“Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

Integration of circadian and phototransduction pathways in the network controlling CAB gene transcription in Arabidopsis

The transcription of CAB genes, encoding the chlorophyll a/b-binding proteins, is rapidly induced in dark-grown Arabidopsis seedlings following a light pulse. The transient induction is followed by several cycles of a circadian rhythm. Seedlings transferred to continuous light are known to exhibit a robust circadian rhythm of CAB expression. The precise waveform of CAB expression in light–dark cycles, however, reflects a regulatory network that integrates information from photoreceptors, from the circadian clock and possibly from a developmental program. We have used the luciferase reporter system to investigate CAB expression with high time resolution. We demonstrate that CAB expression in light-grown plants exhibits a transient induction following light onset, similar to the response in dark-grown seedlings. The circadian rhythm modulates the magnitude and the kinetics of the response to light, such that the CAB promoter is not light responsive during the subjective night. A signaling pathway from the circadian oscillator must therefore antagonize the phototransduction pathways controlling the CAB promoter. We have further demonstrated that the phase of maximal CAB expression is delayed in light–dark cycles with long photoperiods, due to the entrainment of the circadian oscillator. Under short photoperiods, this pattern of entrainment ensures that dawn coincides with a phase of high light responsiveness, whereas under long photoperiods, the light response at dawn is reduced.

Spatial integration and cortical dynamics.

Cells in adult primary visual cortex are capable of integrating information over much larger portions of the visual field than was originally thought. Moreover, their receptive field properties can be altered by the context within which local features are presented and by changes in visual experience. The substrate for both spatial integration and cortical plasticity is likely to be found in a plexus of long-range horizontal connections, formed by cortical pyramidal cells, which link cells within each cortical area over distances of 6-8 mm. The relationship between horizontal connections and cortical functional architecture suggests a role in visual segmentation and spatial integration. The distribution of lateral interactions within striate cortex was visualized with optical recording, and their functional consequences were explored by using comparable stimuli in human psychophysical experiments and in recordings from alert monkeys. They may represent the substrate for perceptual phenomena such as illusory contours, surface fill-in, and contour saliency. The dynamic nature of receptive field properties and cortical architecture has been seen over time scales ranging from seconds to months. One can induce a remapping of the topography of visual cortex by making focal binocular retinal lesions. Shorter-term plasticity of cortical receptive fields was observed following brief periods of visual stimulation. The mechanisms involved entailed, for the short-term changes, altering the effectiveness of existing cortical connections, and for the long-term changes, sprouting of axon collaterals and synaptogenesis. The mutability of cortical function implies a continual process of calibration and normalization of the perception of visual attributes that is dependent on sensory experience throughout adulthood and might further represent the mechanism of perceptual learning.

Chemotactic signal integration in bacteria.

Chemotactic signaling in Escherichia coli involves transmission of both negative and positive signals. In order to examine mechanisms of signal processing, behavioral responses to dual inputs have been measured by using photoactivable "caged" compounds, computer video analysis, and chemoreceptor deletion mutants. Signaling from Tar and Tsr, two receptors that sense amino acids and pH, was studied. In a Tar deletion mutant the photoactivated release of protons, a Tsr repellent, and of serine, a Tsr attractant, in separate experiments at pH 7.0 resulted in tumbling (negative) or smooth-swimming (positive) responses in ca. 50 and 140 ms, respectively. Simultaneous photorelease of protons and serine resulted in a single tumbling or smooth-swimming response, depending on the relative amounts of the two effectors. In contrast, in wild-type E. coli, proton release at pH 7.0 resulted in a biphasic response that was attributed to Tsr-mediated tumbling followed by Tar-mediated smooth-swimming. In wild-type E. coli at more alkaline pH values the Tar-mediated signal was stronger than the Tsr signal, resulting in a strong smooth-swimming response preceded by a diminished tumbling response. These observations imply that (i) a single receptor time-averages the binding of different chemotactic ligands generating a single response; (ii) ligand binding to different receptors can result in a nonintegrated response with the tumbling response preceding the smooth-swimming response; (iii) however, chemotactic signals of different intensities derived from different receptors can also result in an apparently integrated response; and (iv) the different chemotactic responses to protons at neutral and alkaline pH may contribute to E. coli migration toward neutrality.