Gene circuit designs for noisy excitable dynamics

Certain cellular processes take the form of activity pulses that can be interpreted in terms of noise-driven excitable dynamics. Here we present an overview of different gene circuit architectures that exhibit excitable pulses of protein expression, when subject to molecular noise. Different types of excitable dynamics can occur depending on the bifurcation structure leading to the specific excitable phase-space topology. The bifurcation structure is not, however, linked to a particular circuit architecture. Thus a given gene circuit design can sustain different classes of excitable dynamics depending on the system parameters.     

Organization of excitable dynamics in hierarchical biological networks

This study investigates the contributions of network topology features to the dynamic behavior of hierarchically organized excitable networks. Representatives of different types of hierarchical networks as well as two biological neural networks are explored with a three-state model of node activation for systematically varying levels of random background network stimulation. The results demonstrate that two principal topological aspects of hierarchical networks, node centrality and network modularity, correlate with the network activity patterns at different levels of spontaneous network activation. The approach also shows that the dynamic behavior of the cerebral cortical systems network in the cat is dominated by the network's modular organization, while the activation behavior of the cellular neuronal network of Caenorhabditis elegans is strongly influenced by hub nodes. These findings indicate the interaction of multiple topological features and dynamic states in the function of complex biological networks.     

Spatio-temporal protein dynamics in single living cells

The development and application of single cell optical imaging has identified dynamic and oscillatory signalling processes in individual cells. This requires single cell analyses since the processes may otherwise be masked by the population average. These oscillations range in timing from seconds/minutes (e.g. calcium) to minutes/hours (e.g. NF-kappaB, Notch/Wnt and p53) and hours/days (e.g. circadian clock and cell cycle). Quantitative live cell measurement of the protein processes underlying these complex networks will allow characterisation of the core mechanisms that drive these signalling pathways and control cell function. Ultimately, such studies can be applied to develop predictive models of whole tissues and organisms.     

Bursting excitable cell models by a slow Ca2+ current

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.     

Gap junctions in excitable cells

Gap junction channels are an integral part of the conduction or propagation of an action potential from cell to cell. Gap junctions have rather unique gating and permeability properties which permit the movement of molecules from cell to cell. These molecules may not be directly linked to action potentials but can alter nonjunctional processes within cells, which in turn can affect conduction velocity. The data described in this review reveal that, for the majority of excitable cells, there are two limiting factors, with respect to gap junctions, that affect the conduction/propagation of action potentials. These are (1) the total number of channels and (2) the selective permeability of the channels. Interestingly, voltage dependence and the time course of voltage inactivation (kinetics) are not rate limiting steps under normal physiological conditions for any of the connexins studied so far. Only specialized rectifying electrical synapses utilize strong voltage dependence and rapid kinetics to permit or deny the continued propagation of an action potential.     

Spontaneous secondary spiking in excitable cells

Kepler & Marder (1993, Biol. Cybern.68, 209-214) proposed a model describing the electrical activity of a crab neuron in which a train of directly induced action potentials is sometimes followed by one or more spontaneous action potentials, referred to as spontaneous secondary spikes. We reduce their five-dimensional model to three dimensions in two different ways in order to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a qualitative explanation of the phenomenon using phase-plane and bifurcation analysis. We demonstrate that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model. The exact dynamics of the excitable system are not important, but the relative time constant and amplitude of the slow inward current are crucial. Using the numerical bifurcation analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential for all values of the bifurcation parameter. At a critical value of the bifurcation parameter, a stable tonic firing mode arises via a saddle-node of periodics bifurcation. Whether or not the models can exhibit transient or continuous spontaneous spiking depends on their position in parameter space relative to this saddle-node of periodics.     

Exploring dynamics in living cells by tracking single particles

In the last years, significant advances in microscopy techniques and the introduction of a novel technology to label living cells with genetically encoded fluorescent proteins revolutionized the field of Cell Biology. Our understanding on cell dynamics built from snapshots on fixed specimens has evolved thanks to our actual capability to monitor in real time the evolution of processes in living cells. Among these new tools, single particle tracking techniques were developed to observe and follow individual particles. Hence, we are starting to unravel the mechanisms driving the motion of a wide variety of cellular components ranging from organelles to protein molecules by following their way through the cell. In this review, we introduce the single particle tracking technology to new users. We briefly describe the instrumentation and explain some of the algorithms commonly used to locate and track particles. Also, we present some common tools used to analyze trajectories and illustrate with some examples the applications of single particle tracking to study dynamics in living cells.     

A general approach to modeling conduction and concentration dynamics in excitable cells of concentric cylindrical geometry.

This paper discusses mathematical approaches for modeling the propagation of the action potential and ion concentration dynamics in a general class of excitable cells and cell assemblies of concentric cylindrical geometry. Examples include myelinated and unmyelinated axons, single strands of interconnected cardiac cells and outer hair cells. A key feature in some of the cells is the presence of a small working volume such as the periaxonal space between the myelin sheath and the axon in the myelinated axon and the extracisternal space between the plasma membrane and the subsurface cisterna of the outer hair cell. Proper treatment of these cell types requires a modeling approach which can readily address these anatomical properties and the non-uniform biophysical properties of the concentric membranes and the ionic composition of the volumes between the membranes. An electrodiffusion approach is first developed in which the Nernst-Planck equation is used to characterize axial ion fluxes. It is then demonstrated that this "full" model can be stepwise reduced, eventually becoming equivalent to the standard cable equation formulation. This is done in a manner that permits direct comparisons between the full and simplified models by running simulations using a single parameter set. An intermediate approach where the contributions of the axial currents to ion concentration changes and the effect of varying ion concentrations on solution conductivities are ignored is derived and is found adequate in many cases. Two application examples are given: a "cardiac strand" model, for which the intermediate formulation is shown sufficient and a model of the myelinated axon, for which the full electrodiffusion formulation is clearly necessary. The latter finding is due to spatial inhomogeneities in the anatomy and distribution of ion channels and transporters in the myelinated axon and the restricted periaxonal space between the myelin sheath and the axon.     

Computational model for autophagic vesicle dynamics in single cells

Macroautophagy (autophagy) is a cellular recycling program essential for homeostasis and survival during cytotoxic stress. This process, which has an emerging role in disease etiology and treatment, is executed in four stages through the coordinated action of more than 30 proteins. An effective strategy for studying complicated cellular processes, such as autophagy, involves the construction and analysis of mathematical or computational models. When developed and refined from experimental knowledge, these models can be used to interrogate signaling pathways, formulate novel hypotheses about systems, and make predictions about cell signaling changes induced by specific interventions. Here, we present the development of a computational model describing autophagic vesicle dynamics in a mammalian system. We used time-resolved, live-cell microscopy to measure the synthesis and turnover of autophagic vesicles in single cells. The stochastically simulated model was consistent with data acquired during conditions of both basal and chemically-induced autophagy. The model was tested by genetic modulation of autophagic machinery and found to accurately predict vesicle dynamics observed experimentally. Furthermore, the model generated an unforeseen prediction about vesicle size that is consistent with both published findings and our experimental observations. Taken together, this model is accurate and useful and can serve as the foundation for future efforts aimed at quantitative characterization of autophagy.     

Imaging dynamics of endogenous mitochondrial RNA in single living cells

We developed genetically encoded RNA probes for characterizing localization and dynamics of mitochondrial RNA (mtRNA) in single living cells. The probes consist of two RNA-binding domains of PUMILIO1, each connected with split fragments of a fluorescent protein capable of reconstituting upon binding to a target RNA. We designed the probes to specifically recognize a 16-base sequence of mtRNA encoding NADH dehydrogenase subunit 6 (ND6) and to be targeted into the mitochondrial matrix, which allowed real-time imaging of ND6 mtRNA localization in living cells. We showed that ND6 mtRNA is localized within mitochondria and concentrated particularly on mitochondrial DNA (mtDNA). Movement of the ND6 mtRNA is restricted but oxidative stress induces the mtRNA to disperse in the mitochondria and gradually decompose. These probes provide a means to study spatial and temporal mRNA dynamics in intracellular compartments in living mammalian cells.     

Investigation of the effects of continuous-wave,pulse- and amplitude-modulated microwaves on single excitable cells of chara corallina

Single internodal excitable cells of Chara corallina were exposed to CW, pulse-modulated and sinusoidally modulated S-band microwave fields in a temperature-controled waveguide exposure chamber. All electrical measurements were made external to the waveguide (ie, under no impressed microwave field). The dependent variables measured before, during, and after exposure to the S-band microwave fields included: resting potential, amplitude of the action potential, rise and decay time of the action potential, conduction velocity, and excitability. Cells maintained at 22 ± 0.1 °C during exposure showed no consistent or statistically significant microwave-dependent alterations in any of the dependent variables.     

A numerical method to model excitable cells

We have extended a fast, stable, and accurate method for the numerical solution of cable equations to include changes in geometry and membrane properties in order to model a single excitable cell realistically. In addition, by including the provision that the radius may be a function of distance along an axis, we have achieved a general and powerful method for simulating a cell with any number of branched processes, any or all of which may be nonuniform in diameter, and with no restriction on the branching pattern.     

Large-scale cortical dynamics of sleep slow waves

Slow waves constitute the main signature of sleep in the electroencephalogram (EEG). They reflect alternating periods of neuronal hyperpolarization and depolarization in cortical networks. While recent findings have demonstrated their functional role in shaping and strengthening neuronal networks, a large-scale characterization of these two processes remains elusive in the human brain. In this study, by using simultaneous scalp EEG and intracranial recordings in 10 epileptic subjects, we examined the dynamics of hyperpolarization and depolarization waves over a large extent of the human cortex. We report that both hyperpolarization and depolarization processes can occur with two different characteristic time durations which are consistent across all subjects. For both hyperpolarization and depolarization waves, their average speed over the cortex was estimated to be approximately 1 m/s. Finally, we characterized their propagation pathways by studying the preferential trajectories between most involved intracranial contacts. For both waves, although single events could begin in almost all investigated sites across the entire cortex, we found that the majority of the preferential starting locations were located in frontal regions of the brain while they had a tendency to end in posterior and temporal regions.     

The purification of ion channels from excitable cells

Conclusion It should be emphasized that the problem of isolating a gene for which the gene product has been defined solely by genetic criteria is not trivial, but this approach is potentially very powerful. Not only does it provide a means for determining the sequence and structure of ion channels for which there are no biochemical probes, but it also provides a system in which specific mutations in the channel gene can be correlated with changes in channel function. Although this approach is limited to organisms that are readily manipulated by genetic methods, the results of these studies should be widely applicable. The biophysical properties of ion channels are generally similar from one species to another, suggesting that the structures of the channels have been highly conserved. The combination of biochemical, biophysical and genetic techniques will undoubtedly be exploited more fully in future studies of ion channel structure and function.