A synergetic theory of quadrupedal gaits and gait transitions

We present a theoretical analysis of the patterns of interlimb co-ordination in the gaits of quadrupedal locomotion. Introducing as collective variables a set of relative phases that describe the co-ordination patterns, we classify gaits by their symmetry properties, which can be expressed as invariances under groups of transformations. We define dynamics of the collective variables, on which we impose symmetry restrictions. The stable observable gait patterns correspond to atractors of these dynamics. A non-trivial consequence of this theoretical viewpoint is that gait transitions can take the form of non-equilibrium phase transitions that are accompanied by loss of stability. We show how various types of such phase transitions involving hysteresis, slowing down and fluctuation enhancement can occur. Also the difference between smooth and abrupt transitions is given theoretical foundation. While existing experimental evidence is consistent with the theory developed here, we propose new experimental measures that can serve to test the present theoretical framework. Finally, the influence of underlying symmetries of the dynamics on the nature of the gait patterns and their stability is analyzed. For example, breaking of a front-hind symmetry can lead to a change from absolute to relative co-ordination in the sense of von Holst (1939, Ergebnisse der Physiologie 42, 228). Also, differential stability of straight and reverse gaits results from thus lowering the symmetry.     

Effect of a light-induced pH gradient on purple-to-blue and purple-to-red transitions of bacteriorhodopsin

Bacteriorhodopsin-containing vesicles that were able to alkalize the extravesicular medium by greater than 1.5 pH units under illumination, i.e., inside-out vesicles, were reconstituted by reverse-phase evaporation with Halobacterium halobium polar lipids or exogenous phospholipids. Acid titration of a dark-adapted sample was accompanied by a color change from purple to blue (pKa = 2.5-4.5 in 0.15 M K2SO4), and alkali titration resulted in the formation of a red species absorbing maximally at 480 nm (pKa = 7 to greater than 9), the pKa values and the extents of these color changes being dependent on the nature of lipid. When a vesicle suspension at neutral or weakly acidic pH was irradiated by continuous light so that a large pH gradient was generated across the membrane, either a purple-to-blue or a purple-to-red transition took place. The light-induced purple-to-red transition was significant in an unbuffered vesicle suspension and correlated with the pH change in the extravesicular medium. The result suggests that the purple-to-red transition is driven from the extravesicular side, i.e., from the C-terminal membrane surface. In the presence of buffer molecules outside, the dominant color change induced in the light was the purple-to-blue transition, which seemed to be due to a large decrease in the intravesicular pH. But an apparently inconsistent result was obtained when the extravesicular medium was acidified by a HCl pulse, which was accompanied by a rapid color change to blue. We arrived at the following explanation: The two bR isomers, one containing all-trans-retinal and the other 13-cis-retinal, respond differently to pH changes in the extravesicular and the intravesicular medium. In this relation, full light adaptation was not achieved when the light-induced purple-to-blue transition was significant; i.e., only the 13-cis isomer is likely to respond to a pH change at the N-terminal membrane surface.     

Water potential gradient in a tall sequoiadendron

With an elevator installed in a 90-meter tall Sequoiadendron to collect the samples, xylem pressure potential measurements were made approximately every 15 meters along 60 meters of the tree's height. The measured gradient was about −0.8 bar per 10 meters of height, i.e., less than the hydrostatic gradient. Correction of the xylem pressure potential data by calibration against a thermocouple psychrometer confirmed this result. Similar gradients are described in the literature in tall conifers at times of low transpiration, although a different sampling technique was used. If the data in the present study and those supporting it are typical, they imply a re-evaluation of either the use of the pressure chamber to estimate water potential or the present theories describing water transport in tall trees.     

Mitochondrial membrane potential probes and the proton gradient: a practical usage guide

Fluorescent probes for monitoring mitochondrial membrane potential are frequently used for assessing mitochondrial function, particularly in the context of cell fate determination in biological and biomedical research. However, valid interpretation of results obtained with such probes requires careful consideration of numerous controls, as well as possible effects of non-protonic charges on dye behavior. In this context, we provide an overview of some of the important technical considerations, controls, and parallel complementary assays that can be employed to help ensure appropriate interpretation of results, thus providing a practical usage guide for monitoring mitochondrial membrane potentials with cationic probes. In total, this review will help illustrate both the strengths and potential pitfalls of common mitochondrial membrane potential dyes, and highlight best-usage approaches for their efficacious application in life sciences research.     

Transmural potential and a profile of intestinal motility in dogs]

Relationships between motor patterns of the jejunum and transmural potential changes were studied in the dog chronically fitted with transparietal electrodes and with an intraluminal catheter. The transmural potential increases above 13 mV at the beginning of the irregular spiking activity of a myoelectric complex and reaches 6 mV at the end of a phase of regular spiking activity. Oscillations of 3 mV are recorded during the phases of irregular spiking activity. A sustained increase of the potential occurred after feeding : 10 mV for a diet rich in protides and fat, 25 mV when glucides are predominant.     

Evaluation of a patient-specific cost function to predict the influence of foot path on the knee adduction torque during gait

A large external knee adduction torque during gait has been correlated with the progression of knee osteoarthritis (OA). Though foot path changes (e.g. toeing out) can reduce the adduction torque, no method currently exists to predict whether an optimal foot path exists for a specific patient. This study evaluates a patient-specific optimization cost function to predict how foot path changes influence both adduction torque peaks. Video motion and ground reaction data were collected from a patient with knee OA performing normal, toe out, and wide stance gait. Joint and inertial parameters in a dynamic, 27 degree-of-freedom, full-body gait model were calibrated to the patient's normal gait data. The model was then used in gait optimizations that predicted how the patient's adduction torque peaks would change due to changes in foot path. The cost function tracked the patient's normal gait data using weight factors calibrated to toe out gait and tested using wide stance gait. For both gait motions, the same cost function weights predicted the change in both adduction torque peaks to within 7% error. With further development, this approach may eventually permit the design of patient-specific rehabilitation procedures such as an optimal foot path for patients with knee OA.     

Modifications in choline transport activity as a function of membrane potential and the sodium gradient

Advantage was taken of a preparation of proteoliposomes made using Torpedo presynaptic membranes in which both the internal and external media can be controlled to investigate the effects of membrane potential and the Na+ gradient on choline transport activity. Under control conditions, Na+ outside and K+ inside, choline was concentrated by proteoliposomes and this phenomenon was sensitive to hemicholinium-3 and high levels of external choline. While proteoliposomes showed no permeability towards K+ spontaneously, in the presence of valinomycin a transmembrane potential was developed. The rate of transport was higher, the greater the inside negative potential. Both the affinity and the maximal velocity of high affinity transport rose in the presence of a potential. Likewise, the affinity and velocity of this transporter increased with increasing external Na+. Increasing internal Na+, on the other hand, caused a decrease in affinity and had little effect on the maximal velocity. The low affinity component was much less, if at all, affected by these changes. These results are consistent with a model of high affinity choline transport in which Na+ binds before choline and the carrier-Na+-choline complex is positively charged. However, these results do not provide a direct explanation for choline transport activation by nerve activity, underlining the need to study the effects of parameters other than membrane potential and the Na+ gradient on choline transport activity.     

Measurement of a growth-induced water potential gradient in tall fescue leaves

Spatial distribution of cell turgor pressure, cell osmotic pressure and relative elemental growth rate were measured in growing tall fescue leaves ( Festuca arundinacea ). Cell turgor pressure (measured with a pressure probe) was c . 0.55 MPa in expanding cells but increased steeply (+0.3 MPa) in cells where elongation had stopped. However, cell osmotic pressure (measured with a picolitre osmometer) was almost constant at 0.85 MPa throughout the leaf. The water potential difference between the growth zone and the mature zone (0.3 MPa) was interpreted as a growth-induced water potential gradient. This and further implications for the mechanism of growth control are discussed.     

Proton electrochemical gradient and phosphate potential in mitochondria

The paper reports an analysis of the relationship between deltamuH the proton electrochemical potential difference, and deltaGp, the phosphate potential. Depression of deltamuH and deltaGp has been obtained by titration with: (a) carbonylcyanide trifluoromethoxyphenylhydrazone; (b) nigericin (+ valinomycin); (c) KCl (+ valinomycin); and (d) rotenone. The uncoupler depresses deltamuH more than nigericin (+ valinomycin), KCl (+ valinomycin) and rotenone at equivalent deltaGp. The deltaGp/deltamuH ratio is about 3 at high values of deltamuH. When deltaGp and deltamuH are depressed by nigericin (4 valinomycin) the deltaGp/deltamuH ratio remains constant. When deltaGp and deltamuH are depressed by uncouplers, the deltaGp/deltamuH ratio increases hyperbolically tending to infinity while deltamuH tends to zero. The absence of constant proportionality between deltaGp and deltamuH indicates that the proton gradients driving ATP synthesis presumably operate within microscopic environments.     

Influence of the heart on the vertical gradient of transpulmonary pressure in dogs

Estimations were made of the vertical gradient of transpulmonary pressure (VGTP) from measurements of esophageal pressure in nine head-up dogs at functional residual capacity (FRC) when alive, when dead, and after total bilateral pneumothorax. The VGTP of 0.4 cmH2O/cm height in the alive state was abolished by pneumothorax, and roentgenograms showed that the heart moved in a caudal-dorsal direction. There was a small but significant increase in the VGTP on going from FRC to near total lung capacity (TLC) in alive head-up dogs. In eight dead head-up dogs heart weight was increased by replacing various amounts of heart blood with Hg. The VGTP was significantly increased from 0.28 to 0.51 cmH2O/cm height. The fractional increase in the VGTP was similar to the fractional increase in heart weight. In five dogs extrapolation to zero heart weight gave an average VGTP of 0.14 cmH2O/cm height. We conclude that the lungs help support the heart in the head-up dog and that the VGTP is in part determined by the pressure distribution required for this support.     

Functional traits composition predict macrophytes community productivity along a water depth gradient in a freshwater lake

Functional trait composition of plant communities has been proposed as a helpful key for understanding the mechanisms of biodiversity effects on ecosystem functioning. In this study, we applied a step‐wise modeling procedure to test the relative effects of taxonomic diversity, functional identity, and functional diversity on macrophytes community productivity along water depth gradient. We sampled 42 plots and 1513 individual plants and measured 16 functional traits and abundance of 17 macrophyte species. Results showed that there was a significant decrease in taxonomic diversity, functional identity (i.e., stem dry mass content, leaf [C] and leaf [N]), and functional diversity (i.e., floating leaf, mean Julian flowering date and rooting depth) with increasing water depth. For the multiple‐trait functional diversity (FD) indices, functional richness decreased, while functional divergence increased with water depth gradient. Macrophyte community productivity was strongly determined by functional trait composition within community, but not significantly affected by taxonomic diversity. Community‐weighted means (CWM) showed a two times higher explanatory power relative to FD indices in determining variations in community productivity. For nine of sixteen traits, CWM and FD showed significant correlations with community productivity, although the strength and direction of those relations depended on selected trait. Furthermore, functional composition in a community affected productivity through either additive or opposite effects of CWM and FD, depending on the particular traits being considered. Our results suggested both mechanisms of mass ratio and niche complementarity can operate simultaneously on variations in community productivity, and considering both CWM and FD would lead to a more profound understanding of traits–productivity relationships.     

Electric conductivity of tissue measured in the physiological potential gradient

An ability to measure the electric conductivity of tissue in physiological potential gradient is proposed. It has been shown that the resistance of plant leaf during the development of the injured potential is constant.     

Estimation of the pH gradient and donnan potential in de-energized heart mitochondria

The transmembrane pH gradient maintained by nonrespiring, uncoupled heart mitochondria has been estimated using the distribution of methylamine and of 5,5-dimethyl-2,4-oxazolidinedione (DMO) and compared with the delta pH reported by the fluorescent probe 2,7-biscarboxyethyl-5(6)-carboxyfluorescein (BCECF). Under these conditions the protonmotive force approaches zero and the membrane potential (delta psi) should equal 59 delta pH (P. Mitchell and J. Moyle (1969) Eur. J. Biochem. 7, 471-484). The delta pH reported by DMO corresponds closely to that estimated by BCECF and is consistent with a Donnan potential of no greater than about -30 mV (interior negative) for nonenergized mitochondria in a sucrose medium. This potential appears to result from the presence of immobile negative charges in the matrix and is eliminated by addition of 10 to 25 mM KCl. Measurements of delta pH using the methylamine and of delta tsi using the distribution of 42K+ in the presence of valinomycin result in an apparent overestimation of these parameters due to binding of these components to negative sites on the membrane. Increasing ionic strength decreases this contribution of surface potential, but significant binding can still be detected in 100 mM KCl. These studies suggest that 42K+ (or 86Rb+) is far from an ideal probe for measuring delta tsi in respiring mitochondria and may significantly overestimate this parameter, especially in sucrose media.     

Hydrotropism in roots: sensing of a gradient in water potential by the root cap

Roots of the agravitropic pea (Pisum sativum L.) mutant, ageotropum, responded to a gradient in water potential as small as 0.5 MPa by growing toward the higher water potential. This positive response occurred when a sorbitol-containing agar block was unilaterally applied to the root cap but not when applied to the elongation region. Unilateral application of higher concentrations of sorbitol to the elongation region caused root curvature toward the sorbitol source, presumably because of growth reduction on the water-stressed side. The control blocks of plain agar applied to either the root cap or the elongation region did not cause significant curvature of the roots. These results demonstrate that hydrotropism in roots occurs following perception of a gradient in water potential by the root cap.