Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior activation by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at ± , the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Environmental gradients in a simulation model of a beech-yellow-poplar stand

A simulation model for growth and succession of a hypothetical American-beech–yellow-poplar forest has been developed to study changes in response of this simple community under conditions of a slowly varying climate. As the temperature, through a model parameter equal to the number of growing degree days (GDD), is increased, a sharp discontinuity in response of the model is noted at approximately 4500 GDD. A similar discontinuity is observed at about 4800 GDD as the temperature is slowly decreased. This hysteretic response with width of 300 GDD can be compared for consistency against the rather sharp boundaries which occur between forest communities along smooth environmental gradients. Although many environmental gradients are responsible for transitions is natural systems, the model calculations described indicate that changes in a single environmental variable, temperature, can account for transition zones by affecting competitive ability.     

The interleukin-15/interleukin-15 receptor system as a model for juxtacrine and reverse signaling

Interleukin-15 (IL-15) is a pleiotropic cytokine of the 4 alpha-helix bundle family, which binds to a receptor complex that displays common elements with the IL-2 receptor and a unique high-affinity alpha chain. This review focuses on juxtacrine and reverse signaling levels in the IL-15/IL-15R system. Specifically, we discuss how agonistic stimulation of membrane-bound IL-15 induces phosphorylation of members of the MAP kinase family and of focal adhesion kinase (FAK), thereby upregulating processes including cytokine secretion, cell adhesion and migration. In addition, we explore IL-15 trans-presentation and intracellular signaling, and define promising molecular targets for future pharmacological intervention in infectious diseases and immunological disorders. These frontiers in IL-15/IL-15Ralpha research serve as highly instructive examples for key concepts, unsolved problems and therapeutic opportunities in juxtacrine and reverse signaling in general.     

Processing and juxtacrine activity of membrane-anchored betacellulin

Betacellulin (BTC) was originally isolated as a secreted growth factor from a mouse pancreatic beta-tumor cell line, whereas the cDNA sequence predicts that BTC is synthesized as a larger transmembrane protein. In the present study, we have characterized the membrane-anchored forms of BTC, using Chinese hamster ovary (CHO) cells, mouse fibroblast A9 cells, and a human breast cancer cell line MCF-7, all of which were stably transfected with human BTC cDNA. A9 and MCF-7 transfectants produced membrane-anchored BTC isoforms of 21, 25, 29, and 40 kDa on the cell surface, as well as a secreted BTC isoform. CHO transfectants secreted little BTC but accumulated the membrane-anchored isoforms. The cleavage of the membrane-anchored forms to release a secreted from of BTC was not enhanced by biological mediators such as a phorbol ester, which stimulates the cleavage of other membrane-anchored growth factors. The membrane-anchored forms of BTC expressed on the transfected cells induced the insulin production and/or promoted the growth in subclones of AR42J rat pancreatic cells. These results suggest that the membrane-anchored BTC can function as a juxtacrine factor in regulating the growth and differentiation of pancreatic endocrine cells.     

On travelling wave solutions in a model for the Belousov-Zhabotinskii reaction.

DNA fragments partially digested by a 3′- or 5′-specific nuclease to produce single chain ends of opposite polarity will form a ring if the ends contain complementary sequences and are allowed to anneal. The frequency of rings can then be used as an assay to determine where and how identical repetitious sequences are arranged in the DNA. Thomas et al. (1973b) showed that all eucaryote chromosomes studied contain similar if not identical repetitious sequences clustered into regions called g-regions. To account for the observed ring frequency under different experimental conditions Thomas, Zimm &; Dancis (1973c) derived equations for two possible models of g-region organization. In the pure tandem model, the repetitious sequences are contiguous and occupy the entire g-region. In the intermittent repetition model, the repetitious sequences are simple copolymers and are irregularly arranged among non-repetitious sequences which are heterogenous in length. In the present paper, the results of Thomas et al. (1973c) are extended to cover the fractional tandem model. In this model, adjacent repetitious sequences are separated by non-repetitious sequences of uniform length. In addition, the equations for both the pure tandem and intermittent repetition models are shown to be special cases of the fractional tandem model but not vice versa.The capabilities and limitations of an analysis of ring formation are demonstrated using data from Drosophila. Although it is not possible to rule out any of the three models, the analysis can limit the ranges of the parameters describing each of the models that are consistent with the data. Previous conclusions that the data could only be explained by a pure tandem model which lacks any intervening unique sequences (Bick, Huang &; Thomas, 1973; Thomas et al., 1973b), are shown to be incorrect, in part because the equations for the fractional tandem model had not then been derived. Thus ring theory equations can be used to show the presence of clusters of similar if not identical sequences from ring-forming experiments, but they may not be able to determine the exact spacing and arrangement of these sequences within the clusters.     

A model of herd grazing as a travelling wave,chemotaxis and stability

Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.     

The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model

This paper concerns the minimal speed of traveling wave fronts for a two-species diffusion-competition model of the Lotka-Volterra type. An earlier paper used this model to discuss the speed of invasion of the gray squirrel by estimating the model parameters from field data, and predicted its speed by the use of a heuristic analytical argument. We discuss the conditions which assure the validity of their argument and show numerically the existence of the realistic range of parameter values for which their heuristic argument does not hold. Especially for the case of the strong interaction of two competing species compared with the intraspecific competition, we show that all parameters appearing in the system affect the minimal speed of invasion. Dedicated to the Memory of Akira Okubo     

Hormone gradients and cartilage cell kinetics

We present a model of growth control in mammalian cartilage growth plates by hormones. The model is based on the distribution of insulin-like growth factors I and II (IGF-I and IGF-II) and their receptors, and assumes that a hormone-receptor complex of IGF controls cells proliferation. A system of differential equations is derived and solved with simplifications in extreme cases, for the one-dimensional time independent case. Even if opposite extremes, such as proliferation control by factors extrinsic to the cell versus intrinsic to the cell, are assumed, similar distributions of hormones and proliferating cells are produced. Hence, choice between alternative models of growth control must be based on empirical observations. On the positive side, similarities between our model for cartilage growth and other models for differentiation and proliferation are evident and might be exploited for unifying these systems on an abstract level.     

Tumor cell haptotaxis on covalently immobilized linear and exponential gradients of a cell adhesion peptide

The movement of cells up an adhesive substratum gradient has been proposed as a mechanism for directing cell migration during development and metastasis. Critical evaluation of this hypothesis (haptotaxis) benefits from the use of quantifiable, stable substratum gradients of biologically relevant adhesion molecules. We report covalent derivatization of polyacrylamide surfaces with quantifiable gradients of a nonapeptide containing the adhesive Arg-Gly-Asp sequence. Cell migration was studied by seeding derivatized surfaces evenly with B16F10 murine melanoma cells. Within 8 hr, cells on gradients redistributed markedly; higher cell densities were found at gel positions having higher immobilized peptide densities. In contrast, cells seeded on control gels with uniform concentrations of adhesive peptide did not redistribute. Redistribution occurred on gradients in both serum-free and serum-containing media. Experiments with uniform density peptide-derivatized gels demonstrated that redistribution on gradients was not due to preferential initial cell attachment or preferential growth on the higher density of immobilized peptide, but must have been due to cell translocation. Cells on exponential gradients of immobilized peptide migrated to a position on the gel surface corresponding to the highest immobilized peptide density, while cells on linear gradients of the same peptide migrated to a position of intermediate peptide density. These data suggest that the B16F10 cells respond to proportional changes in immobilized peptide density rather than to absolute changes, implying a sensing mechanism which utilizes adaptation. These results demonstrate that (1) a gradient of a small adhesive peptide is sufficient to generate redistribution of cell populations and (2) controlled quantifiable substratum gradients can be produced and used to probe the underlying cellular mechanisms of this behavior.     

Schlafen-3 decreases cancer stem cell marker expression and autocrine/juxtacrine signaling in FOLFOX-resistant colon cancer cells

We have previously demonstrated that expression of the novel gene schlafen-3 (Slfn-3) correlates with intestinal epithelial cell differentiation (Patel VB, Yu Y, Das JK, Patel BB, Majumdar AP. Biochem Biophys Res Commun 388: 752-756, 2009). The present investigation was undertaken to examine whether Slfn-3 plays a role in regulating differentiation of FOLFOX-resistant (5-fluorouracil + oxaliplatin) colon cancer cells that are highly enriched in cancer stem cells (CSCs). Transfection of Slfn-3 in FOLFOX-resistant colon cancer HCT-116 cells resulted in increase of alkaline phosphatase activity, a marker of intestinal differentiation. Additionally, Slfn-3 transfection resulted in reduction of mRNA and protein levels of the CSC markers CD44, CD133, CD166, and aldehyde dehydrogenase 1 in both FOLFOX-resistant HCT-116 and HT-29 cells. This was accompanied by decreased formation of tumorosphere/colonosphere (an in vitro model of tumor growth) in stem cell medium and inhibition of expression of the chemotherapeutic drug transporter protein ABCG2. Additionally, Slfn-3 transfection of FOLFOX-resistant HCT-116 and HT-29 cells reduced Hoechst 33342 dye exclusion. Finally, Slfn-3 transfection inhibited the expression of transforming growth factor-α in both FOLFOX-resistant colon cancer cells, but stimulated apoptosis in response to additional FOLFOX treatment. In summary, our data demonstrate that Slfn-3 expression inhibits multiple characteristics of CSC-enriched, FOLFOX-resistant colon cancer cells, including induction of differentiation and reduction in tumorosphere/colonosphere formation, drug transporter activity, and autocrine stimulation of proliferation. Thus Slfn-3 expression may render colon CSCs more susceptible to cancer chemotherapeutics.     

pH gradients and cell polarity inPelvetia embryos

Summary Endogenous pH profiles were measured around single fertilized eggs of the brown algaPelvetia during the earliest stages of development. Profiles were constructed by measuring the pH near the cell surface at several positions using a pH sensitive microelectrode. Transcellular pH differences in the medium surrounding zygotes were detected soon after fertilization, as the developmental axis was being formed. The future rhizoid end of the cell was relatively alkaline and the presumptive thallus was acidic. At germination and throughout the first 5 d of embryogenesis, the apex of the elongating rhizoid was alkaline with respect to more distal regions. However, conditions that dissipated or reversed this extracellular pH gradient had little or no effect on polarization or growth, indicating that the gradient was not essential for early development.Inhibition of respiratory electron transport by cyanide and antimycin A eliminated the pH gradient, while uncouplers of oxidative phosphorylation [2,4-dinitrophenol (DNP) and carbonylcyanide m-chlorophenylhydrazone (CCCP)] stimulated acidification of the thallus regions. Proton ATPase inhibitors had no effect. Acidification, therefore, is not generated by ATP-dependent proton pumps in the plasma membrane, and instead probably reflects secretion of metabolic acids. Localized metabolism may establish an internal pH gradient that controls regional differentiation, and we are presently investigating this possibility.Abbreviations ASW artificial seawater - CCCP carbonylcyanide m-chlorophenylhydrazone - CD cytochalasin D - DNP 2,4-dinitrophenol     

Adaptation, oscillations and relay in a model for cAMP secretion in cellular slime molds

This work presents a unified theory for adaptation and for quiescence-excitable-oscillatory transitions in the cyclic AMP secretion system of the cellular slime mold amoeba Dictyostelium discoideum.     

Fabrication of protein gradients for cell culture using a miniature squeegee

We present a straightforward method to create spatial gradients of substrate bound protein for live cell studies using only mechanical parts. Protein concentration gradients on a micron scale can be fabricated in several minutes for a relatively low cost using a method that is generally applicable to any protein and substrate combination. We describe the details of the device construction, and provide examples of mammalian cells grown on substrates patterned with protein concentration gradients using this technique.     

Null model analysis of communities on gradients

Aim I employed a novel null model and metric to uncover unusual species co‐occurrence patterns in a herpetofaunal assemblage of 49 species collected at discrete elevations along a gradient. Location Mount Kupe, Cameroon. Methods Using a construction algorithm that started from a matrix of 0s, a sample null space of 25,000 unique null matrices was generated by simultaneously conserving (1) the number of occurrences of each species, (2) site richness and (3) species range spans derived from the observed incidence matrix. I then compared the number of times each pair of confamilial species co‐occurred in the null space with the same number derived from the observed incidence matrix. Two cases dealing with embedded absences in species ranges were tested: (1) embedded absences were maintained, and (2) embedded absences were assumed to be sampling omissions and were replaced by presences. Results In the observed absence/presence assemblage there were 147 possible confamilial species pairs. Therefore, 5% or eight were expected by chance alone to have co‐occurrence patterns that differed from chance expectations by chance alone. Of these confamilial species pairs, 38 were congeneric and so 5% or two were expected to differ from chance expectations. For case (1) 16, and for case (2) 17 confamilial species pairs’ co‐occurrence patterns differed significantly from chance expectations. For case (1) nine congeneric species pairs, and for case (2) 10 congeneric pairs differed significantly from chance expectations. For case (1) four, and for case (2) five congeneric species pairs formed checkerboards (patterns of mutual exclusion). Results from case (1) were a proper subset of case (2) indicating that sampling omissions did not alter greatly the results. Main conclusions I have demonstrated that null models are valuable tools to analyse ecological communities provided that proper models are employed. The choice of the appropriate null space to analyse distributions is critical. The null model employed to analyse birds on islands of an archipelago can be adapted to analyse species along gradients provided an additional range constraint is added to the null model. Moreover, added precision to results can be obtained by analysing each species pair separately, particularly those in the same family or genus, as opposed to applying a community‐wide metric to the faunal assemblage. My results support some of the speculations of previous authors who were unable to demonstrate their suspicions analytically.