting the mechanism by which BiP targets proIGF2 and E-peptide oligomers we discover that electrostatic attraction is a powerful driving force for oligomer recognition. We identify the specific BiP binding sites on proIGF2 and as monomers they bind BiP with characteristically weak affinity in the low micromolar range, but electrostatic attraction to E-peptide oligomers boosts the affinity to the low nanomolar level. The dominant role of electrostatics is manifested kinetically as a steering force that accelerates the binding of BiP to E-peptide oligomers by approximately two orders of magnitude as compared against monomeric peptides. Electrostatic targeting of Hsp70 provides an explanation for why preferential binding has been observed for some aggregated clients but not others, as all the currently-documented cases in which Hsp70 binds aggregates with high-affinity involve clients that have an opposite charge to Hsp70.

The use of vector formalism in the analysis of hydrophobic and electric driving forces in biological assemblies

Hydrophobic forces are known to have a crucial part not only in the conformation of the three-dimensional structure of proteins, but also in the build-up of DNA–protein complexes. Electric forces also play an important role both in the tertiary as well in the quaternary structure of macromolecular associations. Sometimes both hydrophobic and electric interactions add up their strengths to accomplish these structures but in most cases they act in opposite directions. This fact, together with being overall interactions with different ranges, provides a nuanced equilibrium also modulated by the need to comply with steric hindrances and geometric frustration effects. This review focuses on the utility of using the hydrophobic and electrical dipole moment vectors to describe the interactions that give rise to the structures of biological macromolecules. Although different definitions of both electric dipole and hydrophobic moments have been described in the literature, results obtained in biological assemblies demonstrate the principle of the biological membrane model. According to this model, postulated by our group, biological macromolecules tend to associate by aligning their hydrophobic moments in a similar manner to phospholipids in a membrane. Examples of both closed and open structures are used to assess the predictability of our model. We seek agreement between our results with those described in the current literature. The review ends with possible future projections using this formalism.

Hysteresis in Hepatitis B Virus (HBV) Requires Assembly of Near-Perfect Capsids

The hepatitis B virus (HBV) must release its contents to initiate infection, making capsid disassembly critical to the viral life cycle. Capsid assembly proceeds through a cascade of weak interactions between copies of capsid protein (Cp) to yield uniform particles. However, there is a hysteresis to capsid dissociation that allows capsids to persist under conditions where they could not assemble. In this study, we have sought to define the basis of hysteresis by examining urea-induced dissociation of in vitro-assembled HBV capsids. In general, capsid samples show a mixture of two pools, differentiated by stability. Labile capsid dissociation corresponds to an ∼5 μM pseudocritical concentration of assembly (pcc), the same as that observed in assembly reactions. Dissociation of the stable pool corresponds to a subfemtomolar pcc, indicative of hysteresis. The fraction of stable capsids in an assembly reaction increases with the integrity of the Cp preparation and when association is performed at a higher ionic strength, which modifies the Cp conformation. Labile complexes are more prevalent when assembly conditions yield many kinetically trapped (incomplete and overgrown) products. Cp isolated from stable capsids reassembles into a mixture of stable and labile capsids. These results suggest that hysteresis arises from an ideal capsid lattice, even when some of the substituents in that lattice have defects. Consistent with structural studies that show a subtle difference between Cp dimers and Cp in capsid, we propose that hysteresis arises when HBV capsids undergo a lattice-dependent structural transition.

Polyelectrolyte Encapsulation and Confinement within Protein Cage-Inspired Nanocompartments

Protein cages are nanocompartments with a well-defined structure and monodisperse size. They are composed of several individual subunits and can be categorized as viral and non-viral protein cages. Native viral cages often exhibit a cationic interior, which binds the anionic nucleic acid genome through electrostatic interactions leading to efficient encapsulation. Non-viral cages can carry various cargo, ranging from small molecules to inorganic nanoparticles. Both cage types can be functionalized at targeted locations through genetic engineering or chemical modification to entrap materials through interactions that are inaccessible to wild-type cages. Moreover, the limited number of constitutional subunits ease the modification efforts, because a single modification on the subunit can lead to multiple functional sites on the cage surface. Increasing efforts have also been dedicated to the assembly of protein cage-mimicking structures or templated protein coatings. This review focuses on native and modified protein cages that have been used to encapsulate and package polyelectrolyte cargos and on the electrostatic interactions that are the driving force for the assembly of such structures. Selective encapsulation can protect the payload from the surroundings, shield the potential toxicity or even enhance the intended performance of the payload, which is appealing in drug or gene delivery and imaging.

Relationships between RNA topology and nucleocapsid structure in a model icosahedral virus

The process of genome packaging in most of viruses is poorly understood, notably the role of the genome itself in the nucleocapsid structure. For simple icosahedral single-stranded RNA viruses, the branched topology due to the RNA secondary structure is thought to lower the free energy required to complete a virion. We investigate the structure of nucleocapsids packaging RNA segments with various degrees of compactness by small-angle x-ray scattering and cryotransmission electron microscopy. The structural differences are mild even though compact RNA segments lead on average to better-ordered and more uniform particles across the sample. Numerical calculations confirm that the free energy is lowered for the RNA segments displaying the larger number of branch points. The effect is, however, opposite with synthetic polyelectrolytes, in which a star topology gives rise to more disorder in the capsids than a linear topology. If RNA compactness and size account in part for the proper assembly of the nucleocapsid and the genome selectivity, other factors most likely related to the host cell environment during viral assembly must come into play as well.

Percolation Theory Reveals Biophysical Properties of Virus-like Particles

The viral protein containers that encapsulate a virus' genetic material are repurposed as virus-like particles in a host of nanotechnology applications, including cargo delivery, storage, catalysis, and vaccination. These viral architectures have evolved to sit on the knife's edge between stability, to provide adequate protection for their genetic cargoes, and instability, to enable their efficient and timely release in the host cell environment upon environmental cues. By introducing a percolation theory for viral capsids, we demonstrate that the geometric characteristics of a viral capsid in terms of its subunit layout and intersubunit interaction network are key for its disassembly behavior. A comparative analysis of all alternative homogeneously tiled capsid structures of the same stoichiometry identifies evolutionary drivers favoring specific viral geometries in nature and offers a guide for virus-like particle design in nanotechnology.

Experiment and FEM Analysis of Silica Nanoparticle-Based Impedance Immunosensor for Sensitivity Enhancement

This article investigates the impact of incorporating silica nanoparticles of varying diameters in label free impedance immunosensor. It has been observed that even if the surface area improvement has been adjusted to be similar for all the diameters, the sensitivity is enhanced by five times at a particular diameter of 100 nm due to the optimum combination of intersection with electric field lines and surface convexity. This study has enabled the detection of 0.1 fM Hep-B surface antigen with a reliable sensitivity of around 75%. Further, it has been observed that the SNR corresponding to 0.1 fM is 20 dB only for 100 nm particle. This SNR is comparable to a recent report on Hep-B virus detection but the limit of detection in the proposed sensor is lowered by more than three orders of magnitude.

Controlling Protein Nanocage Assembly with Hydrostatic Pressure

Controlling the assembly and disassembly of nanoscale protein cages for the capture and internalization of protein or non-proteinaceous components is fundamentally important to a diverse range of bionanotechnological applications. Here, we study the reversible, pressure-induced dissociation of a natural protein nanocage, E. coli bacterioferritin (Bfr), using synchrotron radiation small-angle X-ray scattering (SAXS) and circular dichroism (CD). We demonstrate that hydrostatic pressures of 450 MPa are sufficient to completely dissociate the Bfr 24-mer into protein dimers, and the reversibility and kinetics of the reassembly process can be controlled by selecting appropriate buffer conditions. We also demonstrate that the heme B prosthetic group present at the subunit dimer interface influences the stability and pressure lability of the cage, despite its location being discrete from the interdimer interface that is key to cage assembly. This indicates a major cage-stabilizing role for heme within this family of ferritins.

Nonsymmetrical Dynamics of the HBV Capsid Assembly and Disassembly Evidenced by Their Transient Species

As with many protein multimers studied in biophysics, the assembly and disassembly dynamical pathways of hepatitis B virus (HBV) capsid proteins are not symmetrical. Using time-resolved small-angle X-ray scattering and singular value decomposition analysis, we have investigated these processes in vitro by a rapid change of salinity or chaotropicity. Along the assembly pathway, the classical nucleation-growth mechanism is followed by a slow relaxation phase during which capsid-like transient species self-organize in accordance with the theoretical prediction that the capture of the few last subunits is slow. By contrast, the disassembly proceeds through unexpected, fractal-branched clusters of subunits that eventually vanish over a much longer time scale. On the one hand, our findings confirm and extend previous views as to the hysteresis phenomena observed and theorized in capsid formation and dissociation. On the other hand, they uncover specifics that may directly relate to the functions of HBV subunits in the viral cycle.

Molecular recognition mechanisms directing the self-assembly of biological structures

Self-assembling may be defined as the spontaneous association of material units into structures that are often capable of cyclic reorganization and functional behavior. Various molecular recognition processes stabilize assemblies of polymers and biological structures. The present article analyzes cases in which chemical, shape and other recognition mechanisms are individually or cooperatively operative. Simpler self-assembling theories reported in the literature are highlighted. Detailed processes for which chemical recognition is the prevailing, enthalpy-driven, process include the non-ideal component of miscibility, supramolecular polymerization, host-guest complexes and template polymerization. Also discussed are systems such as liquid crystalline closed polymers, ternary mesogenic systems and rigid crystalline polymers for which shape recognition is the prevailing entropy-driven process. Other recognition mechanisms include ion condensation effects, hydrophobic bonding and growth-coupled-to-orientation. Combinations of various recognition mechanisms are particularly evident in biological structures. Self-assembling mechanisms involved in the genesis of some biological systems can be scientifically identified, but much more needs to be known to describe the "engineered" assembling modes that support complex functional organs.

Computational studies of shape control of charged deformable nanocontainers

Biological matter is often compartmentalized by soft membranes that dynamically change their shape in response to chemical and mechanical cues. Deformable soft-matter-based nanoscale membranes or nanocontainers that mimic this behavior can be used as drug-delivery carriers that can adapt to evolving physiological conditions, or as dynamic building blocks for the design of novel hierarchical materials via assembly engineering. Here, we connect the intrinsic features of charged deformable nanocontainers such as their size, charge, surface tension, and elasticity with their equilibrium shapes for a wide range of solution conditions using molecular dynamics simulations. These links identify the fundamental mechanisms that establish the chemical and materials design control strategies for modulating the equilibrium shape of these nanocontainers. We show that flexible nanocontainers of radii ranging from 10-20 nm exhibit sphere-to-rod-to-disc shape transitions yielding rods and discs over a wide range of aspect ratio λ (0.3 < λ < 5). The shape transitions can be controlled by tuning salt and/or surfactant concentration as well as material elastic parameters. The shape changes are driven by reduction in the global electrostatic energy and are associated with dramatic changes in local surface elastic energy distributions. To illustrate the shape transition mechanisms, exact analytical calculations for idealized spheroidal nanocontainers in salt-free conditions are performed. Explicit counterion simulations near nanocontainers and associated Manning model calculations provide an assessment of the stability of observed shape deformations in the event of ion condensation.

Rapidly Forming Early Intermediate Structures Dictate the Pathway of Capsid Assembly

There are ∼10³⁰ possible intermediates on the assembly path from hepatitis B capsid protein dimers to the 120-dimer capsid. If every intermediate was tested, assembly would often get stuck in an entropic trap and essentially every capsid would follow a unique assembly path. Yet, capsids assemble rapidly with minimal trapped intermediates, a realization of the Levinthal paradox. To understand the fundamental mechanisms of capsid assembly, it is critical to resolve the early stages of the reaction. We have used time-resolved small angle X-ray scattering, which is sensitive to solute size and shape and has millisecond temporal resolution. Scattering curves were fit to a thermodynamically curated library of assembly intermediates, using the principle of maximum entropy. Maximum entropy also provides a physical rationale for the selection of species. We found that the capsid assembly pathway was exquisitely sensitive to initial assembly conditions. With the mildest conditions tested, the reaction appeared to be two-state from dimer to 120-dimer capsid with some dimers-of-dimers and trimers-of-dimers. In slightly more aggressive conditions, we observed transient accumulation of a decamer-of-dimers and the appearance of 90-dimer capsids. In conditions where there is measurable kinetic trapping, we found that highly diverse early intermediates accumulated within a fraction of a second and propagated into long-lived kinetically trapped states (≥90-mer). In all cases, intermediates between 35 and 90 subunits did not accumulate. These results are consistent with the presence of low barrier paths that connect early and late intermediates and direct the ultimate assembly path to late intermediates where assembly can be paused.

Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number

The Caspar–Klug (CK) classification of viruses is discussed by parallel examination of geometry of icosahedral geodesic domes, fullerenes, and viruses. The underlying symmetry of all structures is explained and thoroughly visually represented. Euler’s theorem on polyhedra is used to calculate the number of vertices, edges, and faces in domes, number of atoms, bonds, and pentagonal and hexagonal rings in fullerenes, and number of proteins and protein–protein contacts in viruses. The T-number, the characteristic for the CK classification, is defined and discussed. The superposition of fullerene and dome designs is used to obtain a representation of a CK virus with all the proteins indicated. Some modifications of the CK classifications are sketched, including elongation of the CK blueprint, fusion of two CK blueprints, dodecahedral view of the CK shapes, and generalized CK designs without a clearly visible geometry of the icosahedron. These are compared to cases of existing viruses.

How a Virus Circumvents Energy Barriers to Form Symmetric Shells

Previous self-assembly experiments on a model icosahedral plant virus have shown that, under physiological conditions, capsid proteins initially bind to the genome through an en masse mechanism and form nucleoprotein complexes in a disordered state, which raises the question as to how virions are assembled into a highly ordered structure in the host cell. Using small-angle X-ray scattering, we find out that a disorder-order transition occurs under physiological conditions upon an increase in capsid protein concentrations. Our cryo-transmission electron microscopy reveals closed spherical shells containing in vitro transcribed viral RNA even at pH 7.5, in marked contrast with the previous observations. We use Monte Carlo simulations to explain this disorder-order transition and find that, as the shell grows, the structures of disordered intermediates in which the distribution of pentamers does not belong to the icosahedral subgroups become energetically so unfavorable that the caps can easily dissociate and reassemble, overcoming the energy barriers for the formation of perfect icosahedral shells. In addition, we monitor the growth of capsids under the condition that the nucleation and growth is the dominant pathway and show that the key for the disorder-order transition in both en masse and nucleation and growth pathways lies in the strength of elastic energy compared to the other forces in the system including protein-protein interactions and the chemical potential of free subunits. Our findings explain, at least in part, why perfect virions with icosahedral order form under different conditions including physiological ones.

Thermodynamics of amyloid fibril formation from chemical depolymerization

Amyloid fibrils are homo-molecular protein polymers that play an important role in disease and biological function. While much is known about their kinetics and mechanisms of formation, the origin and magnitude of their thermodynamic stability has received significantly less attention. This is despite the fact that the thermodynamic stability of amyloid fibrils is an important determinant of their lifetimes and processing in vivo. Here we use depolymerization by chemical denaturants of amyloid fibrils of two different proteins (PI3K-SH3 and glucagon) at different concentrations and show that the previously applied isodesmic linear polymerization model is an oversimplification that does not capture the concentration dependence of chemical depolymerization of amyloid fibrils. We show that cooperative polymerization, which is compatible with the picture of amyloid formation as a nucleated polymerization process, is able to quantitatively describe the thermodynamic data. We use this combined experimental and conceptual framework in order to probe the ionic strength dependence of amyloid fibril stability. In combination with previously published data on the ionic strength dependence of amyloid fibril growth kinetics, our results provide strong evidence for the product-like nature of the transition state of amyloid fibril growth.

Of capsid structure and stability: The partnership between charged residues of E-loop and P-domain of the bacteriophage P22 coat protein

Tailed dsDNA bacteriophages and herpesviruses form capsids using coat proteins that have the HK97 fold. In these viruses, the coat proteins first assemble into procapsids, which subsequently mature during DNA packaging. Generally interactions between the coat protein E-loop of one subunit and the P-domain of an adjacent subunit help stabilize both capsomers and capsids. Based on a recent 3.3 Å cryo-EM structure of the bacteriophage P22 virion, E-loop amino acids E52, E59 and E72 were suggested to stabilize the capsid through intra-capsomer salt bridges with the P-domain residues R102, R109 and K118. The glutamic acid residues were each mutated to alanine to test this hypothesis. The substitutions resulted in a WT phenotype and did not destabilize capsids; rather, the alanine substituted coat proteins increased the stability of procapsids and virions. These results indicate that different types of interactions must be used between the E-loop and P-domain to stabilize phage P22 procapsids and virions.

Assembly Reactions of Hepatitis B Capsid Protein into Capsid Nanoparticles Follow a Narrow Path through a Complex Reaction Landscape

For many viruses, capsids (biological nanoparticles) assemble to protect genetic material and dissociate to release their cargo. To understand these contradictory properties, we analyzed capsid assembly for hepatitis B virus; an endemic pathogen with an icosahedral, 120-homodimer capsid. We used solution X-ray scattering to examine trapped and equilibrated assembly reactions. To fit experimental results, we generated a library of distinct intermediates, selected by umbrella sampling of Monte Carlo simulations. The number of possible capsid intermediates is immense, ∼10³⁰, yet assembly reactions are rapid and completed with high fidelity. If the huge number of possible intermediates were actually present, maximum entropy analysis shows that assembly reactions would be blocked by an entropic barrier, resulting in incomplete nanoparticles. When an energetic term was applied to select the stable species that dominated the reaction mixture, we found only a few hundred intermediates, mapping out a narrow path through the immense reaction landscape. This is a solution to a viral application of the Levinthal paradox. With the correct energetic term, the match between predicted intermediates and scattering data was striking. The grand canonical free energy landscape for assembly, calibrated by our experimental results, supports a detailed analysis of this complex reaction. There is a narrow range of energies that supports on-path assembly. If association energy is too weak or too strong, progressively more intermediates will be entropically blocked, spilling into paths leading to dissociation or trapped incomplete nanoparticles, respectively. These results are relevant to many viruses and provide a basis for simplifying assembly models and identifying new targets for antiviral intervention. They provide a basis for understanding and designing biological and abiological self-assembly reactions.

A dimorphism shift of hepatitis B virus capsids in response to ionic conditions

The dimorphism of HBV capsids (coexistence of T = 3 and T = 4 capsids) was found to be regulatable by controlling the rate of capsid nucleation using cations such as K+ or Ca2+: a quick addition of highly concentrated monovalent and/or multivalent counter-cations resulted in a morphism transition from a thermodynamically more stable, T = 4 capsid-dominant state (>80% of total capsids) to a new state containing ∼1 : 1 amounts of T = 3 and T = 4 capsids. These results suggested that the salts with strong charge screening ability could narrow the difference in nucleation energy barriers between the two states, which were not inter-convertible once formed. The effect of salts was more significant than other factors such as pH or protein concentration in achieving such a dimorphism shift. The general mechanism of HBV capsid dimorphism described here provides a new perspective in understanding the virus assembly during infection and directing the design of non-infectious capsids for nanotechnology applications.

Nanoscale insight into silk-like protein self-assembly: effect of design and number of repeat units

By means of replica exchange molecular dynamics simulations we investigate how the length of a silk-like, alternating diblock oligopeptide influences its secondary and quaternary structure. We carry out simulations for two protein sizes consisting of three and five blocks, and study the stability of a single protein, a dimer, a trimer and a tetramer. Initial configurations of our simulations are β-roll and β-sheet structures. We find that for the triblock the secondary and quaternary structures upto and including the tetramer are unstable: the proteins melt into random coil structures and the aggregates disassemble either completely or partially. We attribute this to the competition between conformational entropy of the proteins and the formation of hydrogen bonds and hydrophobic interactions between proteins. This is confirmed by our simulations on the pentablock proteins, where we find that, as the number of monomers in the aggregate increases, individual monomers form more hydrogen bonds whereas their solvent accessible surface area decreases. For the pentablock β-sheet protein, the monomer and the dimer melt as well, although for the β-roll protein only the monomer melts. For both trimers and tetramers remain stable. Apparently, for these the entropy loss of forming β-rolls and β-sheets is compensated for in the free-energy gain due to the hydrogen-bonding and hydrophobic interactions. We also find that the middle monomers in the trimers and tetramers are conformationally much more stable than the ones on the top and the bottom. Interestingly, the latter are more stable on the tetramer than on the trimer, suggesting that as the number of monomers increases protein-protein interactions cooperatively stabilize the assembly. According to our simulations, the β-roll and β-sheet aggregates must be approximately equally stable.

Electrostatics-Driven Inflation of Elastic Icosahedral Shells as a Model for Swelling of Viruses

We develop a clear theoretical description of radial swelling in virus-like particles that delineates the importance of electrostatic contributions to swelling in the absence of any conformational changes. The model couples the elastic parameters of the capsid-represented as a continuous elastic shell-to the electrostatic pressure acting on it. We show that different modifications of the electrostatic interactions brought about by, for instance, changes in pH or solution ionic strength are often sufficient to achieve the experimentally observed swelling (∼10% of the capsid radius). Additionally, we derive analytical expressions for the electrostatics-driven radial swelling of virus-like particles that enable one to quickly estimate the magnitudes of physical quantities involved.

Nonequilibrium self-assembly dynamics of icosahedral viral capsids packaging genome or polyelectrolyte

The survival of viruses partly relies on their ability to self-assemble inside host cells. Although coarse-grained simulations have identified different pathways leading to assembled virions from their components, experimental evidence is severely lacking. Here, we use time-resolved small-angle X-ray scattering to uncover the nonequilibrium self-assembly dynamics of icosahedral viral capsids packaging their full RNA genome. We reveal the formation of amorphous complexes via an en masse pathway and their relaxation into virions via a synchronous pathway. The binding energy of capsid subunits on the genome is moderate (~7k_BT₀, with k_B the Boltzmann constant and T₀ = 298 K, the room temperature), while the energy barrier separating the complexes and the virions is high (~ 20k_BT₀). A synthetic polyelectrolyte can lower this barrier so that filled capsids are formed in conditions where virions cannot build up. We propose a representation of the dynamics on a free energy landscape.

The mechanism by which virus capsules assemble around RNA to package their genetic material is not clear. Here, the authors observed the assembly of the cowpea chlorotic mottle virus capsid around viral RNA or poly(styrene sulfonic acid) using time-resolved small-angle X-ray scattering measurements.

Hydrophobic-Interaction-Induced Stiffening of α-Synuclein Fibril Networks

In water, networks of semiflexible fibrils of the protein α-synuclein stiffen significantly with increasing temperature. We make plausible that this reversible stiffening is a result of hydrophobic contacts between the fibrils that become more prominent with increasing temperature. The good agreement of our experimentally observed temperature dependence of the storage modulus of the network with a scaling theory linking network elasticity with reversible cross-linking enables us to quantify the endothermic binding enthalpy and estimate the effective size of hydrophobic patches on the fibril surface. Our findings may not only shed light on the role of amyloid deposits in disease conditions, but can also inspire new approaches for the design of thermoresponsive materials.

Role of Protein Charge Density on Hepatitis B Virus Capsid Formation

The role of electrostatic interactions in the viral capsid assembly process was studied by comparing the assembly process of a truncated hepatitis B virus capsid protein Cp149 with its mutant protein D2N/D4N, which has the same conformational structure but four fewer charges per dimer. The capsid protein self-assembly was investigated under a wide range of protein surface charge densities by changing the protein concentration, buffer pH, and solution ionic strength. Lowering the protein charge density favored the capsid formation. However, lowering charge beyond a certain point resulted in capsid aggregation and precipitation. Interestingly, both the wild-type and D2N/D4N mutant displayed identical assembly profiles when their charge densities matched each other. These results indicated that the charge density was optimized by nature to ensure an efficient and effective capsid proliferation under the physiological pH and ionic strength.

All-atom molecular dynamics of the HBV capsid reveals insights into biological function and cryo-EM resolution limits

The hepatitis B virus capsid represents a promising therapeutic target. Experiments suggest the capsid must be flexible to function; however, capsid structure and dynamics have not been thoroughly characterized in the absence of icosahedral symmetry constraints. Here, all-atom molecular dynamics simulations are leveraged to investigate the capsid without symmetry bias, enabling study of capsid flexibility and its implications for biological function and cryo-EM resolution limits. Simulation results confirm flexibility and reveal a propensity for asymmetric distortion. The capsid’s influence on ionic species suggests a mechanism for modulating the display of cellular signals and implicates the capsid’s triangular pores as the location of signal exposure. A theoretical image reconstruction performed using simulated conformations indicates how capsid flexibility may limit the resolution of cryo-EM. Overall, the present work provides functional insight beyond what is accessible to experimental methods and raises important considerations regarding asymmetry in structural studies of icosahedral virus capsids.

Self assembled linear polymeric chains with tuneable semiflexibility using isotropic interactions

We propose a two-body spherically symmetric (isotropic) potential such that particles interacting by the potential self-assemble into linear semiflexible polymeric chains without branching. By suitable control of the potential parameters, we can control the persistence length of the polymer and can even introduce a controlled number of branches. Thus we show how to achieve effective directional interactions starting from spherically symmetric potentials. The self-assembled polymers have an exponential distribution of chain lengths akin to what is observed for worm-like micellar systems. On increasing particle density, the polymeric chains self-organize to an ordered line-hexagonal phase where every chain is surrounded by six parallel chains, the transition is first order. On further increase in monomer density, the order is destroyed and we get a branched gel-like phase. This potential can be used to model semi-flexible equilibrium polymers with tunable semiflexibility and excluded volume. The use of the potential is computationally cheap and hence can be used to simulate and probe equilibrium polymer dynamics with long chains. The potential also gives a plausible method of tuning colloidal interactions in experiments such that one can obtain self-assembling polymeric chains made up of colloids and probe polymer dynamics using an optical microscope. Furthermore, we show how a modified potential leads to the observation of an intermediate nematic phase of self-assembled chains in between the low density disordered phase and the line-ordered hexagonal phase.

The different faces of mass action in virus assembly

The spontaneous encapsulation of genomic and non-genomic polyanions by coat proteins of simple icosahedral viruses is driven, in the first instance, by electrostatic interactions with polycationic RNA binding domains on these proteins. The efficiency with which the polyanions can be encapsulated in vitro, and presumably also in vivo, must in addition be governed by the loss of translational and mixing entropy associated with co-assembly, at least if this co-assembly constitutes a reversible process. These forms of entropy counteract the impact of attractive interactions between the constituents and hence they counteract complexation. By invoking mass action-type arguments and a simple model describing electrostatic interactions, we show how these forms of entropy might settle the competition between negatively charged polymers of different molecular weights for co-assembly with the coat proteins. In direct competition, mass action turns out to strongly work against the encapsulation of RNAs that are significantly shorter, which is typically the case for non-viral (host) RNAs. We also find that coat proteins favor forming virus particles over nonspecific binding to other proteins in the cytosol even if these are present in vast excess. Our results rationalize a number of recent in vitro co-assembly experiments showing that short polyanions are less effective at attracting virus coat proteins to form virus-like particles than long ones do, even if both are present at equal weight concentrations in the assembly mixture.

Minimum energy paths for conformational changes of viral capsids

In this work we study conformational changes of viral capsids using techniques of large deviations theory for stochastic differential equations. The viral capsid is a model of a complex system in which many units-the proteins forming the capsomers-interact by weak forces to form a structure with exceptional mechanical resistance. The destabilization of such a structure is interesting both, per se, since it is related either to infection or maturation processes and because it yields insights into the stability of complex structures in which the constitutive elements interact by weak attractive forces. We focus here on a simplified model of a dodecahedral viral capsid and assume that the capsomers are rigid plaquettes with one degree of freedom each. We compute the most probable transition path from the closed capsid to the final configuration using minimum energy paths and discuss the stability of intermediate states.

Cellular and Nuclear Forces: An Overview

Biological cells sample their surrounding microenvironments using nanoscale force sensors on the cell surfaces. These surface-based force and stress sensors generate physical and chemical responses inside the cell. The inherently well-connected cytoskeleton and its physical contacts with the force elements on the nuclear membrane lead these physicochemical responses to cascade all the way inside the cell nucleus, physically altering the nuclear state. These physical alterations of the cell nucleus, through yet-unknown complex steps elicit physical and functional response from the chromatin in the form of altered gene expression profiles. This mechanism of force/stress sensing by the cell and then its nuclear response has been shown to play a vital role in maintaining robust cellular homeostasis, controlling gene expression profiles during developmental phases as well as cell differentiation. Over the last few years, there has been appreciable progress toward identification of the molecular players responsible for force sensing. However, the actual sensing mechanism of cell surface bound force sensors and more importantly cascading of the signals, both physical (via cytosolic force sensing elements such as microtubule and actin framework) and chemical (cascade of biochemical signaling from cell surface to nuclear surface and further to the chromatin), inside the cell is poorly understood. In this chapter, we present a review of the currently known molecular players in cellular as well as nuclear force sensing repertoire and their possible mechanistic aspects. We also introduce various biophysical concepts that are used to describe the force/stress sensing and response of a cell. We hope this will help asking clearer questions and designing pointed experiments for better understanding of the force-dependent design principles of the cell surface, nuclear surface, and gene expression.

Investigating the thermal dissociation of viral capsid by lattice model

The dissociation of icosahedral viral capsids was investigated by a homogeneous and a heterogeneous lattice model. In thermal dissociation experiments with cowpea chlorotic mottle virus and probed by small-angle neutron scattering, we observed a slight shrinkage of viral capsids, which can be related to the strengthening of the hydrophobic interaction between subunits at increasing temperature. By considering the temperature dependence of hydrophobic interaction in the homogeneous lattice model, we were able to give a better estimate of the effective charge. In the heterogeneous lattice model, two sets of lattice sites represented different capsid subunits with asymmetric interaction strengths. In that case, the dissociation of capsids was found to shift from a sharp one-step transition to a gradual two-step transition by weakening the hydrophobic interaction between AB and CC subunits. We anticipate that such lattice models will shed further light on the statistical mechanics underlying virus assembly and disassembly.

Temperature-Induced, Selective Assembly of Supramolecular Colloids in Water

In this article, we report the synthesis and physical characterization of colloidal polystyrene particles that carry water-soluble supramolecular N,N',N″,-trialkyl-benzene-1,3,5-tricarboxamides (BTAs) on their surface. These molecules are known to assemble into one-dimensional supramolecular polymers via noncovalent interactions. By tethering the BTAs to charge-stabilized particles, the clustering behavior of the resulting colloids was dictated by a balance between interparticle electrostatic repulsion and the BTA-mediated attractions. Through careful tuning of the dispersing medium's ionic strength, a regime was found in which particle aggregation could be reversibly induced upon heating the dispersion. These findings clearly indicate that hydrophobic interactions, which become stronger upon heating, play an important role during the clustering process. Besides the thermoreversible nature of the generated hydrophobic interparticle attractions, we found the clustering to be selective, that is, the BTA-functionalized colloids do not interact with nonfunctionalized hydrophobic polystyrene particles. This selectivity in the association process can be rationalized by the preferred stacking of the surface-tethered BTAs. These selective intermolecular/particle bonds are likely stabilized by the formation of hydrogen bonds, as previously observed for analogous molecular BTA assemblies. The resulting driving force responsible for particle clustering is therefore dual in nature and depends on both hydrophobic attractions and hydrogen bonding.

Identification of a major intermediate along the self-assembly pathway of an icosahedral viral capsid by using an analytical model of a spherical patch

Viruses are astonishing edifices in which hundreds of molecular building blocks fit into the final structure with pinpoint accuracy. We established a robust kinetic model accounting for the in vitro self-assembly of a capsid shell derived from an icosahedral plant virus by using time-resolved small-angle X-ray scattering (TR-SAXS) data at high spatiotemporal resolution. By implementing an analytical model of a spherical patch into a global fitting algorithm, we managed to identify a major intermediate species along the self-assembly pathway. With a series of data collected at different protein concentrations, we showed that free dimers self-assembled into a capsid through an intermediate resembling a half-capsid. The typical lifetime of the intermediate was a few seconds and yet the presence of so large an oligomer was not reported before. The progress in instrumental detection along with the development of powerful algorithms for data processing contribute to shedding light on nonequilibrium processes in highly complex systems such as viruses.

A Thermodynamic Model for Genome Packaging in Hepatitis B Virus

Understanding the fundamentals of genome packaging in viral capsids is important for finding effective antiviral strategies and for utilizing benign viral particles for gene therapy. While the structure of encapsidated genomic materials has been routinely characterized with experimental techniques such as cryo-electron microscopy and x-ray diffraction, much less is known about the molecular driving forces underlying genome assembly in an intracellular environment and its in vivo interactions with the capsid proteins. Here we study the thermodynamic basis of the pregenomic RNA encapsidation in human Hepatitis B virus in vivo using a coarse-grained molecular model that captures the essential components of nonspecific intermolecular interactions. The thermodynamic model is used to examine how the electrostatic interaction between the packaged RNA and the highly charged C-terminal domains (CTD) of capsid proteins regulate the nucleocapsid formation. The theoretical model predicts optimal RNA content in Hepatitis B virus nucleocapsids with different CTD lengths in good agreement with mutagenesis measurements, confirming the predominant role of electrostatic interactions and molecular excluded-volume effects in genome packaging. We find that the amount of encapsidated RNA is not linearly correlated with the net charge of CTD tails as suggested by earlier theoretical studies. Our thermodynamic analysis of the nucleocapsid structure and stability indicates that ∼10% of the CTD residues are free from complexation with RNA, resulting in partially exposed CTD tails. The thermodynamic model also predicts the free energy of complex formation between macromolecules, which corroborates experimental results for the impact of CTD truncation on the nucleocapsid stability.

Energetically favoured defects in dense packings of particles on spherical surfaces

The dense packing of interacting particles on spheres has proved to be a useful model for virus capsids and colloidosomes. Indeed, icosahedral symmetry observed in virus capsids corresponds to potential energy minima that occur for magic numbers of, e.g., 12, 32 and 72 identical Lennard-Jones particles, for which the packing has exactly the minimum number of twelve five-fold defects. It is unclear, however, how stable these structures are against thermal agitation. We investigate this property by means of basin-hopping global optimisation and Langevin dynamics for particle numbers between ten and one hundred. An important measure is the number and type of point defects, that is, particles that do not have six nearest neighbours. We find that small icosahedral structures are the most robust against thermal fluctuations, exhibiting fewer excess defects and rearrangements for a wide temperature range. Furthermore, we provide evidence that excess defects appearing at low non-zero temperatures lower the potential energy at the expense of entropy. At higher temperatures defects are, as expected, thermally excited and thus entropically stabilised. If we replace the Lennard-Jones potential by a very short-ranged (Morse) potential, which is arguably more appropriate for colloids and virus capsid proteins, we find that the same particle numbers give a minimum in the potential energy, although for larger particle numbers these minima correspond to different packings. Furthermore, defects are more difficult to excite thermally for the short-ranged potential, suggesting that the short-ranged interaction further stabilises equilibrium structures.

Modeling Effects of RNA on Capsid Assembly Pathways via Coarse-Grained Stochastic Simulation

The environment of a living cell is vastly different from that of an in vitro reaction system, an issue that presents great challenges to the use of in vitro models, or computer simulations based on them, for understanding biochemistry in vivo. Virus capsids make an excellent model system for such questions because they typically have few distinct components, making them amenable to in vitro and modeling studies, yet their assembly can involve complex networks of possible reactions that cannot be resolved in detail by any current experimental technology. We previously fit kinetic simulation parameters to bulk in vitro assembly data to yield a close match between simulated and real data, and then used the simulations to study features of assembly that cannot be monitored experimentally. The present work seeks to project how assembly in these simulations fit to in vitro data would be altered by computationally adding features of the cellular environment to the system, specifically the presence of nucleic acid about which many capsids assemble. The major challenge of such work is computational: simulating fine-scale assembly pathways on the scale and in the parameter domains of real viruses is far too computationally costly to allow for explicit models of nucleic acid interaction. We bypass that limitation by applying analytical models of nucleic acid effects to adjust kinetic rate parameters learned from in vitro data to see how these adjustments, singly or in combination, might affect fine-scale assembly progress. The resulting simulations exhibit surprising behavioral complexity, with distinct effects often acting synergistically to drive efficient assembly and alter pathways relative to the in vitro model. The work demonstrates how computer simulations can help us understand how assembly might differ between the in vitro and in vivo environments and what features of the cellular environment account for these differences.

Allosteric Control of Icosahedral Capsid Assembly

During the life cycle of a virus, viral proteins and other components self-assemble to form an ordered protein shell called a capsid. This assembly process is subject to multiple competing constraints, including the need to form a thermostable shell while avoiding kinetic traps. It has been proposed that viral assembly satisfies these constraints through allosteric regulation, including the interconversion of capsid proteins among conformations with different propensities for assembly. In this article, we use computational and theoretical modeling to explore how such allostery affects the assembly of icosahedral shells. We simulate assembly under a wide range of protein concentrations, protein binding affinities, and two different mechanisms of allosteric control. We find that above a threshold strength of allosteric control, assembly becomes robust over a broad range of subunit binding affinities and concentrations, allowing the formation of highly thermostable capsids. Our results suggest that allostery can significantly shift the range of protein binding affinities that lead to successful assembly and thus should be taken into account in models that are used to estimate interaction parameters from experimental data.

Effects of long-range interactions on curvature energies of viral shells

We formulate a theory of the effects of long-range interactions on the surface tension and spontaneous curvature of proteinaceous shells based on the general Deryaguin-Landau-Verwey-Overbeek mesoscale approach to colloid stability. We derive the full renormalization formulas for the elastic properties of the shell and consider in detail the renormalization of the spontaneous curvature as a function of the corresponding Hamaker coefficient, inner and outer capsid charges, and bathing solution properties. The renormalized spontaneous curvature is found to be a nonmonotonic function of several parameters describing the system.