Mathematical model for force and energy of virion-cell interactions during full engulfment in HIV: Impact of virion maturation and host cell morphology.

Viral endocytosis involves elastic cell deformation, driven by chemical adhesion energy, and depends on physical interactions between the virion and cell membrane. These interactions are not easy to quantify experimentally. Hence, this study aimed to develop a mathematical model of the interactions of HIV particles with host cells and explore the effects of mechanical and morphological parameters during full virion engulfment. The invagination force and engulfment energy were described as viscoelastic and linear-elastic functions of radius and elastic modulus of virion and cell, ligand-receptor energy density and engulfment depth. The influence of changes in the virion-cell contact geometry representing different immune cells and ultrastructural membrane features and the decrease in virion radius and shedding of gp120 proteins during maturation on invagination force and engulfment energy was investigated. A low invagination force and high ligand-receptor energy are associated with high virion entry ability. The required invagination force was the same for immune cells of different sizes but lower for a local convex geometry of the cell membrane at the virion length scale. This suggests that localized membrane features of immune cells play a role in viral entry ability. The available engulfment energy decreased during virion maturation, indicating the involvement of additional biological or biochemical changes in viral entry. The developed mathematical model offers potential for the mechanobiological assessment of the invagination of enveloped viruses towards improving the prevention and treatment of viral infections.

Intracellular mechanics and TBX3 expression jointly dictate the spreading mode of melanoma cells in 3D environments.

Cell stiffness and T-box transcription factor 3 (TBX3) expression have been identified as biomarkers of melanoma metastasis in 2D environments. This study aimed to determine how mechanical and biochemical properties of melanoma cells change during cluster formation in 3D environments. Vertical growth phase (VGP) and metastatic (MET) melanoma cells were embedded in 3D collagen matrices of 2 and 4 mg/ml collagen concentrations, representing low and high matrix stiffness. Mitochondrial fluctuation, intracellular stiffness, and TBX3 expression were quantified before and during cluster formation. In isolated cells, mitochondrial fluctuation decreased and intracellular stiffness increased with increase in disease stage from VGP to MET and increased matrix stiffness. TBX3 was highly expressed in soft matrices but diminished in stiff matrices for VGP and MET cells. Cluster formation of VGP cells was excessive in soft matrices but limited in stiff matrices, whereas for MET cells it was limited in soft and stiff matrices. In soft matrices, VGP cells did not change the intracellular properties, whereas MET cells exhibited increased mitochondrial fluctuation and decreased TBX3 expression. In stiff matrices, mitochondrial fluctuation and TBX3 expression increased in VGP and MET, and intracellular stiffness increased in VGP but decreased in MET cells. The findings suggest that soft extracellular environments are more favourable for tumour growth, and high TBX3 levels mediate collective cell migration and tumour growth in the earlier VGP disease stage but play a lesser role in the later metastatic stage of melanoma.

An analytical model describing the mechanics of erythrocyte membrane wrapping during active invasion of a plasmodium falciparum merozoite.

The invasion of a merozoite into an erythrocyte by membrane wrapping is a hallmark of malaria pathogenesis. The invasion involves biomechanical interactions whereby the merozoite exerts actomyosin-based forces to push itself into and through the erythrocyte membrane while concurrently inducing biochemical damage to the erythrocyte membrane. Whereas the biochemical damage process has been investigated, the detailed mechanistic understanding of the invasion mechanics remains limited. Thus, the current study aimed to develop a mathematical model describing the mechanical factors involved in the merozoite invasion into an erythrocyte and explore the invasion mechanics. A shell theory model was developed comprising constitutive, equilibrium and governing equations of the deformable erythrocyte membrane to predict membrane mechanics during the wrapping of an entire non-deformable ellipsoidal merozoite. Predicted parameters include principal erythrocyte membrane deformations and stresses, wrapping and indentation forces, and indentation work. The numerical investigations considered two limits for the erythrocyte membrane deformation during wrapping (4% and 51% areal strain) and erythrocyte membrane phosphorylation (decrease of membrane elastic modulus from 1 to 0.5 kPa). For an intact erythrocyte, the maximum indentation force was 1 and 8.5 pN, and the indentation work was 1.92 × 10-18 and 1.40 × 10-17 J for 4% and 51% areal membrane strain. Phosphorylation damage in the erythrocyte membrane reduced the required indentation work by 50% to 0.97 × 10-18 and 0.70 × 10-17 J for 4% and 51% areal strain. The current study demonstrated the developed model's feasibility to provide new knowledge on the physical mechanisms of the merozoite invasion process that contribute to the invasion efficiency towards the discovery of new invasion-blocking anti-malaria drugs.

Mathematical model of mechano-sensing and mechanically induced collective motility of cells on planar elastic substrates.

Cells mechanically interact with their environment to sense, for example, topography, elasticity and mechanical cues from other cells. Mechano-sensing has profound effects on cellular behaviour, including motility. The current study aims to develop a mathematical model of cellular mechano-sensing on planar elastic substrates and demonstrate the model's predictive capabilities for the motility of individual cells in a colony. In the model, a cell is assumed to transmit an adhesion force, derived from a dynamic focal adhesion integrin density, that locally deforms a substrate, and to sense substrate deformation originating from neighbouring cells. The substrate deformation from multiple cells is expressed as total strain energy density with a spatially varying gradient. The magnitude and direction of the gradient at the cell location define the cell motion. Cell-substrate friction, partial motion randomness, and cell death and division are included. The substrate deformation by a single cell and the motility of two cells are presented for several substrate elasticities and thicknesses. The collective motility of 25 cells on a uniform substrate mimicking the closure of a circular wound of 200 µm is predicted for deterministic and random motion. Cell motility on substrates with varying elasticity and thickness is explored for four cells and 15 cells, the latter again mimicking wound closure. Wound closure by 45 cells is used to demonstrate the simulation of cell death and division during migration. The mathematical model can adequately simulate the mechanically induced collective cell motility on planar elastic substrates. The model is suitable for extension to other cell and substrates shapes and the inclusion of chemotactic cues, offering the potential to complement in vitro and in vivo studies.

On the role of mechanical signals on sprouting angiogenesis through computer modeling approaches.

Sprouting angiogenesis, the formation of new vessels from preexisting vasculature, is an essential process in the regeneration of new tissues as well as in the development of some diseases like cancer. Although early studies identified chemical signaling as the main driver of this process, many recent studies have shown a strong role of mechanical signals in the formation of new capillaries. Different types of mechanical signals (e.g., external forces, cell traction forces, and blood flow-induced shear forces) have been shown to play distinct roles in the process; however, their interplay remains still largely unknown. During the last decades, mathematical and computational modeling approaches have been developed to investigate and better understand the mechanisms behind mechanically driven angiogenesis. In this manuscript, we review computational models of angiogenesis with a focus on models investigating the role of mechanics on the process. Our aim is not to provide a detailed review on model methodology but to describe what we have learnt from these models. We classify models according to the mechanical signals being investigated and describe how models have looked into their role on the angiogenic process. We show that a better understanding of the mechanobiology of the angiogenic process will require the development of computer models that incorporate the interactions between the multiple mechanical signals and their effect on cellular responses, since they all seem to play a key in sprout patterning. In the end, we describe some of the remaining challenges of computational modeling of angiogenesis and discuss potential avenues for future research.

In silico stress fibre content affects peak strain in cytoplasm and nucleus but not in the membrane for uniaxial substrate stretch.

Existing in silico models for single cell mechanics feature limited representations of cytoskeletal structures that contribute substantially to the mechanics of a cell. We propose a micromechanical hierarchical approach to capture the mechanical contribution of actin stress fibres. For a cell-specific fibroblast geometry with membrane, cytoplasm and nucleus, the Mori-Tanaka homogenization method was employed to describe cytoplasmic inhomogeneities and constitutive contribution of actin stress fibres. The homogenization was implemented in a finite element model of the fibroblast attached to a substrate through focal adhesions. Strain in cell membrane, cytoplasm and nucleus due to uniaxial substrate stretch was assessed for different stress fibre volume fractions and different elastic modulus of the substrate. A considerable decrease of the peak strain with increasing stress fibre content was observed in cytoplasm and nucleus but not the membrane, whereas the peak strain in cytoplasm, nucleus and membrane increased for increasing elastic modulus of the substrate. Finite element mesh of reconstructed human fibroblast and intracellular strain distribution in cell subjected to substrate stretch.

Cytoskeletal tubulin competes with actin to increase deformability of metastatic melanoma cells.

The formation of membrane protrusions during migration is reliant upon the cells' cytoskeletal structure and stiffness. It has been reported that actin disruption blocks protrusion and decreases cell stiffness whereas microtubule disruption blocks protrusion but increases stiffness in several cell types. In melanoma, cell migration is of concern as this cancer spreads unusually rapidly during early tumour development. The aim of this study was to characterise motility, structural properties and stiffness of human melanoma cells at radial growth phase (RGP), vertical growth phase (VGP), and metastatic stage (MET) in two-dimensional in vitro environments. Wound assays, western blotting and mitochondrial particle tracking were used to assess cell migration, cytoskeletal content and intracellular fluidity. Our results indicate that cell motility increase with increasing disease stage. Despite their different motility, RGP and VGP cells exhibit similar fluidity, actin and tubulin levels. MET cells, however, display increased fluidity which was associated with increased actin and tubulin content. Our findings demonstrate an interplay between actin and microtubule activity and their role in increasing motility of cells while minimizing cell stiffness at advanced disease stage. In earlier disease stages, cell stiffness may however not serve as an indicator of migratory capabilities.

Cell focal adhesion clustering leads to decreased and homogenized basal strains.

The subendothelial matrix of the artery is a complex mechanical environment where endothelial cells respond to and affect changes upon the underlying substrate. Our recent work has demonstrated that endothelial cell strain heterogeneity increases on a more heterogeneous underlying subendothelial matrix, and these cells display increased focal adhesion presence on stiffer substrate areas. However, the impact of these grouped focal adhesions on endothelial cell strains has not been explored. Here, we use finite element modeling to investigate the effects of microscale stiffness heterogeneities and focal adhesion location and stiffness on endothelial cell strains. Shear stress applied to the apical cell layer demonstrated a minimal effect on cell strain values, while substrate stretch had a greater effect on cell strain in the cell-substrate model. The addition of focal adhesions into the computational model (cell-FA-substrate model) predicted a decrease and homogenization of the cell strains. For simulations including focal adhesions, stiffer and more distributed adhesions caused increased and more heterogeneous endothelial cell strains. Overall, our data indicate that cells may group focal adhesions to minimize and homogenize their basal strains.

Suitability of developed composite materials for meniscal replacement: Mechanical, friction and wear evaluation.

The meniscus is a complex and frequently damaged tissue which requires a substitute capable of reproducing similar biomechanical functions. This study aims to develop a synthetic meniscal substitute that can mimic the function of the native meniscus. Medical grade silicones reinforced with nylon were fabricated using compression moulding and evaluated for mechanical and tribological properties. The optimal properties were obtained with tensile modulus increased considerably from 10.7 ±â€¯2.9 MPa to 114.6 ±â€¯20.9 MPa while compressive modulus was found to reduce from 2.5 ±â€¯0.6 MPa to 0.7 ±â€¯0.3 MPa. Using a tribometer, the coefficient of friction of 0.08 ±â€¯0.02 was measured at the end of the 100,000 cycles. The developed composite could be an auspicious substitute for the native meniscus and the knowledge gained from this study is useful as it enhances the understanding of a potentially suitable material for meniscal implants.

Analytical modeling of the mechanics of early invasion of a merozoite into a human erythrocyte.

In this study, we used a continuum model based on contact mechanics to understand the mechanics of merozoite invasion into human erythrocytes. This model allows us to evaluate the indentation force and work as well as the contact pressure between the merozoite and erythrocyte for an early stage of invasion (γ = 10%). The model predicted an indentation force of 1.3e -11N and an indentation work of 1e -18J. The present analytical model can be considered as a useful tool not only for investigations in mechanobiology and biomechanics but also to explore novel therapeutic targets for malaria and other parasite infections.

Cellular mechanosensitivity to substrate stiffness decreases with increasing dissimilarity to cell stiffness.

Computational modelling has received increasing attention to investigate multi-scale coupled problems in micro-heterogeneous biological structures such as cells. In the current study, we investigated for a single cell the effects of (1) different cell-substrate attachment (2) and different substrate modulus [Formula: see text] on intracellular deformations. A fibroblast was geometrically reconstructed from confocal micrographs. Finite element models of the cell on a planar substrate were developed. Intracellular deformations due to substrate stretch of [Formula: see text], were assessed for: (1) cell-substrate attachment implemented as full basal contact (FC) and 124 focal adhesions (FA), respectively, and [Formula: see text]140 KPa and (2) [Formula: see text], 140, 1000, and 10,000 KPa, respectively, and FA attachment. The largest strains in cytosol, nucleus and cell membrane were higher for FC (1.35[Formula: see text], 0.235[Formula: see text] and 0.6[Formula: see text]) than for FA attachment (0.0952[Formula: see text], 0.0472[Formula: see text] and 0.05[Formula: see text]). For increasing [Formula: see text], the largest maximum principal strain was 4.4[Formula: see text], 5[Formula: see text], 5.3[Formula: see text] and 5.3[Formula: see text] in the membrane, 9.5[Formula: see text], 1.1[Formula: see text], 1.2[Formula: see text] and 1.2[Formula: see text] in the cytosol, and 4.5[Formula: see text], 5.3[Formula: see text], 5.7[Formula: see text] and 5.7[Formula: see text] in the nucleus. The results show (1) the importance of representing FA in cell models and (2) higher cellular mechanical sensitivity for substrate stiffness changes in the range of cell stiffness. The latter indicates that matching substrate stiffness to cell stiffness, and moderate variation of the former is very effective for controlled variation of cell deformation. The developed methodology is useful for parametric studies on cellular mechanics to obtain quantitative data of subcellular strains and stresses that cannot easily be measured experimentally.

Self-healing of hierarchical materials.

We present a theoretical and numerical analysis of the mechanical behavior of self-healing materials using an analytical model and numerical calculations both based on a Hierarchical Fiber Bundle Model, and applying them to graphene- or carbon-nanotube-based materials. The self-healing process can be described essentially through a single parameter, that is, the healing rate, but numerical simulations also highlight the influence of the location of the healing process on the overall strengthening and toughening of the material. The role of hierarchy is discussed, showing that full-scale hierarchical structures can in fact acquire more favorable properties than smaller, nonhierarchical ones through interaction with the self-healing process, thus inverting the common notion in fracture mechanics that specimen strength increases with decreasing size. Further, the study demonstrates that the developed analytical and numerical tools can be useful to develop strategies for the optimization of strength and toughness of synthetic bioinspired materials.

Hierarchical fiber bundle model to investigate the complex architectures of biological materials.

The mechanics of fiber bundles has been widely studied in the literature, and fiber bundle models in particular have provided a wealth of useful analytical and numerical results for modeling ordinary materials. These models, however, are inadequate to treat bioinspired nanostructured materials, where hierarchy, multiscale, and complex properties play a decisive role in determining the overall mechanical characteristics. Here, we develop an ad hoc hierarchical theory designed to tackle these complex architectures, thus allowing the determination of the strength of macroscopic hierarchical materials from the properties of their constituents at the nanoscale. The roles of finite size, twisting angle, and friction are also included. Size effects on the statistical distribution of fiber strengths naturally emerge without invoking best-fit or unknown parameters. A comparison between the developed theory and various experimental results on synthetic and natural materials yields considerable agreement.

Investigating the role of hierarchy on the strength of composite materials: evidence of a crucial synergy between hierarchy and material mixing.

Natural materials are often organized in complex hierarchical architectures to optimize mechanical properties. Artificial bio-inspired materials, however, have thus far failed to successfully mimic how these architectures improve material characteristics, for example strength. Here, a method is proposed for evaluating the role of hierarchy on structural strength. To do this, we consider different hierarchical architectures of fiber bundles through analytical multiscale calculations based on a fiber bundle model at each hierarchical level. In general, we find that an increase in the number of hierarchy levels leads to a decrease in the strength of material. However, when a composite bundle with two different types of fibers is considered, an improvement in the mean strength is obtained for some specific hierarchical architectures, indicating that both hierarchy and material "mixing" are necessary ingredients to obtain improved mechanical properties. Results are promising for the improvement and "tuning" of the strength of bio-inspired materials.