Stiffening response of a cellular tensegrity model

Living cells exhibit, as most biological tissues, a stiffening (strain-hardening) response which reflects the nonlinearity of the stress-strain relationship. Tensegrity structures have been proposed as a comprehensive model of such a cell's mechanical response. Based on a theoretical model of a 30-element tensegrity structure, we propose a quantitative analysis of its nonlinear mechanical behavior under static conditions and large deformations. This study provides theoretical foundation to the passage from large-scale tensegrity models to microscale living cells, as well as the comparison between results obtained in biological specimens of different sizes. We found two non-dimensional parameters (L*-normalized element length and T*-normalized elastic tension) which govern the mechanical response of the structure for three types of loading tested (extension, compression and shear). The linear strain-hardening is uniquely observed for extension but differed for the two other types of loading tested. The stiffening response of the theoretical model was compared and discussed with the living cells stiffening response observed by different methods (shear flow experiments, micromanipulation and magnetocytometry).     

A cellular model of lung elasticity

The mechanics of the lung parenchyma is studied using models comprised of line members interconnected to form 3-D cellular structures. The mechanical properties are represented as elastic constants of a continuum. These are determined by perturbing each individual cell from a reference state by an increment in stress which is superimposed upon the uniform stretching forces initially present in the members due to the transpulmonary pressure. A force balance on the distorted structure, together with a force-deformation law for the members, leads to a calculation of the strain increments of the members. Predictions based on the analysis of the 3-D isotropic dodecahedron are in good agreement with experimental values for the Young's, shear, and bulk moduli reported in the literature. The model provides an explanation for the dependence of the elastic moduli on transpulmonary pressure, the geometrical details of the structure, and the stress-strain law of the tissue.     

A cellular tensegrity model to analyse the structural viscoelasticity of the cytoskeleton

This study describes the viscoelastic properties of a refined cellular-tensegrity model composed of six rigid bars connected to a continuous network of 24 viscoelastic pre-stretched cables (Voigt bodies) in order to analyse the role of the cytoskeleton spatial rearrangement on the viscoelastic response of living adherent cells. This structural contribution was determined from the relationships between the global viscoelastic properties of the tensegrity model, i.e., normalized viscosity modulus (eta(*)), normalized elasticity modulus (E(*)), and the physical properties of the constitutive elements, i.e., their normalized length (L(*)) and normalized initial internal tension (T(*)). We used a numerical method to simulate the deformation of the structure in response to different types of loading, while varying by several orders of magnitude L(*) and T(*). The numerical results obtained reveal that eta(*) remains almost independent of changes in T(*) (eta(*) proportional, variant T(*+0.1)), whereas E(*) increases with approximately the square root of the internal tension T(*) (from E(*) proportional, variant T(*+0.3) to E(*) proportional, variant T(*+0.7)). Moreover, structural viscosity eta(*) and elasticity E(*) are both inversely proportional to the square of the size of the structure (eta(*) proportional, variant L(*-2) and E(*) proportional, variant L(*-2)). These structural properties appear consistent with cytoskeleton (CSK) mechanical properties measured experimentally by various methods which are specific to the CSK micromanipulation in living adherent cells. Present results suggest, for the first time, that the effect of structural rearrangement of CSK elements on global CSK behavior is characterized by a faster cellular mechanical response relatively to the CSK element response, which thus contributes to the solidification process observed in adherent cells. In extending to the viscoelastic properties the analysis of the mechanical response of the cellular 30-element tensegrity model, the present study contributes to the understanding of recent results on the cellular-dynamic response and allows to reunify the scattered data reported for the viscoelastic properties of living adherent cells.     

Experimental tests of the cellular tensegrity hypothesis

The tensegrity model depicts the cytoskeleton (CSK) as a prestressed network of interconnected filaments. The prestress is generated by the CSK contractile apparatus and is partly balanced by traction at the cell-substrate interface and partly by CSK internal compression elements such as microtubules (MTs). A key feature of tensegrity is that the shear modulus (G) must increase in proportion with the prestress. Here we have tested that prediction as well as the idea that compression of MTs balance a portion of the cell prestress. Airway smooth muscle cells were studied. Traction microscopy was used to calculate traction. Because traction must be balanced by the stress within the cell, the prestress could be computed. Cell G was measured by oscillatory magnetic cytometry. The prestress was modulated using graded concentrations of contracting (histamine) or relaxing (isoproterenol) agonists and by disrupting MTs by colchicine. It was found that G increased in proportion with the prestress and that compression of MTs balanced a significant, but a relatively small fraction of the prestress. Taken together, these results do not disprove other models of cell deformability, nor they prove tensegrity. However, they do support a priori predictions of tensegrity. As such, it may not be necessary to invoke more complex mechanisms to explain these central features of cell deformability.     

Computational model for the cell-mechanical response of the osteocyte cytoskeleton based on self-stabilizing tensegrity structures

The mechanism by which mechanical stimulation on osteocytes results in biochemical signals that initiate the remodeling process inside living bone tissue is largely unknown. Even the type of stimulation acting on these cells is not yet clearly identified. However, the cytoskeleton of osteocytes is suggested to play a major role in the mechanosensory process due to the direct connection to the nucleus. In this paper, a computational approach to model and simulate the cell structure of osteocytes based on self-stabilizing tensegrity structures is suggested. The computational model of the cell consists of the major components with respect to mechanical aspects: the integrins that connect the cell with the extracellular bone matrix, and different types of protein fibers (microtubules and intermediate filaments) that form the cytoskeleton, the membrane-cytoskeleton (microfilaments), the nucleus and the centrosome. The proposed geometrical cell models represent the cell in its physiological environment which is necessary in order to give a statement on the cell behavior in vivo. Studies on the mechanical response of osteocytes after physiological loading and in particular the mechanical response of the nucleus show that the load acting on the nucleus is rising with increasing deformation applied to the integrins.     

A new molecular model for collagen elasticity based on synchrotron X-ray scattering evidence.

Collagen is the most abundant structural protein in vertebrates. The specific shape of its stress-strain curve is crucial for the function of a number of organs. Although the macroscopic mechanical behavior of collagen is well known, there is still no explanation of the elastic process at the supramolecular level. We have performed in situ synchrotron x-ray scattering experiments, which show that the amount of lateral molecular order increases upon stretching of collagen fibers. In strain cycling experiments the relation between strain and diffuse equatorial scattering was found to be linear in the heel region of the stress-strain curve. A new molecular model for collagen elasticity is proposed, which, based on the existence of thermally activated molecular kinks, reproduces this linearity and gives a simple explanation for the form of the stress-strain curve of collagen.     

A quantitative study of chromosomal elasticity and its influence on chromosome movement

Summary Chromosome elasticity and movement have been studied in living cells in two distinct situations: early anaphase stretch due to opposed external forces, and drag stretch — an elongation due to frictional resistance or drag on a chromosome being pulled toward one pole. Drag stretch provides a simultaneous display of both friction and elasticity and shows that chromosomes in living cells are elastic up to approximately six-fold increases in length.Neither early anaphase stretch nor drag stretch produce detectable alterations in the velocity of chromosome movement. A simple mechanical model is described which permits interpretation of this result for drag stretch: no matter how extensive, drag stretch should produce no change in the force required to maintain a given velocity of movement and hence should not alter movement velocity. Early anaphase stretch is a very different proposition, and additional assumptions leading to a quantitative model are necessary for its interpretation. Nevertheless it is reasonably certain that the amount of stretch actually seen in these circumstances would influence chromosome movement if the applied force were not increased over that necessary in the absence of stretch. It is concluded that the mitotic forces are continually adjusted to produce a standard velocity of movement even when an unusual hindrance to movement exists. The implications of this are considered, particularly in regard to the stretching and rupture of dikinetochoric (dicentric) bridges in anaphase.The quantitative version of the mechanical model for elasticity and movement can be applied to the drag stretch data, and permits calculation of the ratio between frictional and elastic coefficients. The chief assumptions are that the elasticity is Hookian, and the frictional resistance Newtonian in character. The model has not been critically tested, but it is consonant with existing data.This investigation was supported in part by research grant number RG-8480 from the Division of General Medical Sciences, United States Public Health Service.     

A tensegrity model of the cytoskeleton in spread and round cells.

Measurements on adherent cells have shown that spreading affects their mechanics. Highly spread cells are stiffer than less spread cells. The stiffness increases approximately linearly with increasing applied stress and more so in highly spread cells than in less spread cells. In this study, a six-strut tensegrity model of the cytoskeleton is used to analyze the effect of spreading on cellular mechanics. Two configurations are considered: a round configuration where a spherically shaped model is anchored to a flat rigid surface at three joints, and a spread configuration, where three additional joints of the model are attached to the surface. In both configurations a pulling force is applied at a free joint, distal from the anchoring surface, and the corresponding deformation is determined from equations of equilibrium. The model stiffness is obtained as the ratio of applied force to deformation. It is found that the stiffness changes with spreading consistently with the observations in cells. These findings suggest the possibility that the spreading-induced changes of the mechanical properties of the cell are the result of the concomitant changes in force distribution and microstructural geometry of the cytoskeleton.     

A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth

We report numerical results from a 2D cellular automaton (CA) model describing the dynamics of the in vitro cultivated multicellular spheroid obtained from EMT6/Ro (mammary carcinoma) cell line. Significantly, the CA model relaxes the often assumed one-to-one correspondence between cells and CA sites so as to correctly model the peripheral mitotic boundary region, and to enable the study of necrosis in large avascular tumours. By full calibration and scaling to available experimental data, the model produces with good accuracy experimentally comparable data on a range of bulk tumour kinetics and necrosis measures. Our main finding is that the metabolic production of H⁺ ions is not sufficient to cause central necrosis prior to the sub-viable nutrient-deficient stage of tumour development being reached. Thus, the model suggests that an additional process is required to explain the experimentally observable onset of necrosis prior to the non-viable nutrient-deficient point being reached.     

A quantitative approach for polymerase chain reactions based on a hidden Markov model

Polymerase chain reaction (PCR) is a major DNA amplification technology from molecular biology. The quantitative analysis of PCR aims at determining the initial amount of the DNA molecules from the observation of typically several PCR amplifications curves. The mainstream observation scheme of the DNA amplification during PCR involves fluorescence intensity measurements. Under the classical assumption that the measured fluorescence intensity is proportional to the amount of present DNA molecules, and under the assumption that these measurements are corrupted by an additive Gaussian noise, we analyze a single amplification curve using a hidden Markov model(HMM). The unknown parameters of the HMM may be separated into two parts. On the one hand, the parameters from the amplification process are the initial number of the DNA molecules and the replication efficiency, which is the probability of one molecule to be duplicated. On the other hand, the parameters from the observational scheme are the scale parameter allowing to convert the fluorescence intensity into the number of DNA molecules and the mean and variance characterizing the Gaussian noise. We use the maximum likelihood estimation procedure to infer the unknown parameters of the model from the exponential phase of a single amplification curve, the main parameter of interest for quantitative PCR being the initial amount of the DNA molecules. An illustrative example is provided. This research was financed by the Swedish foundation for Strategic Research through the Gothenburg Mathematical Modelling Centre.     

A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system

A quantitative, physiology-based model of the ascending arousal system is developed, using continuum neuronal population modeling, which involves averaging properties such as firing rates across neurons in each population. The model includes the ventrolateral preoptic area (VLPO), where circadian and homeostatic drives enter the system, the monoaminergic and cholinergic nuclei of the ascending arousal system, and their interconnections. The human sleep-wake cycle is governed by the activities of these nuclei, which modulate the behavioral state of the brain via diffuse neuromodulatory projections. The model parameters are not free since they correspond to physiological observables. Approximate parameter bounds are obtained by requiring consistency with physiological and behavioral measures, and the model replicates the human sleep-wake cycle, with physiologically reasonable voltages and firing rates. Mutual inhibition between the wake-promoting monoaminergic group and sleep-promoting VLPO causes ;;flip-flop' behavior, with most time spent in 2 stable steady states corresponding to wake and sleep, with transitions between them on a timescale of a few minutes. The model predicts hysteresis in the sleep-wake cycle, with a region of bistability of the wake and sleep states. Reducing the monoaminergic-VLPO mutual inhibition results in a smaller hysteresis loop. This makes the model more prone to wake-sleep transitions in both directions and makes the states less distinguishable, as in narcolepsy. The model behavior is robust across the constrained parameter ranges, but with sufficient flexibility to describe a wide range of observed phenomena.     

A quantitative model of nucleosome dynamics

The expression, replication and repair of eukaryotic genomes require the fundamental organizing unit of chromatin, the nucleosome, to be unwrapped and disassembled. We have developed a quantitative model of nucleosome dynamics which provides a fundamental understanding of these DNA processes. We calibrated this model using results from high precision single molecule nucleosome unzipping experiments, and then tested its predictions for experiments in which nucleosomes are disassembled by the DNA mismatch recognition complex hMSH2-hMSH6. We found that this calibrated model quantitatively describes hMSH2-hMSH6 induced disassembly rates of nucleosomes with two separate DNA sequences and four distinct histone modification states. In addition, this model provides mechanistic insight into nucleosome disassembly by hMSH2-hMSH6 and the influence of histone modifications on this disassembly reaction. This model''s precise agreement with current experiments suggests that it can be applied more generally to provide important mechanistic understanding of the numerous nucleosome alterations that occur during DNA processing.     

An epigenetic model for pigment patterning based on mechanical and cellular interactions

Pigment patterning in animals generally occurs during early developmental stages and has ecological, physiological, ethological, and evolutionary significance. Despite the relative simplicity of color patterns, their emergence depends upon multilevel complex processes. Thus, theoretical models have become necessary tools to further understand how such patterns emerge. Recent studies have reevaluated the importance of epigenetic, as well as genetic factors in developmental pattern formation. Yet epigenetic phenomena, specially those related to physical constraints that might be involved in the emergence of color patterns, have not been fully studied. In this article, we propose a model of color patterning in which epigenetic aspects such as cell migration, cell-tissue interactions, and physical and mechanical phenomena are central. This model considers that motile cells embedded in a fibrous, viscoelastic matrix-mesenchyme-can deform it in such a way that tension tracks are formed. We postulate that these tracks act, in turn, as guides for subsequent cell migration and establishment, generating long-range phenomenological interactions. We aim to describe some general aspects of this developmental phenomenon with a rather simple mathematical model. Then we discuss our model in the context of available experimental and morphological evidence for reptiles, amphibians, and fishes, and compare it with other patterning models. We also put forward novel testable predictions derived from our model, regarding, for instance, the localization of the postulated tension tracks, and we propose new experiments. Finally, we discuss how the proposed mechanism could constitute a dynamic patterning module accounting for pattern formation in many animal lineages.     

A note on the quantitative properties of McGhee-von Hippel model

Exact closed-form expressions are presented for the properties of the McGhee-von Hippel model of non-specific binding of large ligands to one-dimensional homogeneous lattices. These properties include the midpoint location and the slope at the middle point of the binding isotherms (v varies with ln L plots), the location and magnitude of the maximum, as well as the location of the inflection point, in the Scatchard plots (v/L varies with v plots).     

Evolutionary development of tensegrity structures

Contributions from the emerging fields of molecular genetics and evo-devo (evolutionary developmental biology) are greatly benefiting the field of evolutionary computation, initiating a promise of renewal in the traditional methodology. While direct encoding has constituted a dominant paradigm, indirect ways to encode the solutions have been reported, yet little attention has been paid to the benefits of the proposed methods to real problems. In this work, we study the biological properties that emerge by means of using indirect encodings in the context of form-finding problems. A novel indirect encoding model for artificial development has been defined and applied to an engineering structural-design problem, specifically to the discovery of tensegrity structures. This model has been compared with a direct encoding scheme. While the direct encoding performs similarly well to the proposed method, indirect-based results typically outperform the direct-based results in aspects not directly linked to the nature of the problem itself, but to the emergence of properties found in biological organisms, like organicity, generalization capacity, or modularity aspects which are highly valuable in engineering.     

Hydrostatic pressure sensation in cells: integration into the tensegrity model.

Hydrostatic pressure (HP) is a mechanical stimulus that has received relatively little attention in the field of the cell biology of mechanotransduction. Generalized models, such as the tensegrity model, do not provide a detailed explanation of how HP might be detected. This is significant, because HP is an important mechanical stimulus, directing cell behaviour in a variety of tissues, including cartilage, bone, airways, and the vasculature. HP sensitivity may also be an important factor in certain clinical situations, as well as under unique environmental conditions such as microgravity. While downstream cellular effects have been well characterized, the initial HP sensation mechanism remains unclear. In vitro evidence shows that HP affects cytoskeletal polymerization, an effect that may be crucial in triggering the cellular response. The balance between free monomers and cytoskeletal polymers is shifted by alterations in HP, which could initiate a cellular response by releasing and (or) activating cytoskeleton-associated proteins. This new model fits well with the basic tenets of the existing tensegrity model, including mechanisms in which cellular HP sensitivity could be tuned to accommodate variable levels of stress.     

A cellular automaton model for microcarrier cultures

In order to achieve high cell densities anchoragedependent cells are commonly cultured on microcarriers, where spatial restrictions to cell growth complicates the determination of the growth kinetics. To design and operate large-scale bioreactors for microcarrier cultures, the effect of this spatial restriction to growth, referred to as contact inhibition, must be decoupled from the growth kinetics. In this article, a cellular automaton approach is recommended to model the growth of anchorage-dependent cells on microcarriers. The proposed model is simple to apply yet provides an accurate representation of contact-inhibited cell growth on microcarriers. The distribution of the number of neighboring cells per cell, microcarrier surface areas, and inoculation densities are taken into account with this model. When compared with experimental data for Vero and MRC-5 microcarrier cultures, the cellular automaton predictions were very good. Furthermore, the model can be used to generate contact-inhibition growth curves to decouple the effect of contact-inhibition from growth kinetics. With this information, the accurate determination of kinetic parameters, such as nutrient uptake rates, and the effects of other environmental factors, such as toxin levels, may be determined. (c) 1994 John Wiley & Sons, Inc.