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.     

Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis I: The Single Cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area.     

Superposition of modes in a caricature of a model for morphogenesis

In a model proposed for cell pattern formation by Nagorcka et al. (J. Theor. Biol. 1987) linear analysis revealed the possibility of an initially spatially uniform cell density going unstable to perturbations of two distinct spatial modes. Here we examine a simple one-dimensional caricature of their model which exhibits similar linear behaviour and present a nonlinear analysis which shows the possibility of superposition of modes subject to appropriate parameter values and initial conditions.     

Logistic Proliferation of Cells in Scratch Assays is Delayed

Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction–diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation; however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay; at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.     

<Emphasis Type="Italic">Apis mellifera</Emphasis> and <Emphasis Type="Italic">Osmia cornuta</Emphasis> as carriers for the secondary spread of <Emphasis Type="Italic">Bacillus subtilis</Emphasis> on apple flowers

The efficiency of two pollinators, Apis mellifera L. (Hymenoptera: Apidae) and the mason bee Osmia cornuta (Latreille) (Hymenoptera: Megachilidae), as carriers of biocontrol agents (BCA) from flower to flower (secondary colonisation) was investigated on apple cv ‘Golden Delicious’. The BCA tested was Bacillus subtilis, strain BD170 (Biopro®) developed for the control of the ‘fire blight’ caused by Erwinia amylovora (Burril) Winslow et al. The two insect species were studied as secondary BCA carriers on apple plants in pots under net screened tunnels. Their behaviour and capacity to deposit the BCA in the most receptive flower parts were compared both by washing, diluting and plating the flower organs on a recovery medium and by means of PCR analyses based on a molecular marker. O. cornuta showed better performances with respect to A. mellifera. For the field trials, pollinators were introduced in four apple orchards. During apple’s flowering, the BD170 (100 g hl^?l) was sprayed once in two fields, and twice in the others. The pollinators’ efficacy in carrying the BCA from sprayed flowers to the stigmas of newly opened ones at different times after the spray treatment was evaluated. The detection of the BCA was performed by PCR analysis. The percentages of positive PCR flower samples were higher in the internal treated areas of the fields with respect to the external untreated ones, but the high colonisation level found in the latter and in the flowers opened in both areas several days after the treatment(s) demonstrated that pollinators can play an important role as secondary carriers.     

A model invalidation-based approach for elucidating biological signalling pathways,applied to the chemotaxis pathway in R. sphaeroides

Background  

Developing methods for understanding the connectivity of signalling pathways is a major challenge in biological research. For this purpose, mathematical models are routinely developed based on experimental observations, which also allow the prediction of the system behaviour under different experimental conditions. Often, however, the same experimental data can be represented by several competing network models.