The intact permeability of the Cobourg limestone is estimated to be in the range of K ∈  m2. This paper discusses the factors influencing the measurement of the intact permeability of the Cobourg limestone and presents an upscaling approach for estimating this parameter. The procedure first involves the dissection of a cuboidal sample of the rock measuring, 80 mm × 120 mm × 300 mm, into ten 8 mm-thick slabs. Digital imaging and mapping of the larger surfaces of these sections are used to create, from both surface image extrusion and surface image interpolation techniques, the fabric within the dissected regions. The estimated permeabilities of the lighter and darker regions are used in the computational models of the computer-generated fabric to estimate the effective permeability of the rock. These results are complemented by estimates derived from mathematical theories for estimating permeabilities of multiphasic composites.The Cobourg limestone is a heterogeneous argillaceous rock consisting of  The proposals for the DGR call for its siting to be approximately 680 m below ground level within the Cobourg limestone formation. The Cobourg limestone host rock is overlain by Upper Ordovician-age siltstone and gray shale extending to a thickness of approximately 200 m and underlain by argillaceous limestone and gray shale, approximately 150 m thick. Despite its low clay content, the Cobourg limestone is nominally referred to as an argillaceous limestone. A characteristic visual feature of the Cobourg limestone is the extraordinary heterogeneity of the fabric, consisting of lighter nodular regions of calcite and dolomite separated by darker argillaceous partings of a similar composition but with quartz and a low clay content  to be sited near the eastern shores of Lake Huron in southern Ontario, CanadaCalcite ~82.4%; Dolomite ~1.3%; Quartz ~7.5%; Clay ~ 8.8%, with a Porosity ~ 0.001; (ii) the darker argillaceous partings consist of: Calcite ~ 69.5%; Dolomite ~ 4.2%; Quartz ~ 16.9%; Clay 9.4%, with a Porosity ~ 0.006. ~ In the aforementioned, the clay fraction includes varying proportions of Illite, Kaolinite and Montmorillonite. The partings give the appearance of a distinct heterogeneity and on occasions, the presence of nominal stratifications and raises important issues related to the dimensions of the representative volume elements (RVE) that should be tested to obtain meaningful intact permeability estimates, necessary for assessment of a DGR constructed in the Cobourg formation. SEM images of the Cobourg limestone distinctly indicate the presence of the calcite phase, the argillaceous phase, the micro-texture boundary and a well-developed micro-porosity network in the argillaceous phase that contributes to the higher porosity. Previous studies3 indicate that a suitable REV for the Cobourg limestone should be a cuboidal region on the order of 75 mm in dimension.The mineralogical compositions obtained from XRD techniques performed at the McGill Institute for Advanced Materials are summarized for completeness: (i) the lighter grey nodular phase consists of: in situ stress state and the geomechanical properties of the rock in terms of its deformability, damage in terms of micro-crack development, failure, fracture and movement of discontinuous regions. In the HDZ zone, the rock can experience gross fracturing leading to a fragmented medium. In the EDZ zone there can be micro-crack development that can lead to alterations of the geomechanical and transport properties. In the EdZ zone, the rock is largely undisturbed and the permeability of this region is of interest to modelling the long-term containment that can be provided by the undisturbed rock mass.It is understood that the creation of a DGR will perturb the geostatic stress state at the repository location and will lead to the creation of Highly Damaged Zones (HDZ), Excavation Damaged Zones (EDZ) and Excavation disturbed Zones (EdZ), all with altered permeability characteristics due to the generation of visible cracks, micro-cracks and other defects at various scales Fig. . The HDZintact Cobourg limestone is a critical parameter in a concept assessment exercise.When the long-term extent of these damaged zones are established and designated as zones of compromised retardation potential for the migration of radionuclides, the primary natural geological barrier that will provide for the retardation of any released radionuclides from the stored waste to the environment is the intact Cobourg limestone. In this sense, the fluid transport characteristics of et al.4, Golder Associates5, Intera6, NWMO1, Selvadurai2 Mazurek7 who used both laboratory steady state tests and in situ hydraulic pulse tests to estimate the permeability. Vilks and Miller8 conducted steady state axial flow permeability tests on samples measuring 25 mm in diameter and 10 mm in length. This sample size is appropriate if the presence of fabric heterogeneities can be disregarded. The experiments reported by Selvadurai et al.9 were performed on Cobourg samples measuring 108 mm in diameter and 200 mm in length. The influence of isotropic stress increase and isotropic stress cycling up to 20 MPa, on permeability evolution was investigated. The permeability was estimated using hydraulic pulse tests. The study showed that, in contrast to the conventional reduction in permeability with increasing isotropic compression, the heterogeneous fabric contributed to an increase in the residual permeability  × 10−22 m2. When the confining pressure was increased to 20 MPa, the permeability in the two samples increased respectively in the range  × 10−22 m2. When the confining pressure was reduced to 5 MPa, the permeability increased to the range  × 10−22 m2). Selvadurai and Jenner10 conducted both steady state and transient radial flow hydraulic pulse tests on unstressed Cobourg cylinders measuring 106.8 mm in diameter with lengths between 116.8 mm and 173.6 mm. The permeabilities were estimated to be 0.01 × 10−19 m2 to 2.0 × 10−19 m2. Nasseri et al.11 performed Thermo-Hydro-Mechanical (THM) experiments on Coburg limestone samples measuring 50 mm in diameter and 125 mm in length. Hydraulic pulse tests were performed on the samples to determine the permeability of the Cobourg at room temperature, with the estimates varying from 0.117 × 10−19 m2 to 1.68 × 10−19 m2. Selvadurai and Najari12 present results of permeability tests conducted on unstressed Cobourg cylinders measuring 150 mm in diameter and 278 mm in length containing a 26 mm diameter cylindrical cavity drilled to a depth of 39 mm from a plane surface. When the pressurized cavity is aligned normal to the nominal partings, the estimated permeability ranged from 0.15 × 10−19 m2 to 1.3 × 10−19 m2. When the pressurized cavity was aligned along the nominal partings, the estimated permeability ranged from 1.04 × 10−19 m2 to 3.9 × 10−19 m2. These results are presented in Table 13 performed hydraulic pulse tests on 85 mm diameter and either 86 mm or 127 mm long samples of the Cobourg limestone, subjected to stress states that can lead to the development of damage. The experiments were performed using an Obert-Hoek Triaxial Cell and an MTS Rock Testing Machine to control the deformations. Similar to the experiments reported by Selvadurai et al.9, the study by Selvadurai and Głowacki13 also points to increases in permeability evolution during the application of confining stresses up to 30 MPa and a permanent increase in the permeability upon unloading of the sample to 5 MPa.Earlier investigations of the permeability of the Cobourg limestone  are due to Raven 14 conducted boundary heating of the experimental configuration to determine the pore pressure generation and decay within the fluid-filled cavity. The cavity pressure rise due to the advancing heat pulse and the subsequent decay can be used to investigate the influence of the Cobourg permeability on the cavity pressure decay response in particular. In a recent investigation by Selvadurai and Głowacki15 the local permeability measurements were obtained using miniature flow entry points to determine the permeability of the lighter and darker facies of the Cobourg limestone. Transient pulse tests and steady state tests were performed to estimate the permeability. The results in Table 3 of15, contain a range of estimates for the permeabilities of the lighter and darker facies. The steady state tests for these vary from 0.27 × 10−19 m2 to 8.0 × 10−19 m2 for the darker material and 0.62 × 10−19 m2 to 6.3 × 10−19 m2 for the lighter material.Selvadurai and Najari−23 m2 to 10−19 m2), the preferred method for estimating permeability is the hydraulic pulse test. The appeal of this method is the rapidity with which the test can be performed to obtain permeability estimates. The hydraulic pulse method is, however, not without its limitations. When steady state flow is used to estimate the permeability of a saturated rock, the only known information relates to the hydraulic potential difference on two surfaces of the sample, the steady flow rate established by the potential difference, the temperature at which the experiment is conducted and the dimensions of the flow domain. In contrast, for hydraulic pulse tests several additional parameters, including the compressibility of the solid material composing the porous fabric, the compressibility of the porous skeleton, the porosity of the rock and the compressibility of the pore fluid, are required to accurately interpret the results. In addition, factors such as saturation of the sample, the presence of air voids both within the pore space of the sample and within the connections, and the residual pressures present in the sample can influence the results of hydraulic pulse tests conducted on low permeability rocks similar to the Cobourg limestone. These aspects are discussed by Selvadurai and Głowacki15 and Selvadurai and Najari16. The bulk permeability estimates for the Cobourg limestone available in the literature are summarized in Table For low permeability rocks . The surface irregularities introduced during the cutting of the 80 mm × 120 mm × 300 mm cuboid could not be removed from the 8 mm thick edges of the plates without damaging of the plates. A typical view of the edges of a cube of Cobourg limestone measuring 130 mm is shown in Fig. 17), where the porosities can range from 0.1 to 0.4, whereas for the Cobourg limestone, the porosities are in the range 0.001 to 0.006. Attention is, however, drawn to the work of Klaver et al.18 dealing with pore geometries in shale, with comparable porosities.In this paper, attention is first focused on the mapping of the spatial distribution of the lighter and darker facies of the Cobourg limestone by dissection of a cuboid measuring 80 mm × 120 mm × 300 mm into thick plate samples measuring 80 mm × 120 mm × 8 mm with a separation of 8 mm. The sizes of the lighter nodular regions tend to vary  consider the volume fractions of the two facies and (ii) their respective isotropic permeabilities and uses the theoretical relationships proposed by Voigt23, Reuss24, Hill19 and Hashin and Shtrikman20 to arrive at an estimate for the effective permeabilities.The estimates for the bulk permeability, as defined by the geometric mean and obtained from an effective property estimation using a composite materials approach are compared with results obtained from experimental techniques. [It should be noted that the arithmetic mean (two estimates) and harmonic mean are needed to calculate the effective permeabilities of the composite solid consisting of the ten thick slabs in the 1 to map the internal fabric of smaller Cobourg limestone cylinders measuring 85 mm diameter and 165 mm in length [Selvadurai2]. The facilities used for that particular XRT imaging had insufficient power to map the fabric of a 150 mm diameter and 300 mm long sample. The 150 mm diameter sample was dissected to discs of approximately 25 mm thickness; however, such a thickness does not allow for the accurate computer-aided reconstruction of the internal fabric of the disc. In a previous research exercise2, a cuboidal sample of the Cobourg limestone measuring 80 mm × 120 mm × 300 mm was saw cut into thick slabs measuring 80 mm × 120 mm × 8 mm. This sample thickness offers a better possibility for reconstructing the interior fabric of the Cobourg limestone. First, each thick plate sample was assigned a sequential position along the axis of the block. Photographic images of all the sections were prepared and catalogued according their position and orientation. Photographic images from the two plane faces of the thick sections were used to generate black and white digital images. These images identified, manually, the zones of the lighter and darker regions corresponding to the two facies of limestone encountered on the plane faces of the thick plates. Examples of the operations as applied to the thick plates of the Cobourg limestone are given in Figs. The Cobourg limestone has a visibly heterogeneous internal fabric that consists of nodular regions of a mixture of calcite and dolomite separated by argillaceous partings, which also consist of calcite and dolomite with fractions of quartz and a small clay fraction that gives the darker colour. The most effective technique for mapping the fabric is through X-ray tomographic (XRT) imaging of the rock. Such an approach was used21) of the heterogeneity. While this is a useful line of enquiry, certainly with respect to the possible influences of in situ stress states and geologic heterogeneity that can be introduced by depositional effects, the value of the extension will be more meaningful in relation to a site-specific location where such a repository could be situated. The measures of quantifiable heterogeneities should ultimately be linked to hydro-mechanical properties that will be of interest to fluid transport.A reviewer suggested the possibility of the numerical quantification  and fills the space between the faces with self-similar images of either increasing or decreasing dimensions. The region to be extruded need not be a closed contour. On occasions, the region to be extruded between the two faces of the slab needs to be further discretized to achieve a suitable replica of the region. It is also possible to use the kriging techniques that are available in the literature22 and these were successfully adopted to estimate the interior distribution of permeability for surface measurements26, but such techniques are perhaps more relevant to the re-construction of the interior distributions for general three-dimensional configurations rather than the plate-like domains that are examined in this study, and the information for the kriging process is provided from multiple surface features. In the context of modelling the interior distribution of the facies within the 8 mm thick slab of the Cobourg limestone, the LOFT command-based solids modelling approach is likely to give 3D visualizations that are expected to be similar to the approach that uses the direct extrusion of the surface features to the mid-plane of the slab.  Figs. The surface imaging of the Cobourg limestone thick plates and the approximate identification of the regions of the lighter and darker facies were used to develop finite element models of each thick plate with a heterogeneous fabric. In order to perform the three-dimensional modelling of each thick plate, it was necessary to represent the fabric of the Cobourg at its interior. In this study, we adopt two approaches: in the first, the images from both surfaces of the thick plate are extruded to the mid-plane of each plate, and in the second, the LOFT command available in the AutoCAD program is used to generate the internal fabric of the lighter/darker facies using a solid modelling. In its intact condition, the Cobourg limestone is a low porosity rock with no visible presence of fractures and other defects. As briefly mentioned previously, the presence of such features in a low porosity rock would contribute to much larger values of the intact permeability compared to the values indicated in Table 2 and the average volume fraction of the darker facies is variable and for the dissected cuboid it is estimated  and the lower limit for the lighter material (with lower porosity) are chosen. i.e.The facies permeability is best estimated by isolating small enough samples that can be tested using one-dimensional steady state flow. This experiment requires specialized procedural techniques for ensuring interface sealing and the prevention of delaminations and defects in the samples. In the experimental technique recently proposed by Selvadurai and Głowacki15 for the respective permeabilities but the intention here is to outline a methodology that can be used as a basis for estimating the influences of heterogeneity. Also, from an applications perspective, the upper limit of permeability for the low porosity facies and the lower limit of permeability for the high porosity facies would provide a bounding of the facies permeabilities. The basic assumption is that the volume fractions can exhibit variations within the sample but the intrinsic facies permeabilities are isotropic measures and considered invariant.The work can certainly be extended to include a range of estimated valuesK*) can be bounded by the Voigt23 and Reuss24 estimates in the formVi are the volume fractions and Ki are the permeabilities. Considering the permeability estimates for the lighter and darker regions given by , takes the formThe bounds can be improved by considering the theoretical developments proposed by Hashin and ShtrikmanKDR > KLR,  embedded in a porous matrix. The internal fabric of the Cobourg limestone, however, does not lend itself to consistent mathematical modelling using geometric primitives.Kz and the permeabilities for the orthogonal directions aligned with the edges of the thick slab are denoted by Kx and Ky.The estimated internal distributions of the lighter and darker facies within each of the Cobourg limestone slabs were used to construct finite element models and these representations were used to estimate the axial and in plane permeabilities of each thick plate. The one-dimensional permeabilities of each slab in three orthogonal directions are determined in the following way: (i) by imposing constant potential Dirichlet boundary conditions, with a potential difference, on two opposite faces of the slab, and (ii) by imposing no flow or null Neumann boundary conditions on the remaining adjacent faces of the slab. This procedure is performed in all three orthogonal directions. The permeabilities of the lighter and darker phases of the Cobourg limestone can be used in the computational models of the fabric to evaluate the effective permeability values for each thick slab in (i) a direction normal to the plane of the plate and (ii) two orthogonal directions in the plane of the plate aligned with the boundaries. The computational modelling of the potential flow problem is performed using the COMSOL™ finite element code and the domains are modelled using tetrahedral elements. The number of elements for each thick plate are in excess of 150, 000 for the extruded geometry and more than 210,000 elements for the LOFT command internal fabric reconstruction. The number of degrees of freedom is generally greater than 280,000. The computational approach was used to estimate the permeabilities of ten adjacent thick plates of the Cobourg limestone obtained by dissecting the original cuboidal sample measuring 80 mm × 120 mm × 300 mm. Table Kz*) can be obtained from the Weighted Harmonic Mean. Considering the Extruded geometry, we haveti are the thicknesses of the plate regions and Kzi are the permeabilities.The computational results for the 10 thick plates of the Cobourg limestone can now be combined to obtain the effective permeabilities for the assemblages. The effective permeability of the cuboid of the Cobourg limestone in the direction normal to the plane of the plates  development of a permeameter that can be used to measure the near surface permeability of the cuboidal region, (ii) a complete mathematical analysis of the permeameter performance under steady flow, (iii) the measurement of surface permeabilities at 54 locations of the cuboidal region, (iv) the development of Kriging techniques to estimate the interior permeabilities of the Indiana limestone cuboid, (v) implementation of the spatial heterogeneity of the permeability in a computational model to estimate the effective permeability of the cuboid in three orthogonal directions, (vi) estimation of the geometric mean and (vii) the use of the geometric mean concept to estimate the fluid flow in multi-dimensional configurations to validate the geometric mean concept.It is clear that the two modes of reconstruction of the interior fabric of the Cobourg limestone yield approximately the same estimates for the effective permeabilities in the three orthogonal directions. For purposes of comparison with the Hashin-Shtrikmangiven by , 9) and and20 esgiven by  for the Extruded fabric, we obtainFor the Cobourg limestone, considering the Morphed fabric, we haveSimilarly, for the K*) can be assigned the following bounds, assuming that the upper limit selected is the lowest computational estimate based on the geometric mean:From these developments, it can be concluded that the effective intact permeability of the heterogeneous Cobourg limestone . The results shown in Table K ∈  m2. Omitting details, it can be shown that the Voigt23 and Reuss24 results for the permeability derived from gas permeability values indicated in 18). ConsMorphed fabric, we haveSimilarly, for the 20 bounds given by . An important observation of the research is that the multiphasic mathematical theories provide estimates that compare very accurately with the computational simulations. If the phasic permeabilities and their volume fractions can be accurately determined, then the effective permeability of the statistically heterogeneous rock can be determined, as a set of bounds, using a convenient analytical result. This estimation procedure also points to factors that should be addressed when conducting hydraulic pulse tests on low permeability heterogeneous porous media30.The estimation of the bulk permeabilities of heterogeneous porous rocks is a non-routine exercise if the rock has high spatial variability and low permeability. In the context of experimentally estimating permeability, the best approach is to conduct steady state permeability tests on samples that capture accurately the RVE of the material. For the Cobourg limestone, the sample dimensions should exceed 75 mm to account for the heterogeneity presented by the nodular nature of the rock. As the sample size increases, the steady state measurement of permeability will take an inordinate amount of time. Measurement and interpretation of permeability therefore has to use transient techniques such as hydraulic pulse tests. Even with hydraulic pulse tests, estimation of the effective compressibility of the porous skeleton, the compressibility of the solid phases constituting the porous skeleton and the porosity are non-routine experiments. The methodologies presented in the paper require the accurate estimation of the permeabilities of the lighter and darker regions of the Cobourg limestone and their respective volume fractions. In certain situations, the identification of the respective volume fractions of the lighter and darker facies can be non-routine because of the high density and low porosity of the Cobourg limestone. Conventional XRT techniques cannot be used to estimate the volume fractions of larger RVEs. The research reported uses the dissection of a cuboidal region and surface imaging techniques to map the fabric of the Cobourg limestone, which is composed of lighter and darker regions. The research draws on developments in theories of multiphasic media to obtain a set of bounds for the effective permeability. These approaches are complemented by novel techniques that re-construct, from surface imaging of the dissections, the interior distribution of the heterogeneous fabric. Computational simulations are used to determine the bulk permeability properties with the input parameters being only the phasic permeabilities. It should be noted that the modelling precludes inter-plate flow to make the computational approach more efficient. The experimental results for the phasic permeabilities can be influenced by the experimental techniques used. Any invasive technique for estimating permeability can introduce sample disturbance and the measured permeability can also be influenced by the experimental technique (e.g. interface leakage). Gas permeability measurements can also be influenced by the sample preparation procedures and the theories and corrections used to interpret the permeability (e.g. corrections for the Klinkenberg


The distribution of pattern across scales has predictive power in the analysis of complex systems. Discontinuity approaches remain a fruitful avenue of research in the quest for quantitative measures of resilience because discontinuity analysis provides an objective means of identifying scales in complex systems and facilitates delineation of hierarchical patterns in processes, structure, and resources. However, current discontinuity methods have been considered too subjective, too complicated and opaque, or have become computationally obsolete; given the ubiquity of discontinuities in ecological and other complex systems, a simple and transparent method for detection is needed. In this study, we present a method to detect discontinuities in census data based on resampling of a neutral model and provide the R code used to run the analyses. This method has the potential for advancing basic and applied ecological research. AccordiMuch of the discontinuity literature is based on the examination of body mass distributions because the scale at which an organism perceives the environment and procures resources is a function of its size Calder, . The bodDiscontinuity research has direct application to ecosystem management (−Mn−1)/Mn)γ, where M is the mean body mass of the species index n in a fully censused community /(Mn)). Although informative, these difference index methods suffer from inherent subjectivity.Numerous methods have been used to identify discontinuities in datasets  = log10(Mn+1)−log10(Mn), where M is the mean body mass of the species n in a fully censused community. The resampling creates a hypothetical distribution of gaps against which the gaps within the real data are tested. The GRI has been the most widely used method to date in discontinuity analysis of census data   and classification and regression trees (CART), including the Bayesian version thereof , has two components: the development of a neutral null distribution and a bootstrapping approach that compares rank‐ordered differences in data to differences generated through resampling the neutral null. We then compare the observed data to the bootstrapped samples and calculate the percentile of the bootstrap distribution to infer the likelihood that the gap is not through random chance.2.1The DD is designed for census data, as opposed to sample data. We use body size as an example, but researchers working in other types of complex systems have chosen proxies that represent an integration of system‐specific drivers, such as firm size or city size  are generated for bandwidth values (h) ranging from 0 to half of the range in the dataset. We used half of the range as this avoids too many edge effects being introduced by the smoothing function and makes the computation more efficient; however, this can be changed if necessary by the user. A lower bound of zero was used as body masses can only be positive so it is not sensible to generate a model that predicts negative body mass. We utilized a bandwidth increment of 0.001 ( = hj)( + 0.001 for j = 1… n). The estimates are easily calculated in R using the base package and the density function (Equation h<∞). K is the standard normal density function h is the bandwidth at index j, against which the masses of vector m, at index i are calculated. A kernel density estimator generates, for a given bandwidth, a smoothed curve that integrates to 1, which shows the “density” of data points at the particular value.Kernel density estimates of logEquation , with thThe output is a set of smoothed kernel density functions of varying bandwidths ranging from 0 to half of the data range. For each of these kernel density estimates, the second derivative test is utilized to determine whether the function is unimodal. The neutral null model is selected as the kernel density estimate with the smallest bandwidth that is unimodal.We have used the Gaussian function as the smoothing function as it speaks to the common ecological expectation that there is a limited range of scales of resource distribution, implying an optimum scale at which an animal can exploit resources. For instance, in the absence of reinforcing process rates, we would not expect multiple scales or hierarchical pattern and process. Instead, an equilibrium of pattern and process would be reached , with the same sample size as the observed data. For instance, if there are 60 species in the community, a sample size of 60 is resampled without replacement 5,000 times. For each draw, the sampled data are rank‐ordered and the differences between each of the rank‐ordered, simulated body masses are calculated. By doing this multiple times, a distribution of gap sizes is generated for each rank‐ordered gap. Each of the observed differences is compared to the resampled distribution of the same rank, and the percentile  of the resampled gap distribution is calculated. The gap percentile is then used as a measure of how likely the observed difference between census points is to those which can be considered randomly sampled from the neutral null distribution. Decisions regarding what percentile to accept as a gap are akin to alpha value assessment in any frequentist method, and inference can be made accordingly.2.4N = 36), boreal forest birds (N = 101), boreal prairie mammals (N = 53), boreal prairie birds (N = 108) , and the method must be verified for incomplete census data (sample data). We tested three hypotheses: The DD algorithm can reliably detect gaps in data that encompasses (a) multiple scales, (b) under conditions of reduced census success, and (c) under conditions of low species richness.To approximate realistic ecological scenarios, we simulated a dataset similar in scale and composition to the Holling  boreal bOne thousand test datasets with differing sample sizes and census success  were generated and tested for discontinuities using the DD. This generated a population of discontinuities from which the effects of sample size and census success could be tested. Because each experimental dataset was an independent simulation, it was necessary to objectively cluster discontinuity values together to measure the effect of changing simulation parameters on the location of discontinuities. If the algorithm reruns an analysis on the same dataset, yet the discontinuities are identified in different places, the algorithm is not robust, by running repeated analysis of the same dataset we explicitly tested for this. To do this, the DD was run on all test datasets, and then, independent estimations of discontinuities were clustered together using the mClust package which uses a Bayes clustering algorithm that selects clusters based on the BIC (Bayes information criteria ~ similar to an Akaike information criteria) value was used to determine the best‐fit number of clusters. The distribution of discontinuities per cluster was then compared under varying conditions of census success and sample size to determine the sensitivity of DD to real‐world constraints in the census:The population of discontinuities was compared among samples with equal sample size. Sample success was simulated by randomly deleting a proportion of the sampled population. Comparisons were made between proportions sampled from 0.75 to 1.The population of discontinuities was compared among samples with unequal sample size. Sample success is assumed to be 1, with sample sizes varying from 20 to 100 samples  in the proxy dataset.3R code for implementing the DD is provided in the The method detects discontinuities accurately and precisely under increased sample sizes and census success , which is a result of comparing the observed data to a unimodal neutral null Figure . A majorHow well the census represents what's in the community and how many species are in a community will affect the precision of the DD in detecting the discontinuities. This is because the variance in the location of the resampled distribution is a function of the number of samples from which the resampling is drawn. If only ten samples are taken from a distribution, the variability in the gaps between individual samples is higher than when there are around 100 samples. Practically, this may limit the application of the DD for species‐poor systems (such as a desert) but, if the census largely accounts for the species in the community, the method will be effective in detecting discontinuities.The DD method provides a quick, easy to use and, most importantly, transparent method to objectively detect discontinuities in census data. Such a method will allow for increased discontinuity research in the various fields of ecology and other complex system science affording the ability to build upon this burgeoning area of study.None declared.CB was the primary author. CB and KN tested the application. DA, TE, AG, KN, CS, SS, and CA provided development discussion, critique of the method, and contributed significantly to the writing of the manuscript. CA provided the source code for original Neutral Null and obtained permission from the author and helped to develop the theory behind the method.https://esajournals.onlinelibrary.wiley.com/doi/abs/10.2307/2937313. Source code supplied as All data used are freely available from Holling : https:/ Click here for additional data file.


We introduce a simple model that describes the average occurrence of point variations in a generic protein sequence. This model is based on the idea that mutations are more likely to be fixed at sites in contact with others that have mutated in the recent past. Therefore, we extend the usual assumptions made in protein coevolution by introducing a time dumping on the effect of a substitution on its surrounding and makes correlated substitutions happen in avalanches localized in space and time. The model correctly predicts the average correlation of substitutions as a function of their distance along the sequence. At the same time, it predicts an among-site distribution of the number of substitutions per site highly compatible with a negative binomial, consistently with experimental data. The promising outcomes achieved with this model encourage the application of the same ideas in the field of pairwise and multiple sequence alignment. At the same time, models of protein evolution that incorporate structural information have been shown to better approximate the real evolutionary process9. These findings suggest an important contribution of structural coevolution in the process of sequence evolution and ensures that multiple mutations can affect the fitness of a protein sequence in a non linear way, phenomenon known as epistasis13. Moreover, it has been observed that the rate at which each site fixes random mutations varies both among sites17 and in time20 making the process of sequence evolution more complex than a simple set of identical and independent Markov processes on the protein sites21. In genetics, this feature is well known. The evolutionary time scales are usually defined according to a “molecular clock” that was originally modeled by taking sequence evolution as a set of independent and identically distributed Poisson processes23. In the last 30 years, several models have tried both to explain the overdispersion of this model26 and to propose more reliable ways to estimate evolutionary times27 both within neutral theory of evolution and invoking natural selection. Recently, it was shown that defining the evolutionary time scales according to the molecular clock is intrinsically biased, especially for proteins belonging to complex organisms, in which most of the sequence codes for complex structural motifs where random mutations are unlikely, and only highly correlated changes are possible28. Violations from the simple framework predicted by molecular clock and neutral theory can be quantified, for example, by estimating the so-called overlap ratio, namely the number of sites mutated more than once divided by the number of sites where at least a mutation took place28.It is commonly recognized that the process of evolution of the DNA sequences coding for proteins tends to conserve protein structure and function more than their sequence. In particular, the coevolution of residues inside a protein sequence entangles substitutions between contacting sites. For example, after a mildly destabilizing mutation at a site, the residues in contact with it may more easily fix mutations in order to adapt to the new situation and re-establish the original structural and functional balance. In the last twenty years, many investigations on coevolving residues have been performed showing that one can infer structural information from residue covariation in large multiple sequence alignments18 theorized that only a limited number of protein sites at a time can fix mutations, and when this happens other residues coupled with them will gain mutational freedom. This phenomenon would produce groups of COncomitantly VARIable codONS (for simplicity covarions) which vary over time in a correlated way. Unfortunately, despite this initial qualitative intuition, most of the quantitative implementations of the covarion model30 had to sacrifice the inclusion of any spatial pattern of covariation for the sake of computability.In the context of substitution rate variability, Fitch and MarkowitzIn the line of thought introduced by the covarion model we here propose a simple model to describe the time evolution of the substitution rates of protein sites involved in structural and functional stability by explicitly including spatial and temporal interdependence. The core idea is that the probability of a mutation to be fixed at a site is enhanced if this site is in spatial proximity with other recently mutated sites, mimicking the mechanism of compensatory mutations. At variance with standard coevolutionary models, we assume that this mechanism acts only for a finite time: if compensatory mutations do not take place in a few generations the substitution rates of the neighboring sites go back to their baseline value as if the initial substitution was effectively forgotten. This, as we will show, allows reproducing several non trivial features observed in protein sequence evolution by a simple model.Our model accurately reproduces the experimental patterns of along-chain conditional probability of substitutions in alignments in the sequence identity range 60–90%. Moreover, the number of substitutions per site produced by our model is well described by a negative binomial distribution, as generally found in phylogenetic analysis. The shape parameter of this negative binomial distribution falls in a realistic range and increases for growing evolutionary time in qualitative accordance with experimental data. This indicates that the model reproduces, at least qualitatively, a realistic distribution of substitution rates.avalanches localized not only in three-dimensional space, as commonly predicted by coevolution, but also in time.Our model predicts that substitutions take place in 34.The simplicity of this model lies in an implementation which neglects the specific sequence, focusing only on contact maps and on the set of times of the last mutation of each site. Our results foster the hypothesis that the variability of substitution rates, both along-chain and in time, is strongly connected with coevolution and that a mutation triggers indeed the acceptance of other mutations in nearby sites only for a limited time. The minimal model described here, even if simple, may provide a handy implementation of combined space and time variation of substitution rates that might be included in more complex models. For example, it may be combined with a model of codon or amino acid substitutions35 hereafter denoted by C, with Ci,k = 1 if sites i and k are in contact and Ci,k = 0 otherwise . Under the working hypothesis that a mutation at any given residue increases the mutation rate of all its spatially close neighbors, we model the substitution rate of site i at time t byWe developed a minimal dynamic model to describe the substitution rates in protein sequence evolution. Here mutations in the neighborhood of recently mutated sites have increased chances to be fixed by natural selection. For simplicity we model only substitutions, i.e. mutations fixed during the evolutionary process, and neglect those mutations which are destined to vanish. We will then indifferently label the time of appearance of a substitution as its time of mutation or time of substitution. This approximation becomes acceptable when dealing, as in our case, with time scales longer than the interval between the appearance and fixation of a mutation. This model accounts for realistic contact probabilities between the protein sites by means of model contact maps extracted from PDB structurestk corresponds to the time of the last mutation at site k. All times are measured in units of an implicit memory time and are thus dimensionless. There are two different terms involved in the right-hand side of Eq. ] is to progressively reduce the impact of a substitution on the rest of the chain as time passes: if no other substitution appears in the neighborhood, after a sufficient amount of time, the substitution rates of that zone recovers their unperturbed value r0. From a biological point of view, this mimics the case in which a mutation keeps being transmitted from generation to generation without the early emergence of any compensatory mutation. In other words, this process mimics the occurrence of neutral mutations, i.e. mutations that are likely to have no significant detrimental effect on the protein structure and function. The inclusion of a memory kernel in Eq. where the sum runs over all protein sites and e of Eq. . The firs by averaging on different realizations of contact maps to compare them with data from real sequences.With this model, we perform a set of simulations in each of which we simulate the temporal evolution of two sequences which split at time zero from a common ancestor and are characterized by an empirical contact map sampled from real PDB structures . The two sequences at time zero are identical and characterized by the same mutational history. With time passing, we keep track of the number of mutated sites and of the number of substitutions per site in the two branches. We then group these simulations by fraction of mutated sites and perform quantitative analysis separately for each sequence identity range r0 in the limit of small r0 . We always use pairwise alignments in which one of the proteins is human, and the analysis is performed only for proteins with relatively high sequence identity with the human reference. This automatically excludes the vast majority of proteins belonging to simple organisms, such as yeast, where evolution can take place by different mechanisms28. For each sequence identity bin we collected at most one alignment per cluster per window of sequence identity to avoid uneven sampling . Each alignment was translated to a binary sequence, with 0 corresponding to two identical paired amino acids (a persistence) and 1 to different paired residues (a substitution).We prepared several sets of ungapped local alignments of at least 80 residues starting from the UniRef databases, we first investigate the along-chain conditional probability of finding a substitution d sites away from another one, For each analyzed sequence identity range mentclass2pt{minim37 predicts to have a fraction of α-helical residues larger than 38% . The orange lines in Fig. d < 20 at 90% of sequence identity. This deviation may be due to our requirement of ungapped alignments , which at low sequence identity seems to select more structured regions with respect to higher sequence identity. With this in mind, it is then natural to expect, at different sequence identities, slightly different contact probabilities and, therefore, slightly different optimal values of J. In practice, the deviations of these values, which we attributed to the structural non uniformity in the dataset, are negligible. For simplicity we therefore consider both r0 and J as constant in the whole range of considered sequence identities.We then compare mentclass2pt{minim38 whose shape parameter α is estimated from the distribution of the number of substitutions per site40. Indeed, when dealing with a mixture of Poisson processes characterized by Γ-distributed rates, the number of substitutions per site is necessarily distributed according to a negative binomial whose shape parameter is the same α. The probability of observing k substitutions at a site is then:In most algorithms for phylogenetic reconstruction based on likelihood maximization, the among-site rate variability is modeled by a Γ-distributionk〉 is the average number of substitutions per site computed from the data and α is the parameter to be estimated.where 〈p which are statistically compatible with a negative binomial, consistently with what inferred for real substitutions from phylogenetic trees40. These simulations are performed with the same values of J = 0.02 and r0 = 0.0004 discussed in the previous section.We here show that also our model produces distributions for k as simulated by our model at four different sequence identities. On top of each histogram we plot its weighted fit with a negative binomial . The negative binomial distribution well reproduces the statistics of the substitutions obtained by our model in all four cases.In Fig. 40. The shape parameter α is obtained by fitting to Eq. α characterizes the distribution of time-averaged substitution rates rather than instantaneous rates. As we reduce the sequence identity, the time interval over which we mediate grows, making the distribution of time-averaged rates diverge progressively from the distribution of instantaneous rates, becoming broader and broader. The solid line in panel a) of Fig. α obtained by fitting to Eq. 41. In particular, to allow a visual qualitative comparison, the panel (b) in Fig. α for decreasing average sequence identity when estimated from the multiple sequence alignments of five Pfam large families . The values of α in panel (b) at growing sequence identity are estimated on smaller and smaller phylogenetic trees; thus these estimates become progressively less reliable, explaining the noisy cloud of points at sequence identity larger than 0.7. The increase of α predicted by our model is larger than the one observed in the examples of phylogenetic trees shown in figure. This indicates that the model introduced in this work reproduces this phenomenon only qualitatively. As we are going to see in the next section, introducing in the model a variability in the spontaneous substitution rate improves the qualitative agreement.It is important to notice that in the proposed model each site has a rate that changes in time. This is very different to what happens in other models, for example those in which substitution rates are Γ-distributed but constant in timeα observed in our model, much bigger than what is found in real protein families, reasonably depends, among other effects, on our simplifying approximation of equal dynamics on all sites . This assumption is known to be too simplistic: indeed in real proteins, flexible regions are typically more prone to fix mutations than structured regions. Moreover, part of the sequence codes for structural regions which are essential for the function of the proteins where the neutral theory cannot be considered a valid description28. In complex organisms, such as the ones considered in this study, the fraction of residues belonging to structural regions can be large. To verify the hypothesis that larger variations of α are due to the approximation of equal dynamics, we process the PDB structures used to build the contact matrices with the software STRIDE42, and classify each residue in one of two categories: structured or unstructured. Then, we make them evolve according to slightly different patterns. Both classes still obey Eq. Jclass2 (different from before) to both classes, but we give them a different capability to fix new mutations (different r0). In particular, this capability will be very small for structured residues and we approximate it to zero. Therefore, structured residues will be characterised by Jclass2 and The significant variation of ons Fig.  in SI, aα variations: these values are shown in dashed line in panel (a) of Fig. α with respect to a model in which all sites are statistically identical. We also compute the overlap ratio28 in the two versions of the model in the analysed sequence identity bins and found that in all cases this value is bigger with the two-class model , showing once again that including information on structural variability makes the model more realistic.When fitting to a negative binomial the histogram of the number of substitutions per site obtained with this two-class model, we find indeed smaller 17 and in time20. Here we show how these two phenomena can be qualitatively described together by a simple model based on the idea that mutations may perturb the stability and functionality of a protein by introducing a frustration which dumps down in time. As a consequence, all sites that are in contact with a mutated one are themselves stimulated to accept mutations in order, for instance, to reduce frustration again. This idea has already emerged in the domain of protein sequence coevolution7: the novel ingredient that we introduce in this work is that this perturbation acts only for a finite time. In fact, if an isolated mutation has persisted for many generations, it is likely that it is neutral and has no effects on the fitness of the protein, so does not need compensation. We presented a model based on this idea, which revisits the covarion model proposed for the first time by18.Protein coevolution studies showed that substitutions exhibit a strong three dimensional spatial correlation due to compensatory mutations, which in turn is reflected on a correlation along the protein sequence. On the other hand, the substitution rates vary significantly among sitesα-helical contact periodicity that vanish when the predicted α-helices are removed from the dataset  is already enough to bring the shape parameter closer to the one observed in real data.Remarkably, the same model also reproduces a distribution of the number of substitutions per site largely consistent with a negative binomial, a distribution also empirically observed in phylogenetics. An overestimate of the shape parameter of this negative binomial distribution increasing with evolutionary time is visible both in real data and in our simulations and, at least in the model, is due to the time variation of the rates. We have also shown that the larger variability of the shape parameter in our model with respect to real data is likely to be due to the approximation of equal dynamics at all sites. This does not take into account the well known fact that in large regions of the proteins belonging to complex organisms random mutations are very unlikely. Indeed, we have shown that a trivial inclusion of differences in the mutation rates among sites  [9606] AND identity:0.5].We retrieved the protein sequences used to test our model from UniRefwater46 in the emboss software package47 and only ungapped parts of at least 80 residues were considered.From each of these clusters we detected the human sequence and aligned it with each of the others. We splitted the full range of 50–100% of sequence identity into windows of 2% of sequence identity each  and then collected at most one alignment per cluster per window of sequence identity, to avoid weighting bigger clusters more. Here we used only the alignments at 60–62%, 70–72%, 80–82% and 90–92%. Sequences were aligned locally by the algorithm Each alignment was translated to a binary sequence, with 0 corresponding to two identical paired amino acids (a persistence) and 1 to different paired residues (a substitution).s, we computed the overall conditional probability of observing a substitution d sites away from another substitution as the fractionNs1(d) and Ns0(d) are respectively the overall observed number of 1 and 0 found at a distance d from another substitution in the considered set of alignments. The first and last 5 residues of each alignments have been neglected to reduce boundary effects.For each window of sequence identity 35, where the highly precise structures of 500 proteins are provided. For each structure, we computed the contact map by defining that two residues are in contact if their alpha-carbons are nearer than 10 Å.As model contact maps we downloaded the top500H database42 on these 500 structures to separate residues involved in secondary structured from those that are not. This separation has been used in the two-class model described in section Two-class model.We then ran STRIDEL whose distribution is compatible with the experimental one and obtained the desired quantities as averages over these different realizations . During each simulation we kept track of the number of mutated sites as well as of the number of substitutions per site (ki). We simulated the evolution of the two sequences until they reach the desired sequence identity. For each length we simulated many contact maps  and averaged on both lengths and contact maps. We assume that consecutive substitutions at the same site can not bring the site back to its initial amino acid type and that independent substitutions in the two branches do not give the same results. In other words, the number of mutated sites is simply the number of sites that mutated at least once in at least one of the two branches. A consequence of this fact is that what we call sequence identity is, more precisely, a lower bound of the sequence identity. However we expect this fact not to dramatically change any of the showed result.Before each simulation we ran an equilibration dynamics, allowing the initial times  see Eq.  to reachd sites away from another one in sequences characterized by sequence identity s byd in the comparison of pairs of simulated sequences evolved with our model up to a sequence identity s. Also in this case the first and last 5 residues of each simulated sequences have been neglected to reduce boundary effects. For each range of sequence identity the simulations were stopped at the average sequence identity of the corresponding set of alignments. The optimization of parameters J and r0 have been accomplished by minimizing the root mean square displacement (RMSD) between the experimental d along the chain shorter than 30 amino acids have been used during this optimization.We computed, according to our model, the conditional probability of finding a substitution a(d) Eq.  and \docα was optimized.The weighted fit of Fig. 48 characterized by large multiple sequence alignments: PF00042, PF00211, PF01523, PF01641 and PF03453. For each of these families we downloaded the full multiple sequence alignment (download on March 7th 2016) and we built its phylogenetic tree (T0) by FastTree249 with the inclusion of the Γ-correction. Given a tree, the level of each of its subtree is given by the number of edges between the root of the tree and the root of the subtree. Thus, starting from the root of this tree (T0), let us label T1 its 1-level subtree containing the largest number of leaves. From the common ancestor of subtree T1, we label T2 its most populated 1-level subtree, and so on until the leaves are reached. For each of these subtrees, we computed both the average sequence identity between the leaves, 〈seqID〉, and a new estimate of α recomputed from the sequences of that subtree only (again by FastTree2). These are the quantities shown in Fig. α is difficult to determine and get progressively noisier and less reliable.We selected five Pfam familiesth 2016 from the NIAID Influenza Research Database44 by selecting protein data of virus type A and subtype H3N2 for the period 1981–2015 in Homo sapiens (complete segments only). Each sequence is characterized by a year, so its temporal evolution can be easily reconstructed. On average, the sequences of the same year are much more similar among themselves than to those of other years50. So, for each year, we computed a consensus sequence which has, at each position, the most common amino acid and we studied the evolution in time of these consensus sequences. From the PDB database51 we retrieved the entry 2WR0, containing one x-ray structure of the homotrimer of Influenza Hemagglutinin.The sequences of Influenza Hemagglutinin used in Fig. PdbTool (https://github.com/christophfeinauer/PdbTool.jl) we mapped each position in the sequence dataset to the corresponding residue on the PDB file.The sequences downloaded from the Influenza Research Database have all the same length and so they can be considered as a multiple sequence alignment. Using the program α-carbons are nearer than 8.5 Å. We mapped this contact map on the indexes of our MSA obtaining matrix Ci,j, with i and j in the range 1 to 566 and Ci,j = 1 if the residues corresponding to sites i and j on the PDB are in contact and Ci,j = 0 otherwise. Not all the 566 residues in the MSA were mapped on a residue of the PDB, so we also added a contact between these sites not mapped on the the PDB file and their along-chain neighbors.From the PDB file we built a contact map between the protein residues by considering in contact two residues whose i and time t by employing a mutation matrix mi,t, where mi,t = 0 corresponds to a match between the amino acids at site i in the consensus sequences at times t and t − 1 and a mi,t = 1 corresponds to a mismatch (implying that a mutation got fixed during that year). Thus, matrix m has size 566 (sites) × 33 (years).We compared the consensus sequences of consecutive years and kept track of mutations at each site m also for one of our simulations on protein 16 pk, chain H (PDB name) after discretizing their times to a number of bins comparable to the number of years available from influenza Hemagglutinin history.We built a similar matrix m, we built an undirected graph whose nodes are labeled  by a site i and a year t and whose links connect nodes whose sites are in contact and whose associated times (year or bin) differ by no more than two units. An avalanche is then defined as a connected component of the graph.From matrix Supplementary Figure 1


Physical‐activity approaches for people with intellectual disabilities (ID) are more likely to be effective and sustainable if they also target direct support professionals' behaviour. However, no tools to measure the behavioural determinants for direct support professionals are available as of yet. This study aims to construct a self‐report tool to measure direct support professionals' behavioural determinants in physical‐activity support for people with ID and to analyse its psychometric properties.The tools' sub‐scales and items corresponded with a proposed conceptual model. A pilot study was carried out to investigate and improve content validity. Construct validity and measurement precision were examined using item response theory models with data from a convenience sample of 247 direct support professionals in the support of people with ID.Results supported the three theory‐driven behaviour scales and indicated reasonable to good construct validity. The marginal reliability for the scales ranged from 0.84 to 0.87, and adequate measurement precision along the latent continua was found.The tool appears to be promising for measuring the behavioural determinants of direct support professionals for the physical‐activity support of people with ID and has potential as a tool for identifying areas to focus on for interventions and policies in the future. There is growing recognition that interventions aimed at promoting the participation in physical activity of people with intellectual disabilities (ID) should also target the physical and social environment of these people  professional supporting a group of people with ID in a living unit and/or activity centre and (2) being directly in contact with people with ID for most of the working time. No reward or incentive was offered for participation. The participants were mainly recruited from 10 residential facilities in the Netherlands. Each facility was allowed to decide how to internally distribute the invitation for participation in this study. An indication of the overall response rate was given by calculating the response rate for the four participating facilities that invited professionals to participate by e‐mail (21.4% response rate). Awareness for this study was raised by online advertising in the other six facilities. In addition, participants were also recruited via a national information platform for direct support professionals and by social media.In total, 395 potential participants visited the online application that introduced the tool . Of these, 363 chose to participate and completed the screening questions (i.e. the inclusion criteria for this study). A total of 28 did not meet our inclusion criteria. A further 50 did meet our inclusion criteria but exited the questionnaire after the screening, and another 38 completed less than half of the items (<21 items).A convenience sample of 247 participants was used in this study. Table et al. et al. Capability represents the professionals' psychological and physical ability to enact a behaviour, which includes having the necessary knowledge and skills. Opportunity is defined as any circumstance in the physical or social environment that influences a behaviour: all factors that are external to the professional. Motivation represents all those brain processes that energise and direct the behaviour of the professional , their own characteristics  and characteristics of their organisations. The evaluation form included questions about the time needed to complete the tool, the clarity and completeness of the instructions at the start and in the course of completing the tool, the clarity and applicability of individual items and their response options in the tool and the completeness of the tool in terms of the physical‐activity support topic. The proposed tool, a demographic questionnaire, and an evaluation form were made available online using The pilot results were discussed with the research group and two field experts, which resulted in some adjustments. One item on education was removed from the sub‐scale ‘capability’ and translated into an organisational characteristic about whether or not they were trained in physical‐activity support and what sort of education they had received, which was then relocated in the demographic questionnaire. Another item on practical support was added to the tool and was attributed to the sub‐scale ‘opportunity’. Based on the pilot results, we also added ‘expected time costs’ to the introduction section and screening questions. Furthermore, we decided to add a question to the demographic questionnaire about the role of the physiotherapist in their organisation.Qualtrics research software was again used to make both the adapted demographic questionnaire and the proposed tool available online. The psychometric properties of the tool were examined in this study.A final 41‐item self‐reported tool was proposed, with seven items covering the capability construct, 16 the opportunity construct and 18 the motivational construct. The descriptive statistics were computed first. Raw item scores were described according to mean (standard deviation), and the frequency scores of the response options were given. Response categories were collapsed for further analyses, in case too few participants had chosen a response option (minimum of 12 ratings for a response option).et al. et al. The psychometric properties were analysed using an item response theory (IRT) model separately for the three sub‐scales proposed. IRT is a statistical theory consisting of mathematical models describing the relationships between the properties of single items of a tool, the underlying construct that a tool proposes to measure and respondents' answers to any item Kline . Compareθ), measured by the item set, is characterised by two parameters: a slope parameter (denoted α) and the thresholds category parameters (denoted as β). α describes how well an item can differentiate along θ and, similar to factor loadings, how well the item relates to the construct measured. A reasonable range for α is from 0.5 to 3.0  β's for each item  and their standard errors were assessed.The information functions Toland  are the θ Toland . In addiθ Toland . For eacpartly agree and agree were, on average, slightly more often (56% of responses) used by the participants, although only the mean score of the item covering social influence by colleagues was higher than 3.0. The mean score of the item covering unforeseen things was the only one in the direction of the disagree point along the continuum (mean score < 2.0). Participants also responded, on average, more frequently with partly agree or agree to the items in the motivation scale (70% of responses). Ten out of 18 items had a mean score higher than 3.0. Three out of 18 had a mean score lower than 2.0.Table α for each item was reasonable. This also indicates that all the items have a satisfactory distinction power. The category threshold parameters range from −2.08 to 1.99. Within each item, the distance between the lowest and highest category threshold parameters is 1.74 to 4.07 units, which means that the capability construct is well covered. In addition, as shown in Table The calibrated graded response model for the capability scale explained 50% of the data variance. Factor loadings ranged from 0.56 to 0.82. The estimated slope parameters for the items in the capability scale range from 1.14 to 2.41 Table  and confIn Fig. The calibrated graded response model for the opportunity scale explained a proportional variance of 0.31, where factor loadings ranged from 0.41 to 0.75. The estimated slope parameters for the items in the opportunity scale range from 0.77 to 1.94, which indicates that all the items have a satisfactory distinction power . These items, however, did not correlate with items from the capability and opportunity scales. Their retention in these scales is warranted as long as the IRT score estimates, which take into account item properties, are used. Furthermore, removing items is only allowed when it does not destroy content validity Toland . The resn = 38) exited the online tool before completion, and this study's design did not allow for the reasons for quitting to be identified. It might be that these direct support professionals did not agree with the content of the tool. In future, we should incorporate the rationale behind the reason for not completing. The same applies to percentage of missing data for some items (range: 0 to 6.4). However, it can be assumed that these limitations did not significantly affect the results presented in the current study. In IRT models, because of the invariance property, a non‐random sample from the population of interest can be used . Changing the content in terms of difficulty of these items could also contribute to the scale's construct validity.The content in terms of the difficulty of some of the items could be adjusted. For example, the category threshold estimates for the response options of The scales in their current state are particularly reliable in determining those who score on the lower levels of capability, opportunity and motivation. To improve the distinctiveness and reliability of the scales, we recommend adding more items to the capability scale with thresholds category above 1.2, to the opportunity scale above 1.8 and to the motivation scale above 1.3. However, additional items are not necessary when the intention is to use the scales in the clinical field to principally identify those direct support professionals who can benefit from an intervention or change in policy.Another recommendation for practical purposes may be to shorten the tool, especially for the opportunity and motivation scales. In reference to the study results, some items both reflect the same concept and have overlapping IIFs (Appendix). In the context of a critical look at content validity, one might consider removing one of the items or merging them. For example, although various aspects were addressed, there are multiple items covering the concept of organisation. Policymakers might choose to merge the items for organisational support, time provided and budget. Alternatively or in addition, policymakers might choose between the item on family expectations and the one on family support, because both function in a similar way in this study's data. However, the psychometric properties would then have to be re‐examined, which can be carried out in close collaboration with researchers.Future psychometric research on the tool should incorporate participant‐centred research methods, such as interviews and behavioural observations. Interviews that investigate the perspectives of direct support professionals for different positions on the continua or with striking combinations will contribute to validation of the tool. Accordingly, this can help to improve our understanding of direct support professional behaviour. Behavioural observations allow researchers to measure the tool's correlation with the actual physical‐activity support for people with ID. In addition, future research should assess the tool's intra‐rater reliability and its sensitivity to change over time. This will enable the use of this tool to monitor and evaluate intervention functions and organisational policy change focused on improving the physical‐activity support.This study focused on the development of a tool to measure the behaviour of direct support professionals and has provided evidence on preliminary content, construct and reliability. The tool can be used to measure the capability, motivations and opportunities afforded to carry out physical‐activity support among direct support professionals who support people with ID. The tool can also be used to measure differences between direct support professionals in terms of their own characteristics, the diversity of the people with whom they work and their environmental context. Moreover, this study's results have addressed theoretical support for the model of direct support professional behaviour in the physical‐activity support for people with ID.This research did not receive any specific grants from funding agencies in the public, commercial or not for‐profit sectors.There was no conflict of interest, and no restrictions were imposed on the publication of results.


Leptodiaptomus sicilis in multiple perspectives together with suspended particles that allow us to analyse the flow field created by the animal. We observe a highly periodic motion of the mouth appendages that is mirrored in oscillations of nearby tracer particles. We propose that the phase shift between the fluid and the particle velocities is sufficient for mechanical detection of the particles entrained in the feeding current. Moreover, we propose that an immersed algal cell may benefit from the excitation by increased uptake of dissolved inorganic compounds.Calanoid copepods are small crustaceans that constitute a major element of aquatic ecosystems. Key to their success is their feeding apparatus consisting of sensor-studded mouth appendages that are in constant motion. These appendages generate a feeding current to enhance the encounter probability with food items. Additionally, sensing enables the organism to determine the position and quality of food particles, and to alter the near-field flow to capture and manipulate the particles for ingestion or rejection. Here we observe a freely swimming copepod  As they are located between autotrophic phytoplankton and higher heterotrophic organisms, they play a crucial role in the biogeochemical cycles on earth. Chemical and mechanical reception are their main way of obtaining information about their surroundings. They are alert swimmers and are known to create flow fields suited for different purposes. Fossilized copepods have been found in bitumen clast from the upper Carboniferous (303 Ma)2. Thus, we may be looking at the 500 millionth generation which gives a lot of room for evolutionary refinement of sensory appendages and the processing apparatus.Living organisms depend on extracting the “right” information from their environments. Carriers of informations are manifold: chemical substances announce the presence of food or conspecifics interested in mating, electromagnetic waves are the source of all vision, and mechanical signals in form of fluid disturbances allow to infer the presence of preys, predators, and again, conspecifics in both air and water. Planktonic copepods are one of the most abundant classes of animal life and occupy nearly every possible location in the worlds marine and freshwater bodies3. To overcome the dominating viscosity due to the low Reynolds number, swimming and flying animals vibrate their appendages at high frequency, e.g. 30–70 Hz in planktonic copepods5. However, vibrating appendages create fluid mechanical signals that can be detected by predators at large distances9. Fish, the predators of planktonic copepods, sense the presence and location using their lateral line system10. Similar receptor systems have been described in various aquatic and terrestrial species, such as reptiles, amphibians, chaetognaths (arrow worms)11, insects12 and arachnids13. Legs of spiders are equipped with trichobothria that are capable of detecting oscillations of the air caused by potential prey several centimetres away. Cockroaches, crickets and mosquitoes have been found to respond to and to produce sound of selected frequencies, for example in order to ease mating15. The hairs and hair-like sensory organs in these arthropods are extremely sensitive, responding to displacements of a few nanometres16. Guided by the mechanosensory pathway in the central nervous system of the crayfish Procambarus17, we suggest that small velocity perturbations can also be captured by small aquatic arthropods.Life in water and with a body size of a few millimetres means “Life at Low Reynolds Number”19. At low Reynolds number, in order to generate effective propulsion it is also required that the power stroke and the recovery stroke of the oscillatory movements should not be symmetric or reciprocal3. This is certainly true for planktonic animals, such as calanoid copepods20. Operating at low Reynolds number in the water, planktonic copepods oscillate their cephalic appendages achieve thrust for active swimming and to generate a feeding current, or more precisely, a scanning current22. Both active swimming and generation of a feeding current increase the encounter rate between the copepod and the suspended food items24. Copepod appendages are equipped with hair-like sensors (setae) with chemical and mechanical sensory functions27. The non-oscillatory far-field component of the feeding current enables the copepod to sense the presence, position and quality of food particles via chemoreception30. As the flow created by the copepod is chaotic in the vicinity of the animal, chemoreception is possible only for objects at greater distances. Near the animal, chemoreception has to be complemented by mechanoreception. Serendipitously, the oscillatory motion of the animal’s appendages needed for swimming, creates an oscillatory velocity field. Embedded particles that do not exactly match the specific weight of the surrounding water follow this oscillatory motion with a velocity shift. The velocity shift has a chance to be detected by the animal if it is above a certain threshold. Earlier observations and numerical simulations30 confirmed the ability of a copepod to detect streamline deformations by a particle in the direction of linear motion and to elicit a jump and capture reaction.Oscillatory movements of animal appendages  are a universal scheme for generating thrust for continuous propulsion by the animal in a fluid medium31. Accounting for the damping of the signal that emerges 200–300 μm away from the sensory apparatus, we conclude that it is sufficient for detection of the particles entrained in the feeding current. Our theoretical considerations are complemented by observations of an adult freshwater calanoid copepod Leptodiaptomus sicilis swimming while sensing its environment for food. Earlier observations of small and translucent animals in water required that these be tethered to hairs to prevent the animals from escaping the field of vision of the camera. This may distort the behaviour of the animal as it may vigorously try to escape. In contrast, we observe a freely swimming copepod together with suspended particles at micrometer resolution. This technique was used recently in a study of the swimming behaviour of the copepod Eurytemora affinis under turbulent flow conditions32. The particles allow us to analyse the oscillatory flow field created by the animal. We observe a highly periodic motion of the mouth appendages that is mirrored in oscillations of the tracer particles in the adjacent regions. Our findings are compatible with a theory of echolocation in freshwater calanoid copepods. We also propose that algal particles in the feeding current stand to profit from exposure to turbulent shear.Here we present fluid-mechanical calculations to determine the phase shift between the velocities of an oscillating fluid and an immersed particle together with its maximal magnitude. Similar work has been recently done to determine the effect of the impulsive motion of a plate on a small sphereLeptodiaptomus sicilis is given in Fig. 34 near the copepod.A general perspective of an untethered copepod f is given byA (estimated from observations) is related to the animal prosome length L = 1 mm by A = 0.15⋅L. For an aspect ratio of the copepod of s = 0.37, according to35 the preferred food particle size is given byIf even small density differences between the particles and the fluid are present, the particles do not follow the flow entirely. In the following we compute the velocity of a tracer particle and its deviation from a surrounding oscillating flow field. The fluid velocity caused by the beating motion of the cephalic appendage with frequency The density ratio of the particle isν = 1.004·10−6 m2 s−1. Following  we calculateτ is the dimensionless period (or Stokes’ number) and η and β are the amplitude ratio and phase shift, respectively. Intuitively, for low frequency (high τ), the particle follows the fluid motion, as it does for very small deviations from the fluid density. The amplitude ratio and the phase shift determine the particle velocityThe kinematic viscosity of water at 20 °C is The maximum relative velocity between the particle and the local water motion isγ in Fig. We plot the maximum velocity difference as a function of the density ratio ω in a direction parallel to itself. The velocity disturbance decays exponentially with the distance from the plate. The thickness of the boundary layer is given by ν is the kinematic viscosity . Thus the penetration depth decreases with the frequency but increases with the kinematic viscosity of the fluid.A Stokes Boundary Layer is formed by a rigid plate oscillating with a frequency Leptodiaptomus sicilis together with synchronously oscillating tracer particles in its vicinity. The 51 s long film can be considered a case of serendipity, as it is extremely difficult to find such an animal and to observe it for so long. A tethered copepod may sense that it is tethered and change its swimming behavior.Located between autotrophs and heterotrophs in the aquatic food web, copepods play an important role and have been studied widely by ecologists. We have observed a freely swimming copepod Nitella, Nitellopsis and Chara have been reported to be 1.01–1.015 g cm−3 39 while the density of the cell wall is considerably larger, namely 1.35 g cm−3. Assuming a particle density of 1.05 g cm−3, the maximal velocity difference is approximately 160 μm s−1 at the site of the particle. Ignoring the fact that we have a different geometry, the distance over which an oscillation at 34 Hz is damped by a factor of e−1 (≈37 %) is 240 μm. This is within the distance of the particles in Fig. μm s−1 would be damped to about 60 μm s−1. This is above the detection threshold of 20 μm s−1 reported for the first antennae16.The maximal velocity difference from Eq.  is in maL and beating frequencies f. Figures of this type go back to the early works of Schröder5, who found a functional relationship between body length and beating frequency in calanoid copepods. We propose that various copepod species of different lengths will select their beating frequencies to minimize the energetic cost but at the same time to make sure that the relative velocity differences are detectable. Many arthropod species are known to “listen” to external mechanical signals, and copepods have long been known to react to reflections of their own swimming wake to orient themselves40. The process of true echolocation differs in that the animal intentionally emits the outgoing signal, which is well known in insectivorous bats and toothed whales. The increased risk of the copepod itself being detected by its predators seems a price worth paying.In Fig. 43. As the flow around the cell changes from laminar to turbulent, the phycosphere gets deformed until it turns into a web of filaments, see e.g. Figure 3 in42. In such an event, the cell also leaves its “stale” diffusion limited surroundings and has access to fresh water with inorganic nutrients. Very recent research has demonstrated that cell chains of the diatom genera Skeletonema and Chaetoceros increased their carbon assimilation in turbulent shear conditions44. The bloom-forming and toxic cyanobacterium Dolichospermum is also able to increase its nutrient uptake in more livid hydrodynamical conditions45.In this interaction, we have so far considered only one of the partners, the copepod. An algal particle is surrounded by a sphere that is enriched in organic molecules leaked by the cell. This phycosphere is the important interface in which exchanges between phytoplankton and bacteria and their surroundings take place48. Here, we propose a new mechanism by which copepods perceive algae in their oscillatory near-field feeding currents. Molecules emitted by organic matter at even very low rates carry information about the quality of the particle as potential foodstuff49, see Figure 3 in8 for a numerical study and Fig. 50, for example by fanning of the wings in moths. In his groundbreaking paper, Aref51 indicated mechanisms by which simple periodic motions of rotating stirrers can create chaotic flow patterns, see also52. This has now been termed “chaotic advection”. Depending on the precise model for agitator motion, there are frequencies that result in larger areas or volumes of the chaotic advection region. The theoretical considerations of Aref51 indicate that too high a beating frequency leads to inefficient stirring, apart from the energetic cost to the animal.Returning to the feeding copepod, not all particles suspended in water are created equal. There are recent debates on the viability of remote chemoreception of algae by copepods that generate feeding currentsLeptodiaptomus sicilis  were caught in Lake Michigan at the Fox Point time series station  near Milwaukee. The animals were stored at 4 °C and used within a day. Copepods were transferred to an observation aquarium with a volume of 125 mL with water at 20 °C and left to acclimatize for 2 h. The untethered copepods were filmed using a Photron camera at 500 Hz together with tracer particles of 15 μm diameter. The concentration of tracer particles was estimated from a frame of the film to be 6 mm−3 which is comparable to an environmental algae concentration. The optical system of beam splitters and prisms used for the multi-view films is described in53. The digital videos were analysed with CellTrack54 and ImageJ55 and the data  were exported to Microsoft Excel for further evaluation.Adult copepods of the species 56 and together with the data are available from the authors upon request. The maxilliped was traced at its first joint while we also corrected for the small net motion of the pivot point. To analyse the appendage oscillations we consider the angle that is formed with the bisector of the sweep angle. Tracer particles immersed in the water were traced simultaneously. Cubic polynomials were fitted to the trajectories without self-intersections (18 trajectories out of 35). Cubic polynomials were chosen since all particle trajectories were oriented nearly horizontally and showed relatively little curvature. The density of particles is quite high in order to better visualize the flow field around the copepod. In nature, the encounters are much rarer. To analyse the particle oscillations we used the Discrete Fourier Transform (DFT). The trajectories were smoothened by eliminating all contributions in the Fourier transformed signal that were smaller than 25 % of the peak energy.The numerical procedures have been implemented in the open source package ScilabSupplementary InformationVideo 1Video 2Video 3Dataset 1


This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.The online version of this article (10.1007/s10898-018-0711-5) contains supplementary material, which is available to authorized users. This paper deals with the problem of finding the smallest value of r such that one can pack Let Packomania  Let mentclass2pt{minimn due to symmetries in the search space. This section introduces a tiling method to split ..\documenA and B be the vertices in C be the vertex in The triangle entclass1pt{minimaentclass1pt{minimay axis of Proposition The tiling produced by Algorithm 2 suffices if one wants to use a complete global optimization approach for the packing problem. On the other hand, it is not suitable for a rigorous approach since the elements in s in the Problem (The elements in the Eqs. –11) are are and  Problem  by the iMarkót and Csendes  propose Friedman  gives anLet mentclass2pt{minimcore i7 processor with a frequency of 2.6 GHz, 6 GB of RAM and Windows 10. We compiled the code using the g++ 7.3 compiler with the option http://www.mat.univie.ac.at/~montanhe/publications/n3.zipWe perform the computational part of the proof in a i-th iteration, we consider instances of form (We prove the theorem in three phases. At the  of form  and defi of form –11) acc acci-th s replaced by the interval The proof considers the scaled version of the problem to ensure the mathematical certainty of our statements. Therefore, the CSPs in this section are of form  with thementclasspt{minimaWe also assume the labeling scheme for the triangles introduced in Sect. Phase 1 In this iteration, we are interested in reducing the search domain of each subproblem and finding triangles which can contain the center of squares with no rotation. The tiling has 36 triangles, but the symmetries in Instances Phase 2 This phase aims to discard as many instances as possible to reduce the number of hard subproblems in the last iteration. There are 630 possible combinations of 36 triangles taken 2 by 2. After eliminating symmetric and previously discarded cases, we obtain 43 instances. We also propagate any reduction in the search domain in the first phase to the subproblems in the second phase. Again, we remove the condition We run the Algorithm 1 with entclass1pt{minimaWe conclude the second phase with 22 infeasible subproblems. Again, any case in the next phase containing a combination found infeasible in this step can be discarded without any processing.Phase 3 In this phase we set the full model in Problem Below is the link to the electronic supplementary material.