Study of Nonlinear MHD Tribological Squeeze Film at Generalized Magnetic Reynolds Numbers Using DTM

In the current article, a combination of the differential transform method (DTM) and Padé approximation method are implemented to solve a system of nonlinear differential equations modelling the flow of a Newtonian magnetic lubricant squeeze film with magnetic induction effects incorporated. Solutions for the transformed radial and tangential momentum as well as solutions for the radial and tangential induced magnetic field conservation equations are determined. The DTM-Padé combined method is observed to demonstrate excellent convergence, stability and versatility in simulating the magnetic squeeze film problem. The effects of involved parameters, i.e. squeeze Reynolds number (N ₁), dimensionless axial magnetic force strength parameter (N ₂), dimensionless tangential magnetic force strength parameter (N ₃), and magnetic Reynolds number (Re _m) are illustrated graphically and discussed in detail. Applications of the study include automotive magneto-rheological shock absorbers, novel aircraft landing gear systems and biological prosthetics.     

Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.     

Left-right asymmetric morphogenesis of the anterior midgut depends on the activation of a non-muscle myosin II in Drosophila

Many animals exhibit stereotypical left-right (LR) asymmetry in their internal organs. The mechanisms of LR axis formation required for the subsequent LR asymmetric development are well understood, especially in some vertebrates. However, the molecular mechanisms underlying LR asymmetric morphogenesis, particularly how mechanical force is integrated into the LR asymmetric morphogenesis of organs, are poorly understood. Here, we identified zipper (zip), encoding a Drosophila non-muscle myosin II (myosin II) heavy chain, as a gene required for LR asymmetric development of the embryonic anterior midgut (AMG). Myosin II is known to directly generate mechanical force in various types of cells during morphogenesis and cell migration. We found that myosin II was involved in two events in the LR asymmetric development of the AMG. First, it introduced an LR bias to the directional position of circular visceral muscle (CVMU) cells, which externally cover the midgut epithelium. Second, it was required for the LR-biased rotation of the AMG. Our results suggest that myosin II in CVMU cells plays a crucial role in generating the force leading to LR asymmetric morphogenesis. Taken together with previous studies in vertebrates, the involvement of myosin II in LR asymmetric morphogenesis might be conserved evolutionarily.     

Near optimal manufacturing flow controller design

Flow control of flexible manufacturing systems (FMSs) addresses an important real-time scheduling requirement of modern manufacturing facilities, which are prone to failures and other controllable or stochastic discrete events affecting production capacity, such as change of setup and maintenance scheduling. Flow controllers are useful both in the coordination of interconnected flexible manufacturing cells through distributed scheduling policies and in the hierarchical decomposition of the planning and scheduling problem of complex manufacturing systems. Optimal flow-control policies are hedging-point policies characterized by a generally intractable system of stochastic partial differential equations. This article proposes a near optimal controller whose design is computationally feasible for realistic-size systems. The design exploits a decomposition of the multiple-part-type problem to many analytically tractable one-part-type problems. The decomposition is achieved by replacing the polyhedra production capacity sets with inscribed hypercubes. Stationary marginal densities of state variables are computed iteratively for successive trial controller designs until the best inscribed hypercubes and the associated optimal hedging points are determined. Computational results are presented for an illustrative example of a failureprone FMS.     

A Highlights from MBoC Selection: Fast Microtubule Dynamics in Meiotic Spindles Measured by Single Molecule Imaging: Evidence That the Spindle Environment Does Not Stabilize Microtubules

Metaphase spindles are steady-state ensembles of microtubules that turn over rapidly and slide poleward in some systems. Since the discovery of dynamic instability in the mid-1980s, models for spindle morphogenesis have proposed that microtubules are stabilized by the spindle environment. We used single molecule imaging to measure tubulin turnover in spindles, and nonspindle assemblies, in Xenopus laevis egg extracts. We observed many events where tubulin molecules spend only a few seconds in polymer and thus are difficult to reconcile with standard models of polymerization dynamics. Our data can be quantitatively explained by a simple, phenomenological model—with only one adjustable parameter—in which the growing and shrinking of microtubule ends is approximated as a biased random walk. Microtubule turnover kinetics did not vary with position in the spindle and were the same in spindles and nonspindle ensembles nucleated by Tetrahymena pellicles. These results argue that the high density of microtubules in spindles compared with bulk cytoplasm is caused by local enhancement of nucleation and not by local stabilization. It follows that the key to understanding spindle morphogenesis will be to elucidate how nucleation is spatially controlled.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

Temperature Effects on Race Determination in Heterodera glycines

Currently there are 16 possible races for Heterodera glycines, and these are differentiated based on ability of a nematode population to develop on a set of four differential soybean genotypes. Because results are based on numbers of nematode females that develop to a specific stage rather than on the reproductive capability of these females, race determinations based on female indices may not represent results obtained after several reproductive cycles of H. glycines. Counting numbers of eggs and juveniles, and then developing corresponding indices, would allow reproduction to be considered in making race determinations. Our objectives were to compare the present race identification scheme for H. glycines based on female indices with those using egg and juvenile indices and to examine the effect of temperature on race designations using female, egg, and juvenile indices. Race designations for H. glycines populations from two locations in Illinois were determined at 20, 27, and 30 °C in a water bath. The numbers of females, eggs, and juveniles (at 19 days) were recorded, and an index based on each life stage was calculated. Race determinations based on female, egg, or juvenile indices were inconsistent when conducted at 20 °C, which demonstrates that this temperature is not suitable for identifying races of H. glycines. However race designations at 27 and 30 °C were consistent for all three indices. This indicates that counting females, eggs, or juveniles should be equally reliable when race determinations are conducted at these two temperatures, and choice of method would depend on investigator preference or research objective.     

New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.     

Investigation of Turing instability for the Gierer–Meinhardt model

The dependence of the emergence of Turing instability for a distributed system of nonlinear differential equations that describe hydra morphogenesis based on the oscillatory properties of the corresponding trajectories of the system was investigated. The limits in the parameter space that provide diffusive instability were obtained. The frequency and amplitude dependences of the resulting spatial self oscillations on the values of the main parameters were investigated. Comparative analysis of the properties of the distributed system and corresponding trajectories of the system was carried out and the analytical conclusions were confirmed by the solutions of the system that were found using MATLAB.

     

Sine-Gordon Equation in (1+2) and (1+3) dimensions: Existence and Classification of Traveling-Wave Solutions

The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its kink solutions (one-dimensional fronts) are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. Although it has been derived over the years for quite a few physical systems that have nothing to do with Special Relativity, the Sine-Gordon equation emerges as a non-linear relativistic wave equation. This opens the way for exploiting the tools of the Theory of Special Relativity. Using no more than the relativistic kinematics of tachyonic momentum vectors, from which the solutions are constructed through the Hirota algorithm, the existence and classification of N-moving-front solutions of the (1+2)- and (1+3)-dimensional equations for all N ≥ 1 are presented. In (1+2) dimensions, each multi-front solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than a limiting speed, c = 1, or are greater than or equal to c. To connect with concepts of the Theory of Special Relativity, c will be called “the speed of light.” In (1+3)-dimensions, multi-front solutions are characterized by spatial structure and by velocity composition. The spatial structure is either planar (rotated (1+2)-dimensional solutions), or genuinely three-dimensional – branes. Planar solutions, propagate rigidly at one velocity, which is lower than, equal to, or higher than c. Branes must contain clusters of fronts whose speed exceeds c = 1. Some branes are “hybrids”: different clusters of fronts propagate at different velocities. Some velocities may be lower than c but some must be equal to, or exceed, c. Finally, the speed of light cannot be approached from within the subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.     

Misshapen decreases integrin levels to promote epithelial motility and planar polarity in Drosophila

Complex organ shapes arise from the coordinate actions of individual cells. The Drosophila egg chamber is an organ-like structure that lengthens along its anterior–posterior axis as it grows. This morphogenesis depends on an unusual form of planar polarity in the organ’s outer epithelial layer, the follicle cells. Interestingly, this epithelium also undergoes a directed migration that causes the egg chamber to rotate around its anterior–posterior axis. However, the functional relationship between planar polarity and migration in this tissue is unknown. We have previously reported that mutations in the Misshapen kinase disrupt follicle cell planar polarity. Here we show that Misshapen’s primary role in this system is to promote individual cell motility. Misshapen decreases integrin levels at the basal surface, which may facilitate detachment of each cell’s trailing edge. These data provide mechanistic insight into Misshapen’s conserved role in cell migration and suggest that follicle cell planar polarity may be an emergent property of individual cell migratory behaviors within the epithelium.     

Ontogenetic constructions of some fossil plants

This article presents a survey of ontogenetic studies in paleobotany, and of biologically relevant mathematical results available from such techniques as finite element analyses, algorithmic systems, and computer simulation. Dynamic representations of growth are possible when the observed cellular arrangements in fossils are mathematically described. Successive computer solutions of parameterizing equations allow for the extrapolation of ontogenetic trends forwards and backwards in time (i.e., more and less mature stages, respectively), as well as the interpolation of missing stages or portions of an organism's development. The hypothetical constructions derived from these techniques may be tested against direct comparisons with the fossil being simulated and/pr proposed modern analogues. Similarly, multicellular organisms or portions of organisms (e.g., leaves, sporangia) may be constructed as arrays of symbols — each symbol representing a cell or group of cells. Development in such models is simulated by providing instructions for cell division, cell death, or alteration in cellular states, e.g., vegetative to reproductive. Illustrative simulations of Parka, Mastopora, Rhynia and Calamites are presented and paleobotanical conclusions concerning their respective growth patterns are drawn.     

Solution of large compartmental models using numerical transform inversion

Compartmental models of biological or physical systems are often described by a system of “stiff” differential equations. In this paper an algorithm for solving a system with linear coefficients is presented that employs numerical inversion of the Laplace transform of the model equations. The inversion algorithms and Gear's backward differentiation method are compared for two stiff test problems and a differential system governing a 27-compartment model of bile acid transport and metabolism. The inversion algorithm is reliable, requires modest computation time on a desktop computer and provides better accuracy than Gear's method, especially for the extremely stiff example.     

Cdc42 GTPase-activating proteins (GAPs) regulate generational inheritance of cell polarity and cell shape in fission yeast

The highly conserved small GTPase Cdc42 regulates polarized cell growth and morphogenesis from yeast to humans. We previously reported that Cdc42 activation exhibits oscillatory dynamics at cell tips of Schizosaccharomyces pombe cells. Mathematical modeling suggests that this dynamic behavior enables a variety of symmetric and asymmetric Cdc42 activation distributions to coexist in cell populations. For individual wild-type cells, however, Cdc42 distribution is initially asymmetrical and becomes more symmetrical as cell volume increases, enabling bipolar growth activation. To explore whether different patterns of Cdc42 activation are possible in vivo, we examined S. pombe rga4∆ mutant cells, lacking the Cdc42 GTPase-activating protein (GAP) Rga4. We found that monopolar rga4∆ mother cells divide asymmetrically leading to the emergence of both symmetric and asymmetric Cdc42 distributions in rga4∆ daughter cells. Motivated by different hypotheses that can mathematically reproduce the unequal fate of daughter cells, we used genetic screening to identify mutants that alter the rga4∆ phenotype. We found that the unequal distribution of active Cdc42 GTPase is consistent with an unequal inheritance of another Cdc42 GAP, Rga6, in the two daughter cells. Our findings highlight the crucial role of Cdc42 GAP localization in maintaining consistent Cdc42 activation and growth patterns across generations.     

Graphical rules for non-steady state enzyme kinetic

The exiting graphical methods in enzyme kinetics can be used only within the scope of steady state reactions. In this paper, two graphical rules are presented to deal with the non-steady state enzyme catalysed reaction systems. According to Rule 1 we can immediately write out the phase concentration of enzyme species. The calculation work such as setting up differential equations, making Laplace transformation, expanding determinants, which are both tedious and liable to error, are completely saved. By means of Rule 2 the secular equations for the consecutive first-order reactions can be written out directly without need of setting up differential equations, expanding determinants, etc., that would otherwise be laborious and prone to errors. In addition, two check formulae are also presented for these two graphical methods, respectively. They are useful in order for avoiding the omission of terms during calculations, especially, for complicated mechanisms.     

Ronin governs the metabolic capacity of the embryonic lineage for post‐implantation development

During implantation, the murine embryo transitions from a “quiet” into an active metabolic/proliferative state, which kick‐starts the growth and morphogenesis of the post‐implantation conceptus. Such transition is also required for embryonic stem cells to be established from mouse blastocysts, but the factors regulating this process are poorly understood. Here, we show that Ronin plays a critical role in the process by enabling active energy production, and the loss of Ronin results in the establishment of a reversible quiescent state in which naïve pluripotency is promoted. In addition, Ronin fine‐tunes the expression of genes that encode ribosomal proteins and is required for proper tissue‐scale organisation of the pluripotent lineage during the transition from blastocyst to egg cylinder stage. Thus, Ronin function is essential for governing the metabolic capacity so that it can support the pluripotent lineage’s high‐energy demands for cell proliferation and morphogenesis.     

Dynamic microtubules produce an asymmetric E-cadherin–Bazooka complex to maintain segment boundaries

Distributing junctional components around the cell periphery is key for epithelial tissue morphogenesis and homeostasis. We discovered that positioning of dynamic microtubules controls the asymmetric accumulation of E-cadherin. Microtubules are oriented preferentially along the dorso-ventral axis in Drosophila melanogaster embryonic epidermal cells, and thus more frequently contact E-cadherin at dorso-ventral cell–cell borders. This inhibits RhoGEF2, reducing membrane recruitment of Rho-kinase, and increasing a specific E-cadherin pool that is mobile when assayed by fluorescence recovery after photobleaching. This mobile E-cadherin is complexed with Bazooka/Par-3, which in turn is required for normal levels of mobile E-cadherin. Mobile E-cadherin–Bazooka prevents formation of multicellular rosette structures and cell motility across the segment border in Drosophila embryos. Altogether, the combined action of dynamic microtubules and Rho signaling determines the level and asymmetric distribution of a mobile E-cadherin–Bazooka complex, which regulates cell behavior during the generation of a patterned epithelium.     

On Comparison of Series and Numerical Solutions for Flow of Eyring-Powell Fluid with Newtonian Heating And Internal Heat Generation/Absorption

In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters.