Navigation through the Plasma Membrane Molecular Landscape Shapes Random Organelle Movement.

Eukaryotic plasma membrane organization theory has long been controversial, in part due to a dearth of suitably high-resolution techniques to probe molecular architecture in situ and integrate information from diverse data streams [1]. Notably, clustered patterning of membrane proteins is a commonly conserved feature across diverse protein families (reviewed in [2]), including the SNAREs [3], SM proteins [4, 5], ion channels [6, 7], and receptors (e.g., [8]). Much effort has gone into analyzing the behavior of secretory organelles [9-13], and understanding the relationship between the membrane and proximal organelles [4, 5, 12, 14] is an essential goal for cell biology as broad concepts or rules may be established. Here we explore the generally accepted model that vesicles at the plasmalemma are guided by cytoskeletal tracks to specific sites on the membrane that have clustered molecular machinery for secretion [15], organized in part by the local lipid composition [16]. To increase our understanding of these fundamental processes, we integrated nanoscopy and spectroscopy of the secretory machinery with organelle tracking data in a mathematical model, iterating with knockdown cell models. We find that repeated routes followed by successive vesicles, the re-use of similar fusion sites, and the apparently distinct vesicle "pools" are all fashioned by the Brownian behavior of organelles overlaid on navigation between non-reactive secretory protein molecular depots patterned at the plasma membrane.

Basis properties of the p, q-sine functions.

We improve the currently known thresholds for basisness of the family of periodically dilated p,q-sine functions. Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. We also determine refined bounds on the Riesz constant associated with this family. These results seal mathematical gaps in the existing literature on the subject.

Effects of noise on models of spiny dendrites.

We study the effects of noise in two models of spiny dendrites. Through the introduction of different types of noise to both the Spike-diffuse-spike (SDS) and Baer-Rinzel (BR) models we investigate the change in behaviour of the travelling wave solution present in both deterministic systems, as noise intensity increases. We show that the speed of wave propagation in both the SDS and BR models respectively differs as the noise intensity in the spine heads increases. In contrast the cable is very robust to noise and as such the speed shows very little variation from the deterministic system. We introduce a space-dependent spine density, ρ(x), to the original Baer-Rinzel model and show how this modified model can mimic behaviour (under influence of noise) of both original systems, through variation of one parameter. We also show that the correlation time and length scales of the noise can enhance propagation of travelling wave solutions where the white noise dominates the underlying signal and produces noise induced phenomena.

Secretory vesicles are preferentially targeted to areas of low molecular SNARE density.

Intercellular communication is commonly mediated by the regulated fusion, or exocytosis, of vesicles with the cell surface. SNARE (soluble N-ethymaleimide sensitive factor attachment protein receptor) proteins are the catalytic core of the secretory machinery, driving vesicle and plasma membrane merger. Plasma membrane SNAREs (tSNAREs) are proposed to reside in dense clusters containing many molecules, thus providing a concentrated reservoir to promote membrane fusion. However, biophysical experiments suggest that a small number of SNAREs are sufficient to drive a single fusion event. Here we show, using molecular imaging, that the majority of tSNARE molecules are spatially separated from secretory vesicles. Furthermore, the motilities of the individual tSNAREs are constrained in membrane micro-domains, maintaining a non-random molecular distribution and limiting the maximum number of molecules encountered by secretory vesicles. Together our results provide a new model for the molecular mechanism of regulated exocytosis and demonstrate the exquisite organization of the plasma membrane at the level of individual molecular machines.

Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments.

In many semi-arid environments, vegetation is self-organised into spatial patterns. The most striking examples of this are on gentle slopes, where striped patterns are typical, running parallel to the contours. Previously, Klausmeier [1999. Regular and irregular patterns in semiarid vegetation. Science 284, 1826-1828.] has proposed a model for vegetation stripes based on competition for water. Here, we present a detailed study of the patterned solutions in the full nonlinear model, using numerical bifurcation analysis of both the pattern odes and the model pdes. We show that patterns exist for a wide range of rainfall levels, and in particular for much lower rainfall than have been considered by previous authors. Moreover, we show that for many rainfall levels, patterns with a variety of different wavelengths are stable, with mode selection dependent on initial conditions. This raises the possibility of hysteresis, and in numerical solutions of the model we show that pattern selection depends on rainfall history in a relatively simple way.

Spatio-temporal filtering properties of a dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework.

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In this paper we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. In the first instance this system is shown to support saltatory waves with the same qualitative features as those observed in a model with Hodgkin-Huxley kinetics in the spine-head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time-varying external input we find that the distribution of spines can play a crucial role in determining spatio-temporal filtering properties. In particular, the SDS model in response to periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune [1999, 'Mechanisms for generating temporal filters in the electrosensory system,' The Journal of Experimental Biology 202: 1281-1289]. Further, we demonstrate the robustness of observed wave properties to natural sources of noise that arise both in the cable and the spine-head, and highlight the possibility of purely noise induced waves and coherent oscillations.