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.     

Cell cytoskeleton and tensegrity

The role of tensegrity architecture of the cytoskeleton in the mechanical behavior of living cells is examined by computational studies. Plane and spatial tensegrity models of the cytoskeleton are considered as well as their non-tensegrity counterparts. Local buckling including deep postbuckling response of the compressed microtubules of the cytoskeleton is considered. The tensioned microfilaments cannot sustain compression. Large deformation of the whole model is accounted and fully nonlinear analysis is performed. It is shown that in the case of local buckling of the microtubules non-tensegrity models exhibit qualitatively the same linear stiffening as their tensegrity counterparts. This result raises the question of experimental validation of the local buckling of microtubules. If the microtubules of real cells are not straight, then tensegrity (in a narrow sense) is not a necessary attribute of the cytoskeleton architecture. If the microtubules are straight then tensegrity is more likely to be the cytoskeletal architecture.     

pH regulation in spread cells and round cells

The aim of this work was to characterize the changes in pH regulation that lead to increased intracellular pH (pHi) in well-spread cells on tissue culture plastic relative to cells on a nonadhesive surface. Bicarbonate was not required for maintenance of a control steady state pHi or of the difference in pHi between round and spread cells. In the absence of bicarbonate, lowering the sodium content of the medium led to decreased pHi and elimination of the difference between round and spread cells. In the presence or absence of bicarbonate, adding ethylisopropyl amiloride lowered pHi and eliminated the difference between round and spread cells. Measurements of recovery from acute acidification in the absence of bicarbonate confirmed that Na+/H+ exchange was enhanced in spread cells. However, recovery from both acidification and alkalinization in the presence of bicarbonate showed that bicarbonate-dependent recovery in both directions, most likely due to sodium-dependent and -independent HCO3-/Cl- exchangers, was also stimulated in spread cells. We conclude that Na+/H+ exchange has a primary role in determining steady state pHi in 3T3 cells in serum and is responsible for the lower pHi in round cells. Bicarbonate-dependent pH regulatory mechanisms are also inhibited in round cells.     

Toward a generalised tensegrity model describing the mechanical behaviour of the cytoskeleton structure

The control of many cell functions including growth, migration and mechanotransduction, depends crucially on stress-induced mechanical changes in cell shape and cytoskeleton (CSK) structure. Quantitative studies have been carried out on 6-bar tensegrity models to analyse several mechanical parameters involved in the mechanical responses of adherent cells (i.e. strain hardening, internal stress and scale effects). In the present study, we attempt to generalize some characteristic mechanical laws governing spherical tensegrity structures, with a view of evaluating the mechanical behaviour of the hierarchical multi-modular CSK-structure. The numerical results obtained by studying four different tensegrity models are presented in terms of power laws and point to the existence of unique and constant relationships between the overall structural stiffness and the local properties (length, number and internal stress) of the constitutive components.     

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.     

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 multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

A quantitative model of cellular elasticity based on tensegrity

A tensegrity structure composed of six struts interconnected with 24 elastic cables is used as a quantitative model of the steady-state elastic response of cells, with the struts and cables representing microtubules and actin filaments, respectively. The model is stretched uniaxially and the Young's modulus (E0) is obtained from the initial slope of the stress versus strain curve of an equivalent continuum. It is found that E0 is directly proportional to the pre-existing tension in the cables (or compression in the struts) and inversely proportional to the cable (or strut) length square. This relationship is used to predict the upper and lower bounds of E0 of cells, assuming that the cable tension equals the yield force of actin (approximately 400 pN) for the upper bound, and that the strut compression equals the critical buckling force of microtubules for the lower bound. The cable (or strut) length is determined from the assumption that model dimensions match the diameter of probes used in standard mechanical tests on cells. Predicted values are compared to reported data for the Young's modulus of various cells. If the probe diameter is greater than or equal to 3 microns, these data are closer to the lower bound than to the upper bound. This, in turn, suggests that microtubules of the CSK carry initial compression that exceeds their critical buckling force (order of 10(0)-10(1) pN), but is much smaller than the yield force of actin. If the probe diameter is less than or equal to 2 microns, experimental data fall outside the region defined by the upper and lower bounds.     

Dynamics of the cytoskeleton in live cells.

Actin filaments, microtubules, and intermediate filaments, have all been found to be dynamic structures in living cells. Recent studies have shed important light on the assembly, disassembly, and mobility of these structures. In addition, a growing emphasis has been placed on the regulation of cytoskeletal activities by various signal transduction pathways.     

Complexity of the tensegrity structure for dynamic energy and force distribution of cytoskeleton during cell spreading

Cytoskeleton plays important roles in intracellular force equilibrium and extracellular force transmission from/to attaching substrate through focal adhesions (FAs). Numerical simulations of intracellular force distribution to describe dynamic cell behaviors are still limited. The tensegrity structure comprises tension-supporting cables and compression-supporting struts that represent the actin filament and microtubule respectively, and has many features consistent with living cells. To simulate the dynamics of intracellular force distribution and total stored energy during cell spreading, the present study employed different complexities of the tensegrity structures by using octahedron tensegrity (OT) and cuboctahedron tensegrity (COT). The spreading was simulated by assigning specific connection nodes for radial displacement and attachment to substrate to form FAs. The traction force on each FA was estimated by summarizing the force carried in sounding cytoskeletal elements. The OT structure consisted of 24 cables and 6 struts and had limitations soon after the beginning of spreading by declining energy stored in struts indicating the abolishment of compression in microtubules. The COT structure, double the amount of cables and struts than the OT structure, provided sufficient spreading area and expressed similar features with documented cell behaviors. The traction force pointed inward on peripheral FAs in the spread out COT structure. The complex structure in COT provided further investigation of various FA number during different spreading stages. Before the middle phase of spreading (half of maximum spreading area), cell attachment with 8 FAs obtained minimized cytoskeletal energy. The maximum number of 12 FAs in the COT structure was required to achieve further spreading. The stored energy in actin filaments increased as cells spread out, while the energy stored in microtubules increased at initial spreading, peaked in middle phase, and then declined as cells reached maximum spreading. The dynamic flows of energy in struts imply that microtubules contribute to structure stabilization.     

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 discrete-time model for the spread of periodic diseases without immunity.

We investigate the properties of a discrete-time model for the spread of an infectious disease that does not confer permanent immunity. The contact rate is assumed to be constant. Our major result is that oscillations occur in the levels of the diseased population. The consequences of vaccination are also studied.     

A model for the spread of an epidemic

A temporally continuous and spatially discrete stochastic model for the spread of an epidemic within some set of holdings is constructed. A recursion formula is given for the probability that a certain set of holdings is infected at a certain moment. Moreover, under an additional condition (which will always be satisfied in practice) a formula for the expected value and the variance of the moment when a certain holding is infected the first time is given.     

Characterization of the cytoskeleton in human normal and otosclerotic osteoblast-like cells.

The localization and distribution of some cytoskeletal protein components were studied by immunostaining methods in normal and ostosclerotic osteoblast-like cells. The protein components investigated were microtubules (beta-tubulin), intermediate filaments (vimentin), microfilaments (actin and myosin) and structural proteins (alpha-actinin and fibronectin). Although the mechanism is not yet clear, the alterations observed in the pathological cells could well play a role in the expression of otosclerosis.     

Interrelations between elastic energy and strain in a tensegrity model: contribution to the analysis of the mechanical response in living cells

Interactions between the physical and physiological properties of cellular sub-units result in changes in the shape and mechanical behaviour of living tissues. To understand the mechanotransmission processes, models are needed to describe the complex interrelations between the elements and the cytoskeletal structure. In this study, we used a 30-element tensegrity structure to analyse the influence of the type of loading on the mechanical response and shape changes of the cell. Our numerical results, expressed in terms of strain energy as a function of the overall deformation of the tensegrity structure, suggest that changes in cell functions during mechanical stimuli for a given potential energy are correlated to the type of loading applied, which determines the resultant changes in cell shape. The analysis of these cellular deformations may explain the large variability in the response of bone cells submitted to different types of mechanical loading.     

A model for the spatial spread of an epidemic

Summary We set up a deterministic model for the spatial spread of an epidemic. Essentially, the model consists of a nonlinear integral equation which has an unique solution. We show that this solution has a temporally asymptotic limit which describes the final state of the epidemic and is the minimal solution of another nonlinear integral equation. We outline the asymptotic behaviour of this minimal solution at a great distance from the epidemic's origin and generalize D. G. Kendall's pandemic threshold theorem (1957).