The effects of pH on force and stiffness development in mouse muscles

Changes in force and stiffness during contractions of mouse extensor digitorum longus and soleus muscles were measured over a range of extracellular pH from 6.4 to 7.4. Muscle stiffness was measured using small amplitude (less than 0.1% of muscle length), high frequency (1.5 kHz) oscillations in length. Twitch force was not significantly affected by changes in pH, but the peak force during repetitive stimulation (2, 3, and 20 pulses) was decreased significantly as the pH was reduced. Changes in muscle stiffness with pH were in the same direction, but smaller in extent. If the number of attached cross-bridges in the muscle can be determined from the measurement of small amplitude, high frequency muscle stiffness, then these findings suggest that (a) the number of cross-bridges between thick and thin filaments declines in low pH and (b) the average force per cross-bridge also declines in low pH. The decline in force per cross-bridge could arise from a reduction in the ability of cross-bridges to generate force during their state of active force production and (or) in an increased percentage of bonds in a low force, "rigor" state.     

The distribution of end-to-end distances of the weakly bending rod model

The weakly bending rod model is an approximation to a worm-like chain in the limit where the ratio L(0)/P of the contour length L(0) to the persistence length P is not too large. The range of validity of the weakly bending rod model is investigated by deriving analytical expressions for its distribution of end-to-end distances P(L) and its moments and numerically comparing the results with corresponding values for the worm-like chain model. No general, closed form analytical expression for either P(L) or the average length of a worm-like chain exists, so those quantities are obtained by Monte Carlo simulations. Exact analytical expressions for and for the worm-like chain are employed in the comparison of the computed moments. Moments calculated for the approximate distributions of Daniels and Yamakawa and Stockmayer are also compared with the others. In addition, P(L) and its moments for the alternative model of Winkler et al. are compared with the others. The weakly bending rod model gives a reasonably good account of both P(L) and , m = 1,2,4, over the range L(0)/P /= 1.0. In contrast, the alternative model of Winkler et al. yields rather poor results in the rodlike domain.     

Structural changes in isometrically contracting insect flight muscle trapped following a mechanical perturbation

The application of rapidly applied length steps to actively contracting muscle is a classic method for synchronizing the response of myosin cross-bridges so that the average response of the ensemble can be measured. Alternatively, electron tomography (ET) is a technique that can report the structure of the individual members of the ensemble. We probed the structure of active myosin motors (cross-bridges) by applying 0.5% changes in length (either a stretch or a release) within 2 ms to isometrically contracting insect flight muscle (IFM) fibers followed after 5-6 ms by rapid freezing against a liquid helium cooled copper mirror. ET of freeze-substituted fibers, embedded and thin-sectioned, provides 3-D cross-bridge images, sorted by multivariate data analysis into ~40 classes, distinct in average structure, population size and lattice distribution. Individual actin subunits are resolved facilitating quasi-atomic modeling of each class average to determine its binding strength (weak or strong) to actin. ~98% of strong-binding acto-myosin attachments present after a length perturbation are confined to "target zones" of only two actin subunits located exactly midway between successive troponin complexes along each long-pitch helical repeat of actin. Significant changes in the types, distribution and structure of actin-myosin attachments occurred in a manner consistent with the mechanical transients. Most dramatic is near disappearance, after either length perturbation, of a class of weak-binding cross-bridges, attached within the target zone, that are highly likely to be precursors of strong-binding cross-bridges. These weak-binding cross-bridges were originally observed in isometrically contracting IFM. Their disappearance following a quick stretch or release can be explained by a recent kinetic model for muscle contraction, as behaviour consistent with their identification as precursors of strong-binding cross-bridges. The results provide a detailed model for contraction in IFM that may be applicable to contraction in other types of muscle.     

The affinity-enhancing roles of flexible linkers in two-domain DNA-binding proteins.

Recently many attempts have been made to design high-affinity DNA-binding proteins by linking two domains. Here a theory for guiding these designs is presented. Flexible linkers may play three types of roles: (a) linking domains which by themselves are unfolded and bind to DNA only as a folded dimer (as in a designed single-chain Arc repressor), (b) connecting domains which can separately bind to DNA (as in the Oct-1 POU domain), and (c) linking a DNA-binding domain with a dimerization domain (as in the lambda repressor). In (a), the linker keeps the protein as a folded dimer so that it is always DNA-binding-competent. In (b), the linker is predicted to enhance DNA-binding affinity over those of the individual domains (with dissociation constants K(A) and K(B)) by p(d(0))/K(B) or p(d(0))/K(A), where p(d(0)) = (3/4pil(p)bL)(3/2) exp(-3d(0)(2)/4l(p)bL)(1 - 5l(p)/4bL +...) is the probability density for the end-to-end vector of the linker with L residues to have a distance d(0). In (c), the linker is predicted to enhance the binding affinity by K(d)(C)/p(d(0)), where K(d)(C) is the dimer dissociation constant for the dimerization domain. The predicted affinity enhancements are found to be actually reached by the Oct-1 POU domain and lambda repressor. However, there is room for improvement in many of the recently designed proteins. The theoretical limits presented should provide a useful guide for current efforts of designing DNA-binding proteins.     

Effect of inorganic phosphate on the force and number of myosin cross-bridges during the isometric contraction of permeabilized muscle fibers from rabbit psoas

The relation between the chemical and mechanical steps of the myosin-actin ATPase reaction that leads to generation of isometric force in fast skeletal muscle was investigated in demembranated fibers of rabbit psoas muscle by determining the effect of the concentration of inorganic phosphate (Pi) on the stiffness of the half-sarcomere (hs) during transient and steady-state conditions of the isometric contraction (temperature 12°C, sarcomere length 2.5 μm). Changes in the hs strain were measured by imposing length steps or small 4 kHz oscillations on the fibers in control solution (without added Pi) and in solution with 3-20 mM added Pi. At the plateau of the isometric contraction in control solution, the hs stiffness is 22.8 ± 1.1 kPa nm⁻¹. Taking the filament compliance into account, the total stiffness of the array of myosin cross-bridges in the hs (e) is 40.7 ± 3.7 kPa nm⁻¹. An increase in [Pi] decreases the stiffness of the cross-bridge array in proportion to the isometric force, indicating that the force of the cross-bridge remains constant independently of [Pi]. The rate constant of isometric force development after a period of unloaded shortening (r_F) is 23.5 ± 1.0 s⁻¹ in control solution and increases monotonically with [Pi], attaining a maximum value of 48.6 ± 0.9 s⁻¹ at 20 mM [Pi], in agreement with the idea that Pi release is a relatively fast step after force generation by the myosin cross-bridge. During isometric force development at any [Pi], e and thus the number of attached cross-bridges increase in proportion to the force, indicating that, independently of the speed of the process that leads to myosin attachment to actin, there is no significant (>1 ms) delay between generation of stiffness and generation of force by the cross-bridges.     

Rate constant of muscle force redevelopment reflects cooperative activation as well as cross-bridge kinetics.

The rate of muscle force redevelopment after release-restretch protocols has previously been interpreted using a simple two-state cross-bridge cycling model with rate constants for transitions between non-force-bearing and force-bearing states, f, and between force-bearing and non-force-bearing states, g. Changes in the rate constant of force redevelopment, as with varying levels of Ca2+ activation, have traditionally been attributed to Ca(2+)-dependent f. The current work adds to this original model a state of unactivated, noncycling cross-bridges. The resulting differential equation for activated, force-bearing cross-bridges, Ncf, was Ncf = -[g+f(K/(K + 1))] Ncf+f(K/(K + 1))NT, where K is an equilibrium constant defining the distribution between cycling and noncycling cross-bridges and NT is the total number of cross-bridges. Cooperativity by which force-bearing cross-bridges participate in their own activation was introduced by making K depend on Ncf. Model results demonstrated that such cooperativity, which tends to enhance force generation at low levels of Ca2+ activation, has a counter-intuitive effect of slowing force redevelopment. These dynamic effects of cooperativity are most pronounced at low Ca2+ activation. As Ca2+ activation increases, the cooperative effects become less important to the dynamics of force redevelopment and, at the highest levels of Ca2+ activation, the dynamics of force redevelopment reflect factors other than cooperative mechanisms. These results expand on earlier interpretations of Ca2+ dependence of force redevelopment; rather than Ca(2+)-dependent f, Ca(2+)-dependent force redevelopment arises from changing expressions of cooperativity between force-bearing cross-bridges and activation.     

Microrheology of human lung epithelial cells measured by atomic force microscopy

Lung epithelial cells are subjected to large cyclic forces from breathing. However, their response to dynamic stresses is poorly defined. We measured the complex shear modulus (G(*)(omega)) of human alveolar (A549) and bronchial (BEAS-2B) epithelial cells over three frequency decades (0.1-100 Hz) and at different loading forces (0.1-0.9 nN) with atomic force microscopy. G(*)(omega) was computed by correcting force-indentation oscillatory data for the tip-cell contact geometry and for the hydrodynamic viscous drag. Both cell types displayed similar viscoelastic properties. The storage modulus G'(omega) increased with frequency following a power law with exponent approximately 0.2. The loss modulus G"(omega) was approximately 2/3 lower and increased similarly to G'(omega) up to approximately 10 Hz, but exhibited a steeper rise at higher frequencies. The cells showed a weak force dependence of G'(omega) and G"(omega). G(*)(omega) conformed to the power-law model with a structural damping coefficient of approximately 0.3, indicating a coupling of elastic and dissipative processes within the cell. Power-law behavior implies a continuum distribution of stress relaxation time constants. This complex dynamics is consistent with the rheology of soft glassy materials close to a glass transition, thereby suggesting that structural disorder and metastability may be fundamental features of cell architecture.     

Effects of actin-myosin kinetics on the calcium sensitivity of regulated thin filaments

Activation of thin filaments in striated muscle occurs when tropomyosin exposes myosin binding sites on actin either through calcium-troponin (Ca-Tn) binding or by actin-myosin (A-M) strong binding. However, the extent to which these binding events contributes to thin filament activation remains unclear. Here we propose a simple analytical model in which strong A-M binding and Ca-Tn binding independently activates the rate of A-M weak-to-strong binding. The model predicts how the level of activation varies with pCa as well as A-M attachment, N·k(att), and detachment, k(det), kinetics. To test the model, we use an in vitro motility assay to measure the myosin-based sliding velocities of thin filaments at different pCa, N·k(att), and k(det) values. We observe that the combined effects of varying pCa, N·k(att), and k(det) are accurately fit by the analytical model. The model and supporting data imply that changes in attachment and detachment kinetics predictably affect the calcium sensitivity of striated muscle mechanics, providing a novel A-M kinetic-based interpretation for perturbations (e.g. disease-related mutations) that alter calcium sensitivity.     

Characterization of cross-bridge elasticity and kinetics of cross-bridge cycling during force development in single smooth muscle cells

Force development in smooth muscle, as in skeletal muscle, is believed to reflect recruitment of force-generating myosin cross-bridges. However, little is known about the events underlying cross-bridge recruitment as the muscle cell approaches peak isometric force and then enters a period of tension maintenance. In the present studies on single smooth muscle cells isolated from the toad (Bufo marinus) stomach muscularis, active muscle stiffness, calculated from the force response to small sinusoidal length changes (0.5% cell length, 250 Hz), was utilized to estimate the relative number of attached cross-bridges. By comparing stiffness during initial force development to stiffness during force redevelopment immediately after a quick release imposed at peak force, we propose that the instantaneous active stiffness of the cell reflects both a linearly elastic cross-bridge element having 1.5 times the compliance of the cross-bridge in frog skeletal muscle and a series elastic component having an exponential length-force relationship. At the onset of force development, the ratio of stiffness to force was 2.5 times greater than at peak isometric force. These data suggest that, upon activation, cross-bridges attach in at least two states (i.e., low-force-producing and high-force-producing) and redistribute to a steady state distribution at peak isometric force. The possibility that the cross-bridge cycling rate was modulated with time was also investigated by analyzing the time course of tension recovery to small, rapid step length changes (0.5% cell length in 2.5 ms) imposed during initial force development, at peak force, and after 15 s of tension maintenance. The rate of tension recovery slowed continuously throughout force development following activation and slowed further as force was maintained. Our results suggest that the kinetics of force production in smooth muscle may involve a redistribution of cross-bridge populations between two attached states and that the average cycling rate of these cross-bridges becomes slower with time during contraction.     

The extensive length-force relationship of porcine airway smooth muscle.

The full functional length range of trachealis muscle was measured to identify a precise reference length and to assess the length changes that the myofilament lattice can accommodate. The initial reference length (L(10%)) was that where rest tension equaled 10% of total force (passive tension plus active force). Total force at this length served as a force reference (F(ref) = 219 +/- 12 kPa, N = 7). Muscles initially adapted at L(10%) for 30-60 min had no rest tension when shortened to <0.9 L(10%). Passive tension rose steeply and linearly with slope 11.2 F(ref)/L(10%) at lengths >1.04 L(10%). Rest tension at 1.1 L(10%) declined by <10% over 1 h. The steep slope and stability of rest tension at long lengths suggest that a parameter of the slope could serve as a precise, reproducible reference length. Active force was nearly constant at lengths 0.33-1.0 L(10%) and declined steeply at lengths between 0.1 and 0.2 L(10%), extrapolating to zero at 0.076 L(10%). Muscles visibly reextended during relaxation at lengths <0.25 L(10%). At long lengths, force extrapolated to zero at 1.175 L(10%). The >15-fold length range (0.076-1.175 L(10%)) for force generation and nearly constant force over a greater than threefold length range is likely produced by several structural accommodations, including filament sliding, an increased number of sliding filaments in series, and increased length of passive structures in series with the sliding filaments. Visible reextension during relaxation suggests that the lattice does not undergo plastic adaptations at lengths <25% L(10%) and that lattice plasticity is limited to a three- to fourfold length range.     

Structural and functional studies of the HAMP domain of EnvZ, an osmosensing transmembrane histidine kinase in Escherichia coli

The HAMP domain plays an essential role in signal transduction not only in histidine kinase but also in a number of other signal-transducing receptor proteins. Here we expressed the EnvZ HAMP domain (Arg(180)-Thr(235)) with the R218K mutation (termed L(RK)) or with L(RK) connected with domain A (Arg(180)-Arg(289)) (termed LA(RK)) of EnvZ, an osmosensing transmembrane histidine kinase in Escherichia coli, by fusing it with protein S. The L(RK) and LA(RK) proteins were purified after removing protein S. The CD analysis of the isolated L protein revealed that it consists of a random structure or is unstructured. This suggests that the EnvZ HAMP domain by itself is unable to form a stable structure and that this structural fragility may be important for its role in signal transduction. Interestingly the substitution of Ala(193) in the EnvZ HAMP domain with valine or leucine in Tez1A1, a chimeric protein of Tar and EnvZ, caused a constitutive OmpC phenotype. The CD analysis of LA(RK)(A193L) revealed that this mutated HAMP domain possesses considerable secondary structures and that the thermostability of this entire LA(RK)(A193L) became substantially lower than that of LA(RK) or just domain A, indicating that the structure of the HAMP domain with the A193L mutation affects the stability of downstream domain A. This results in cooperative thermodenaturation of domain A with the mutated HAMP domain. These results are discussed in light of the recently solved NMR structure of the HAMP domain from a thermophilic bacterium (Hulko, M., Berndt, F., Gruber, M., Linder, J. U., Truffault, V., Schultz, A., Martin, J., Schultz, J. E., Lupas, A. N., and Coles, M. (2006) Cell 126, 929-940).     

Force: velocity relationship in single isolated toad stomach smooth muscle cells

The relationship between force and shortening velocity (F:V) in muscle is believed to reflect both the mechanics of the myosin cross-bridge and the kinetics of its interaction with actin. To date, the F:V for smooth muscle cells has been inferred from F:V data obtained in multicellular tissue preparations. Therefore, to determine F:V in an intact single smooth muscle cell, cells were isolated from the toad (Bufo marinus) stomach muscularis and attached to a force transducer and length displacement device. Cells were electrically stimulated at 20 degrees C and generated 143 mN/mm2 of active force per muscle cross-sectional area. At the peak of contraction, cells were subjected to sudden changes in force (dF = 0.10-0.90 Fmax) and then maintained at the new force level. The force change resulted in a length response in which the cell length (Lcell) rapidly decreased during the force step and then decreased monotonically with a time constant between 75 and 600 ms. The initial length change that coincided with the force step was analyzed and an active cellular compliance of 1.9% cell length was estimated. The maintained force and resultant shortening velocity (V) were fitted to the Hill hyperbola with constants a/Fmax of 0.268 and b of 0.163 Lcell/s. Vmax was also determined by a procedure in which the cell length was slackened and the time of unloaded shortening was recorded (slack test). From the slack test, Vmax was estimated as 0.583 Lcell/s, in agreement with the F:V data. The F:V data were analyzed within the framework of the Huxley model (Huxley. 1957. Progress in Biophysics and Biophysical Chemistry. 7:255-318) for contraction and interpreted to indicate that in smooth muscle, as compared with fast striated muscle, there may exist a greater percentage of attached force-generating cross-bridges.     

An approximate matching algorithm for finding (sub-)optimal sequences in S-attributed grammars

MOTIVATION: S-attributed grammars (a generalization of classical Context-Free grammars) provide a versatile formalism for sequence analysis which allows to express long range constraints: the RNA folding problem is a typical example of application. Efficient algorithms have been developed to solve problems expressed with these tools, which generally compute the optimal attribute of the sequence w.r.t. the grammar. However, it is often more meaningful and/or interesting from the biological point of view to consider almost optimal attributes as well as approximate sequences; we thus need more flexible and powerful algorithms able to perform these generalized analyses. RESULTS: In this paper we present a basic algorithm which, given a grammar G and a sequence omega, computes the optimal attribute for all (approximate) strings omega(') in L(G) such that d(omega, omega(')) < or = M, and whose complexity is O(n(r + 1)) in time and O(n(2)) in space (r is the maximal length of the right-hand side of any production of G). We will also give some extensions and possible improvements of this algorithm.     

The frequency response of smooth muscle stiffness during Ca2+-activated contraction

To investigate the mechanism of smooth muscle contraction, the frequency response of the muscle stiffness of single beta-escin permeabilized smooth muscle cells in the relaxed state was studied. Also, the response was continuously monitored for 3 min from the beginning of the exchange of relaxing solution to activating solution, and then at 5-min intervals for up to 20 min. The frequency response (30 Hz bandwidth, 0.33 Hz (or 0.2 Hz) resolution) was calculated from the Fourier-transformed force and length sampled during a 3-s (or 5-s) constant-amplitude length perturbation of increasing-frequency (1-32 Hz) sine waves. In the relaxed state, a large negative phase angle was observed, which suggests the existence of attached energy generating cross-bridges. As the activation progressed, the muscle stiffness and phase angle steadily increased; these increases gradually extended to higher frequencies, and reached a steady state by 100 s after activation or approximately 40 s after stiffness began to increase. The results suggest that a fixed distribution of cross-bridge states was reached after 40 s of Ca2+ activation and the cross-bridge cycling rate did not change during the period of force maintenance.     

An on/off resonance rotating frame relaxation experiment to monitor millisecond to microsecond timescale dynamics

Rotating frame relaxation experiments in proteins are used to study slow motions on the microsecond to millisecond timescale. An on/off resonance rotating frame relaxation experiment (R(1)(rho)) has been developed that incorporates adiabatic rotations into a R(1)(rho)-R(1) constant relaxation time experiment with weak radio frequency field strengths in order to effectively lock the magnetization over a wide range of (15)N frequencies. The new pulse sequence allows the measurement of a wide range of chemical exchange timescales on the order of 1.0 to 0.05 ms over an asymmetric bandwidth from +1.7omega(l) to -0.5omega(l) in a single experiment. A total bandwidth of +/-l.7omega(l) is obtained by performing the experiment a second time with a reversed adiabatic rotation.     

Evidence for increased low force cross-bridge population in shortening skinned skeletal muscle fibers: implications for actomyosin kinetics.

The dynamic characteristics of the low force myosin cross-bridges were determined in fully calcium-activated skinned rabbit psoas muscle fibers shortening under constant loads (0.04-0.7 x full isometric tension Po). The shortening was interrupted at various times by a ramp stretch (duration, 10 ms; amplitude, up to 1.8% fiber length) and the resulting tension response was recorded. Except for the earlier period of velocity transients, the tension response showed nonlinear dependence on stretch amplitude; i.e., the magnitude of the tension response started to rise disproportionately as the stretch exceeded a critical amplitude, as in the presence of inorganic phosphate (Pi). This result, as well as the result of stiffness measurement, suggests that the low force cross-bridges similar to those observed in the presence of Pi (presumably A.M.ADP.Pi) are significantly populated during shortening. The critical amplitude of the shortening fibers was greater than that of isometrically contracting fibers, suggesting that the low force cross-bridges are more negatively strained during shortening. As the load was reduced from 0.3 to 0.04 P0, the shortening velocity increased more than twofold, but the amount of the negative strain stayed remarkably constant (approximately 3 nm). This This insensitiveness of the negative strain to velocity is best explained if the dissociation of the low force cross-bridges is accelerated approximately in proportion to velocity. Along with previous reports, the results suggest that the actomyosin ATPase cycle in muscle fibers has at least two key reaction steps in which rate constants are sensitively regulated by shortening velocity and that one of them is the dissociation of the low force A.M.ADP.Pi cross-bridges. This step may virtually limit the rate of actomyosin ATPase turnover and help increase efficiency in fibers shortening at high velocities.     

Groucho running

An important determinant of the mechanics of running is the effective vertical stiffness of the body. This stiffness increases with running speed. At any one speed, the stiffness may be reduced in a controlled fashion by running with the knees bent more than usual. In a series of experiments, subjects ran in both normal and flexed postures on a treadmill. In other experiments, they ran down a runway and over a force platform. Results show that running with the knees bent reduces the effective vertical stiffness and diminishes the transmission of mechanical shock from the foot to the skull but requires an increase of as much as 50% in the rate of O2 consumption. A new dimensionless parameter (u omega 0/g) is introduced to distinguish between hard and soft running modes. Here, omega 0 is the natural frequency of a mass-spring system representing the body, g is gravity, and u is the vertical landing velocity. In normal running, this parameter is near unity, but in deep-flexed running, where the aerial phase of the stride cycle almost disappears, u omega 0/g approaches zero.     

Molecular Basis of Force Development by Skeletal Muscles During and After Stretch

When activated skeletal muscles are stretched at slow velocities, force increases in two phases: (i) a fast increase, and (ii) a slow increase. The transition between these phases is commonly associated with the mechanical detachment of cross-bridges from actin. This phenomenon is referred to asforce enhancement during stretch. After the stretch, force decreases and reaches steady-state at levels that are higher than the force produced at the corresponding length during purely isometric contractions. This phenomenon is referred to asresidual force enhancement.The mechanisms behind the increase in force during and after stretch are still a matter of debate, and have physiological implications as human muscles perform stretch contractions continuously during daily activity. This paper briefly reviews the potential mechanisms to explain stretch forces, including an increased number of cross-bridges attached to actin, an increased strain in cross-bridges upon stretch, the influence of passive elements upon activation and sarcomere length non-uniformities.     

Walking and running at resonance

Humans and other animals can temporarily store mechanical energy in elastic oscillations, f(el), of body parts and in pendulum oscillations, f(p) = const sq.rt (g/L), of legs, length L, or other appendages, and thereby reduce the energy consumption of locomotion. However, energy saving only occurs if these oscillations are tuned to the leg propagation frequency f. It has long been known that f is tuned to the pendulum frequency of the free-swinging leg of walkers. During running the leg frequency increases to some new value f = f(r). We propose that in order to maintain resonance the animal, mass M, actively increases its leg pendulum frequency to the new value f(p,r) =const sq.rt (a(y)/L)=f(r), by giving its hips a vertical acceleration a(y)= F(y)/M. The pendulum frequency is increased if the impact force F(y) of the stance foot is larger than Mg, explaining the observation by Alexander and Bennet-Clark (1976) that F(v) becomes larger than Mg when animals start to run. Our model predictions of the running velocity U(r) as function of L, F(v), are in agreement with measurements of these quantities (Farley et al. 1993). The leg's longitudinal elastic oscillation frequency scales as f(el) = const sq.rt (k/M). Experiments by Ferris et al., (1998) show that runners adjust their leg's stiffness, k, when running on surfaces of different elasticity so that the total stiffness k remains constant. Our analysis of their data suggests that the longitudinal oscillations of the stance leg are indeed kept in tune with the running frequency. Therefore we conclude that humans, and by extension all animals, maintain resonance during running. Our model also predicts the Froude number of walking-running transitions, Fr = U(2)/gL approximately 0.5 in good agreement with measurements.