Mechanical models for living cells--a review

As physical entities, living cells possess structural and physical properties that enable them to withstand the physiological environment as well as mechanical stimuli occurring within and outside the body. Any deviation from these properties will not only undermine the physical integrity of the cells, but also their biological functions. As such, a quantitative study in single cell mechanics needs to be conducted. In this review, we will examine some mechanical models that have been developed to characterize mechanical responses of living cells when subjected to both transient and dynamic loads. The mechanical models include the cortical shell-liquid core (or liquid drop) models which are widely applied to suspended cells; the solid model which is generally used for adherent cells; the power-law structural damping model which is more suited for studying the dynamic behavior of adherent cells; and finally, the biphasic model which has been widely used to study musculoskeletal cell mechanics. Based upon these models, future attempts can be made to develop even more detailed and accurate mechanical models of living cells once these three factors are adequately addressed: structural heterogeneity, appropriate constitutive relations for each of the distinct subcellular regions and components, and active forces acting within the cell. More realistic mechanical models of living cells can further contribute towards the study of mechanotransduction in cells.     

Power Dissipation in the Cochlea Can Enhance Frequency Selectivity

The cochlear cavity is filled with viscous fluids, and it is partitioned by a viscoelastic structure called the organ of Corti complex. Acoustic energy propagates toward the apex of the cochlea through vibrations of the organ of Corti complex. The dimensions of the vibrating structures range from a few hundred (e.g., the basilar membrane) to a few micrometers (e.g., the stereocilia bundle). Vibrations of microstructures in viscous fluid are subjected to energy dissipation. Because the viscous dissipation is considered to be detrimental to the function of hearing—sound amplification and frequency tuning—the cochlea uses cellular actuators to overcome the dissipation. Compared to extensive investigations on the cellular actuators, the dissipating mechanisms have not been given appropriate attention, and there is little consensus on damping models. For example, many theoretical studies use an inviscid fluid approximation and lump the viscous effect to viscous damping components. Others neglect viscous dissipation in the organ of Corti but consider fluid viscosity. We have developed a computational model of the cochlea that incorporates viscous fluid dynamics, organ of Corti microstructural mechanics, and electrophysiology of the outer hair cells. The model is validated by comparing with existing measurements, such as the viscoelastic response of the tectorial membrane, and the cochlear input impedance. Using the model, we investigated how dissipation components in the cochlea affect its function. We found that the majority of acoustic energy dissipation of the cochlea occurs within the organ of Corti complex, not in the scalar fluids. Our model suggests that an appropriate dissipation can enhance the tuning quality by reducing the spread of energy provided by the outer hair cells’ somatic motility.     

Computational insights into the O<Subscript>2</Subscript>-evolving complex of photosystem II

Mechanistic investigations of the water-splitting reaction of the oxygen-evolving complex (OEC) of photosystem II (PSII) are fundamentally informed by structural studies. Many physical techniques have provided important insights into the OEC structure and function, including X-ray diffraction (XRD) and extended X-ray absorption fine structure (EXAFS) spectroscopy as well as mass spectrometry (MS), electron paramagnetic resonance (EPR) spectroscopy, and Fourier transform infrared spectroscopy applied in conjunction with mutagenesis studies. However, experimental studies have yet to yield consensus as to the exact configuration of the catalytic metal cluster and its ligation scheme. Computational modeling studies, including density functional (DFT) theory combined with quantum mechanics/molecular mechanics (QM/MM) hybrid methods for explicitly including the influence of the surrounding protein, have proposed chemically satisfactory models of the fully ligated OEC within PSII that are maximally consistent with experimental results. The inorganic core of these models is similar to the crystallographic model upon which they were based, but comprises important modifications due to structural refinement, hydration, and proteinaceous ligation which improve agreement with a wide range of experimental data. The computational models are useful for rationalizing spectroscopic and crystallographic results and for building a complete structure-based mechanism of water-splitting in PSII as described by the intermediate oxidation states of the OEC. This review summarizes these recent advances in QM/MM modeling of PSII within the context of recent experimental studies.     

Modelling cartilage mechanobiology

The growth, maintenance and ossification of cartilage are fundamental to skeletal development and are regulated throughout life by the mechanical cues that are imposed by physical activities. Finite element computer analyses have been used to study the role of local tissue mechanics on endochondral ossification patterns, skeletal morphology and articular cartilage thickness distributions. Using single-phase continuum material representations of cartilage, the results have indicated that local intermittent hydrostatic pressure promotes cartilage maintenance. Cyclic tensile strains (or shear), however, promote cartilage growth and ossification. Because single-phase material models cannot capture fluid exudation in articular cartilage, poroelastic (or biphasic) solid/fluid models are often implemented to study joint mechanics. In the middle and deep layers of articular cartilage where poroelastic analyses predict little fluid exudation, the cartilage phenotype is maintained by cyclic fluid pressure (consistent with the single-phase theory). In superficial articular layers the chondrocytes are exposed to tangential tensile strain in addition to the high fluid pressure. Furthermore, there is fluid exudation and matrix consolidation, leading to cell 'flattening'. As a result, the superficial layer assumes an altered, more fibrous phenotype. These computer model predictions of cartilage mechanobiology are consistent with results of in vitro cell and tissue and molecular biology experiments.     

Effect of fluid boundary conditions on joint contact mechanics and applications to the modeling of osteoarthritic joints

The long-term goal of our research is to understand the mechanism of osteoarthritis (OA) initiation and progress through experimental and theoretical approaches. In previous theoretical models, joint contact mechanics was implemented without consideration of the fluid boundary conditions and with constant permeability. The primary purpose of this study was to investigate the effect of fluid boundary conditions at the articular surfaces on the contact mechanics, in terms of load sharing and fluid flow properties using variable permeability. The tested conditions included totally sealed surfaces, open surfaces, and open surfaces with variable permeability. While the sealed surface model failed to predict relaxation times and load sharing properly, the class of open surface models (open surfaces with constant permeability, and surfaces with variable permeability) gave good agreement with experiments, in terms of relaxation time and load sharing between the solid and the fluid phase. In particular, the variable permeability model was judged to be the most realistic of the three models, from a biological and physical point of view. This model was then used to simulate joint contact in the early and late stages of OA. In the early stages of OA, the model predicted a decrease in peak contact pressure and an increase in contact area, while in the late stages of OA, peak pressures were increased and contact areas were decreased compared to normal. These findings agree well with experimental observations.     

Model sensitivity and use of the comparative finite element method in mammalian jaw mechanics: mandible performance in the gray wolf

Finite Element Analysis (FEA) is a powerful tool gaining use in studies of biological form and function. This method is particularly conducive to studies of extinct and fossilized organisms, as models can be assigned properties that approximate living tissues. In disciplines where model validation is difficult or impossible, the choice of model parameters and their effects on the results become increasingly important, especially in comparing outputs to infer function. To evaluate the extent to which performance measures are affected by initial model input, we tested the sensitivity of bite force, strain energy, and stress to changes in seven parameters that are required in testing craniodental function with FEA. Simulations were performed on FE models of a Gray Wolf (Canis lupus) mandible. Results showed that unilateral bite force outputs are least affected by the relative ratios of the balancing and working muscles, but only ratios above 0.5 provided balancing-working side joint reaction force relationships that are consistent with experimental data. The constraints modeled at the bite point had the greatest effect on bite force output, but the most appropriate constraint may depend on the study question. Strain energy is least affected by variation in bite point constraint, but larger variations in strain energy values are observed in models with different number of tetrahedral elements, masticatory muscle ratios and muscle subgroups present, and number of material properties. These findings indicate that performance measures are differentially affected by variation in initial model parameters. In the absence of validated input values, FE models can nevertheless provide robust comparisons if these parameters are standardized within a given study to minimize variation that arise during the model-building process. Sensitivity tests incorporated into the study design not only aid in the interpretation of simulation results, but can also provide additional insights on form and function.     

That hemodynamics and not material mismatch is of primary concern in bypass graft failure: an experimental argument

The long term patency of end-to-side peripheral artery bypasses are low due to failure of the graft generally at the distal end of the bypass. Both material mismatch between the graft and the host artery and junction hemodynamics are cited as being major factors in disease formation at the junction. This study uses experimental methods to investigate the major differences in fluid dynamics and wall mechanics at the proximal and distal ends for rigid and compliant bypass grafts. Injection moulding was used to produce idealized transparent and compliant models of the graft/ artery junction configuration. An ePTFE graft was then used to stiffen one of the models. These models were then investigated using two-dimensional video extensometry and one-dimensional laser Doppler anemometry to determine the junction deformations and fluid velocity profiles for the rigid and complaint graft anastomotic junctions. Junction strains were evaluated and generally found to be under 5% with a peak stain measured in the stiff graft model junction of 8.3% at 100 mmHg applied pressure. Hemodynamic results were found to yield up to 40% difference in fluid velocities for the stiff/compliant comparison but up to 80% for the proximal/distal end comparisons. Similar strain conditions were assumed for the proximal and distal models while significant differences were noted in their associated hemodynamic changes. In contrasting the fluid dynamics and wall mechanics for the proximal and distal anastomoses, it is evident from the results of this study, that junction hemodynamics are the more variable factor.     

An elongation model of left ventricle deformation in diastole

A numerical method of the left ventricle (LV) deformation, an elongation model, was put forth for the study of LV fluid mechanics in diastole. The LV elongated only along the apical axis, and the motion was controlled by the intraventricular flow rate. Two other LV models, a fixed control volume model and a dilation model, were also used for model comparison and the study of LV fluid mechanics. For clinical sphere indices (SIs, between 1.0 and 2.0), the three models showed little difference in pressure and velocity distributions along the apical axis at E-peak. The energy dissipation was lower at a larger SI in that the jet and vortex development was less limited by the LV cavity in the apical direction. LV deformation of apical elongation may represent the primary feature of LV deformation in comparison with the secondary radial expansion. The elongation model of the LV deformation with an appropriate SI is a reasonable, simple method to study LV fluid mechanics in diastole.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Variable ranges of interactions in polypeptide conformations with a method to complement molecular modeling.

Formulations of conformational weights for helix-coil transitions can be extended to substantially more complex situations than are usually pursued. General rules for matrix multiplication that depend parametrically on the interaction ranges and numbers of rotamers of residues are presented. The orders of the matrices of statistical weights can be increased with chain length, so that an individual matrix element can represent any specified single conformation, as needed. By the appropriate choice of interaction ranges and numbers of available conformers, approximations can be introduced in which: (1) an average of the conformations of any chain segment is obtained, (2) specific residue-residue interactions are excluded, or (3) the conformation of a part of the chain is restricted or fixed. The method is appropriate for treating specific interactions in peptides and could be used together with available experimental information to develop models of conformational transitions. As such, the methods represent a class of calculations aimed at more rigorous calculations built around known features of a molecule. The aim is to facilitate calculations that bridge the gap between nonquantitative molecular model building and more rigorous but less directed molecular mechanics calculations. The method can directly include any desired longer range of interactions, if the interaction range is not too long to make impossible the manipulation of the requisite matrices. An outline is presented of an application to treat salt bridges in the C peptide of ribonuclease A.     

Computational study of the drag and oscillatory movement of biofilm streamers in fast flows

Hydrodynamic conditions have a significant impact on the biofilm lifecycle. Not well understood is the fact that biofilms, in return, also affect the flow pattern. A decade ago, it was already shown experimentally that under fast flows, biofilm streamers form and oscillate with large amplitudes. This work is a first attempt to answer the questions on the mechanisms behind the oscillatory movement of the streamers, and whether this movement together with the special streamlined form of the streamers, have both a physical and biological benefit for biofilms. In this study, a state of the art two‐dimensional fluid–structure interaction model of biofilm streamers is developed, which implements a transient coupling between the fluid and biofilm mechanics. Hereby, it is clearly shown that formation of a Kármán vortex street behind the streamer body is the main source of the periodic oscillation of the streamers. Additionally it is shown that the formation of streamers reduces the fluid forces which biofilm surface experiences. Biotechnol. Bioeng. 2010; 105: 600–610. © 2009 Wiley Periodicals, Inc.     

Non-linear arrhenius plots and the analysis of reaction and motional rates in biological membranes

We have systematically derived the rate-temperature relationships for a variety of models of membrane rate processes (particularly enzymic reactions) in order to predict the Arrhenius plot shape(s) appropriate to each model. We have explicitly considered the fact that most thermotropic changes in biological systems extend over finite and sometimes very broad temperature ranges. The rate-temperature relationships for most of the models considered can be expressed in a common, rather simple mathematical form suitable for application in computer data analysis. Only a few models predict Arrhenius plots with the “biphasic linear” form commonly reported in studies of membrane enzymes. However, many of the models yield plots which can be fitted to two intersecting lines within a quite modest experimental error, especially if the change in the slope of the plot around its “break” corresponds to a change in activation enthalpy of less than 15–20 kcal mol^?1. In general, Arrhenius-type plots of motional and reaction rates in membranes are found to be capable of indicating the midpoint but not the endpoints or the overall width of thermotropic transitions in the state of membrane components. Our findings clearly indicate a need for a more rigorous analysis of Arrhenius plot data in terms of graph shapes other than sets of intersecting lines and for more cautious interpretation of Arrhenius plot “breaks” with regard to their physical basis.     

Accurate prediction of wall shear stress in a stented artery: newtonian versus non-newtonian models

A significant amount of evidence linking wall shear stress to neointimal hyperplasia has been reported in the literature. As a result, numerical and experimental models have been created to study the influence of stent design on wall shear stress. Traditionally, blood has been assumed to behave as a Newtonian fluid, but recently that assumption has been challenged. The use of a linear model; however, can reduce computational cost, and allow the use of Newtonian fluids (e.g., glycerine and water) instead of a blood analog fluid in an experimental setup. Therefore, it is of interest whether a linear model can be used to accurately predict the wall shear stress caused by a non-Newtonian fluid such as blood within a stented arterial segment. The present work compares the resulting wall shear stress obtained using two linear and one nonlinear model under the same flow waveform. All numerical models are fully three-dimensional, transient, and incorporate a realistic stent geometry. It is shown that traditional linear models (based on blood's lowest viscosity limit, 3.5 Pa s) underestimate the wall shear stress within a stented arterial segment, which can lead to an overestimation of the risk of restenosis. The second linear model, which uses a characteristic viscosity (based on an average strain rate, 4.7 Pa s), results in higher wall shear stress levels, but which are still substantially below those of the nonlinear model. It is therefore shown that nonlinear models result in more accurate predictions of wall shear stress within a stented arterial segment.     

The potential for intercellular mechanical interaction: simulations of single chondrocyte versus anatomically based distribution

Computational studies of chondrocyte mechanics, and cell mechanics in general, have typically been performed using single cell models embedded in an extracellular matrix construct. The assumption of a single cell microstructural model may not capture intercellular interactions or accurately reflect the macroscale mechanics of cartilage when higher cell concentrations are considered, as may be the case in many instances. Hence, the goal of this study was to compare cell-level response of single and eleven cell biphasic finite element models, where the latter provided an anatomically based cellular distribution representative of the actual number of cells for a commonly used \(100 \, \upmu \hbox {m}\) edge cubic representative volume in the middle zone of cartilage. Single cell representations incorporated a centered single cell model and eleven location-corrected single cell models, the latter to delineate the role of cell placement in the representative volume element. A stress relaxation test at 10% compressive strain was adopted for all simulations. During transient response, volume- averaged chondrocyte mechanics demonstrated marked differences (up to 60% and typically greater than 10%) for the centered single versus the eleven cell models, yet steady-state loading was similar. Cell location played a marked role, due to inhomogeneity of the displacement and fluid pressure fields at the macroscopic scale. When the single cell representation was corrected for cell location, the transient response was consistent, while steady-state differences on the order of 1–4% were realized, which may be attributed to intercellular mechanical interactions. Anatomical representations of the superficial and deep zones, where cells reside in close proximity, may exhibit greater intercellular interactions, but these have yet to be explored.     

Thematic review series: patient-oriented research. Design and analysis of lipoprotein tracer kinetics studies in humans

Lipoprotein tracer kinetics studies have for many years provided new and important knowledge of the metabolism of lipoproteins. Our understanding of kinetics defects in lipoprotein metabolism has resulted from the use of tracer kinetics studies and mathematical modeling. This review discusses all aspects of the performance of kinetics studies, including the development of hypotheses, experimental design, statistical considerations, tracer administration and sampling schedule, and the development of compartmental models for the interpretation of tracer data. In addition to providing insight into new metabolic pathways, such models provide quantitative information on the effect of interventions on lipoprotein metabolism. Compartment models are useful tools to describe experimental data but can also be used to aid in experimental design and hypothesis generation. The SAAM II program provides an easy-to-use interface with which to develop and test compartmental models against experimental models. The development of a model requires that certain checks be performed to ensure that the model describes the experimental data and that the model parameters can be estimated with precision. In addition to methodologic aspects, several compartment models of apoprotein and lipid metabolism are reviewed.     

Study of the motion of magnetotactic bacteria

Motion of flagellate bacteria is considered from the point of view of rigid body mechanics. As a general case we consider a flagellate coccus magnetotactic bacterium swimming in a fluid in the presence of an external magnetic field. The proposed model generalizes previous approaches to the problem and allows one to access parameters of the motion that can be measured experimentally. The results suggest that the strong helical pattern observed in typical trajectories of magnetotactic bacteria can be a biological advantage complementary to magnetic orientation. In the particular case of zero magnetic interaction the model describes the motion of a non-magnetotactic coccus bacterium swimming in a fluid. Theoretical calculations based on experimental results are compared with the experimental track obtained by dark field optical microscopy. Correspondence to: H. G. P. Lins de Barros     

A multiscale model for red blood cell mechanics

The objective of this article is the derivation of a continuum model for mechanics of red blood cells via multiscale analysis. On the microscopic level, we consider realistic discrete models in terms of energy functionals defined on networks/lattices. Using concepts of Γ-convergence, convergence results as well as explicit homogenisation formulae are derived. Based on a characterisation via energy functionals, appropriate macroscopic stress–strain relationships (constitutive equations) can be determined. Further, mechanical moduli of the derived macroscopic continuum model are directly related to microscopic moduli. As a test case we consider optical tweezers experiments, one of the most common experiments to study mechanical properties of cells. Our simulations of the derived continuum model are based on finite element methods and account explicitly for membrane mechanics and its coupling with bulk mechanics. Since the discretisation of the continuum model can be chosen freely, rather than it is given by the topology of the microscopic cytoskeletal network, the approach allows a significant reduction of computational efforts. Our approach is highly flexible and can be generalised to many other cell models, also including biochemical control.     

A thick walled viscoelastic model for the mechanics of arteries

A knowledge of the mechanics of arteries is of importance in the determination of vessel rheological properties and in the studies of blood flow and certain arterial diseases. Most existing arterial models treat only wave motions; however, other types of motion, in particular those associated with flow development and other end effects, occur in the vascular system. Thus, a model is needed which can be applied to a variety of possible types of motion.

An arterial model is described which includes the effects of thick walls, linear viscoelasticity, and wall tethering. The forms of the displacements and stresses are found independently of the exact form of the applied fluid stresses; thus, the results are applicable to a range of possible dynamical conditions. Displacements and stress states can then be found from experimental or theoretical knowledge of the blood pressure and flow. The results are applied to flow development and wave propagation regions in the arteries.