Deformation analyses in cell and developmental biology. Part I--Formal methodology

This study presents a computational approach for the deformation analyses of problems in cell and developmental biology. Cells and embryos are viewed mechanically as axisymmetric shell-like bodies containing a body of incompressible material. The analysis approach is based on the finite element method. It is comprised of three finite element ingredients: an axisymmetric shell/membrane element valid for modeling finite bending, shearing and stretching; a volume constraint algorithm for modeling the membrane-bound incompressible material; and a contact algorithm for modeling the mechanical interactions between deformable bodies. Part II of this study will demonstrate how these three ingredients can be applied to analyze mechanical experiments on cells. This same method is also useful for simulating embryonic shape changes during development.     

Detection of Interactive Effects in Carcinogenesis

The multistage theory for cancer is used to derive models for independent effects and for interactive effects of two carcinogenic agents. The derivation assumes that a tumor is of single cell origin and that a normal cell is transformed to a cancer cell by undergoing a finite number of irreversible, heritable, mutation-like transitions. The interactions enter the proposed models at the level of single cell transition rates rather than, as is conventional, at the level of animal response proportions. Suggestions are given for the design and analysis of fixed time, binary response, two factor, cancer bioassay experiments.     

Finite element modelling approaches for well-ordered porous metallic materials for orthopaedic applications: cost effectiveness and geometrical considerations

The mechanical properties of well-ordered porous materials are related to their geometrical parameters at the mesoscale. Finite element (FE) analysis is a powerful tool to design well-ordered porous materials by analysing the mechanical behaviour. However, FE models are often computationally expensive. This article aims to develop a cost-effective FE model to simulate well-ordered porous metallic materials for orthopaedic applications. Solid and beam FE modelling approaches are compared, using finite size and infinite media models considering cubic unit cell geometry. The model is then applied to compare two unit cell geometries: cubic and diamond. Models having finite size provide similar results than the infinite media model approach for large sample sizes. In addition, these finite size models also capture the influence of the boundary conditions on the mechanical response for small sample sizes. The beam FE modelling approach showed little computational cost and similar results to the solid FE modelling approach. Diamond unit cell geometry appeared to be more suitable for orthopaedic applications than the cubic unit cell geometry.     

A finite representation model for an asynchronous culture of E. coli

A computer model is described which models an asynchronous population of E. coli by using a large, but finite number of representative single cells. Asynchrony generation and maintenance occurs at the single cell level by modulating the activity of an enzyme responsible for septum formation. Such modulation introduces cycle time imprecision and does not require the introduction of any new parameters into the single-cell model. Based on comparisons to experiment, reasonable predictions are possible for changes of cellular dry weight during exponential growth and turbidostat washout, and overall chemostat cell yields and changes in cell number, glucose concentration, and cell size distribution for a chemostat subject to a step change in dilution rate. Additionally, a correlation between cell RNA content and size is predicted as is an inertial effect when chemostat residence time is decreased under conditions of initially high glucose concentrations. Limitations imposed by the model's finite nature and their solutions are discussed.     

The osmotic migration of cells in a solute gradient.

The effect of a nonuniform solute concentration on the osmotic transport of water through the boundaries of a simple model cell is investigated. A system of two ordinary differential equations is derived for the motion of a single cell in the limit of a fast solute diffusion, and an analytic solution is obtained for one special case. A two-dimensional finite element model has been developed to simulate the more general case (finite diffusion rates, solute gradient induced by a solidification front). It is shown that the cell moves to regions of lower solute concentration due to the uneven flux of water through the cell boundaries. This mechanism has apparently not been discussed previously. The magnitude of this effect is small for red blood cells, the case in which all of the relevant parameters are known. We show, however, that it increases with cell size and membrane permeability, so this effect could be important for larger cells. The finite element model presented should also have other applications in the study of the response of cells to an osmotic stress and for the interaction of cells and solidification fronts. Such investigations are of major relevance for the optimization of cryopreservation processes.     

Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method

In this paper, we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises system of three coupled reaction-diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three-dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements affected by singularities of polar and azimuthal angles, as well as the origin.     

Cell-level finite element studies of viscous cells in planar aggregates

A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available.     

Wall shear stress in backward-facing step flow of a red blood cell suspension.

An experimental investigation of the wall shear stress distribution downstream of a backward-facing step is carried out. The wall shear stress distribution was determined by measuring the deformation of a gel layer, attached to the wall downstream of the step. Speckle pattern interferometry was applied to measure the deformation of the gel layer. The measured deformation, combined with the properties of the gel layer, served as an input for a finite element solid mechanics computation to determine the stress distribution in the gel layer. The wall shear stress, required to generate the measured deformation of the gel layer, was determined from these computations. A Newtonian buffer solution and a non-Newtonian red blood cell suspension were used as measuring fluids. The deformation of the gel layer was determined for a Newtonian buffer solution to evaluate the method and to obtain the properties of the gel layer. Subsequently, the wall shear stress distribution for the non-Newtonian red blood cell suspension was determined for three different flow rates. The inelastic non-Newtonian Carreau-Yasuda model served as constitutive model for the red blood cell suspension. Using this model, the velocity and wall shear stress distribution were computed by means of a finite element fluid mechanics computation. From the comparison between the numerical and the experimental results, it can be concluded that wall shear stresses, induced by the red blood cell suspension, can be modeled accurately by employing a Carreau-Yasuda model.     

Continuum model of cell adhesion and migration

The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.     

Comparison of analytical and inverse finite element approaches to estimate cell viscoelastic properties by micropipette aspiration

The viscoelastic properties of cells are important in predicting cell deformation under mechanical loading and may reflect cell phenotype or pathological transition. Previous studies have demonstrated that viscoelastic parameters estimated by finite element (FE) analyses of micropipette aspiration (MA) data differ from those estimated by the analytical half-space model. However, it is unclear whether these differences are statistically significant, as previous studies have been based on average cell properties or parametric analyses that do not reflect the inherent experimental and biological variability of real experimental data. To determine whether cell material parameters estimated by the half-space model are significantly different from those predicted by the FE method, we implemented an inverse FE method to estimate the viscoelastic parameters of a population of primary porcine aortic valve interstitial cells tested by MA. We found that inherent differences between the analytical and inverse FE estimation methods resulted in statistically significant differences in individual cell properties. However, in cases with small pipette to cell radius ratios and short loading periods, model-dependent differences were masked by experimental and cell-to-cell variability. Analytical models that account for finite cell-size and loading rate further relaxed the experimental conditions for which accurate cell material parameter estimates could be obtained. These data provide practical guidelines for analysis of MA data that account for the wide range of conditions encountered in typical experiments.     

Temperature dependence of the viscoelastic recovery of red cell membrane.

The time-dependent recovery of an elongated red cell is studied as a function of temperature. Before release, the elongated cell is in static equilibrium where external forces are balanced by surface elastic force resultants. Upon release, the cell recovers its initial shape with a time-dependent exponential behavior characteristic of a viscoelastic solid material undergoing large ("finite") deformation. The recovery process is characterized by a time constant, tc, that decreases from approximately 0.27 s at 6 degrees C to 0.06 s at 37 degrees C. From this measurement of the time constant and an independent measurement of the shear modulus of surface elasticity for red cell membrane, the value for the membrane surface viscosity as a function of temperature can be calculated.     

Stem cells and the rate of living

The "rate-of-living theory" is an ancient explanation of longevity which holds that aging occurs due to the exhaustion of some finite substance-breaths, heartbeats, etc. While this theory as originally conceived has been debunked, new work (Ruzankina et al. [2007], in this issue of Cell Stem Cell) suggests that mammals in fact do have a finite number of stem cell replications per life.     

Modeling and simulation of chemomechanics at the cell-matrix interface

Chemomechanical characteristics of the extracellular materials with which cells interact can have a profound impact on cell adhesion and migration. To understand and modulate such complex multiscale processes, a detailed understanding of the feedback between a cell and the adjacent microenvironment is crucial. Here, we use computational modeling and simulation to examine the cell-matrix interaction at both the molecular and continuum lengthscales. Using steered molecular dynamics, we consider how extracellular matrix (ECM) stiffness and extracellular pH influence the interaction between cell surface adhesion receptors and extracellular matrix ligands, and we predict potential consequences for focal adhesion formation and dissolution. Using continuum level finite element simulations and analytical methods to model cell-induced ECM deformation as a function of ECM stiffness and thickness, we consider the implications toward design of synthetic substrata for cell biology experiments that intend to decouple chemical and mechanical cues.Key words: cell adhesion, focal adhesion, steered molecular dynamics, finite element, chemomechanics, multiscale modeling, elasticity theory     

A note on the dispersionless growth law for single cells

A population of initially synchronized cells is considered wherein each cell grows according to a dispersionless growth law and the probability of cell division is determined by cell age. The first and second moments of the distribution of birth volumes are considered as functions of time and it is shown that it is impossible for both moments to approach finite, nonzero limits ast→∞. This implies that the volume distribution of the population will not approach a limiting distribution on any finite, nonzero volume interval and that the population will not attain balanced exponential growth. An illustrative example is worked out in detail. The distribution of birth volumes is also analyzed as a function of generation number and it is found that the logarithm of the birth volume in thejth generation is normally distributed asj→∞, with an unbounded variance. Generalizations and implications of these results are briefly discussed. Work supported by the U.S. Atomic Energy Commission.     

3D finite element analysis of nutrient distributions and cell viability in the intervertebral disc: effects of deformation and degeneration

The intervertebral disc (IVD) receives important nutrients, such as glucose, from surrounding blood vessels. Poor nutritional supply is believed to play a key role in disc degeneration. Several investigators have presented finite element models of the IVD to investigate disc nutrition; however, none has predicted nutrient levels and cell viability in the disc with a realistic 3D geometry and tissue properties coupled to mechanical deformation. Understanding how degeneration and loading affect nutrition and cell viability is necessary for elucidating the mechanisms of disc degeneration and low back pain. The objective of this study was to analyze the effects of disc degeneration and static deformation on glucose distributions and cell viability in the IVD using finite element analysis. A realistic 3D finite element model of the IVD was developed based on mechano-electrochemical mixture theory. In the model, the cellular metabolic activities and viability were related to nutrient concentrations, and transport properties of nutrients were dependent on tissue deformation. The effects of disc degeneration and mechanical compression on glucose concentrations and cell density distributions in the IVD were investigated. To examine effects of disc degeneration, tissue properties were altered to reflect those of degenerated tissue, including reduced water content, fixed charge density, height, and endplate permeability. Two mechanical loading conditions were also investigated: a reference (undeformed) case and a 10% static deformation case. In general, nutrient levels decreased moving away from the nutritional supply at the disc periphery. Minimum glucose levels were at the interface between the nucleus and annulus regions of the disc. Deformation caused a 6.2% decrease in the minimum glucose concentration in the normal IVD, while degeneration resulted in an 80% decrease. Although cell density was not affected in the undeformed normal disc, there was a decrease in cell viability in the degenerated case, in which averaged cell density fell 11% compared with the normal case. This effect was further exacerbated by deformation of the degenerated IVD. Both deformation and disc degeneration altered the glucose distribution in the IVD. For the degenerated case, glucose levels fell below levels necessary for maintaining cell viability, and cell density decreased. This study provides important insight into nutrition-related mechanisms of disc degeneration. Moreover, our model may serve as a powerful tool in the development of new treatments for low back pain.     

Realistic model of a fixed-charge membrane according to the theory of Teorell,Meyer, and Sievers

A realistic model of a fixed-charge membrane was constructed to elucidate the principles involved in the Teorell-Meyer-Sievers theory. Polyelectrolyte solutions (polystyrene sulfonic acid or gelatin) were used in a multicompartment cell to serve as the "membrane." The ionic concentration and potential differences between the compartments served as a finite increment approximation to a continuous membrane phase. The profiles obtained for a positively and negatively charged membrane gave striking support for the theory.     

A Survey of Databases for Analysis of Plant Cell Wall-Related Enzymes

Biofuels derived from plant cell wall lignocellulose have the potential to serve as an alternative source of energy, relieving dependence on finite petroleum reserves and reducing production of climate-changing greenhouse gases. To better elucidate cell wall structure, the plant research community has developed databases to host the accumulated information on plant cell wall-related enzymes. The goal of this review is to provide a comprehensive catalog of these databases, as well as to describe the data and tools that they make available. A Directory of Databases for Plant Cell Wall-Related Enzymes (plantcellwalls.ucdavis.edu) was also developed to provide links to all the databases reviewed in this paper and relevant publications.     

SV40-transformed human fibroblasts: evidence for cellular aging in pre-crisis cells

Pre-crisis SV40-transformed human diploid fibroblast (HDF) cultures have a finite proliferative lifespan, but they do not enter a viable senescent state at end of lifespan. Little is known about either the mechanism for this finite lifespan in SV40-transformed HDF or its relationship to finite lifespan in normal HDF. Recently we proposed that in normal HDF the phenomena of finite lifespan and arrest in a viable senescent state depend on two separate processes: 1) an age-related decrease in the ability of the cells to recognize or respond to serum and/or other mitogens such that the cells become functionally mitogen-deprived at the end of lifespan; and 2) the ability of the cells to enter a viable, G1-arrested state whenever they experience mitogen deprivation. In this paper, data are presented that suggest that pre-crisis SV40-transformed HDF retain the first process described above, but lack the second process. It is shown that SV40-transformed HDF have a progressively decreasing ability to respond to serum as they age, but they continue to traverse the cell cycle at the end of lifespan. Concomitantly, the rate of cell death increases steadily toward the end of lifespan, thereby causing the total population to cease growing and ultimately to decline. Previous studies have shown that when SV40-transformed HDF are environmentally serum deprived, they likewise exhibit continued cell cycle traverse coupled with increased cell death. Thus, these results support the hypothesis that pre-crisis SV40-transformed HDF still undergo the same aging process as do normal HDF, but they end their lifespan in crisis rather than in the normal G1-arrested senescent state because they have lost their ability to enter a viable, G1-arrested state in response to mitogen deprivation.     

A formalism for modelling traction forces and cell shape evolution during cell migration in various biomedical processes

The phenomenological model for cell shape deformation and cell migration Chen (BMM 17:1429–1450, 2018), Vermolen and Gefen (BMM 12:301–323, 2012), is extended with the incorporation of cell traction forces and the evolution of cell equilibrium shapes as a result of cell differentiation. Plastic deformations of the extracellular matrix are modelled using morphoelasticity theory. The resulting partial differential differential equations are solved by the use of the finite element method. The paper treats various biological scenarios that entail cell migration and cell shape evolution. The experimental observations in Mak et al. (LC 13:340–348, 2013), where transmigration of cancer cells through narrow apertures is studied, are reproduced using a Monte Carlo framework.