Long time behaviour and self-similarity in an addition model with slow Input of monomers

We consider a coagulation equation with constant coefficients and a time dependent power law input of monomers. We discuss the asymptotic behaviour of solutions as $t \to \infty$, and we prove solutions converge to a similarity profile along the non-characteristic direction

DNA hybridization to mismatched templates: a chip study

High-density oligonucleotide arrays are among the most rapidly expanding technologies in biology today. In the {\sl GeneChip} system, the reconstruction of the target concentration depends upon the differential signal generated from hybridizing the target RNA to two nearly identical templates: a perfect match (PM) and a single mismatch (MM) probe. It has been observed that a large fraction of MM probes repeatably bind targets better than the PMs, against the usual expectation from sequence-specific hybridization; this is difficult to interpret in terms of the underlying physics. We examine this problem via a statistical analysis of a large set of microarray experiments. We classify the probes according to their signal to noise ($S/N$) ratio, defined as the eccentricity of a (PM, MM) pair's `trajectory' across many experiments. Of those probes having large $S/N$ ($>3$) only a fraction behave consistently with the commonly assumed hybridization model. Our results imply that the physics of DNA hybridization in microarrays is more complex than expected, and they suggest new ways of constructing estimators for the target RNA concentration.Comment: 3 figures 1 tabl

Solving the riddle of the bright mismatches: hybridization in oligonucleotide arrays

HDONA technology is predicated on two ideas. First, the differential between high-affinity (perfect match, PM) and lower-affinity (mismatch, MM) probes is used to minimize cross-hybridization. Second, several short probes along the transcript are combined, introducing redundancy. Both ideas have shown problems in practice: MMs are often brighter than PMs, and it is hard to combine the pairs because their brightness often spans decades. Previous analysis suggested these problems were sequence-related; publication of the probe sequences has permitted us an in-depth study of this issue. Our results suggest that fluorescently labeling the nucleotides interferes with mRNA binding, causing a catch-22 since, to be detected, the target mRNA must both glow and stick to its probe: without labels it cannot be seen even if bound, while with too many it won't bind. We show that this conflict causes much of the complexity of HDONA raw data, suggesting that an accurate physical understanding of hybridization by incorporating sequence information is necessary to perfect microarray analysis.Comment: 4 figure

Relating the microscopic rules in coalescence-fragmentation models to the macroscopic cluster size distributions which emerge

Coalescence-fragmentation problems are of great interest across the physical, biological, and recently social sciences. They are typically studied from the perspective of the rate equations, at the heart of such models are the rules used for coalescence and fragmentation. Here we discuss how changes in these microscopic rules affect the macroscopic cluster-size distribution which emerges from the solution to the rate equation. More generally, our work elucidates the crucial role that the fragmentation rule can play in such dynamical grouping models. We focus on two well-known models whose fragmentation rules lie at opposite extremes setting the models within the broader context of binary coalescence-fragmentation models. Further, we provide a range of generalizations and new analytic results for a well-known model of social group formation [V. M. Eguiluz and M. G. Zimmermann, Phys. Rev. Lett. 85, 5659 (2000)]. We develop analytic perturbation treatment of the original model, and extend the mathematical to the treatment of growing and declining populations

A hierarchical cluster system based on Horton-Strahler rules for river networks

We consider a cluster system in which each cluster is characterized by two parameters: an "order"i, following Horton-Strahler rules, and a "mass"j following the usual additive rule. Denoting by ci,j(t) the concentration of clusters of order i and mass j at time t, we derive a coagulation-like ordinary differential system for the time dynamics of these clusters. Results about the existence and the behavior of solutions as t→∞ are obtained; in particular, we prove that ci,j(t) → 0 and Ni(c(t)) → 0 as t→∞, where the functional Ni(·) measures the total amount of clusters of a given fixed order i. Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that suggest the existence of self-similar solutions to these approximate equations and discuss their possible relevance for an interpretation of Horton's law of river numbers

Solution classes of the matrix second Painlevé hierarchy

We explore the generation of classes of solutions of the matrix second Painlevé hierarchy. This involves the consideration of the application of compositions of auto-Bäcklund transformations to different initial solutions, with the number of distinct solutions obtained for each value of the parameter appearing in the hierarchy depending on the symmetry properties of the chosen initial solution. This paper not only extends our previous results for the matrix second Painlevé equation itself, given in a recent paper, to the matrix second Painlevé hierarchy, but also provides a more detailed account of the underlying process

Kinetics of nucleation and growth: Classical nucleation and helium bubbles in nuclear materials

Discrete kinetic equations describe homogeneous nucleation and many other processes such as the formation and growth of helium bubbles due to self-irradiation in plutonium. A key ingredient in the analysis of these equations is a wave front expansion which is the equivalent of boundary layer theory for discrete equations. This expansion solves approximately the nucleation problem, but it needs to be patched to an outer solution describing sizes not too close to the maximum size for the helium bubble problem. The composite theory yields an integro differential equation for the monomer concentration of single helium atoms which compares well with numerical solution of the full discrete model

A continuum receptor model of hepatic lipoprotein metabolism

A mathematical model describing the uptake of low density lipoprotein (LDL) and very low density lipoprotein (VLDL) particles by a single hepatocyte cell is formulated and solved. The model includes a description of the dynamic change in receptor density on the surface of the cell due to the binding and dissociation of the lipoprotein particles, the subsequent internalisation of bound particles, receptors and unbound receptors, the recycling of receptors to the cell surface, cholesterol dependent de novo receptor formation by the cell and the effect that particle uptake has on the cell's overall cholesterol content. The effect that blocking access to LDL receptors by VLDL, or internalisation of VLDL particles containing different amounts of apolipoprotein E (we will refer to these particles as VLDL-2 and VLDL-3) has on LDL uptake is explored. By comparison with experimental data we find that measures of cell cholesterol content are important in differentiating between the mechanisms by which VLDL is thought to inhibit LDL uptake. We extend our work to show that in the presence of both types of VLDL particle (VLDL-2 and VLDL-3), measuring relative LDL uptake does not allow differentiation between the results of blocking and internalisation of each VLDL particle to be made. Instead by considering the intracellular cholesterol content it is found that internalisation of VLDL-2 and VLDL-3 leads to the highest intracellular cholesterol concentration. A sensitivity analysis of the model reveals that binding, unbinding and internalisation rates, the fraction of receptors recycled and the rate at which the cholesterol dependent free receptors are created by the cell have important implications for the overall uptake dynamics of either VLDL or LDL particles and subsequent intracellular cholesterol concentration. (C) 2008 Elsevier Ltd. All rights reserved