Cell prestress. II. Contribution of microtubules

The tensegritymodel hypothesizes that cytoskeleton-based microtubules (MTs) carrycompression as they balance a portion of cell contractile stress. Totest this hypothesis, we used traction force microscopy to measuretraction at the interface of adhering human airway smooth muscle cellsand a flexible polyacrylamide gel substrate. The prediction is that ifMTs balance a portion of contractile stress, then, upon theirdisruption, the portion of stress balanced by MTs would shift to thesubstrate, thereby causing an increase in traction. Measurements weredone first in maximally activated cells (10 µM histamine) and thenagain after MTs had been disrupted (1 µM colchicine). We found that after disruption of MTs, traction increased on average by ~13%. Because in activated cells colchicine induced neither an increase inintracellular Ca²⁺ nor an increase in myosin light chainphosphorylation as shown previously, we concluded that the observedincrease in traction was a result of load shift from MTs to thesubstrate. In addition, energy stored in the flexible substrate wascalculated as work done by traction on the deformation of thesubstrate. This result was then utilized in an energetic analysis. Weassumed that cytoskeleton-based MTs are slender elastic rods supportedlaterally by intermediate filaments and that MTs buckle as the cellcontracts. Using the post-buckling equilibrium theory of Euler struts,we found that energy stored during buckling of MTs was quantitativelyconsistent with the measured increase in substrate energy afterdisruption of MTs. This is further evidence supporting the idea thatMTs are intracellular compression-bearing elements.

     

A finite element model of cell deformation during magnetic bead twisting.

Magnetic twisting cytometry probes mechanical properties of an adherent cell by applying a torque to a magnetic bead that is tightly bound to the cell surface. Here we have used a three-dimensional finite element model of cell deformation to compute the relationships between the applied torque and resulting bead rotation and lateral bead translation. From the analysis, we computed two coefficients that allow the cell elastic modulus to be estimated from measurements of either bead rotation or lateral bead translation, respectively, if the degree of bead embedding and the cell height are known. Although computed strains in proximity of the bead can be large, the relationships between applied torque and bead rotation or translation remain virtually linear up to bead rotations of 15 degrees, above which geometrical nonlinearities become significant. This appreciable linear range stands in contrast to the intrinsically nonlinear force-displacement relationship that is observed when cells are indented during atomic force microscopy. Finally, these computations support the idea that adhesive forces are sufficient to keep the bead firmly attached to the cell surface throughout the range of working torques.     

Contributions of pulmonary perfusion and ventilation to heterogeneity in VA/Q measured by PET

Treppo, Steven, Srboljub M. Mijailovich, and José G. Venegas. Contributions of pulmonary perfusion and ventilation toheterogeneity in A/measured by PET. J. Appl. Physiol. 82(4): 1163-1176, 1997. To estimate the contributions of the heterogeneity in regionalperfusion () and alveolar ventilation(A) to that of ventilation-perfusionratio (A/), we haverefined positron emission tomography (PET) techniques to image localdistributions of andA per unit of gas volume content(s and sA,respectively) and VA/ indogs. sA was assessed in two ways:1) the washout of ¹³NN tracer after equilibrationby rebreathing (sA_i), and2) the ratio of an apneic image after a bolus intravenousinfusion of ¹³NN-saline solution to an image collectedduring a steady-state intravenous infusion of the same solution(sA_p).sA_p was systematically higher than sA_i in allanimals, and there was a high spatial correlation betweens andsA_p in both body positions(mean correlation was 0.69 prone and 0.81 supine) suggesting thatventilation to well-perfused units was higher than to those poorlyperfused. In the prone position, the spatial distributions ofs, sA_p, and A/ were fairlyuniform with no significant gravitational gradients; however, in thesupine position, these variables were significantly more heterogeneous,mostly because of significant gravitational gradients (15, 5.5, and10%/cm, respectively) accounting for 73, 33, and 66% of thecorresponding coefficient of variation (CV)² values. Weconclude that, in the prone position, gravitational forces in blood andlung tissues are largely balanced out by dorsoventral differences inlung structure. In the supine position, effects of gravity andstructure become additive, resulting in substantial gravitationalgradients in s andsA_p, with the higherheterogeneity inA/ caused by agravitational gradient in s, only partially compensated by that in sA.

     

The Hill model for binding myosin S1 to regulated actin is not equivalent to the McKillop-Geeves model

The Hill two-state cooperativity model and the McKillop-Geeves (McK-G) three-state model predict very similar binding traces of myosin subfragment 1 (S1) binding to regulated actin filaments in the presence and absence of calcium, and both fit the experimental data reasonably well [Chen et al., Biophys. J., 80, 2338-2349]. Here, we compared the Hill model and the McK-G model for binding myosin S1 to regulated actin against three sets of experimental data: the titration of regulated actin with S1 and the kinetics of S1 binding of regulated actin with either excess S1 to actin or excess actin to S1. Each data set was collected for a wide range of specified calcium concentrations. Both models were able to generate reasonable fits to the time course data and to titration data. The McK-G model can fit all three data sets with the same calcium-concentration-sensitive parameters. Only K(B) and K(T) show significant calcium dependence, and the parameters have a classic pCa curve. A unique set of the Hill model parameters was extremely difficult to estimate from the best fits of multiple sets of data. In summary, the McK-G cooperativity model more uniquely resolves parameters estimated from kinetic and titration data than the Hill model, predicts a sigmoidal dependence of key parameters with calcium concentration, and is simpler and more suitable for practical use.     

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.     

Multiscale modeling of twitch contractions in cardiac trabeculae

Understanding the dynamics of a cardiac muscle twitch contraction is complex because it requires a detailed understanding of the kinetic processes of the Ca²⁺ transient, thin-filament activation, and the myosin–actin cross-bridge chemomechanical cycle. Each of these steps has been well defined individually, but understanding how all three of the processes operate in combination is a far more complex problem. Computational modeling has the potential to provide detailed insight into each of these processes, how the dynamics of each process affect the complexity of contractile behavior, and how perturbations such as mutations in sarcomere proteins affect the complex interactions of all of these processes. The mechanisms involved in relaxation of tension during a cardiac twitch have been particularly difficult to discern due to nonhomogeneous sarcomere lengthening during relaxation. Here we use the multiscale MUSICO platform to model rat trabecular twitches. Validation of computational models is dependent on being able to simulate different experimental datasets, but there has been a paucity of data that can provide all of the required parameters in a single experiment, such as simultaneous measurements of force, intracellular Ca²⁺ transients, and sarcomere length dynamics. In this study, we used data from different studies collected under similar experimental conditions to provide information for all the required parameters. Our simulations established that twitches either in an isometric sarcomere or in fixed-length, multiple-sarcomere trabeculae replicate the experimental observations if models incorporate a length–tension relationship for the nonlinear series elasticity of muscle preparations and a scheme for thick-filament regulation. The thick-filament regulation assumes an off state in which myosin heads are parked onto the thick-filament backbone and are unable to interact with actin, a state analogous to the super-relaxed state. Including these two mechanisms provided simulations that accurately predict twitch contractions over a range of different conditions.     

Cooperative [Ca]-Dependent Regulation of the Rate of Myosin Binding to Actin: Solution Data and the Tropomyosin Chain Model

The regulation of muscle contraction by calcium involves interactions among actin filaments, myosin-S1, tropomyosin (Tm), and troponin (Tn). We have extended our previous model in which the TmTn regulatory units are treated as a continuous flexible chain, and applied it to transient kinetic data. We have measured the time course of myosin-S1 binding to actin-Tm-Tn filaments in solution at various calcium levels with [actin]/[myosin] ratios of 10 and 0.1, which exhibit modest slowing as [Ca²⁺] is reduced and a lag phase at low calcium. These observations can be explained if myosin binds to actin in two steps, where the first step is rate-limiting and blocked by TmTnI at low calcium, and the second step is fast, reversible, and controlled by the neighboring configuration of coupled tropomyosin-troponin units. The model can describe the calcium dependence of the observed myosin binding reactions and predicts cooperative calcium binding to TnC with competition between actin and Ca-TnC for the binding of TnI. Implications for theories of thin-filament regulation in muscle are discussed.     

The elastic properties of the structurally characterized myosin II S2 subdomain: a molecular dynamics and normal mode analysis

The elastic properties (stretching and bending moduli) of myosin are expected to play an important role in its function. Of particular interest is the extended α-helical coiled-coil portion of the molecule. Since there is no high resolution structure for the entire coiled-coil, a study is made of the scallop myosin II S2 subdomain for which an x-ray structure is available (Protein Data Bank 1nkn). We estimate the stretching and bending moduli of the S2 subdomain with an atomic level model by use of molecular simulations. Results were obtained from nonequilibrium molecular dynamics simulations in the presence of an external force, from the fluctuations in equilibrium molecular dynamics simulations and from normal modes. In addition, a poly-Ala (78 amino acid residues) α-helix model was examined to test the methodology and because of its interest as part of the lever arm. As expected, both the α-helix and coiled-coil S2 subdomain are very stiff for stretching along the main axis, with the stretching stiffness constant in the range 60-80 pN/nm (scaled to the 60 nm long S2). Both molecules are much more flexible for bending with a lateral stiffness of ∼0.010pN/nm for the S2 and 0.0055pN/nm for the α-helix (scaled to 60 nm). These results are expected to be useful in estimating cross-bridge elasticity, which is required for understanding the strain-dependent transitions in the actomyosin cycle and for the development of three-dimensional models of muscle contraction.     

Cooperative [Ca²+]-dependent regulation of the rate of myosin binding to actin: solution data and the tropomyosin chain model

The regulation of muscle contraction by calcium involves interactions among actin filaments, myosin-S1, tropomyosin (Tm), and troponin (Tn). We have extended our previous model in which the TmTn regulatory units are treated as a continuous flexible chain, and applied it to transient kinetic data. We have measured the time course of myosin-S1 binding to actin-Tm-Tn filaments in solution at various calcium levels with [actin]/[myosin] ratios of 10 and 0.1, which exhibit modest slowing as [Ca(2+)] is reduced and a lag phase at low calcium. These observations can be explained if myosin binds to actin in two steps, where the first step is rate-limiting and blocked by TmTnI at low calcium, and the second step is fast, reversible, and controlled by the neighboring configuration of coupled tropomyosin-troponin units. The model can describe the calcium dependence of the observed myosin binding reactions and predicts cooperative calcium binding to TnC with competition between actin and Ca-TnC for the binding of TnI. Implications for theories of thin-filament regulation in muscle are discussed.     

Cooperative regulation of myosin-S1 binding to actin filaments by a continuous flexible Tm–Tn chain

The regulation of striated muscle contraction involves cooperative interactions between actin filaments, myosin-S1 (S1), tropomyosin (Tm), troponin (Tn), and calcium. These interactions are modeled by treating overlapping tropomyosins as a continuous flexible chain (CFC), weakly confined by electrostatic interactions with actin. The CFC is displaced locally in opposite directions on the actin surface by the binding of either S1 or Troponin I (TnI) to actin. The apparent rate constants for myosin and TnI binding to and detachment from actin are then intrinsically coupled via the CFC model to the presence of neighboring bound S1s and TnIs. Monte Carlo simulations at prescribed values of the CFC stiffness, the CFC??s degree of azimuthal confinement, and the angular displacements caused by the bound proteins were able to predict the stopped-flow transients of S1 binding to regulated F-actin. The transients collected over a large range of calcium concentrations could be well described by adjusting a single calcium-dependent parameter, the rate constant of TnI detachment from actin, k _?I. The resulting equilibrium constant $ K_{\text{B}} \equiv 1/K_{\text{I}} $ varied sigmoidally with the free calcium, increasing from 0.12 at low calcium (pCa >7) to 12 at high calcium (pCa <5.5) with a Hill coefficient of ~2.15. The similarity of the curves for excess-actin and excess-myosin data confirms their allosteric relationship. The spatially explicit calculations confirmed variable sizes for the cooperative units and clustering of bound myosins at low calcium concentrations. Moreover, inclusion of negative cooperativity between myosin units predicted the observed slowing of myosin binding at excess-myosin concentrations.