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Recent studies have used transition matrix elasticity analysis to investigate the relative role of survival (L), growth (G) and fecundity (F) in determining the estimated rate of population increase for perennial plants. The relative importance of these three variables has then been used as a framework for comparing patterns of plant life history in a triangular parameter space. Here we analyse the ways in which the number of life-cycle stages chosen to describe a species (transition matrix dimensionality) might influence the interpretation of such comparisons. Because transition matrix elements describing survival ("stasis") and growth are not independent, the number of stages used to describe a species influences their relative contribution to the population growth rate. Reduction in the number of stages increases the apparent importance of stasis relative to growth, since each becomes broader and fewer individuals make the transition to the next stage per unit time period. Analysis of a test matrix for a hypothetical tree species divided into 4-32 life-cycle stages confirms this. If the number of stages were defined in relation to species longevity so that mean residence time in each stage were approximately constant, then the elasticity of G would reflect the importance of relative growth rate to λ. An alternative, and simpler, approach to ensure comparability of results between species may be to use the same number of stages regardless of species longevity. Published studies for both herbaceous and woody species have tended to use relatively few stages to describe life cycles (herbs: n=45, [Formula: see text]; woody plants: n=21, [Formula: see text]) and so approximate this approach. By using the same number of stages regardless of longevities, the position of species along the G-L side of the triangular parameter space largely reflects differences in longevity. The extent of variation in elasticity for L, G and F within and between species may also be related to factors such as successional status and habitat. For example, the shade-tolerant woody species, Araucaria cunninghamii, shows greater importance for stasis (L), while the gap-phase congener species, Araucaria hunsteinii, shows higher values for G (although values are likely to vary with the stage of stand development).
Saito, A, Namiki, Y, and Okada, K. Elasticity of the flexor carpi ulnaris muscle after an increased number of pitches correlates with increased medial elbow joint space suppression. J Strength Cond Res 35(9): 2564-2571, 2021-This study aimed to measure the medial elbow joint space and elasticity of the forearm flexor-pronator muscles in repetitive pitching and to determine which of the forearm flexor-pronator muscles contribute to elbow valgus stability during pitching. Twenty-six collegiate baseball players performed 7 sets of 15 pitches. The medial elbow joint space and elasticity of the pronator teres, flexor carpi radialis, flexor digitorum superficialis (FDS), and flexor carpi ulnaris (FCU) were measured using ultrasonography before pitching and after every 15 pitches. Correlations among the rate of change of these parameters were analyzed using Pearson's correlation coefficients. The medial elbow joint space increased after 60 or more pitches compared with that before pitching (all p < 0.001; effect size [ES]: 0.44-1.22). FDS and FCU elasticity increased after 45 and 60 pitches or more in contrast to that before pitching, respectively (FDS: p = 0.047 and p < 0.001, respectively; ES: 1.05-1.42, FCU: p = 0.011 and p < 0.001, respectively; ES: 1.11-1.48). After 75 or more pitches, the rate of change of FCU elasticity correlated negatively with that of the medial elbow joint space (r = -0.395, r = -0.454, and r = -0.404, after 75, 90, and 105 pitches, respectively). Increased FCU elasticity after repetitive pitching correlated with suppression of the increase of the medial elbow joint space. The FCU may be the primary dynamic stabilizer against the elbow valgus force, and evaluation of the FCU elasticity may be important for preventing elbow injuries.
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Colloidal gels formed by arrested phase separation are found widely in agriculture, biotechnology, and advanced manufacturing; yet, the emergence of elasticity and the nature of the arrested state in these abundant materials remains unresolved. Here, the quantitative agreement between integrated experimental, computational, and graph theoretic approaches are used to understand the arrested state and the origins of the gel elastic response. The micro-structural source of elasticity is identified by the l-balanced graph partition of the gels into minimally interconnected clusters that act as rigid, load bearing units. The number density of cluster-cluster connections grows with increasing attraction, and explains the emergence of elasticity in the network through the classic Cauchy-Born theory. Clusters are amorphous and iso-static. The internal cluster concentration maps onto the known attractive glass line of sticky colloids at low attraction strengths and extends it to higher strengths and lower particle volume fractions.
In soft solids composed of colloidal suspensions, emulsions, foams, and biomaterials, elasticity is governed by the spatial distribution and interactions among amorphous mesoscale structures1. Identifying and understanding the behavior of these fundamental building blocks are the underlying challenges for developing structure-property relations that are essential to controlling and tailoring the mechanics of such materials. Among soft solids, an important and ubiquitous class are colloidal gels, in which attractive interactions between suspended colloidal particles drive a thermodynamic instability that promotes aggregation, arresting in a space spanning network structure possessing unique mechanical and transport properties2,3. Commonly, phase separation is induced by the addition of non-adsorbing polymer to a suspension of repulsive colloids by the well-known depletion interaction4,5,6,7. Depletion gels are often found in industrial processes and products where fine solids are dispersed in polymer solutions, including agrochemicals, consumer care products, and pharmaceuticals, and have frequently served as model experimental systems8,9.
In applications, the rheology of a gel is its principal material property of interest, including its elasticity10 and yielding11. At low volume fractions and strong interaction energies between particles, colloidal gels are effectively modeled as fractal flocs formed through diffusion-controlled aggregation processes, which grow together to form a percolating microstructure12. The flocs are the principal load bearing units of the gel and theories connecting the floc architecture to the gel modulus remain a state-of-the-art description13,14. Yet, there exists no definitive micro-structural theory for the elasticity of colloidal gels at higher volume fractions and lower strengths of interaction. What are the fundamental structural units imparting elasticity to the network, and what physical principles govern their formation?
Linear elasticity in depletion gels has been postulated to result from the spatial organization of particles into locally dense clusters, each cluster acting as a rigid, mechanical unit that propagates the elastic deformation10,15,16,17. For instance, the work of Zaccone, Wu, and Del Gado16 theorized that the role of cluster-cluster contacts were central to gel elasticity. The authors showed that one model of cluster microstructure, based on contact number distributions for hard spheres and the Cauchy-Born theory for the affine elastic response of amorphous solids18, can fit experimental measurements of the shear modulus. Similar models with different approaches to enumerating clusters and cluster contacts based on mode coupling theory (MCT) were also explored by Ramakrishnan and co-workers10. Signatures of clustering are evident in light scattering10 and confocal microscopy19, which probe long-range fluctuations in colloid number density, and active microrheology, which examines the elastic deformation in response to a local perturbation20.
The structure, rheology, and interparticle interactions for the depletion gels are summarized in Fig. 1. In experiments, a recently developed model depletion gel is employed, which enables the rheology, microstructure, and particle interactions to be measured in concert for complete determination of the microscopic properties and macroscopic elastic response22. In simulations, a high-performance Brownian dynamics simulation algorithm generates representative depletion gel structures and enables us to compute the elastic modulus of those gels23,24.
Visual inspection of gel structures measured experimentally and computed in simulations offers no clear delineation of cluster and inter-cluster bonds beyond the correlation length ξ associated with the number density fluctuations. Laser tweezer experiments have identified rigidly clustered regions with similar characteristic length scale through mechanical interrogation of the network by an oscillatory driving force20. This form of mechanical interrogation is purely local and identifies only single clusters, and cannot, in a practical way, provide a statistical description of the cluster number density or their inter-connectedness. Yet, these two quantities: the number density of rigid clusters, nc, and the average number of cluster-cluster contacts per cluster, zc, are sufficient to determine the remaining parameter for the Cauchy-Born theory: the number density of elastically active bonds, \(n_e = \frac12n_\mathrmcz_\mathrmc\). An alternative means of identifying rigid clusters and their connectedness is needed. In the present work, we employ a graph theoretic approach27,28.
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