From microscopic to macroscopic descriptions of cell\ud migration on growing domains

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs

Mathematical description of bacterial traveling pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Trail formation based on directed pheromone deposition

We propose an Individual-Based Model of ant-trail formation. The ants are modeled as self-propelled particles which deposit directed pheromones and interact with them through alignment interaction. The directed pheromones intend to model pieces of trails, while the alignment interaction translates the tendency for an ant to follow a trail when it meets it. Thanks to adequate quantitative descriptors of the trail patterns, the existence of a phase transition as the ant-pheromone interaction frequency is increased can be evidenced. Finally, we propose both kinetic and fluid descriptions of this model and analyze the capabilities of the fluid model to develop trail patterns. We observe that the development of patterns by fluid models require extra trail amplification mechanisms that are not needed at the Individual-Based Model level

Zaštitno djelovanje selenija protiv prekomjerne ekspresije apoptotskih gena povezanih s karcinomom u štakora izloženih o-krezolu

Cresols are monomethyl derivatives of phenol frequently used as solvents and intermediates in the production of disinfectants, fragrances, pesticides, dyes, and explosives, which is probably why they are widely distributed in the environment. General population may be exposed to cresols mainly through inhalation of contaminated air. In this study we evaluated the toxicological effects of o-cresol on differential gene expression profile of rat liver and prostate. Experiments were conducted on 80 male rats, 60 of which were exposed to o-cresol (1.5 g kg-1, 5 g kg-1, or 15 g kg-1) through feed for 8 weeks. Three groups of rats were supplemented with 0.1 mg kg-1 selenium (Se, in the form of, sodium selenite) in addition to o-cresol to evaluate its effectiveness against o-cresol toxicity. Control group received neither o-cresol nor Se, while one group received Se alone. Survival was similar between the exposed and control animals. Rats exposed to 15 g kg-1 of o-cresol showed a 16 % loss in body weight by the end of the study, which may have been related to o-cresol making feed unpalatable at this concentration. Liver and prostate tissue samples were collected at the end of the treatment. mRNA analysis revealed that apoptotic genes (CYP3A, COX-2, PPARγ, BAX, BCL2, AKT-1, and PKCα) related to cancer were up-regulated in liver and prostate tissues isolated from groups exposed to 5 g kg-1 and 15 g kg-1 o-cresol in comparison to control. Changes in gene expression profile were prevented when rats were supplemented with Se. The exact mechanisms underlying its protective effect remain to be clarified by future studies.Krezoli su monometilni derivati fenola koji se često rabe kao otapala te kao posrednici u proizvodnji dezinfekcijskih sredstava, mirisa, pesticida, boja i eksploziva. Otuda i njihova rasprostranjenost u okolišu. Opća je populacija izložena krezolima uglavnom putem zraka. U ovome se toksikološkom istraživanju ocijenilo djelovanje o-krezola, jednoga od tri krezolova izomera, na ekspresiju gena u tkivima jetre i prostate mužjaka štakora. Istraživanje je provedeno na 80 mužjaka, od kojih je 60 tijekom osam tjedana bilo izloženo o-krezolu (1,5 g kg-1, 5 g kg-1, odnosno 15 g kg-1) preko krmiva. Tri skupine štakora primale su uz o-krezol nadomjestak selenija u dozi od 0.1 mg kg-1 (Se, u obliku natrijeva selenita) radi ocjene njegove djelotvornosti protiv toksičnosti o-krezola. Kontrolna skupina nije primala ni o-krezol ni Se, dok je jedna skupina primala samo Se. Preživljenje je bilo podjednako u svih skupina životinja. Štakori izloženi najvišoj dozi o-krezola (15 g kg-1) imali su 16 % manju tjelesnu masu od kontrolne skupine na kraju ispitivanja, što može biti povezano s lošim okusom krmiva zbog primjese visoke doze o-krezola. S istekom osmotjednoga izlaganja o-krezolu životinje su eutanazirane te su prikupljeni uzorci tkiva jetre i prostate. Analiza m-RNA pokazala je značajno povišenu ekspresiju apoptotskih gena CYP3A, COX-2, PPARγ, BAX, BCL2, AKT-1 i PKCα, koji su povezani s nastankom karcinoma u skupinama štakora izloženim o-krezolu (5 g kg-1 i 15 g kg-1 u odnosu na kontrolu. Ova je prekomjerna ekspresija poništena u štakora koji su primali selenij. Još nisu jasni mehanizmi iza ovoga zaštitnog djelovanja, na što će odgovoriti buduća istraživanja

Large scale dynamics of the Persistent Turning Walker model of fish behavior

International audienceThis paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results

The effect of salts on the liquid–liquid phase equilibria of PEG600 + salt aqueous two-phase systems

Six new ATPSs were prepared by combining polyethylene glycol PEG600 with potassium citrate, dipotassium hydrogen phosphate, sodium formate, potassium formate, sodium sulfate, and lithium sulfate. Complete phase diagrams, including the binodal curve and three tie-lines, were determined at 23 °C. The experimental data obtained for the binodal curve were successfully adjusted to the Merchuk equation, and the reliability of tie-line data was confirmed using the equations suggested by Othmer–Tobias and Bancroft. The ability of each ion to induce ATPS formation was investigated. Na+ proved to be more effective in ATPS formation than K+ and Li+. For potassium salts, the order observed for the effectiveness of the anions was: HPO42– > C6H5O73– > HCO2–. Regarding the sodium salts, it was found that SO42– is clearly more effective than HCO2–. The position of the ions in the Hofmeister series and their free energy of hydration (ΔGhyd) were used to explain the ability of the ions to induce PEG salting-out. Furthermore, the effective excluded volume (EEV) of the salts was determined and the following order was found: Na2SO4 > K2HPO4 > Li2SO4 > K3C6H5O7 > NaCHO2 > KCHO2. Similar order was obtained when analyzing the size of the heterogeneous regions, suggesting the practical use of EEV as a comparison parameter between different ATPSs.This work is partially supported by project PEst-C/EQB/LA0020/2011, financed by FEDER through COMPETE-Programa Operacional Factores de Competitividade and by FCT-Fundacao para a Ciencia e a Tecnologia. Sara Silverio acknowledges her Ph.D. grant from FCT (SFRH/BD/43439/2008)

Modelling collective cell behaviour

The classical mean-field approach to modelling biological systems makes a number of simplifying assumptions which typically lead to coupled systems of reaction-diffusion partial differential equations. While these models have been very useful in allowing us to gain important insights into the behaviour of many biological systems, recent experimental advances in our ability to track and quantify cell behaviour now allow us to build more realistic models which relax some of the assumptions previously made. This brief review aims to illustrate the type of models obtained using this approach