Dimitrije Stamenović

Mechanical behavior in living cells consistent with the tensegrity model

Alternative models of cell mechanics depict the living cell as a simple mechanical continuum, porous filament gel, tensed cortical membrane, or tensegrity network that maintains a stabilizing prestress through incorporation of discrete structural elements that bear compression. Real-time microscopic analysis of cells containing GFP-labeled microtubules and associated mitochondria revealed that living cells behave like discrete structures composed of an interconnected network of actin microfilaments and microtubules when mechanical stresses are applied to cell surface integrin receptors. Quantitation of cell tractional forces and cellular prestress by using traction force microscopy confirmed that microtubules bear compression and are responsible for a significant portion of the cytoskeletal prestress that determines cell shape stability under conditions in which myosin light chain phosphorylation and intracellular calcium remained unchanged. Quantitative measurements of both static and dynamic mechanical behaviors in cells also were consistent with specific a priori predictions of the tensegrity model. These findings suggest that tensegrity represents a unified model of cell mechanics that may help to explain how mechanical behaviors emerge through collective interactions among different cytoskeletal filaments and extracellular adhesions in living cells.

Tensegrity, cellular biophysics, and the mechanics of living systems

The recent convergence between physics and biology has led many physicists to enter the fields of cell and developmental biology. One of the most exciting areas of interest has been the emerging field of mechanobiology that centers on how cells control their mechanical properties, and how physical forces regulate cellular biochemical responses, a process that is known as mechanotransduction. In this article, we review the central role that tensegrity (tensional integrity) architecture, which depends on tensile prestress for its mechanical stability, plays in biology. We describe how tensional prestress is a critical governor of cell mechanics and function, and how use of tensegrity by cells contributes to mechanotransduction. Theoretical tensegrity models are also described that predict both quantitative and qualitative behaviors of living cells, and these theoretical descriptions are placed in context of other physical models of the cell. In addition, we describe how tensegrity is used at multiple size scales in the hierarchy of life—from individual molecules to whole living organisms—to both stabilize three-dimensional form and to channel forces from the macroscale to the nanoscale, thereby facilitating mechanochemical conversion at the molecular level.

A Computational Tensegrity Model Predicts Dynamic Rheological Behaviors in Living Cells

Rheological properties of living cells play a key role in the control of cell shape, growth, movement, and contractility, yet little is known about how these properties are governed. Past approaches to understanding cell mechanics focused on the contributions of membranes, the viscous cytoplasm, and the individual filamentous biopolymers that are found within the cytoskeleton. In contrast, recent work has revealed that the dynamic mechanical behavior of cells depends on generic system properties, rather than on a single molecular property of the cell. In this paper, we show that a mathematical model of cell mechanics that depicts the intracellular cytoskeleton as a tensegrity structure composed of a prestressed network of interconnected microfilaments, microtubules, and intermediate filaments, and that has previously explained static cellular properties, also can predict fundamental dynamic behaviors of living cells.

Fractional derivatives embody essential features of cell rheological behavior.

AbstractMechanical moduli of cultured airway smooth muscle cells (Fabry, B., et al. Phys. Rev. Lett. 87:148102, 2001) reveal that the frequency dependence of cell rheological behavior conforms to a weak power-law relationship over a wide range of frequency (10-2 -103 Hz).Such a behavior cannot be accounted for by standard viscoelastic models characterized by a discrete number of time constants that have been commonly used in previous studies of cell viscoelasticity. Fractional calculus, by contrast, provides a natural framework for describing weak power-law relationships and requires no assumptions about the type of material, the time constant distribution or the time/frequency interval in which rheological observations are made. In this study, we developed a rheological model of the cell using methods of fractional calculus. We used a least-squares technique to fit the model to data previously obtained from measurements on airway smooth muscle cells. The fit provided an excellent correspondence to the data, and the estimated values of model parameters were physically plausible. The model leads to a novel and unexpected empirical link between dynamic viscoelastic behavior of the cytoskeleton and the static contractile stress that it bears.

Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis

Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good fit to the observed unconfined compression transients when the out-of-plane Poissons ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (deltaz) is the same in confined and unconfined compression, which implies that the radial stress deltar = 0 in confined compression. In our study, we further investigated the ability of the transversely isotropic model to describe confined and unconfined stress relaxation behavior of calf cartilage. A series of confined and unconfined stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, approximately 3.3 mm height) in a displacement-controlled compression apparatus capable of measuring delta(z) and delta(r). In equilibrium, delta(r) > 0 and delta(z) in confined compression was greater than in unconfined compression. Transient data at each strain were fitted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good fits to the unconfined transients, the estimated parameters overpredicted the measured delta(r). Conversely, if the model was constrained to match equilibrium delta(r), the fits were poor. These findings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage.

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 Youngs 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 Youngs 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.

A Prestressed Cable Network Model of the Adherent Cell Cytoskeleton

A prestressed cable network is used to model the deformability of the adherent cell actin cytoskeleton. The overall and microstructural model geometries and cable mechanical properties were assigned values based on observations from living cells and mechanical measurements on isolated actin filaments, respectively. The models were deformed to mimic cell poking (CP), magnetic twisting cytometry (MTC) and magnetic bead microrheometry (MBM) measurements on living adherent cells. The models qualitatively and quantitatively captured the fibroblast cell response to the deformation imposed by CP while exhibiting only some qualitative features of the cell response to MTC and MBM. The model for CP revealed that the tensed peripheral actin filaments provide the key resistance to indentation. The actin filament tension that provides mechanical integrity to the network was estimated at approximately 158 pN, and the nonlinear mechanical response during CP originates from filament kinematics. The MTC and MBM simulations revealed that the model is incomplete, however, these simulations show cable tension as a key determinant of the model response.

Tensegrity-guided self assembly: from molecules to living cells

One of the wonders of life is that all cells undergo continual turnover, and sustain their structure and function through continuous molecular self assembly. However, this dynamic renewal process is commonly viewed from the ‘bottom-up’, by focusing on the properties and interaction functions of individual molecular components. In reality, all cells form from other cells using preexisting structures, such as the cytoskeleton, as orienting scaffolds that guide replication and formation of new cellular components. In this article, we take a ‘top-down’ approach and describe how living cells may use hierarchical tensegrity principles to stabilize the shape and structure of their internal subcomponents at multiple size scales. We also explain how use of this form of architecture that depends on tensional prestress for shape stability could provide a mechanism to focus mechanical forces on the molecular components that comprise these structures, and thereby control their biochemical activities and self assembly behavior in living cells. In this manner, self assembly of load-bearing structures in cells proceeds in particular patterns that precisely match the forces they need to bear. This also explains how cells seamlessly integrate structure and function at all size scales, a process that is fundamental to all living materials.

Lung Parenchymal Mechanics

The lung parenchyma comprises a large number of thin-walled alveoli, forming an enormous surface area, which serves to maintain proper gas exchange. The alveoli are held open by the transpulmonary pressure, or prestress, which is balanced by tissues forces and alveolar surface film forces. Gas exchange efficiency is thus inextricably linked to three fundamental features of the lung: parenchymal architecture, prestress, and the mechanical properties of the parenchyma. The prestress is a key determinant of lung deformability that influences many phenomena including local ventilation, regional blood flow, tissue stiffness, smooth muscle contractility, and alveolar stability. The main pathway for stress transmission is through the extracellular matrix. Thus, the mechanical properties of the matrix play a key role both in lung function and biology. These mechanical properties in turn are determined by the constituents of the tissue, including elastin, collagen, and proteoglycans. In addition, the macroscopic mechanical properties are also influenced by the surface tension and, to some extent, the contractile state of the adherent cells. This chapter focuses on the biomechanical properties of the main constituents of the parenchyma in the presence of prestress and how these properties define normal function or change in disease. An integrated view of lung mechanics is presented and the utility of parenchymal mechanics at the bedside as well as its possible future role in lung physiology and medicine are discussed.

A micropatterning and image processing approach to simplify measurement of cellular traction forces

Quantification of the traction forces that cells apply to their surroundings has been critical to the advancement of our understanding of cancer, development and basic cell biology. This field was made possible through the development of engineered cell culture systems that permit optical measurement of cell-mediated displacements and computational algorithms that allow conversion of these displacements into stresses and forces. Here, we present a novel advancement of traction force microscopy on polyacrylamide (PAA) gels that addresses limitations of existing technologies. Through an indirect patterning technique, we generated PAA gels with fluorescent 1 μm dot markers in a regularized array. This improves existing traction measurements since (i) multiple fields of view can be measured in one experiment without the need for cell removal; (ii) traction vectors are modeled as discrete point forces, and not as a continuous field, using an extremely simple computational algorithm that we have made available online; and (iii) the pattern transfer technique is amenable to any of the published techniques for producing patterns on glass. In the future, this technique will be used for measuring traction forces on complex patterns with multiple, spatially distinct ligands in systems for applying strain to the substrate, and in sandwich cultures that generate quasi-three-dimensional environments for cells.