On tensegrity in cell mechanics

All models are wrong, but some are useful. This famous saying mirrors the situation in cell mechanics as well. It looks like no particular model of the cell deformability can be unconditionally preferred over others and different models reveal different aspects of the mechanical behavior of living cells. The purpose of the present work is to discuss the so-called tensegrity models of the cell cytoskeleton. It seems that the role of the cytoskeleton in the overall mechanical response of the cell was not appreciated until Donald Ingber put a strong emphasis on it. It was fortunate that Ingber linked the cytoskeletal structure to the fascinating art of tensegrity architecture. This link sparked interest and argument among biologists, physicists, mathematicians, and engineers. At some point the enthusiasm regarding tensegrity perhaps became overwhelming and as a reaction to that some skepticism built up. To demystify Ingber's ideas the present work aims at pinpointing the meaning of tensegrity and its role in our understanding of the importance of the cytoskeleton for the cell deformability and motility. It should be noted also that this paper emphasizes basic ideas rather than carefully follows the chronology of the development of tensegrity models. The latter can be found in the comprehensive review by Dimitrije Stamenovic (2006) to which the present work is complementary.     

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).     

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.     

Models of cytoskeletal mechanics of adherent cells

 Adherent cells sense their mechanical environment, which, in turn, regulates their functions. During the past decade, a growing body of evidence has indicated that a deformable, solid-state intracellular structure known as the cytoskeleton (CSK) plays a major role in transmitting and distributing mechanical stresses within the cell as well as in their conversion into a chemical response. Therefore in order to understand mechanical regulation and control of cellular functions, one needs to understand mechanisms that determine how the CSK changes its shape and mechanics in response to stress. In this survey, we examined commonly used structurally based models of the CSK. In particular, we focused on two classes of these models: open-cell foam networks and stress-supported structures. We identified the underlying mechanisms that determine deformability of those models and compare model predictions with data previously obtained from mechanical tests on cultured living adherent cells at steady state. We concluded that stress-supported structures appear more suitable for describing cell deformability because this class of structures can explain the central role that the cytoskeletal contractile prestress plays in cellular mechanics. Received: 2 January 2002 / Accepted: 27 February 2002     

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.     

How is a tissue built?

Tissues change in many ways in the period that they are part of a living organism. They are created in fairly repeatable structural patterns, and we know that the patterns are due to both the genes and the (mechanical) environment, but we do not know exactly what part or percentage of a particular pattern to consider the genes, or the environment, responsible for. We do not know much about the beginning of tissue construction (morphogenesis) and we do not know the methods of tissue construction. When the tissue structure is altered to accommodate a new loading, we do not know how the decision is made for the structural reconstruction. We do know that tissues grow or reconstruct themselves without ceasing to continue with their structural function, but we do not understand the processes that permit them to accomplish this. Tissues change their structures to altered mechanical environments, but we are not sure how. Tissues heal themselves and we understand little of the structural mechanics of the process. With the objective of describing the interesting unsolved mechanics problems associated with these biological processes, some aspects of the formation, growth, and adaptation of living tissues are reviewed. The emphasis is on ideas and models. Beyond the objective is the hope that the work will stimulate new ideas and new observations in the mechanical and chemical aspects of developmental biology.     

The role of cellular active stresses in shaping the zebrafish body axis

Tissue remodelling and organ shaping during morphogenesis are products of mechanical forces generated at the cellular level. These cell-scale forces can be coordinated across the tissue via information provided by biochemical and mechanical cues. Such coordination leads to the generation of complex tissue shape during morphogenesis. In this short review, we elaborate the role of cellular active stresses in vertebrate axis morphogenesis, primarily using examples from postgastrulation development of the zebrafish embryo.     

The Mechanochemical Basis of Cell and Tissue Regulation

This article is a summary of a lecture presented at a symposium on "Mechanics and Chemistry of Biosystems' in honor of Professor Y.C. Fung that convened at the University of California, Irvine in February 2004. The article reviews work from our laboratory that focuses on the mechanism by which mechanical and chemical signals interplay to control how individual cells decide whether to grow, differentiate, move, or die, and thereby promote pattern formation during tissue morphogenesis. Pursuit of this challenge has required development and application of new microtechnologies, theoretical formulations, computational models and bioinformatics tools. These approaches have been used to apply controlled mechanical stresses to specific cell surface molecules and to measure mechanical and biochemical responses; to control cell shape independently of chemical factors; and to handle the structural, hierarchical and informational complexity of living cells. Results of these studies have changed our view of how cells and tissues control their shape and mechanical properties, and have led to the discovery that integrins and the cytoskeleton play a central role in cellular mechanotransduction. Recognition of these critical links between mechanics and cellular biochemistry should lead to novel strategies for the development of new drugs and engineered tissues, as well as biomimetic microdevices and nanotechnologies that more effectively function within the context of living tissues.     

Ciona intestinalis notochord as a new model to investigate the cellular and molecular mechanisms of tubulogenesis

Biological tubes are a prevalent structural design across living organisms. They provide essential functions during the development and adult life of an organism. Increasing progress has been made recently in delineating the cellular and molecular mechanisms underlying tubulogenesis. This review aims to introduce ascidian notochord morphogenesis as an interesting model system to study the cell biology of tube formation, to a wider cell and developmental biology community. We present fundamental morphological and cellular events involved in notochord morphogenesis, compare and contrast them with other more established tubulogenesis model systems, and point out some unique features, including bipolarity of the notochord cells, and using cell shape changes and cell rearrangement to connect lumens. We highlight some initial findings in the molecular mechanisms of notochord morphogenesis. Based on these findings, we present intriguing problems and put forth hypotheses that can be addressed in future studies.     

Problems of biochemical organization]

Biological organization has been defined as a unity of structure, function and regulation. Biological organization of hierarchical multilevel biological systems is represented by a hierarchy of functioning controllable structures. The hierarchy of levels of material organization predetermines the existence of a hierarchy of regulatory mechanisms. Biochemical organization involves the levels of material organization corresponding to biomacromolecules, supramolecular complexes and cellular organelles. The levels of biomacromolecules and supramolecular structures effectuating elementary functions and controlled by basic regulatory mechanisms occupy key positions in biological systems. These levels play the role of standard functional blocks; their combination leads to hierarchically higher structural levels (cell, tissue, organ, systems of organs, organism) performing more complex functions and controlled by hierarchically more important regulatory mechanisms. The peculiarities of regulation of biological systems that are due to the existence of a hierarchy of regulatory mechanisms are discussed.     

Can morphogenesis be understood in terms of physical rules?

Because the morphogenesis of biological systems is not fully understood, researches from various points of view are necessary. The present author has recently made computer simulations with his colleagues to construct branching systems of human organs, such as the lung airway and the liver blood vessels. In the simulations certain rules are assumed to govern bifurcating processes of the systems. These rules are expressed in terms of physical and geometrical concepts, such as minimum energy consumption and uniform filling of branches in the space of organs. Results of computer simulation are quite similar to real structures. However, actual mechanisms of morphogenesis, i.e. effects of genes or proteins, are not considered in these studies. In this article, the present work is discussed in relation to the concept of biological pattern formation by Meinhardt and a recent study by Miura and Shiota on lung growth.     

The origin of cellular life

This essay presents a scenario of the origin of life that is based on analysis of biological architecture and mechanical design at the microstructural level. My thesis is that the same architectural and energetic constraints that shape cells today also guided the evolution of the first cells and that the molecular scaffolds that support solid-phase biochemistry in modern cells represent living microfossils of past life forms. This concept emerged from the discovery that cells mechanically stabilize themselves using tensegrity architecture and that these same building rules guide hierarchical self-assembly at all size scales (Sci. Amer 278:48-57;1998). When combined with other fundamental design principles (e.g., energy minimization, topological constraints, structural hierarchies, autocatalytic sets, solid-state biochemistry), tensegrity provides a physical basis to explain how atomic and molecular elements progressively self-assembled to create hierarchical structures with increasingly complex functions, including living cells that can self-reproduce.     

Determination of mechanical stress distribution in Drosophila wing discs using photoelasticity

Morphogenesis, the process by which all complex biological structures are formed, is driven by an intricate interplay between genes, growth, as well as intra- and intercellular forces. While the expression of different genes changes the mechanical properties and shapes of cells, growth exerts forces in response to which tissues, organs and more complex structures are shaped. This is exemplified by a number of recent findings for instance in meristem formation in Arabidopsis and tracheal tube formation in Drosophila. However, growth not only generates forces, mechanical forces can also have an effect on growth rates, as is seen in mammalian tissues or bone growth. In fact, mechanical forces can influence the expression levels of patterning genes, allowing control of morphogenesis via mechanical feedback. In order to study the connections between mechanical stress, growth control and morphogenesis, information about the distribution of stress in a tissue is invaluable. Here, we applied stress-birefringence to the wing imaginal disc of Drosophila melanogaster, a commonly used model system for organ growth and patterning, in order to assess the stress distribution present in this tissue. For this purpose, stress-related differences in retardance are measured using a custom-built optical set-up. Applying this method, we found that the stresses are inhomogeneously distributed in the wing disc, with maximum compression in the centre of the wing pouch. This compression increases with wing disc size, showing that mechanical forces vary with the age of the tissue. These results are discussed in light of recent models proposing mechanical regulation of wing disc growth.     

Hox genes as synchronized temporal regulators: implications for morphological innovation

This special issue on the development and evolution of the amniote integument begins with a discussion of the adaptations to terrestrial conditions, the acquisition of water-impermeability of the reptilian integument, and the initial formation of filamentous integumentary appendages that prepare the way towards avian flight. Recent feather fossils are reviewed, and a definition of feathers is developed. Hierarchical models are proposed for the formation of complex structures, such as feathers. Molecular signals that alter the phenotype of integumentary appendages at different levels of the hierarchy are presented. Tissue interactions and the roles of keratins in evolution are discussed and linked to their bio-mechanical properties. The role of mechanical forces on patterning is explored. Elaborate extant feather variants are introduced. The regeneration/gene mis-expression protocol for the chicken feather is established as a testable model for the study of biological structures. The adaptations of the mammalian distal limb end organs to terrestrial, arboreal and aquatic conditions are discussed. The development and cycling of hair are reviewed from a molecular perspective. These contributions reveal that the structure and function of diverse integumentary appendages are variations that are superimposed on a common theme, and that their formation is modular, hierarchical and cyclical. They further reveal that these mechanisms can be understood at the molecular level, and that an integrative and organismal approach to studying integumentary appendages is called for. We propose that future research should foster interdisciplinary approaches, pursue understanding at the cellular and molecular level, analyze interactions between the environment and genome, and recognize the contributions of variation in morphogenesis and evolution.     

A physical (electromagnetic) model of differentiation. 1. Basic considerations

There are a number of biological phenomena and events that cannot yet be adequately described, such as cell growth and differentiation, which may be controlled by physical factors. Fr?hlich (1980) has discussed the principles of dissipative structures as applied to electromagnetic interactions in relation to basic couplings in biological systems. Recently, increasing evidence of photon storage and ultraweak photon emission from living systems, particularly from DNA, has suggested the concept of an electromagnetic model of differentiation, based on the known quantum optical properties of nucleic acids. This model has the advantage over all ideas so far published, that it is (1) simple; (2) universally applicable to events in living matter, because it is consistent with both the quantum mechanical and the thermodynamic properties on the one hand, and the known biological and biochemical data and phenomena at the other hand; (3) it not only describes the phenomena and events in terms of pure mathematical parameters, but it can also explain them; and (4) it escapes the difficulty of finding basic control mechanisms, which themselves do not need a regulator, ad infinitum.     

The role of carbonyl-amine reaction in biological systems and food technology (review of the literature)

Modern ideas about the role of the carbonyl-amine reaction in biochemical systems are reviewed. Some details recently found of the mechanisms of the initial steps of the reactions are considered. The pathways of the formation of numerous by-products are discussed which are very important for taste and aroma of some foodstuffs. Data are presented on nonenzymatic glycosylation of proteins in living organisms, which causes disturbances in their vital functions, diseases and aging. Possible chemical structures of "links" between sugars and protein polypeptide chains, as well as nucleic acids derived from glycosylation are considered. References concerning properties of melanoidins are presented.     

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.     

On the work of the developmental biophysics laboratory of the embryology department of Moscow State University

The laboratory is engaged in morphomechanics—the study of self-organization of mechanical forces that create the shape and structure of the embryonic primordia. As part of its work, the laboratory described pulsating modes of mechanical stresses in hydroids, identified and mapped mechanical stresses in the tissues of amphibian embryos, and studied morphogenetic reorganization caused by the relaxation and reorientation of tensions. The role of mechanical stresses in maintaining the orderly architectonics of the embryo is shown. Mechano-dependent genes are detected. Microstrains of embryonic tissues and stress gradients associated with them are described. A model of hyper-recovery of mechanical stresses as a possible driving force of morphogenesis is proposed.