An adaptive model for synchrony in the firefly Pteroptyx malaccae

We describe a new model for synchronization of neuronal oscillators that is based on the observation that certain species of fireflies are able to alter their free-running period. We show that by adding adaptation to standard oscillator models it is possible to observe the frequency alteration. One consequence of this is the perfect synchrony between coupled oscillators. Stability and some analytic results are included along with numerical simulations.This work was partially supported by NSF Grant DMS9002028 and the Mathematical Research Branch of The National Institutes of Health     

A quantitative model for linking Na+/Ca2+ exchanger to SERCA during refilling of the sarcoplasmic reticulum to sustain [Ca2+] oscillations in vascular smooth muscle

We have developed a quantitative model for the creation of cytoplasmic Ca²⁺ gradients near the inner surface of the plasma membrane (PM). In particular we simulated the refilling of the sarcoplasmic reticulum (SR) via PM–SR junctions during asynchronous [Ca²⁺]_i oscillations in smooth muscle cells of the rabbit inferior vena cava. We have combined confocal microscopy data on the [Ca²⁺]_i oscillations, force transduction data from cell contraction studies and electron microscopic images to build a basis for computational simulations that model the transport of calcium ions from Na⁺/Ca²⁺ exchangers (NCX) on the PM to sarcoplasmic/endoplasmic reticulum Ca²⁺ ATPase (SERCA) pumps on the SR as a three-dimensional random walk through the PM–SR junctional cytoplasmic spaces. Electron microscopic ultrastructural images of the smooth muscle cells were elaborated with software algorithms to produce a very clear and dimensionally accurate picture of the PM–SR junctions. From this study, we conclude that it is plausible and possible for enough Ca²⁺ to pass through the PM–SR junctions to replete the SR during the regenerative Ca²⁺ release, which underlies agonist induced asynchronous Ca²⁺ oscillations in vascular smooth muscle.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Yeast glycolytic oscillations that are not controlled by a single oscillophore: a new definition of oscillophore strength

Biochemical oscillations, such as glycolytic oscillations, are often believed to be caused by a single so-called ‘oscillophore’. The main characteristics of yeast glycolytic oscillations, such as frequency and amplitude, are however controlled by several enzymes. In this paper, we develop a method to quantify to which extent any enzyme determines the occurrence of oscillations. Principles extrapolated from metabolic control analysis are applied to calculate the control exerted by individual enzymes on the real and imaginary parts of the eigenvalues of the Jacobian matrix. We propose that the control exerted by an enzyme on the real part of the smallest eigenvalue, in terms of absolute value, quantifies to which extent that enzyme contributes to the emergence of instability. Likewise the control exerted by an enzyme on the imaginary part of complex eigenvalues may serve to quantify the extent to which that enzyme contributes to the tendency of the system to oscillate. The method was applied both to a core model and to a realistic model of yeast glycolytic oscillations. Both the control over stability and the control over oscillatory tendency were distributed among several enzymes, of which glucose transport, pyruvate decarboxylase and ATP utilization were the most important. The distributions of control were different for stability and oscillatory tendency, showing that control of instability does not imply control of oscillatory tendency nor vice versa. The control coefficients summed up to 1, suggesting the existence of a new summation theorem. These results constitute proof that glycolytic oscillations in yeast are not caused by a single oscillophore and provide a new, subtle, definition for the oscillophore strength of an enzyme.     

Mammalian freeze-dried sperm can maintain their calcium oscillation-inducing ability when microinjected into mouse eggs

Mammalian freeze-dried sperm can maintain their genetic integrity and event support full development to term when microinjected into mature oocytes. However, it is unknown whether freeze-dried sperm can still maintain their calcium oscillation-inducing capability. Here, we microinjected mouse and bovine freeze-dried sperm into mouse MII oocytes and examined their calcium oscillation-inducing ability following intracytoplasmic sperm injection (ICSI). Two pieces of information are revealed. First, nearly all oocytes injected with a freeze-dried mouse sperm head or a bovine sperm showed fertilization-like calcium oscillations, indicating that freeze-drying treatment does not affect the activity of the sperm factor responsible for calcium oscillations. Second, freeze-dried sperm exhibited high resistance to external temperature increase. This is shown by the finding that the freeze-dried sperm can maintain their calcium oscillation-inducing capacity even following exposure to 100 degrees C for 3 h. We therefore conclude that mammalian sperm can maintain their calcium oscillation-inducing capability following freeze-drying, rehydration, and ICSI treatments.     

Oscillatory Activity of P-Type Membrane Adenosine Triphosphatases: a Kinetic Model

A kinetic model for membrane P-type adenosine triphosphatases is considered, the main application being to the erythrocyte Ca²⁺-ATPase. It is shown that a simple modification of the known catalytic mechanism of the ATPase by addition of a self-inhibition step and the steady calcium influx leads to damped oscillations in the system discussed. In this way, the model can explain the kinetic experimental results obtained for the purified enzyme in solution as well as for the enzyme incorporated into liposome membranes. The estimated kinetic parameters are close to the experimental ones. Alternative changes in time, demonstrated by the kinetic model for the conformational enzyme states, E₁ and E₂, confirm the model of two alternatively functioning gates in the ion pumping Ca²⁺-ATPase.__________Translated from Biokhimiya, Vol. 70, No. 4, 2005, pp. 533–538.Original Russian Text Copyright © 2005 by Goldstein, Mayevsky, Zakrjevskaya.     

Brevetoxin activation of voltage-gated sodium channels regulates Ca dynamics and ERK1/2 phosphorylation in murine neocortical neurons

Voltage-gated sodium channels (VGSC) are involved in the generation of action potentials in neurons. Brevetoxins (PbTx) are potent allosteric enhancers of VGSC function and are associated with the periodic 'red tide' blooms. Using PbTx-2 as a probe, we have characterized the effects of activation of VGSC on Ca(2+) dynamics and extracellular signal-regulated kinases 1/2 (ERK1/2) signaling in neocortical neurons. Neocortical neurons exhibit synchronized spontaneous Ca(2+) oscillations, which are mediated by glutamatergic signaling. PbTx-2 (100 nm) increased the amplitude and reduced the frequency of basal Ca(2+) oscillations. This modulatory effect on Ca(2+) oscillations produced a sustained rise in ERK1/2 activation. At 300 nm, PbTx-2 disrupted oscillatory activity leading to a sustained increase in intracellular Ca(2+) ([Ca(2+)](i)) and induced a biphasic, activation followed by dephosphorylation, regulation of ERK1/2. PbTx-2-induced ERK1/2 activation was Ca(2+) dependent and was mediated by Ca(2+) entry through manifold routes. PbTx-2 treatment also increased cAMP responsive element binding protein (CREB) phosphorylation and increased gene expression of brain-derived neurotrophic factor (BDNF). These findings indicate that brevetoxins, by influencing the activation of key signaling proteins, can alter physiologic events involved in survival in neocortical neurons, as well as forms of synaptic plasticity associated with development and learning.     

A dynamic model of saliva secretion

We construct a mathematical model of the parotid acinar cell with the aim of investigating how the distribution of K⁺ and Cl⁻ channels affects saliva production. Secretion of fluid is initiated by Ca²⁺ signals acting on Ca²⁺ dependent K⁺ and Cl⁻ channels. The opening of these channels facilitates the movement of Cl⁻ ions into the lumen which water follows by osmosis. We use recent results into both the release of Ca²⁺ from internal stores via the inositol (1,4,5)-trisphosphate receptor (IP₃R) and IP₃ dynamics to create a physiologically realistic Ca²⁺ model which is able to recreate important experimentally observed behaviours seen in parotid acinar cells. We formulate an equivalent electrical circuit diagram for the movement of ions responsible for water flow which enables us to calculate and include distinct apical and basal membrane potentials to the model. We show that maximum saliva production occurs when a small amount of K⁺ conductance is located at the apical membrane, with the majority in the basal membrane. The maximum fluid output is found to coincide with a minimum in the apical membrane potential. The traditional model whereby all Cl⁻ channels are located in the apical membrane is shown to be the most efficient Cl⁻ channel distribution.