Regulatory inhibition of biological tissue mineralization by calcium phosphate through post-nucleation shielding by fetuin-A.

In vertebrates, insufficient availability of calcium and inorganic phosphate ions in extracellular fluids leads to loss of bone density and neuronal hyper-excitability. To counteract this problem, calcium ions are usually present at high concentrations throughout bodily fluids-at concentrations exceeding the saturation point. This condition leads to the opposite situation where unwanted mineral sedimentation may occur. Remarkably, ectopic or out-of-place sedimentation into soft tissues is rare, in spite of the thermodynamic driving factors. This fortunate fact is due to the presence of auto-regulatory proteins that are found in abundance in bodily fluids. Yet, many important inflammatory disorders such as atherosclerosis and osteoarthritis are associated with this undesired calcification. Hence, it is important to gain an understanding of the regulatory process and the conditions under which it can go awry. In this manuscript, we extend mean-field continuum classical nucleationtheory of the growth of clusters to encompass surface shielding. We use this formulation to study the regulation of sedimentation of calcium phosphate salts in biological tissues through the mechanism of post-nuclear shielding of nascent mineral particles by binding proteins. We develop a mathematical description of this phenomenon using a countable system of hyperbolic partial differential equations. A critical concentration of regulatory protein is identified as a function of the physical parameters that describe the system.

A mathematical model of the metabolic and perfusion effects on cortical spreading depression.

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism-aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium-ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD--vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization--found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

Mathematical approaches to modeling of cortical spreading depression.

Migraine with aura (MwA) is a debilitating disease that afflicts about 25%-30% of migraine sufferers. During MwA, a visual illusion propagates in the visual field, then disappears, and is followed by a sustained headache. MwA was conjectured by Lashley to be related to some neurological phenomenon. A few years later, Leão observed electrophysiological waves in the brain that are now known as cortical spreading depression (CSD). CSD waves were soon conjectured to be the neurological phenomenon underlying MwA that had been suggested by Lashley. However, the confirmation of the link between MwA and CSD was not made until 2001 by Hadjikhani et al. [Proc. Natl. Acad. Sci. U.S.A. 98, 4687-4692 (2001)] using functional MRI techniques. Despite the fact that CSD has been studied continuously since its discovery in 1944, our detailed understandings of the interactions between the mechanisms underlying CSD waves have remained elusive. The connection between MwA and CSD makes the understanding of CSD even more compelling and urgent. In addition to all of the information gleaned from the many experimental studies on CSD since its discovery, mathematical modeling studies provide a general and in some sense more precise alternative method for exploring a variety of mechanisms, which may be important to develop a comprehensive picture of the diverse mechanisms leading to CSD wave instigation and propagation. Some of the mechanisms that are believed to be important include ion diffusion, membrane ionic currents, osmotic effects, spatial buffering, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. Discrete and continuum models of CSD consist of coupled nonlinear differential equations for the ion concentrations. In this review of the current quantitative understanding of CSD, we focus on these modeling paradigms and various mechanisms that are felt to be important for CSD.

A continuum neuronal model for the instigation and propagation of cortical spreading depression.

Cortical spreading depression (CSD) waves can occur in the cortices of various brain structures and are associated with the spread of depression of the electroencephalogram signal. In this paper, we present a continuum neuronal model for the instigation and spreading of CSD. Our model assumes that the brain-cell microenvironment can be treated as a porous medium consisting of extra- and intracellular compartments. The main mechanisms in our model for the transport of ions into and out of neurons are cross-membrane ionic currents and (active) pumps, coupled with diffusion in the extracellular space. To demonstrate the applicability of our model, we have carried out extensive numerical simulations under different initial conditions and inclusion of various mechanisms. Our results show that CSD waves can be instigated by injecting cross-membrane ionic currents or by applying KCl in the extracellular space. Furthermore, the estimated speeds of CSD waves are within the experimentally observed range. Effects of specific ion channels, background ion concentrations, extracellular volume fractions, and cell swelling on the propagation speed of CSD are also investigated.

Restricted diffusion in cellular media: (1+1)-dimensional model.

We consider the diffusion of molecules in a one-dimensional medium consisting of a large number of cells separated from the extra-cellular space by permeable membranes. The extra-cellular space is completely connected and allows unrestricted diffusion of the molecules. Furthermore, the molecules can diffuse within a given cell, i.e., the intra-cellular space; however, direct diffusion from one cell to another cell cannot occur. There is a movement of molecules across the permeable membranes between the intra- and extra-cellular spaces. Molecules from one cell can cross the permeable membrane into the extra-cellular space, then diffuse through the extra-cellular space, and eventually enter the intra-cellular space of a second cell. Here, we develop a simple set of model equations to describe this phenomenon and obtain the solutions using an eigenfunction expansion. We show that the solutions obtained using this method are particularly convenient for interpreting data from experiments that use techniques from nuclear magnetic resonance imaging.

Traveling waves in coupled reaction-diffusion models with degenerate sources.

We consider a general system of coupled nonlinear diffusion equations that are characterized by having degenerate source terms and thereby not having isolated rest states. Using a general form of physically relevant source terms, we derive conditions that are required to trigger traveling waves when a stable uniform steady-state solution is perturbed by a highly localized disturbance. We show that the degeneracy in the source terms implies that traveling waves have a number of surprising properties that are not present for systems with nondegenerate source terms. We also show that such systems can lead to a pair of waves that initially propagate outwards from the disturbance, slow down, and reverse direction before ultimately colliding and annihilating each other.

Prediction of mRNA polyadenylation sites by support vector machine.

mRNA polyadenylation is responsible for the 3' end formation of most mRNAs in eukaryotic cells and is linked to termination of transcription. Prediction of mRNA polyadenylation sites [poly(A) sites] can help identify genes, define gene boundaries, and elucidate regulatory mechanisms. Current methods for poly(A) site prediction achieve moderate sensitivity and specificity. Here, we present a method using support vector machine for poly(A) site prediction. Using 15 cis-regulatory elements that are over-represented in various regions surrounding poly(A) sites, this method achieves higher sensitivity and similar specificity when compared with polyadq, a common tool for poly(A) site prediction. In addition, we found that while the polyadenylation signal AAUAAA and U-rich elements are primary determinants for poly(A) site prediction, other elements contribute to both sensitivity and specificity of the prediction, indicating a combinatorial mechanism involving multiple elements when choosing poly(A) sites in human cells.

Spatial buffering mechanism: mathematical model and computer simulations.

It is generally accepted that the spatial buffering mechanism is important to buffer extracellular-space potassium in the brain-cell microenviron- ment. In the past, this phenomenon, generally associated with glial cells, has been treated analytically and numerically using a simplified one-dimensional description. The present study extends the previous research by using a novel numerical scheme for the analysis of potassium buffering mechanisms in the extracellular brain-cell microenvironment. In particular, a lattice-cellular automaton was employed to simulate a detailed two-compartment model of a two-dimensional brain-cell system. With this numerical approach, the present study elaborates upon previous theoretical work on spatial buffering (SB) by incorporating a more realistic structure of the brain-cell microenvironment, which was not feasible earlier. We use the experimental paradigm consisting of iontophoretic injection of KCl to study the SB mechanism. Our simulations confirmed the results reported in the literature obtained by an averaged model. The results also show that the additional effects captured by a simplified two- dimensional geometry do not alter significantly the conclusions obtained from the averaged model. The details of applying such a numerical method to the study of ion movements in cellular environments, as well as its potential for future study, are discussed.

Membrane resonance and stochastic resonance modulate firing patterns of thalamocortical neurons.

We examined the interactions of subthreshold membrane resonance and stochastic resonance using whole-cell patch clamp recordings in thalamocortical neurons of rat brain slices, as well as with a Hodgkin-Huxley-type mathematical model of thalamocortical neurons. The neurons exhibited the subthreshold resonance when stimulated with small amplitude sine wave currents of varying frequency, and stochastic resonance when noise was added to sine wave inputs. Stochastic resonance was manifest as a maximum in signal-to-noise ratio of output response to subthreshold periodic input combined with noise. Stochastic resonance in conjunction with subthreshold resonance resulted in action potential patterns that showed frequency selectivity for periodic inputs. Stochastic resonance was maximal near subthreshold resonance frequency and a high noise level was required for detection of high frequency signals. We speculate that combined membrane and stochastic resonances have physiological utility in coupling synaptic activity to preferred firing frequency and in network synchronization under noise.

The influences of Ih on temporal summation in hippocampal CA1 pyramidal neurons: a modeling study.

Recent experimental and theoretical studies have found that active dendritic ionic currents can compensate for the effects of electrotonic attenuation. In particular, temporal summation, the percentage increase in peak somatic voltage responses invoked by a synaptic input train, is independent of location of the synaptic input in hippocampal CA1 pyramidal neurons under normal conditions. This independence, known as normalization of temporal summation, is destroyed when the hyperpolarization-activated current, Ih, is blocked [Magee JC (1999a), Nature Neurosci. 2: 508-514]. Using a compartmental model derived from morphological recordings of hippocampal CA1 pyramidal neurons, we examined the hypothesis that Ih was primarily responsible for normalization of temporal summation. We concluded that this hypothesis was incomplete. With a model that included Ih, the persistent Na(+) current (INaP), and the transient A-type K+ current (IA), however, we observed normalization of temporal summation across a wide range of synaptic input frequencies, in keeping with experimental observations.

Resonances and noise in a stochastic Hindmarsh-Rose model of thalamic neurons.

Thalamic neurons exhibit subthreshold resonance when stimulated with small sine wave signals of varying frequency and stochastic resonance when noise is added to these signals. We study a stochastic Hindmarsh-Rose model using Monte-Carlo simulations to investigate how noise, in conjunction with subthreshold resonance, leads to a preferred frequency in the firing pattern. The resulting stochastic resonance (SR) exhibits a preferred firing frequency that is approximately exponential in its dependence on the noise amplitude. In similar experiments, frequency dependent SR is found in the reliability of detection of alpha-function inputs under noise, which are more realistic inputs for neurons. A mathematical analysis of the equations reveals that the frequency preference arises from the dynamics of the slow variable. Noise can then transfer the resonance over the firing threshold because of the proximity of the fast subsystem to a Hopf bifurcation point. Our results may have implications for the behavior of thalamic neurons in a network, with noise switching the membrane potential between different resonance modes.

Spermine modulates neuronal excitability and NMDA receptors in juvenile gerbil auditory thalamus.

Medial geniculate body (MGB) neurons process synaptic inputs from auditory cortex. Corticothalamic stimulation evokes glutamatergic excitatory postsynaptic potentials (EPSPs) that vary markedly in amplitude and duration during development. The EPSP decay phase is prolonged during second postnatal week but then shortens, significantly, until adulthood. The EPSP prolongation depends on spermine interactions with a polyamine-sensitive site on receptors for N-methyl-D-aspartate (NMDA). We examined effects of spermine application on EPSPs, firing modes, and membrane properties in gerbil MGB neurons during the P14 period of highest polyamine sensitivity. Spermine slowed EPSP decay and promoted firing on EPSPs, without changing passive membrane properties. Spermine increased membrane rectification on depolarization, which is mediated by tetrodotoxin (TTX)-sensitive, persistent Na(+) conductance. As a result, spermine lowered threshold and increased tonic firing evoked with current injection by up to approximately 150%. These effects were concentration-dependent (ED(50)=100 microM), reversible, and eliminated by NMDA receptor antagonist, 2-amino-5-phosphonovalerate (APV). In contrast, spermine increased dV/dt of the low threshold Ca(2+) spike (LTS) and burst firing, evoked from hyperpolarized potentials. LTS enhancement was greater at -55 mV than at hyperpolarized potentials and did not result from persistent Na(+) conductance or glutamate receptor mechanisms. In summary, spermine increased excitability by modulating NMDA receptors in juvenile gerbil neurons.