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Altered tissue mechanics is an important signature of invasive solid tumors. While the phenomena have been extensively studied by measuring the bulk rheology of the extracellular matrix (ECM) surrounding tumors, micromechanical remodeling at the cellular scale remains poorly understood. By combining holographic optical tweezers and confocal microscopy on in vitro tumor models, we show that the micromechanics of collagen ECM surrounding an invading tumor demonstrate directional anisotropy, spatial heterogeneity and significant variations in time as tumors invade. To test the cellular mechanisms of ECM micromechanical remodeling, we construct a simple computational model and verify its predictions with experiments. We find that collective force generation of a tumor stiffens the ECM and leads to anisotropic local mechanics such that the extension direction is more rigid than the compression direction. ECM degradation by cell-secreted matrix metalloproteinase softens the ECM, and active traction forces from individual disseminated cells re-stiffen the matrix. Together, these results identify plausible biophysical mechanisms responsible for the remodeled ECM micromechanics surrounding an invading tumor. 

Abstract Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume v ex ( K ) for a general convex body K that applies in any space dimension. While our main interests concern the rotationally-averaged exclusion volume of a convex body with respect to another convex body, we also describe some results for the exclusion volumes for convex bodies with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume v ex ( K )/ v ( K ) among all convex bodies, whether randomly oriented or uniformly oriented, for any d , where v ( K ) is the volume of K . When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any d with a large- d asymptotic scaling behavior of 2 2 d / d 3/2 , which is to be contrasted with the corresponding scaling of 2 d for the sphere. We present explicit formulas for quermassintegrals W 0 ( K ), …, W d ( K ) for many different nonspherical convex bodies, including cubes, parallelepipeds, regular simplices, cross-polytopes, cylinders, spherocylinders, ellipsoids as well as lower-dimensional bodies, such as hyperplates and line segments. These results are utilized to determine the rotationally-averaged exclusion volume v ex ( K ) for these convex-body shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, we show that, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal v ex ( K )/ v ( K ) with a scaling behavior of 2 1.6618… d . Subsequently, we apply these results to determine the corresponding second virial coefficient B 2 ( K ) of the aforementioned hard hyperparticles. Our results are also applied to compute estimates of the continuum percolation threshold η c derived previously by the authors for systems of identical overlapping convex bodies. We conjecture that overlapping spheres possess the maximal value of η c among all identical nonzero-volume convex overlapping bodies for d ⩾ 2, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of η c for d ⩾ 2. 

We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives:min–avg  minimizes the average tour cost andmin–max  minimizes the maximum tour cost over all days. Formin–max  we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. Formin–avg  we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.

 

Stand-alone devices for tactile speech reception serve a need as communication aids for persons with profound sensory impairments as well as in applications such as human-computer interfaces and remote communication when the normal auditory and visual channels are compromised or overloaded. The current research is concerned with perceptual evaluations of a phoneme-based tactile speech communication device in which a unique tactile code was assigned to each of the 24 consonants and 15 vowels of English. The tactile phonemic display was conveyed through an array of 24 tactors that stimulated the dorsal and ventral surfaces of the forearm. Experiments examined the recognition of individual words as a function of the inter-phoneme interval (Study 1) and two-word phrases as a function of the inter-word interval (Study 2). Following an average training period of 4.3 hrs on phoneme and word recognition tasks, mean scores for the recognition of individual words in Study 1 ranged from 87.7% correct to 74.3% correct as the inter-phoneme interval decreased from 300 to 0 ms. In Study 2, following an average of 2.5 hours of training on the two-word phrase task, both words in the phrase were identified with an accuracy of 75% correct using an inter-word interval of 1 sec and an inter-phoneme interval of 150 ms. Effective transmission rates achieved on this task were estimated to be on the order of 30 to 35 words/min. 

Although catenanes comprising two ring-shaped components can be made in large quantities by templation, the preparation of three-dimensional (3D) catenanes with cage-shaped components is still in its infancy. Here, we report the design and syntheses of two 3D catenanes by a sequence of S N 2 reactions in one pot. The resulting triply mechanically interlocked molecules were fully characterized in both the solution and solid states. Mechanistic studies have revealed that a suit[3]ane, which contains a threefold symmetric cage component as the suit and a tribromide component as the body, is formed at elevated temperatures. This suit[3]ane was identified as the key reactive intermediate for the selective formation of the two 3D catenanes which do not represent thermodynamic minima. We foresee a future in which this particular synthetic strategy guides the rational design and production of mechanically interlocked molecules under kinetic control.