Simple living organisms (“cells”) are known to move both by diffusion and by chemotaxis. The latter denotes directed movement towards regions of higher concentration of a chemical substance secreted by the cells. We perform a mathematical analysis of this phenomenon, using the Keller-Segal model.

In this series of lectures, starting from a single logistic equation, I will describe the interactions between diffusion and spatial heterogeneity in Lotka-Volterra competition systems. Various phenomena in mathematical ecology will be compared in both homogeneous and inhomogeneous environments. Directed movements will be considered as well. Two-component Gross-Pitaevskii functionals Professor ...

The image-mean square displacement technique applies the calculation of the mean square displacement commonly used in single-molecule tracking to images without resolving single particles. The image-mean square displacement plot obtained is similar to the mean square displacement plot obtained using the single-particle tracking technique. This plot is then used to reconstruct the protein diffus...

The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these method...

Dynamic patterning of specific proteins is essential for the spatiotemporal regulation of many important intracellular processes in procaryotes, eucaryotes, and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article we review quantitative models for intracellular Min protein patterns in ...

Various challenges are faced when animalcules such as bacteria, protozoa, algae, or sperms move autonomously in aqueous media at low Reynolds number. These active agents are subject to strong stochastic fluctuations, that compete with the directed motion. So far most studies consider the lowest order moments of the displacements only, while more general spatio-temporal information on the stocha...

It has been difficult to establish whether pre-messenger ribonucleoprotein (pre-mRNP) particles move from the gene towards the periphery of the nucleus in a directed or random manner. Two recent in vivo studies indicate that most pre-mRNP particles move randomly in the nucleus, apparently by free diffusion.

Semi-supervised learning algorithms are an indispensable tool when labeled examples are scarce and there are many unlabeled examples [Blum and Chawla 2001, Zhu et. al. 2003]. With graph-based methods, entities (examples) correspond to nodes in a graph and edges correspond to related entities. The graph structure is used to infer implicit pairwise affinity values (kernel) which are used to compu...

The mechanism by which homologous chromosomes pair during meiosis, as a prelude to recombination, has long been mysterious. At meiosis, the telomeres in many organisms attach to the nuclear envelope and move together to form the telomere bouquet, perhaps to facilitate the homologous search. It is believed that diffusion alone is not sufficient to account for the formation of the bouquet, and th...

We describe an 8-spot confocal setup for high-throughput smFRET assays and illustrate its performance with two characteristic experiments. First, measurements on a series of freely diffusing doubly-labeled dsDNA samples allow us to demonstrate that data acquired in multiple spots in parallel can be properly corrected and result in measured sample characteristics consistent with those obtained w...