Nuclear membranes control symbiotic calcium signaling of legumes.

Nuclear-associated oscillations in calcium act as a secondary messenger in the symbiotic signaling pathway of legumes. These are decoded by a nuclear-localized calcium and calmodulin-dependent protein kinase, the activation of which is sufficient to drive downstream responses. This implies that the calcium oscillations within the nucleus are the predominant signals for legume symbiosis. However, the mechanisms that allow targeted release of calcium in the nuclear region have not been defined. Here we show that symbiosis-induced calcium changes occur in both the nucleoplasm and the perinuclear cytoplasm and seem to originate from the nuclear membranes. Reaction diffusion simulations suggest that spike generation within the nucleoplasm is not possible through transmission of a calcium wave from the cytoplasm alone and that calcium is likely to be released across the inner nuclear membrane to allow nuclear calcium changes. In agreement with this, we found that the cation channel DMI1, which is essential for symbiotic calcium oscillations, is preferentially located on the inner nuclear membrane, implying an essential function for the inner nuclear membrane in symbiotic calcium signaling. Furthermore, a sarco/endoplasmic reticulum calcium ATPase (SERCA) essential for symbiotic calcium oscillations is targeted to the inner nuclear membrane, as well as the outer nuclear membrane and endoplasmic reticulum (ER). We propose that release of calcium across the inner nuclear membrane allows targeted release of the ER calcium store, and efficient reloading of this calcium store necessitates the capture of calcium from the nucleoplasm and nuclear-associated cytoplasm.

Buffering capacity explains signal variation in symbiotic calcium oscillations.

Legumes form symbioses with rhizobial bacteria and arbuscular mycorrhizal fungi that aid plant nutrition. A critical component in the establishment of these symbioses is nuclear-localized calcium (Ca(2+)) oscillations. Different components on the nuclear envelope have been identified as being required for the generation of the Ca(2+) oscillations. Among these an ion channel, Doesn't Make Infections1, is preferentially localized on the inner nuclear envelope and a Ca(2+) ATPase is localized on both the inner and outer nuclear envelopes. Doesn't Make Infections1 is conserved across plants and has a weak but broad similarity to bacterial potassium channels. A possible role for this cation channel could be hyperpolarization of the nuclear envelope to counterbalance the charge caused by the influx of Ca(2+) into the nucleus. Ca(2+) channels and Ca(2+) pumps are needed for the release and reuptake of Ca(2+) from the internal store, which is hypothesized to be the nuclear envelope lumen and endoplasmic reticulum, but the release mechanism of Ca(2+) remains to be identified and characterized. Here, we develop a mathematical model based on these components to describe the observed symbiotic Ca(2+) oscillations. This model can recapitulate Ca(2+) oscillations, and with the inclusion of Ca(2+)-binding proteins it offers a simple explanation for several previously unexplained phenomena. These include long periods of frequency variation, changes in spike shape, and the initiation and termination of oscillations. The model also predicts that an increase in buffering capacity in the nucleoplasm would cause a period of rapid oscillations. This phenomenon was observed experimentally by adding more of the inducing signal.

Regime shifts in ecological systems can occur with no warning.

Predicting regime shifts - drastic changes in dynamic behaviour - is a key challenge in ecology and other fields. Here we show that the class of ecological systems that will exhibit leading indicators of regime shifts is limited, and that there is a set of ecological models and, therefore, also likely to be a class of natural systems for which there will be no forewarning of a regime change. We first describe how nonlinearities in combination with environmental variability lead to model descriptions that will not have smooth potentials, concluding that many ecological systems are described by systems without smooth potentials and thus will not show typical leading indicators of regime shifts. We then illustrate the impact of these general arguments by numerically examining the dynamics of several model ecological systems under slowly changing conditions. Our results offer a cautionary note about the generality of forecasting sudden changes in ecosystems.

Nuclear pores enable sustained perinuclear calcium oscillations.

BACKGROUND: Calcium signalling relies on the flux of calcium ions across membranes yet how signals in different compartments are related remains unclear. In particular, similar calcium signals on both sides of the nuclear envelope have been reported and attributed to passive diffusion through nuclear pores. However, observed differing cytosolic and nucleosolic calcium signatures suggest that the signalling machinery in these compartments can act independently. RESULTS: We adapt the fire-diffuse-fire model to investigate the generation of perinuclear calcium oscillations. We demonstrate that autonomous spatio-temporal calcium patterns are still possible in the presence of nuclear and cytosolic coupling via nuclear pores. The presence or absence of this autonomy is dependent upon the strength of the coupling and the maximum firing rate of an individual calcium channel. In all cases, coupling through the nuclear pores enables robust signalling with respect to changes in the diffusion constant. CONCLUSIONS: We show that contradictory interpretations of experimental data with respect to the autonomy of nuclear calcium oscillations can be reconciled within one model, with different observations being a consequence of varying nuclear pore permeabilities for calcium and refractory conditions of channels. Furthermore, our results provide an explanation for why calcium oscillations on both sides of the nuclear envelope may be beneficial for sustained perinuclear signaling.

Nonlinear time series analysis of nodulation factor induced calcium oscillations: evidence for deterministic chaos?

Legume plants form beneficial symbiotic interactions with nitrogen fixing bacteria (called rhizobia), with the rhizobia being accommodated in unique structures on the roots of the host plant. The legume/rhizobial symbiosis is responsible for a significant proportion of the global biologically available nitrogen. The initiation of this symbiosis is governed by a characteristic calcium oscillation within the plant root hair cells and this signal is activated by the rhizobia. Recent analyses on calcium time series data have suggested that stochastic effects have a large role to play in defining the nature of the oscillations. The use of multiple nonlinear time series techniques, however, suggests an alternative interpretation, namely deterministic chaos. We provide an extensive, nonlinear time series analysis on the nature of this calcium oscillation response. We build up evidence through a series of techniques that test for determinism, quantify linear and nonlinear components, and measure the local divergence of the system. Chaos is common in nature and it seems plausible that properties of chaotic dynamics might be exploited by biological systems to control processes within the cell. Systems possessing chaotic control mechanisms are more robust in the sense that the enhanced flexibility allows more rapid response to environmental changes with less energetic costs. The desired behaviour could be most efficiently targeted in this manner, supporting some intriguing speculations about nonlinear mechanisms in biological signaling.

Sudden shifts in ecological systems: intermittency and transients in the coupled Ricker population model.

Many real ecological systems show sudden changes in behavior, phenomena sometimes categorized as regime shifts in the literature. The relative importance of exogenous versus endogenous forces producing regime shifts is an important question. These forces' role in generating variability over time in ecological systems has been explored using tools from dynamical systems. We use similar ideas to look at transients in simple ecological models as a way of understanding regime shifts. Based in part on the theory of crises, we carefully analyze a simple two patch spatial model and begin to understand from a mathematical point of view what produces transient behavior in ecological systems. In particular, since the tools are essentially qualitative, we are able to suggest that transient behavior should be ubiquitous in systems with overcompensatory local dynamics, and thus should be typical of many ecological systems.

Iterative techniques for computing the linearized manifolds of quasiperiodic tori.

We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba [Nonlinearity 14, 943 (2001)], the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first-order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomeli and Meiss [Nonlinearity 16, 1573 (2003)] for a volume-preserving mapping.