Properties of solutions for a chemotaxis system

 We investigate mathematically the system of equations proposed by Chaplain and Stuart [2], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus. In particular, we characterize the steady state endothelial cell density function, and give conditions on the chemotactic parameter k and cell proliferation parameter b that ensure that migration/ proliferation either does or does not occur in steady state. The time dependent problem is also treated. Received 12 September 1995; received in revised form 6 August 1996     

Analysis of a densitometry assay for bacterial chemotaxis

Analytic expressions are derived for density profiles of chemotactic bacteria moving in a discontinuously exponential chemoattractant gradient. The results are used to indicate how motility coefficients, in addition to chemotactic response velocities, may be obtained from the densitometry assay developed by Dahlquist, Lovely &; Koshland (1972).     

Protein phosphorylation in the bacterial chemotaxis system

Bacterial chemotaxis involves the detection of changes in concentration of specific chemicals in the environment of the cell as a function of time. This process is mediated by a series of cell surface receptors that interact with and activate intracellular protein phosphorylation. Five cytoplasmic proteins essential for chemotaxis have been shown to be involved in a coupled system of protein phosphorylation. Ligand binding to cell surface receptors affects the rate of autophosphorylation of the CheA protein. In the absence of an attractant bound to receptor and in the presence of the CheW protein, the rate of CheA autophosphorylation is markedly increased. Phosphorylated CheA can transfer phosphate to the CheY or CheB proteins; phosphorylation of these "effector" proteins may increase their activity. The CheY protein is thought to regulate flagellar rotation and thus control swimming behavior. The CheB protein modifies the cell surface receptor and thus regulates receptor function. Finally, another chemotaxis protein, CheZ, acts to specifically dephosphorylate CheY-phosphate. This system shows marked similarity to the 2-component sensor-regulator systems found to control specific gene expression in a variety of bacteria.     

External optimal control of self-organisation dynamics in a chemotaxis reaction diffusion system

Detailed quantitative understanding and specific external control of cellular behaviour are general long-term goals of modem bioscience research activities in systems biology. Pattern formation and self-organisation processes both in single cells and in distributed cell populations are phenomena which are highly significant for the functionality of life, because life requires to maintain a highly organised spatiotemporal system structure. In particular chemotaxis is crucial for various biological aspects of intercellular signalling and cell aggregation. As an example for model based control of self-organising biological systems, we describe numerical optimal control of E. coli bacterial chemotaxis based on a 1-D two-component partial differential equation (PDE) model of reaction diffusion type. We present a numerical scheme to force cell aggregation patterns to particular desired results by applying a boundary influx control of chemoattractant without interfering with the system itself. Optimal controls are numerically computed by using a specially tailored interior point optimisation technique applied to a direct collocation discretisation of the control function and the PDE constraint. The objective to be minimised is the deviation of a desired cell distribution from the cell density, which results from the dynamics of the controlled system.     

Spatial organization of the bacterial chemotaxis system

Sensory complexes in bacterial chemotaxis are organized in large clusters, building complex signal-processing machinery. Interactions among chemoreceptors are the main determinant of cluster formation and create an allosteric network that is able to integrate and amplify stimuli, before transmitting the signal to downstream proteins. Association of the other proteins with the receptor cluster creates a signalling scaffold, which enhances the efficiency and specificity of the pathway. Clusters localize to specific locations inside the cell, perhaps to ensure their proper distribution during cell division. Clustering is conserved among all studied prokaryotic chemotaxis systems and exemplifies a growing number of bacterial pathways with a reported sub-cellular spatial organization. Moreover, because allostery provides a simple mechanism to achieve very high response sensitivity, it is probable that clustering-based signal amplification is not limited to bacterial chemotaxis but also exists in other prokaryotic and eukaryotic pathways.     

Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.This work has been supported by the Deutsche Forschungsgemeinschaft     

Structural analysis of bacterial chemotaxis proteins: components of a dynamic signaling system

Most motile bacteria are capable of directing their movement in response to chemical gradients, a behavior known as chemotaxis. The signal transduction system that mediates chemotaxis in enteric bacteria consists of a set of six cytoplasmic proteins that couple stimuli sensed by a family of transmembrane receptors to behavioral responses generated by the flagellar motors. Signal transduction occurs via a phosphotransfer pathway involving a histidine protein kinase, CheA, and a response regulator protein, CheY, that in its phosphorylated state, modulates the direction of flagellar rotation. Two auxiliary proteins, CheW and CheZ, and two receptor modification enzymes, methylesterase CheB and methyltransferase CheR, influence the flux of phosphoryl groups within this central pathway. This paper focuses on structural characteristics of the four signaling proteins (CheA, CheY, CheB, and CheR) for which NMR or x-ray crystal structures have been determined. The proteins are examined with respect to their signaling activities that involve reversible protein modifications and transient assembly of macromolecular complexes. A variety of data suggest conformational flexibility of these proteins, a feature consistent with their multiple roles in a dynamic signaling pathway.     

Separation of propagating and non propagating components in surface EMG

Surface electromyogram (EMG) detected by electrode arrays along the muscle fibre direction can be approximated by the sum of propagating and non propagating components. A technique to separate propagating and non propagating components in surface EMG signals is developed. The first step is an adaptive filter, which allows obtaining an estimation of the delay between signals detected at different channels and a first estimate of propagating and non propagating components; the second step is used to optimise the estimation of the two components. The method is applicable to signals with one propagating and one non propagating component. It was optimised on simulated signals, and then applied to single motor unit action potentials (MUAP) and to electrically elicited EMG (M-waves).

The new method was first tested on phenomenological signals constituted by the sum of a propagating and a non propagating signal and then applied to simulated and experimental EMG signals. Simulated signals were generated by a cylindrical, layered volume conductor model. Experimental signals were monopolar surface EMG signals collected from the abductor pollicis brevis muscle and M-waves recorded during transcutaneous electrical stimulation of the biceps muscle. The technique may find different applications: in single motor unit (MU) studies (a) for decreasing the variability and bias of CV estimates due to the presence of the non propagating components, (b) for estimating automatically the length of the muscle fibres from only three detected channels and (c) for removal of the stimulation artifact M-waves.     

Quantitative analysis of cell motility and chemotaxis in Dictyostelium discoideum by using an image processing system and a novel chemotaxis chamber providing stationary chemical gradients

An image processing system was programmed to automatically track and digitize the movement of amebae under phase-contrast microscopy. The amebae moved in a novel chemotaxis chamber designed to provide stable linear attractant gradients in a thin agarose gel. The gradients were established by pumping attractant and buffer solutions through semipermeable hollow fibers embedded in the agarose gel. Gradients were established within 30 min and shown to be stable for at least a further 90 min. By using this system it is possible to collect detailed data on the movement of large numbers of individual amebae in defined attractant gradients. We used the system to study motility and chemotaxis by a score of Dictyostelium discoideum wild-type and mutant strains, including "streamer" mutants which are generally regarded as being altered in chemotaxis. None of the mutants were altered in chemotaxis in the optimal cAMP gradient of 25 nM/mm, with a midpoint of 25 nM. The dependence of chemotaxis on cAMP concentration, gradient steepness, and temporal changes in the gradient were investigated. We also analyzed the relationship between turning behavior and the direction of travel during chemotaxis in stable gradients. The results suggest that during chemotaxis D. discoideum amebae spatially integrate information about local increases in cAMP concentration at various points on the cell surface.     

Bacterial chemotaxis: a field in motion

Many different types of studies are being combined to provide an increasingly detailed picture of the bacterial chemotaxis system. The structures of periplasmic receptors and a cytoplasmic response regulator, along with structures of domains of a membrane receptor, a receptor-modifying enzyme and a cytoplasmic histidine kinase, have been determined. These structures provide a basis for other work which is likely to open up new structural avenues.     

Competitive exclusion in a two-species chemotaxis model

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka–Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.     

Analysis of the signal transduction properties of a module of spatial sensing in eukaryotic chemotaxis

The movement of cells in response to a gradient in chemical concentration—known as chemotaxis—is crucial for the proper functioning of uni-and multicellular organisms. How a cell senses the chemical concentration gradient surrounding it, and what signal is transmitted to its motion apparatus is known as gradient sensing. The ability of a cell to sense gradients persists even when the cell is immobilized (i.e., its motion apparatus is deactivated). This suggests that important features of gradient sensing can be studied in isolation, decoupling this phenomenon from the movement of the cell. A mathematical model for gradient sensing in Dictyostelium cells and neutrophils was recently proposed. This consists of an adaptation/spatial sensing module. This spatial sensing module feeds into an amplification module, magnifying the effects of the former. In this paper, we analyze the spatial sensing module in detail and examine its signal transduction properties. We examine the response of this module to several inputs of experimental and biological relevance.     

Modelling run-and-tumble chemotaxis in a shear flow

The biased random walk undergone by chemotactic bacteria such as Escherichia coli will be influenced at the microscopic level by flow in the ambient medium. In this paper, we model swimming bacteria being advected and rotated by a simple shear flow. Under certain scaling assumptions, we obtain an advection—diffusion equation for cell density, when the chemotactic response is small, which shows a coupling between the rotation and chemotaxis. We also present an alternative method for calculating the chemotactic flux in an unbounded region which is valid for more general chemotactic responses.     

Diversity pattern in Sesamum mutants selected for a semi-arid cropping system

Summary Due to the complex requirements of moisture stress, substantial genetic diversity with a wide array of character combinations and effective simultaneous selection for several variables is necessary for improving the productivity and adaptation of a component crop in order for it to fit into a cropping system under semi-arid tropical conditions. Sesamum indicum L. is grown in Venezuela after rice/sorghum/or maize under such conditions. A mutation breeding program was undertaken using six locally adapted varieties to develop genotypes suitable for the above system. The diversity pattern for nine variables was assessed by multivariate analysis in 301 M₄ progenies. Analysis of the characteristic roots and principal components in three methods of selection, i.e., M₂ bulks (A), individual plant selection throughout (B), and selection in M₃ for single variable (C), revealed differences in the pattern of variation between varieties, selection methods, and varieties x methods interactions. Method B was superior to the others and gave 17 of the 21 best M₅ progenies. Piritu and CF varieties yielded the most productive progenies in M₅ and M₆. Diversity was large and selection was effective for such developmental traits as earliness and synchrony, combined with multiple disease resistance, which could be related to their importance by multivariate analyses. Considerable differences in the variety of character combinations among the high yielding. M₅ progenies of CF and Piritu suggested possible further yield improvement. The superior response of Piritu and CF over other varieties in yield and adaptation was due to major changes in plant type and character associations. Multilocation testing of M₅ generations revealed that the mutant progenies had a 40%–100% yield superiority over the parents; this was combined with earliness, synchrony, and multiple disease resistance, and was confirmed in the M₆ generation grown on a commercial scale. This study showed that multivariate analysis is an effective tool for assessing diversity patterns, choice of appropriate variety, and selection methodology in order to make rapid progress in meeting the complex requirements of semi-arid cropping systems.Project supported by the International Atomic Energy Agency, Vienna, Austria     

A Central pattern generator to control a pyloric-based system

A central pattern generator (CPG) is built to control a mechanical device (plant) inspired by the pyloric chamber of the lobster. Conductance-based models are used to construct the neurons of the CPG. The plant has an associated function that measures the amount of food flowing through it per unit of time. We search for the best set of solutions that give a high positive flow of food in the maximization function. The plant is symmetric and the model neurons are identical to avoid any bias in the space of solutions. We find that the solution is not unique and that three neurons are sufficient to produce positive flow. We propose an effective principle for CPGs (effective on-off connectivity) and a few predictions to be corroborated in the pyloric system of the lobster.     

Modeling microbial chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) has proven to be a useful experimental tool for studying population-level microbial growth and chemotaxis. A mathematical model capable of reproducing the population-level patterns formed as a result of cellular growth and chemotaxis in the DGC has been developed. The model consists of coupled partial differential balance equations for cells, chemoattractants, and a nutrient, which are solved simultaneously by the alternating direction implicit method. Modeling simulation results were compared with population-level migration patterns of Escherichia coli growing on glycerol and responding to a gradient of the chemoattractant aspartate for two different initial conditions. To accurately reproduce the experimental results, a second chemoattractant equation was necessary. The second chemoattractant has been identified as oxygen by directly measuring oxygen gradients in the DGC. Important trends observed experimentally and reproduced by the model include the formation of a chemotactic wave, a reduction in the wave velocity as it encounters higher chemoattractant concentrations, and chemotaxis in response to two different chemoattractants simultaneously. The model was also used to study the relative magnitude of cell fluxes due to random motility and chemotaxis, and the suppression of chemotaxis due to receptor saturation. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 191-205, 1997.