Dynamic similarity during human running: About Froude and Strouhal dimensionless numbers

Dynamic similarity is a widely used concept in the fluid mechanics field, and consists in placing two different-sized systems in equivalent experimental conditions. This enables removal of the effects of size and prediction of the behavior of a full size system from a scale model. The aim of this study was to test whether the Froude number (Nfr) or the Strouhal number (Str) could be used as a criterion for dynamic similarity during running. Fifteen male subjects ran barefoot on a runway in three experimental conditions (i) all subjects ran at the same speed V=3.5 m s^?1; (ii) the speed was determined from Nfr; (iii) the stride frequency was determined from Str. Antero-posterior (Fy) and vertical (Fz) ground reaction force components were assessed. The similarity between the subjects was analysed from scale factor sets computed from anthropometric and kinetic data. The use of Str implied strong inter-subject similarity for temporal parameters (mean r=0.96, time to Fz peak, time to Fy braking peak, Fy zero fore–aft shear, time to Fy propulsive peak) while Nfr induced fewer and lower similarities (mean r=0.75, Fy zero fore–aft shear, time to Fy propulsive peak, Fy braking impulse) that only concerned antero-posterior parameters. This study brought experimental evidence that neither Nfr nor Str were sufficient for dynamic similarity during running, but that each of them made its own contribution. These findings suggested that the concomitant use of Nfr and Str should be assessed to induce inter-subject dynamic similarity during running.     

Criteria for dynamic similarity in bouncing gaits

Animals of different sizes tend to move in a dynamically similar manner when travelling at speeds corresponding to equal values of a dimensionless parameter (DP) called the Froude number. Consequently, the Froude number has been widely used for defining equivalent speeds and predicting speeds of locomotion by extinct species and on other planets. However, experiments using simulated reduced gravity have demonstrated that equality of the Froude number does not guarantee dynamic similarity. This has cast doubt upon the usefulness of the Froude number in locomotion research. Here we use dimensional analysis of the planar spring-mass model, combined with Buckingham's Pi-Theorem, to demonstrate that four DPs must be equal for dynamic similarity in bouncing gaits such as trotting, hopping and bipedal running. This can be reduced to three DPs by applying the constraint of maintaining a constant average speed of locomotion. Sensitivity analysis indicates that all of these DPs are important for predicting dynamic similarity. We show that the reason humans do not run in a dynamically similar manner at equal Froude number in different levels of simulated reduced gravity is that dimensionless leg stiffness decreases as gravity increases. The reason that the Froude number can predict dynamic similarity in Earth gravity is that dimensionless leg stiffness and dimensionless vertical landing speed are both independent of size. In conclusion, although equal Froude number is not sufficient for dynamic similarity, it is a necessary condition. Therefore, to detect fundamental differences in locomotion, animals of different sizes should be compared at equal Froude number, so that they can be as close to dynamic similarity as possible. More generally, the concept of dynamic similarity provides a powerful framework within which similarities and differences in locomotion can be interpreted.     

Biomechanical maturation of joint dynamics during early childhood: Updated conclusions

Dynamic parameters have been commonly explored to characterize the biomechanical maturation of children's gaits, i.e., age-revealing joint moment and power patterns similar to adult patterns. However, the literature revealed a large disparity of conclusions about maturation depending on the study, which was most likely due to an inappropriate scaling strategy and uncontrolled walking speed. With the first years of independent walking, a large growth in height and a large variability of dimensionless walking speed are observed. Moreover, the dynamic parameters were not well studied during early childhood.     

Humans control stride-to-stride stepping movements differently for walking and running,independent of speed

As humans walk or run, external (environmental) and internal (physiological) disturbances induce variability. How humans regulate this variability from stride-to-stride can be critical to maintaining balance. One cannot infer what is “controlled” based on analyses of variability alone. Assessing control requires quantifying how deviations are corrected across consecutive movements. Here, we assessed walking and running, each at two speeds. We hypothesized differences in speed would drive changes in variability, while adopting different gaits would drive changes in how people regulated stepping. Ten healthy adults walked/ran on a treadmill under four conditions: walk or run at comfortable speed, and walk or run at their predicted walk-to-run transition speed. Time series of relevant stride parameters were analyzed to quantify variability and stride-to-stride error-correction dynamics within a Goal-Equivalent Manifold (GEM) framework. In all conditions, participants’ stride-to-stride control respected a constant-speed GEM strategy. At each consecutively faster speed, variability tangent to the GEM increased (p ≤ 0.031), while variability perpendicular to the GEM decreased (p ≤ 0.044). There were no differences (p ≥ 0.999) between gaits at the transition speed. Differences in speed determined how stepping variability was structured, independent of gait, confirming our first hypothesis. For running versus walking, measures of GEM-relevant statistical persistence were significantly less (p ≤ 0.004), but showed minimal-to-no speed differences (0.069 ≤ p ≤ 0.718). When running, people corrected deviations both more quickly and more directly, each indicating tighter control. Thus, differences in gait determined how stride-to-stride fluctuations were regulated, independent of speed, confirming our second hypothesis.     

Spatio-temporal gait characteristics of the hind-limb cycles during voluntary bipedal and quadrupedal walking in bonobos (Pan paniscus)

Spatio-temporal gait characteristics (step and stride length, stride frequency, duty factor) were determined for the hind-limb cycles of nine bonobos (Pan paniscus) walking quadrupedally and bipedally at a range of speeds. The data were recalculated to dimensionless quantities according to the principle of dynamic similarity. Lower leg length was used as the reference length. Interindividual variability in speed modulation strategy of bonobos appears to be low. Compared to quadrupedal walking, bipedal bonobos use smaller steps to attain a given speed (differences increase with speed), resulting in shorter strides at a higher frequency. In the context of the ("hybrid") dynamic pattern approach to locomotion (Latach, 1998) we argue that, despite these absolute differences, intended walking speed is the basic control variable which elicits both quadrupedal and bipedal walking kinematics in a similar way. Differences in the initial status of the dynamic system may be responsible for the differences in step length between both gaits. Comparison with data deduced from the literature shows that the effects of walking speed on stride length and frequency are similar in bonobos, common chimpanzees, and humans. This suggests that (at least) within extant homininae, spatio-temporal gait characteristics are highly comparable, and this in spite of obvious differences in mass distribution and bipedal posture.     

Shaking a leg and hot to trot: the effects of body size and temperature on running speed in ants

Abstract.  1. Data were compiled from the literature and our own studies on 24 ant species to characterise the effects of body size and temperature on forager running speed.
2. Running speed increases with temperature in a manner consistent with the effects of temperature on metabolic rate and the kinetic properties of muscles.
3. The exponent of the body mass-running speed allometry ranged from 0.14 to 0.34 with a central tendency of approximately 0.25. This body mass scaling is consistent with both the model of elastic similarity, and a model combining dynamic similarity with available metabolic power.
4. Even after controlling for body size or temperature, a substantial amount of inter-specific variation in running speed remains. Species with certain lifestyles [e.g. nomadic group predators, species which forage at extreme (>60 °C) temperatures] may have been selected for faster running speeds.
5. Although ants have a similar scaling exponent to mammals for the running speed allometry, they run slower than predicted compared with a hypothetical mammal of similar size. This may in part reflect physiological differences between invertebrates and vertebrates.     

A comparative collision-based analysis of human gait

This study compares human walking and running, and places them within the context of other mammalian gaits. We use a collision-based approach to analyse the fundamental dynamics of the centre of mass (CoM) according to three angles derived from the instantaneous force and velocity vectors. These dimensionless angles permit comparisons across gait, species and size. The collision angle Φ, which is equivalent to the dimensionless mechanical cost of transport CoT_mech, is found to be three times greater during running than walking of humans. This threefold difference is consistent with previous studies of walking versus trotting of quadrupeds, albeit tends to be greater in the gaits of humans and hopping bipeds than in quadrupeds. Plotting the collision angle Φ together with the angles of the CoM force vector Θ and velocity vector Λ results in the functional grouping of bipedal and quadrupedal gaits according to their CoM dynamics—walking, galloping and ambling are distinguished as separate gaits that employ collision reduction, whereas trotting, running and hopping employ little collision reduction and represent more of a continuum that is influenced by dimensionless speed. Comparable with quadrupedal mammals, collision fraction (the ratio of actual to potential collision) is 0.51 during walking and 0.89 during running, indicating substantial collision reduction during walking, but not running, of humans.     

The evolution of thermal performance curves in semi-aquatic newts: Thermal specialists on land and thermal generalists in water?

The position and shape of thermal performance curves (TPCs, the functions relating temperature to physiological performance) for ecologically relevant functions will directly affect the fitness of ectotherms and therefore should be under strong selection. However, thermodynamic considerations predict that relationships between the different components of the TPC will confound its evolutionary optimization. For instance, the “jack-of-all-temperatures” hypothesis predicts a trade-off between the breadth of the TPC and the maximal performance capacity; the “warmer is better” hypothesis suggests that low thermal optima will come with low absolute performances. Semi-aquatic organisms face the additional challenge of having to adjust their TPCs to two environments that are likely to differ in mean temperature and thermal variability. In this paper, we examine how parameters of the TPCs for maximal running and swimming speed have co-evolved in the semi-aquatic newt genus Triturus. We consider evolutionary relationships between the width and the height of the TPCs, the optimal temperatures and maximal performance. Phylogenetic comparative analyses reveal that in Triturus, swimming and running differ substantially in the (co-)variation of TPC parameters. Whereas evolutionary changes in the TPC for swimming primarily concern the shape of the curve (generalist versus specialist), most interspecific variation in running speed TPCs involves shifts in overall performance across temperatures.     

Running on flat turns: experiments, theory, and applications

Theoretical and experimental results are presented which demonstrate the mechanical effects of running along a circular turn. The theory is a simple one-parameter model, requiring only the top speed Vo of the runner as an input. The dimensionless parameter (Rg/v2o), a reciprocal Froude number or dimensionless parameter (Rg/v20), a reciprocal Froude number or dimensionless radius, appears as a natural result of the theory. This radial Froude number allows for the comparison of the theory and experiment for a large number of individuals on the same set of axes. The parameters of speed, foot contact time, ballistic air time, step length, stride length, and stride time are all predicted and measured for 23 different subjects. The agreement between theory and experiment is good. Exact solutions and approximate asymptotic results for the speed-radius relation are presented. Applications are made to the practical problem of the design of indoor and outdoor running tracks for athletic competition.     

Pairs of Genetically Unrelated Look-Alikes

Relationships of physical resemblance to personality similarity and social affiliation have generated considerable discussion among behavioral science researchers. A “twin-like” experimental design (involving genetically unrelated look-alikes, U-LAs) explores associations among resemblance in appearance, the Big Five personality traits, self-esteem, and social attraction within an evolutionary framework. The Personality for Professionals Inventory (PfPI), NEO/NEO-FFI-3, Rosenberg Self-Esteem Scale, and a Social Relationship Survey were variously completed by 45 U-LA pairs, identified from the “I’m Not a Look-Alike” project, Mentorn Media, and personal referrals. The mean U-LA intraclass correlations were negligible for all Big Five personality traits on the PfPI and NEO/NEO-FFI-3 (r_i =??.02 and???.04, respectively). In contrast, mean r_i values of .53 and .15 for monozygotic (MZA) and dizygotic (DZA) reared-apart twins, respectively, have been reported for these personality measures. The U-LA self-esteem correlation (r_i =??.18) was also below the correlations reported for MZ and DZ reared-together twins (r_{i =} .31 and .13, respectively). Finally, far fewer U-LAs expressed close social relationships (20%) than MZA (80%) and DZA (65%) twins. The present study extends earlier findings indicating that appearance is not meaningfully related to personality similarity and social relatedness. The criticism that MZ twins are alike in personality because their matched looks invite similar treatment by others is refuted. A more judicious interpretation is reactive genotype-environment correlation, namely that MZ twins’ similar personalities evoke similar reactions from others. MZ twins’ close social relations most likely derive from their perceptions of genetically based within-pair similarities that are lacking in U-LAs.     

Using body size to understand the structural design of animals: quadrupedal locomotion.

Many parameters of gait and performance, including stride frequency, stride length, maximum speed, and rate of O2 uptake are experimentally found to be power-law functions of body weight in running quadrupeds. All of these parameters are reasonably easy to measure except maximum speed, where the question arises whether one means top sprinting speed or top speed for sustained running. Moreover, differences in training and motivation make comparisons of top speed difficult. The problem is circumvented by comparing animals running at the transition between trotting and galloping, a physiologically similar speed. Theoretical models are proposed which preserve either geometric similarity, elastic similarity, or static stress similarity between animals of large and small body weights. The model postulating elastic similarity provides the best correlation with published data on body and bone proportions, body surface area, resting metabolic rate, and basal heart and lung frequencies. It also makes the most successful prediction of stride frequency, stride length, limb excursion angles, and the metabolic power required for running at the trot-gallop transition in quadrupeds ranging in size from mice to horses.     

Walking at the preferred stride frequency maximizes local dynamic stability of knee motion

Healthy humans display a preference for walking at a stride frequency dependent on the inertial properties of their legs. Walking at preferred stride frequency (PSF) is predicted to maximize local dynamic stability, whereby sensitivity to intrinsic perturbations arising from natural variability inherent in biological motion is minimized. Previous studies testing this prediction have employed different variability measures, but none have directly quantified local dynamic stability by computing maximum finite-time Lyapunov exponent (λ^Max), which quantifies the rate of divergence of nearby trajectories in state space. Here, ten healthy adults walked 45 m overground while sagittal motion of both knees was recorded via electrogoniometers. An auditory metronome prescribed 7 different frequencies relative to each individual's PSF (PSF; ±5, ±10, ±15 strides/min). Stride frequencies were performed under both freely adopted speed (FS) and controlled speed (CS: set at the speed of PSF trials) conditions. Local dynamic stability was maximal (λ^Max was minimal) at the PSF, becoming less stable for higher and lower stride frequencies. This occurred under both FS and CS conditions, although controlling speed further reduced local dynamic stability at non-preferred stride frequencies. In contrast, measures of variability revealed effects of stride frequency and speed conditions that were distinct from λ^Max. In particular, movement regularity computed by approximate entropy (ApEn) increased for slower walking speeds, appearing to depend on speed rather than stride frequency. The cadence freely adopted by humans has the benefit of maximizing local dynamic stability, which can be interpreted as humans tuning to their resonant frequency of walking.     

Fractal fluctuations in spatiotemporal variables when walking on a self-paced treadmill

This study investigated the fractal dynamic properties of stride time (ST), stride length (SL) and stride speed (SS) during walking on a self-paced treadmill (STM) in which the belt speed is automatically controlled by the walking speed. Twelve healthy young subjects participated in the study. The subjects walked at their preferred walking speed under four conditions: STM, STM with a metronome (STM+met), fixed-speed (conventional) treadmill (FTM), and FTM with a metronome (FTM+met). To compare the fractal dynamics between conditions, the mean, variability, and fractal dynamics of ST, SL, and SS were compared. Moreover, the relationship among the variables was examined under each walking condition using three types of surrogates. The mean values of all variables did not differ between the two treadmills, and the variability of all variables was generally larger for STM than for FTM. The use of a metronome resulted in a decrease in variability in ST and SS for all conditions. The fractal dynamic characteristics of SS were maintained with STM, in contrast to FTM, and only the fractal dynamic characteristics of ST disappeared when using a metronome. In addition, the fractal dynamic patterns of the cross-correlated surrogate results were identical to those of all variables for the two treadmills. In terms of the fractal dynamic properties, STM walking was generally closer to overground walking than FTM walking. Although further research is needed, the present results will be useful in research on gait fractal dynamics and rehabilitation.     

Intra-limb coordination while walking is affected by cognitive load and walking speed

Knowledge about intra-limb coordination (ILC) during challenging walking conditions provides insight into the adaptability of central nervous system (CNS) for controlling human gait. We assessed the effects of cognitive load and speed on the pattern and variability of the ILC in young people during walking. Thirty healthy young people (19 female and 11 male) participated in this study. They were asked to perform 9 walking trials on a treadmill, including walking at three paces (preferred, slower and faster) either without a cognitive task (single-task walking) or while subtracting 1?s or 3?s from a random three-digit number (simple and complex dual-task walking, respectively). Deviation phase (DP) and mean absolute relative phase (MARP) values—indicators of variability and phase dynamic of ILC, respectively—were calculated using the data collected by a motion capture system. We used a two-way repeated measure analysis of variance for statistical analysis. The results showed that cognitive load had a significant main effect on DP of right shank–foot and thigh–shank, left shank–foot and pelvis–thigh (p<0.05), and MARP of both thigh–shank segments (p<0.01). In addition, the main effect of walking speed was significant on DP of all segments in each side and MARP of both thigh–shank and pelvis–thigh segments (p<0.001). The interaction of cognitive load and walking speed was only significant for MARP values of left shank–foot and right pelvis–thigh (p<0.05 and p<0.001, respectively). We suggest that cognitive load and speed could significantly affect the ILC and variability and phase dynamic during walking.     

A new dimensionless number highlighted from mechanical energy exchange during running

This study aimed to highlight a new dimensionless number from mechanical energy transfer occurring at the centre of gravity (Cg) during running. We built two different-sized spring-mass models (SMM #1 and SMM #2). SMM #1 was built from the previously published data, and SMM #2 was built to be dynamically similar to SMM #1. The potential gravitational energy (E(P)), kinetic energy (E(K)), and potential elastic energy (E(E)) were taken into account to test our hypothesis. For both SMM #1 and SMM #2, N(Mo-Dela)=(E(P)+E(K))/E(E) reached the same mean value and was constant (4.1+/-0.7) between 30% and 70% of contact time. Values of N(Mo-Dela) obtained out of this time interval were due to the absence of E(E) at initial and final times of the simulation. This phenomenon does not occur during in vivo running because a leg muscle's pre-activation enables potential elastic energy storage prior to ground contact. Our findings also revealed that two different-sized spring-mass models bouncing with equal N(Mo-Dela) values moved in a dynamically similar fashion. N(Mo-Dela), which can be expressed by the combination of Strouhal and Froude numbers, could be of great interest in order to study animal and human locomotion under Earth's gravity or to induce dynamic similarity between different-sized individuals during bouncing gaits.     

Cold-induced genetic instability in micropropagated Pistacia lentiscus L. plantlets

Genetic stability of plants during in vitro propagation and conservation is one of the important aspects of plant biotechnology. In the present study, micropropagated P. lentiscus L. shoot cultures, which are cultivated for the mastic resin, have been cold stored up to 12 months at 4 °C in the dark for different durations (2, 4, 6, 8, 10 and 12 months) and genetic alterations in cold storage conditions were evaluated. Growth parameters such as proliferation rate, shoot numbers per explant, shoot lengths and shoot forming capacity were also calculated. Since the highest proliferation rate (100 %) was obtained in 6?month-stored shoot cultures without any severe influence of cold stress on proliferation ability, amplified fragment length polymorphism (AFLP) and inter-retrotransposon amplified polymorphism (IRAP) marker systems were used to determine genetic stability in the plantlets after this storage period. Totally, 702 scorable bands were produced by 10 AFLP primer pairs. Genetic similarity value of the non-stored (control) plant and cold-stored clones ranged from 0.66 to 0.84 with a mean of 0.74. In the case of IRAP, 159 bands were produced by 8 IRAP primers. Genetic similarity value of the non-stored plant and cold-stored clones varied from 0.65 to 0.83 and the average genetic similarity value was determined as 0.72. The genetic similarity indices revealed that genetic variability was similar in both techniques. Our results showed that tissue culture and especially cold storage of P. lentiscus L. may result transposons activation, thus could cause genetic instability.     

Effects of Stochasticity in Early Life History on Steepness and Population Growth Rate Estimates: An Illustration on Atlantic Bluefin Tuna

The intrinsic population growth rate (r) of the surplus production function used in the biomass dynamic model and the steepness (h) of the stock-recruitment relationship used in age-structured population dynamics models are two key parameters in fish stock assessment. There is generally insufficient information in the data to estimate these parameters that thus have to be constrained. We developed methods to directly estimate the probability distributions of r and h for the Atlantic bluefin tuna (Thunnus thynnus, Scombridae), using all available biological and ecological information. We examined the existing literature to define appropriate probability distributions of key life history parameters associated with intrinsic growth rate and steepness, paying particular attention to the natural mortality for early life history stages. The estimated probability distribution of the population intrinsic growth rate was weakly informative, with an estimated mean r = 0.77 (±0.53) and an interquartile range of (0.34, 1.12). The estimated distribution of h was more informative, but also strongly asymmetric with an estimated mean h = 0.89 (±0.20) and a median of 0.99. We note that these two key demographic parameters strongly depend on the distribution of early life history mortality rate (M₀), which is known to exhibit high year-to-year variations. This variability results in a widely spread distribution of M₀ that affects the distribution of the intrinsic population growth rate and further makes the spawning stock biomass an inadequate proxy to predict recruitment levels.     

Fluid pressures at the shoe–floor–contaminant interface during slips: Effects of tread & implications on slip severity

Previous research on slip and fall accidents has suggested that pressurized fluid between the shoe and floor is responsible for initiating slips yet this effect has not been verified experimentally. This study aimed to (1) measure hydrodynamic pressures during slipping for treaded and untreaded conditions; (2) determine the effects of fluid pressure on slip severity; and (3) quantify how fluid pressures vary with instantaneous resultant slipping speed, position on the shoe surface, and throughout the progression of the slip. Eighteen subjects walked on known dry and unexpected slippery floors, while wearing treaded and untreaded shoes. Fluid pressure sensors, embedded in the floor, recorded hydrodynamic pressures during slipping. The maximum fluid pressures (mean+/−standard deviation) were significantly higher for the untreaded conditions (124+/−75 kPa) than the treaded conditions (1.1+/−0.29 kPa). Maximum fluid pressures were positively correlated with peak slipping speed (r=0.87), suggesting that higher fluid pressures, which are associated with untreaded conditions, resulted in more severe slips. Instantaneous resultant slipping speed and position of sensor relative to the shoe sole and walking direction explained 41% of the fluid pressure variability. Fluid pressures were primarily observed for untreaded conditions. This study confirms that fluid pressures are relevant to slipping events, consistent with fluid dynamics theory (i.e. the Reynolds equation), and can be modified with shoe tread design. The results suggest that the occurrence and severity of unexpected slips can be reduced by designing shoes/floors that reduce underfoot fluid pressures.     

Analysis of genetic diversity in Phytophthora colocasiae causing leaf blight of taro (Colocasia esculenta) using AFLP and RAPD markers

The oomycetous fungus Phytophthora colocasiae that causes taro leaf blight is one of the most devastating diseases of taro and is widely distributed in India. Molecular and cultural techniques were employed for assessing and exploiting the genetic variability among isolates of P. colocasiae obtained from different geographical regions of India. Analysis of the 5.8-ITS region revealed detectable intraspecific variation among isolates. Ten random amplified polymorphic DNA (RAPD) and eight amplified fragment length polymorphism (AFLP) primers produced 198 and 510 reproducible fragments, respectively. AFLP produced 100 % polymorphism, whereas RAPD showed 93.5 % polymorphism. The average value of the number of observed alleles, the number of effective alleles, mean Nei’s genetic diversity, and Shannon’s information index were 2.00–1.94, 1.53–1.36, 0.31–0.24, and 0.47–0.40, respectively, for two DNA markers used. Analysis of molecular variance (AMOVA) for both markers produced similar results with the majority (85 %, AFLP; 89 %, RAPD) of the diversity present within population of P. colocasiae. Dendrograms based on two molecular data using the unweighted pair group method with arithmetic mean (UPGMA) was incongruent and classified the P. colocasiae isolates into one and two major clusters. Cophenetic correlation coefficient between dendrogram and original similarity matrix were significant for RAPD (r?=?0.904) and AFLP (r?=?0.825). The results of this study displayed a high level of genetic variation among the isolates irrespective of the geographical origin. The possible mechanisms and implications of this genetic variation are discussed.     

Hypervariable genes—experimental error or hidden dynamics

In a homogeneous group of samples, not all genes of high variability stem from experimental errors in microarray experiments. These expression variations can be attributed to many factors including natural biological oscillations or metabolic processes. The behavior of these genes can tease out important clues about naturally occurring dynamic processes in the organism or experimental system under study. We developed a statistical procedure for the selection of genes with high variability denoted hypervariable (HV) genes. After the exclusion of low expressed genes and a stabilizing log-transformation, the majority of genes have comparable residual variability. Based on an F-test, HV genes are selected as having a statistically significant difference from the majority of variability stabilized genes measured by the ‘reference group’. A novel F-test clustering technique, further noted as ‘F-means clustering’, groups HV genes with similar variability patterns, presumably from their participation in a common dynamic biological process. F-means clustering establishes, for the first time, groups of co-expressed HV genes and is illustrated with microarray data from patients with juvenile rheumatoid arthritis and healthy controls.