A mathematical model of pacemaker activity recorded from mouse small intestine

Ca2+ dynamics in interstitial cells: foundational mechanisms for the motor patterns in the gastrointestinal tract

The gastrointestinal (GI) tract displays multiple motor patterns that move nutrients and wastes through the body. Smooth muscle cells (SMCs) provide the forces necessary for GI motility, but interstitial cells, electrically coupled to SMCs, tune SMC excitability, transduce inputs from enteric motor neurons, and generate pacemaker activity that underlies major motor patterns, such as peristalsis and segmentation. The interstitial cells regulating SMCs are interstitial cells of Cajal (ICC) and PDGF receptor (PDGFR)α+ cells. Together these cells form the SIP syncytium. ICC and PDGFRα+ cells express signature Ca2+-dependent conductances: ICC express Ca2+-activated Cl- channels, encoded by Ano1, that generate inward current, and PDGFRα+ cells express Ca2+-activated K+ channels, encoded by Kcnn3, that generate outward current. The open probabilities of interstitial cell conductances are controlled by Ca2+ release from the endoplasmic reticulum. The resulting Ca2+ transients occur spontaneously in a stochastic manner. Ca2+ transients in ICC induce spontaneous transient inward currents and spontaneous transient depolarizations (STDs). Neurotransmission increases or decreases Ca2+ transients, and the resulting depolarizing or hyperpolarizing responses conduct to other cells in the SIP syncytium. In pacemaker ICC, STDs activate voltage-dependent Ca2+ influx, which initiates a cluster of Ca2+ transients and sustains activation of ANO1 channels and depolarization during slow waves. Regulation of GI motility has traditionally been described as neurogenic and myogenic. Recent advances in understanding Ca2+ handling mechanisms in interstitial cells and how these mechanisms influence motor patterns of the GI tract suggest that the term "myogenic" should be replaced by the term "SIPgenic," as this review discusses.

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

Computational models of autonomic regulation in gastric motility: Progress, challenges, and future directions

The stomach is extensively innervated by the vagus nerve and the enteric nervous system. The mechanisms through which this innervation affects gastric motility are being unraveled, motivating the first concerted steps towards the incorporation autonomic regulation into computational models of gastric motility. Computational modeling has been valuable in advancing clinical treatment of other organs, such as the heart. However, to date, computational models of gastric motility have made simplifying assumptions about the link between gastric electrophysiology and motility. Advances in experimental neuroscience mean that these assumptions can be reviewed, and detailed models of autonomic regulation can be incorporated into computational models. This review covers these advances, as well as a vision for the utility of computational models of gastric motility. Diseases of the nervous system, such as Parkinson’s disease, can originate from the brain-gut axis and result in pathological gastric motility. Computational models are a valuable tool for understanding the mechanisms of disease and how treatment may affect gastric motility. This review also covers recent advances in experimental neuroscience that are fundamental to the development of physiology-driven computational models. A vision for the future of computational modeling of gastric motility is proposed and modeling approaches employed for existing mathematical models of autonomic regulation of other gastrointestinal organs and other organ systems are discussed.

Three-dimensional multi-field modelling of gastric arrhythmias and their effects on antral contractions

The contraction activation of smooth muscle in the stomach wall (SW) is coordinated by slow electrical waves. The interstitial cells of Cajal (ICC), specialised pacemaker cells, initiate and propagate these slow waves. By establishing an electrically coupled network, each ICC adjusts its intrinsic pacing frequency to a single dominant frequency, to be a key aspect in modelling the electrophysiology of gastric tissue. In terms of modelling, additional fields associated with electrical activation, such as voltage-dependent calcium influx and the resulting deformation, have hardly been considered so far. Here we present a three-dimensional model of the electro-chemomechanical activation of gastric smooth muscle contractions. To reduce computational costs, an adaptive multi-scale discretisation strategy for the temporal resolution of the electric field is used. The model incorporates a biophysically based model of gastric ICC pacemaker activity that aims to simulate stable entrainment and physiological conduction velocities of the electrical slow waves. Together with the simulation of concomitant gastric contractions and the inclusion of a mechanical feedback mechanism, the model is used to study dysrhythmias of gastric slow waves induced by abnormal stretching of the antral SW. The model is able to predict the formation of stretch-induced gastric arrhythmias, such as the emergence of an ectopic pacemaker in the gastric antrum. The results show that the ectopic event is accompanied by smooth muscle contraction and, although it disrupts the normal propagation pattern of gastric slow electrical waves, it can also catalyse the process of handling indigestible materials that might otherwise injure the gastric SW.

Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells

Spatiotemporal patterns are common in biological systems. For electrically coupled cells, previous studies of pattern formation have mainly used applied current as the primary bifurcation parameter. The purpose of this paper is to show that applied current is not needed to generate spatiotemporal patterns for smooth muscle cells. The patterns can be generated solely by external mechanical stimulation (transmural pressure). To do this we study a reaction-diffusion system involving the Morris-Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system without coupling - in particular Type I and Type II excitability both occur. We show the patterns are not due to a Turing instability and that the spatially extended model exhibits spatiotemporal chaos. We also use travelling wave coordinates to analyse travelling waves.

Current applications of mathematical models of the interstitial cells of Cajal in the gastrointestinal tract

The interstitial cells of Cajal (ICC) form interconnected networks throughout the gastrointestinal (GI) tract. ICC act as the pacemaker cells that initiate the rhythmic bioelectrical slow waves and intermediary between the GI musculature and nerves, both of which are critical to GI motility. Disruptions to the number of ICC and the integrity of ICC networks have been identified as a key pathophysiological mechanism in a number of clinically challenging GI disorders. The current analyses of ICC generally rely on either functional recordings taken directly from excised tissue or morphological analysis based on images of labeled ICC, where the structural-functional relationship is investigated in an associative manner rather than mechanistically. On the other hand, computational physiology has played a significant role in facilitating our understanding of a number of physiological systems in both health and disease, and investigations in the GI field are beginning to incorporate several mathematical models of the ICC. The main aim of this review is to present the major modeling advances in GI electrophysiology, in order to introduce a multi-scale framework for mathematically quantifying the functional consequences of ICC degradation at both cellular and tissue scales. The outcomes will inform future investigators utilizing modeling techniques in their studies. This article is categorized under: Metabolic Diseases > Computational Models.

The Cardiac Pacemaker Story-Fundamental Role of the Na+/Ca2+ Exchanger in Spontaneous Automaticity

The electrophysiological mechanism of the sinus node automaticity was previously considered exclusively regulated by the so-called "funny current". However, parallel investigations increasingly emphasized the importance of the Ca²⁺-homeostasis and Na⁺/Ca²⁺ exchanger (NCX). Recently, increasing experimental evidence, as well as insight through mechanistic in silico modeling demonstrates the crucial role of the exchanger in sinus node pacemaking. NCX had a key role in the exciting story of discovery of sinus node pacemaking mechanisms, which recently settled with a consensus on the coupled-clock mechanism after decades of debate. This review focuses on the role of the Na⁺/Ca²⁺ exchanger from the early results and concepts to recent advances and attempts to give a balanced summary of the characteristics of the local, spontaneous, and rhythmic Ca²⁺ releases, the molecular control of the NCX and its role in the fight-or-flight response. Transgenic animal models and pharmacological manipulation of intracellular Ca²⁺ concentration and/or NCX demonstrate the pivotal function of the exchanger in sinus node automaticity. We also highlight where specific hypotheses regarding NCX function have been derived from computational modeling and require experimental validation. Nonselectivity of NCX inhibitors and the complex interplay of processes involved in Ca²⁺ handling render the design and interpretation of these experiments challenging.

Stretch modulation of cardiac contractility: importance of myocyte calcium during the slow force response

The mechanical response of the heart to myocardial stretch has been understood since the work of muscle physiologists more than 100 years ago, whereby an increase in ventricular chamber filling during diastole increases the subsequent force of contraction. The stretch-induced increase in contraction is biphasic. There is an abrupt increase in the force that coincides with the stretch (the rapid response), which is then followed by a slower response that develops over several minutes (the slow force response, or SFR). The SFR is associated with a progressive increase in the magnitude of the Ca²⁺ transient, the event that initiates myocyte cross-bridge cycling and force development. However, the mechanisms underlying the stretch-dependent increase in the Ca²⁺ transient are still debated. This review outlines recent literature on the SFR and summarizes the different stretch-activated Ca²⁺ entry pathways. The SFR might result from a combination of several different cellular mechanisms initiated in response to activation of different cellular stretch sensors.

A Formal Approach for Scalable Simulation of Gastric ICC Electrophysiology

OBJECTIVE

Efficient and accurate organ models are crucial for closed-loop validation of implantable medical devices. This paper investigates bio-electric slow wave modeling of the stomach, so that gastric electrical stimulator (GES) can be validated and verified prior to implantation. In particular, we consider high-fidelity, scalable, and efficient modeling of the pacemaker, Interstitial cells of Cajal (ICC), based on the formal hybrid input output automata (HIOA) framework.

METHODS

Our work is founded in formal methods, a collection of mathematically sound techniques originating in computer science for the design and validation of safety-critical systems. We modeled each ICC cell using an HIOA. We also introduce an HIOA path model to capture the electrical propagation delay between cells in a network. The resultant network of ICC cells can simulate normal and diseased action potential propagation patterns, making it useful for device validation.

RESULTS

The simulated slow wave of a single ICC cell had high correlation ( ≈ 0.9) with the corresponding biophysical models.

CONCLUSIONS

The proposed model is able to simulate the slow wave activity of a network of ICC cells with high-fidelity for device validation.

SIGNIFICANCE

The proposed HIOA model is significantly more efficient than the corresponding biophysical models, scales to larger networks of ICC cells, and is capable of simulating varying propagation patterns. This has the potential to enable verification and validation of implantable GESs in closed-loop with gastrointestinal models in the future.

Computational motility models of neurogastroenterology and neuromodulation

The success of neuromodulation therapies, particularly in the brain, spinal cord, and peripheral nerves, has been greatly aided by computational, biophysical models. However, treating gastrointestinal disorders with electrical stimulation has been much less explored, partly because the mode of action of such treatments is unclear, and selection of stimulation parameters is often empirical. Progress in gut neuromodulation is limited by the comparative lack of biophysical models capable of simulating neuromodulation of gastrointestinal function. Here, we review the recently developed biophysical models of electrically-active cells in the gastrointestinal system that contribute to motility. Biophysical models are replacing phenomenologically-defined models due to advancements in electrophysiological characterization of key players in the gut: enteric neurons, smooth muscle fibers, and interstitial cells of Cajal. In this review, we explore existing biophysically-defined cellular and network models that contribute to gastrointestinal motility. We focus on recent models that are laying the groundwork for modeling electrical stimulation of the gastrointestinal system. Developing models of gut neuromodulation will improve our mechanistic understanding of these treatments, leading to better parameterization, selectivity, and efficacy of neuromodulation to treat gastrointestinal disorders. Such models may have direct clinical translation to current neuromodulation therapies, such as sacral nerve stimulation.

Investigation of the role of ion channels in human pancreatic β-cell hubs: A mathematical modeling study

In many cellular networks, the structure of the network follows a scale-free organization, where a limited number of cells are strongly coupled to other cells. These cells are called hub cells and their critical roles are well accepted. Despite their importance, there have been only a few studies investigating the characteristic features of these cells. In this paper, a computational approach is proposed to study the possible role of different ion channels in distinguishing between the hub and non-hub cells. The results show that the P/Q-type and T-type calcium channels may have an especial role in the β-cell hubs because the high-level expressions of these channels make a pancreatic β-cell more potent to force other coupled cells to follow it. In addition, in order to consider the variation of the coupling strength with voltage, a novel mathematical model is proposed for the gap junction coupling between the pancreatic β-cells. The proposed approach is validated based on the data from the literature.

Problems with extracellular recording of electrical activity in gastrointestinal muscle

Motility patterns of the gastrointestinal tract are important for efficient processing of nutrients and waste. Peristalsis and segmentation are based on rhythmic electrical slow waves that generate the phasic contractions fundamental to gastrointestinal motility. Slow waves are generated and propagated actively by interstitial cells of Cajal (ICC), and these events conduct to smooth muscle cells to elicit excitation-contraction coupling. Extracellular electrical recording has been utilized to characterize slow-wave generation and propagation and abnormalities that might be responsible for gastrointestinal motility disorders. Electrode array recording and digital processing are being used to generate data for models of electrical propagation in normal and pathophysiological conditions. Here, we discuss techniques of extracellular recording as applied to gastrointestinal organs and how mechanical artefacts might contaminate these recordings and confound their interpretation. Without rigorous controls for movement, current interpretations of extracellular recordings might ascribe inaccurate behaviours and electrical anomalies to ICC networks and gastrointestinal muscles, bringing into question the findings and validity of models of gastrointestinal electrophysiology developed from these recordings.

The virtual intestine: in silico modeling of small intestinal electrophysiology and motility and the applications

The intestine comprises a long hollow muscular tube organized in anatomically and functionally discrete compartments, which digest and absorb nutrients and water from ingested food. The intestine also plays key roles in the elimination of waste and protection from infection. Critical to all of these functions is the intricate, highly coordinated motion of the intestinal tract, known as motility, which is coregulated by hormonal, neural, electrophysiological and other factors. The Virtual Intestine encapsulates a series of mathematical models of intestinal function in health and disease, with a current focus on motility, and particularly electrophysiology. The Virtual Intestine is being cohesively established across multiple physiological scales, from sub/cellular functions to whole organ levels, facilitating quantitative evaluations that present an integrative in silico framework. The models are also now finding broad physiological applications, including in evaluating hypotheses of slow wave pacemaker mechanisms, smooth muscle electrophysiology, structure-function relationships, and electromechanical coupling. Clinical applications are also beginning to follow, including in the pathophysiology of motility disorders, diagnosing intestinal ischemia, and visualizing colonic dysfunction. These advances illustrate the emerging potential of the Virtual Intestine to effectively address multiscale research challenges in interdisciplinary gastrointestinal sciences.

A simplified biophysical cell model for gastric slow wave entrainment simulation

Gastric electrical activity, also termed slow wave activity, is generated by a class of pacemaker cells called the interstitial cells of Cajal (ICC), which are organized with decreasing intrinsic frequencies along the stomach. In the healthy stomach, slow waves of different intrinsic frequencies converge to a single frequency with a constant phase-lag, in a process called entrainment. The main aim of this study was to develop a simplified biophysical ICC model that is capable of modeling the self-excitatory behavior and standard morphology of gastric slow waves. Entrainment of gastric slow waves was simulated in a one-dimensional (1D) model, with a linear gradient of intrinsic slow wave frequencies. In a coupled 1D model, the simulated slow waves were entrained to a single frequency; whereas in an uncoupled 1D model, the simulated slow waves occurred at different frequencies, resulting in loss of entrainment. The new cell model presents an option for future large multi-scale simulations of gastric slow waves in intact ICC network and diseased conditions where the loss of entrainment may lead to slow wave dysrhythmias and diminished gastric motility.

Modeling of stochastic behavior of pacemaker potential in interstitial cells of Cajal

It is widely accepted that interstitial cells of Cajal (ICCs) generate pacemaker potentials to propagate slow waves along the whole gastrointestinal tract. Previously, we constructed a biophysically based model of ICCs in mouse small intestine to explain the pacemaker mechanism. Our previous model, however, could not explain non-uniformity of pacemaker potentials and random occurrence of unitary potentials, thus we updated our model. The inositol 1,4,5-trisphosphate (IP3)-mediated Ca(2+) mobilization is a key event to drive the cycle of pacemaker activity and was updated to reproduce its stochastic behavior. The stochasticity was embodied by simulating random opening and closing of individual IP3-mediated Ca(2+) channel. The updated model reproduces the stochastic features of pacemaker potentials in ICCs. Reproduced pacemaker potentials are not uniform in duration and interval. The resting and peak potentials are -75.5 ± 1.1 mV and -0.8 ± 0.5 mV, respectively (n = 55). Frequency of pacemaker potential is 14.3 ± 0.4 min(-1) (n = 10). Width at half-maximal amplitude of pacemaker potential is 902 ± 6 ms (n = 55). There are random events of unitary potential-like depolarization. Finally, we compared our updated model with a recently published model to speculate which ion channel is the best candidate to drive pacemaker depolarization. In conclusion, our updated mathematical model could now reproduce stochastic features of pacemaker activity in ICCs.

Interstitial cells: regulators of smooth muscle function

Smooth muscles are complex tissues containing a variety of cells in addition to muscle cells. Interstitial cells of mesenchymal origin interact with and form electrical connectivity with smooth muscle cells in many organs, and these cells provide important regulatory functions. For example, in the gastrointestinal tract, interstitial cells of Cajal (ICC) and PDGFRα(+) cells have been described, in detail, and represent distinct classes of cells with unique ultrastructure, molecular phenotypes, and functions. Smooth muscle cells are electrically coupled to ICC and PDGFRα(+) cells, forming an integrated unit called the SIP syncytium. SIP cells express a variety of receptors and ion channels, and conductance changes in any type of SIP cell affect the excitability and responses of the syncytium. SIP cells are known to provide pacemaker activity, propagation pathways for slow waves, transduction of inputs from motor neurons, and mechanosensitivity. Loss of interstitial cells has been associated with motor disorders of the gut. Interstitial cells are also found in a variety of other smooth muscles; however, in most cases, the physiological and pathophysiological roles for these cells have not been clearly defined. This review describes structural, functional, and molecular features of interstitial cells and discusses their contributions in determining the behaviors of smooth muscle tissues.

Mitochondrial calcium handling within the interstitial cells of Cajal

The interstitial cells of Cajal (ICC) drive rhythmic pacemaking contractions in the gastrointestinal system. The ICC generate pacemaking signals by membrane depolarizations associated with the release of intracellular calcium (Ca(2+)) in the endoplasmic reticulum (ER) through inositol-trisphosphate (IP3) receptors (IP3R) and uptake by mitochondria (MT). This Ca(2+) dynamic is hypothesized to generate pacemaking signals by calibrating ER Ca(2+) store depletions and membrane depolarization with ER store-operated Ca(2+) entry mechanisms. Using a biophysically based spatio-temporal model of integrated Ca(2+) transport in the ICC, we determined the feasibility of ER depletion timescale correspondence with experimentally observed pacemaking frequencies while considering the impact of IP3R Ca(2+) release and MT uptake on bulk cytosolic Ca(2+) levels because persistent elevations of free intracellular Ca(2+) are toxic to the cell. MT densities and distributions are varied in the model geometry to observe MT influence on free cytosolic Ca(2+) and the resulting frequencies of ER Ca(2+) store depletions, as well as the sarco-endoplasmic reticulum Ca(2+) ATP-ase (SERCA) and IP3 agonist concentrations. Our simulations show that high MT densities observed in the ICC are more relevant to ER establishing Ca(2+) depletion frequencies than protection of the cytosol from elevated free Ca(2+), whereas the SERCA pump is more relevant to containing cytosolic Ca(2+) elevations. Our results further suggest that the level of IP3 agonist stimulating ER Ca(2+) release, subsequent MT uptake, and eventual activation of ER store-operated Ca(2+) entry may determine frequencies of rhythmic pacemaking exhibited by the ICC across species and tissue types.

Computational modeling of anoctamin 1 calcium-activated chloride channels as pacemaker channels in interstitial cells of Cajal

Interstitial cells of Cajal (ICC) act as pacemaker cells in the gastrointestinal tract by generating electrical slow waves to regulate rhythmic smooth muscle contractions. Intrinsic Ca(2+) oscillations in ICC appear to produce the slow waves by activating pacemaker currents, currently thought to be carried by the Ca(2+)-activated Cl(-) channel anoctamin 1 (Ano1). In this article we present a novel model of small intestinal ICC pacemaker activity that incorporates store-operated Ca(2+) entry and a new model of Ano1 current. A series of simulations were carried out with the ICC model to investigate current controversies about the reversal potential of the Ano1 Cl(-) current in ICC and to predict the characteristics of the other ion channels that are necessary to generate slow waves. The model results show that Ano1 is a plausible pacemaker channel when coupled to a store-operated Ca(2+) channel but suggest that small cyclical depolarizations may still occur in ICC in Ano1 knockout mice. The results predict that voltage-dependent Ca(2+) current is likely to be negligible during the slow wave plateau phase. The model shows that the Cl(-) equilibrium potential is an important modulator of slow wave morphology, highlighting the need for a better understanding of Cl(-) dynamics in ICC.

Calcium and IP3 dynamics in cardiac myocytes: experimental and computational perspectives and approaches

Calcium plays a crucial role in excitation-contraction coupling (ECC), but it is also a pivotal second messenger activating Ca²⁺-dependent transcription factors in a process termed excitation-transcription coupling (ETC). Evidence accumulated over the past decade indicates a pivotal role of inositol 1,4,5-trisphosphate receptor (IP₃R)-mediated Ca²⁺ release in the regulation of cytosolic and nuclear Ca²⁺ signals. IP₃ is generated by stimulation of plasma membrane receptors that couple to phospholipase C (PLC), liberating IP₃ from phosphatidylinositol 4,5-bisphosphate (PIP₂). An intriguing aspect of IP₃ signaling is the presence of the entire PIP₂-PLC-IP₃ signaling cascade as well as the presence of IP₃Rs at the inner and outer membranes of the nuclear envelope (NE) which functions as a Ca²⁺ store. The observation that the nucleus is surrounded by its own putative Ca²⁺ store raises the possibility that nuclear IP₃-dependent Ca²⁺ release plays a critical role in ETC. This provides a potential mechanism of regulation that acts locally and autonomously from the global cytosolic Ca²⁺ signal underlying ECC. Moreover, there is evidence that: (i) the sarcoplasmic reticulum (SR) and NE are a single contiguous Ca²⁺ store; (ii) the nuclear pore complex is the major gateway for Ca²⁺ and macromolecules to pass between the cytosol and the nucleoplasm; (iii) the inner membrane of the NE hosts key Ca²⁺ handling proteins including the Na⁺/Ca²⁺ exchanger (NCX)/GM1 complex, ryanodine receptors (RyRs), nicotinic acid adenine dinucleotide phosphate receptors (NAADPRs), Na⁺/K⁺ ATPase, and Na⁺/H⁺ exchanger. Thus, it appears that the nucleus represents a Ca²⁺ signaling domain equipped with its own ion channels and transporters that allow for complex local Ca²⁺ signals. Many experimental and modeling approaches have been used for the study of intracellular Ca²⁺ signaling but the key to the understanding of the dual role of Ca²⁺ mediating ECC and ECT lays in quantitative differences of local [Ca²⁺] in the nuclear and cytosolic compartment. In this review, we discuss the state of knowledge regarding the origin and the physiological implications of nuclear Ca²⁺ transients in different cardiac cell types (adult atrial and ventricular myocytes) as well as experimental and mathematical approaches to study Ca²⁺ and IP₃ signaling in the cytosol and nucleus. In particular, we focus on the concept that highly localized Ca²⁺ signals are required to translocate and activate Ca²⁺-dependent transcription factors (e.g., nuclear factor of activated T-cells, NFAT; histone deacetylase, HDAC) through phosphorylation/dephosphorylation processes.

Spatial analysis of slowly oscillating electric activity in the gut of mice using low impedance arrayed microelectrodes

Smooth and elaborate gut motility is based on cellular cooperation, including smooth muscle, enteric neurons and special interstitial cells acting as pacemaker cells. Therefore, spatial characterization of electric activity in tissues containing these electric excitable cells is required for a precise understanding of gut motility. Furthermore, tools to evaluate spatial electric activity in a small area would be useful for the investigation of model animals. We thus employed a microelectrode array (MEA) system to simultaneously measure a set of 8×8 field potentials in a square area of ∼1 mm². The size of each recording electrode was 50×50 µm², however the surface area was increased by fixing platinum black particles. The impedance of microelectrode was sufficiently low to apply a high-pass filter of 0.1 Hz. Mapping of spectral power, and auto-correlation and cross-correlation parameters characterized the spatial properties of spontaneous electric activity in the ileum of wild-type (WT) and W/W^v mice, the latter serving as a model of impaired network of pacemaking interstitial cells. Namely, electric activities measured varied in both size and cooperativity in W/W^v mice, despite the small area. In the ileum of WT mice, procedures suppressing the excitability of smooth muscle and neurons altered the propagation of spontaneous electric activity, but had little change in the period of oscillations. In conclusion, MEA with low impedance electrodes enables to measure slowly oscillating electric activity, and is useful to evaluate both histological and functional changes in the spatio-temporal property of gut electric activity.

A quantitative model of human jejunal smooth muscle cell electrophysiology

Recently, a number of ion channel mutations have been identified in the smooth muscle cells of the human jejunum. Although these are potentially significant in understanding diseases that are currently of unknown etiology, no suitable computational cell model exists to evaluate the effects of such mutations. Here, therefore, a biophysically based single cell model of human jejunal smooth muscle electrophysiology is presented. The resulting cellular description is able to reproduce experimentally recorded slow wave activity and produces realistic responses to a number of perturbations, providing a solid platform on which the causes of intestinal myopathies can be investigated.

Mathematical Distinction in Action Potential between Primo-Vessels and Smooth Muscle

We studied the action potential of Primo-vessels in rats to determine the electrophysiological characteristics of these structures. We introduced a mathematical analysis method, a normalized Fourier transform that displays the sine and cosine components separately, to compare the action potentials of Primo-vessels with those for the smooth muscle. We found that Primo-vessels generated two types of action potential pulses that differed from those of smooth muscle: (1) Type I pulse had rapid depolarizing and repolarizing phases, and (2) Type II pulse had a rapid depolarizing phase and a gradually slowing repolarizing phase.

Small intestinal model for electrically propelled capsule endoscopy

The aim of this research is to propose a small intestine model for electrically propelled capsule endoscopy. The electrical stimulus can cause contraction of the small intestine and propel the capsule along the lumen. The proposed model considered the drag and friction from the small intestine using a thin walled model and Stokes' drag equation. Further, contraction force from the small intestine was modeled by using regression analysis. From the proposed model, the acceleration and velocity of various exterior shapes of capsule were calculated, and two exterior shapes of capsules were proposed based on the internal volume of the capsules. The proposed capsules were fabricated and animal experiments were conducted. One of the proposed capsules showed an average (SD) velocity in forward direction of 2.91 ± 0.99 mm/s and 2.23 ± 0.78 mm/s in the backward direction, which was 5.2 times faster than that obtained in previous research. The proposed model can predict locomotion of the capsule based on various exterior shapes of the capsule.

Quantification of gastrointestinal sodium channelopathy

Na(v)1.5 sodium channels, encoded by SCN5A, have been identified in human gastrointestinal interstitial cells of Cajal (ICC) and smooth muscle cells (SMC). A recent study found a novel, rare missense R76C mutation of the sodium channel interacting protein telethonin in a patient with primary intestinal pseudo-obstruction. The presence of a mutation in a patient with a motility disorder, however, does not automatically imply a cause-effect relationship between the two. Patch clamp experiments on HEK-293 cells previously established that the R76C mutation altered Na(v)1.5 channel function. Here the process through which these data were quantified to create stationary Markov state models of wild-type and R76C channel function is described. The resulting channel descriptions were included in whole cell ICC and SMC computational models and simulations were performed to assess the cellular effects of the R76C mutation. The simulated ICC slow wave was decreased in duration and the resting membrane potential in the SMC was depolarized. Thus, the R76C mutation was sufficient to alter ICC and SMC cell electrophysiology. However, the cause-effect relationship between R76C and intestinal pseudo-obstruction remains an open question.

Biophysically based modeling of the interstitial cells of cajal: current status and future perspectives

Gastrointestinal motility research is progressing rapidly, leading to significant advances in the last 15 years in understanding the cellular mechanisms underlying motility, following the discovery of the central role played by the interstitial cells of Cajal (ICC). As experimental knowledge of ICC physiology has expanded, biophysically based modeling has become a valuable tool for integrating experimental data, for testing hypotheses on ICC pacemaker mechanisms, and for applications in in silico studies including in multiscale models. This review is focused on the cellular electrophysiology of ICC. Recent evidence from both experimental and modeling domains have called aspects of the existing pacemaker theories into question. Therefore, current experimental knowledge of ICC pacemaker mechanisms is examined in depth, and current theories of ICC pacemaking are evaluated and further developed. Existing biophysically based ICC models and their physiological foundations are then critiqued in light of the recent advances in experimental knowledge, and opportunities to improve these models are identified. The review concludes by examining several potential clinical applications of biophysically based ICC modeling from the subcellular through to the organ level, including ion channelopathies and ICC network degradation.

Gastrointestinal system

The functions of the gastrointestinal (GI) tract include digestion, absorption, excretion, and protection. In this review, we focus on the electrical activity of the stomach and small intestine, which underlies the motility of these organs, and where the most detailed systems descriptions and computational models have been based to date. Much of this discussion is also applicable to the rest of the GI tract. This review covers four major spatial scales: cell, tissue, organ, and torso, and discusses the methods of investigation and the challenges associated with each. We begin by describing the origin of the electrical activity in the interstitial cells of Cajal, and its spread to smooth muscle cells. The spread of electrical activity through the stomach and small intestine is then described, followed by the resultant electrical and magnetic activity that may be recorded on the body surface. A number of common and highly symptomatic GI conditions involve abnormal electrical and/or motor activity, which are often termed functional disorders. In the last section of this review we address approaches being used to characterize and diagnose abnormalities in the electrical activity and how these might be applied in the clinical setting. The understanding of electrophysiology and motility of the GI system remains a challenging field, and the review discusses how biophysically based mathematical models can help to bridge gaps in our current knowledge, through integration of otherwise separate concepts.

Tissue-specific mathematical models of slow wave entrainment in wild-type and 5-HT(2B) knockout mice with altered interstitial cells of Cajal networks

Gastrointestinal slow waves are generated within networks of interstitial cells of Cajal (ICCs). In the intact tissue, slow waves are entrained to neighboring ICCs with higher intrinsic frequencies, leading to active propagation of slow waves. Degradation of ICC networks in humans is associated with motility disorders; however, the pathophysiological mechanisms of this relationship are uncertain. A recently developed biophysically based mathematical model of ICC was adopted and updated to simulate entrainment of slow waves. Simulated slow wave propagation was successfully entrained in a one-dimensional model, which contained a gradient of intrinsic frequencies. Slow wave propagation was then simulated in tissue models which contained a realistic two-dimensional microstructure of the myenteric ICC networks translated from wild-type (WT) and 5-HT(2B) knockout (degraded) mouse jejunum. The results showed that the peak current density in the WT model was 0.49 muA mm(-2) higher than the 5-HT(2B) knockout model, and the intracellular Ca(2+) density after 400 ms was 0.26 mM mm(-2) higher in the WT model. In conclusion, tissue-specific models of slow waves are presented, and simulations quantitatively demonstrated physiological differences between WT and 5-HT(2B) knockout models. This study provides a framework for evaluating how ICC network degradation may impair slow wave propagation and ultimately motility and transit.

Biophysically based mathematical modeling of interstitial cells of Cajal slow wave activity generated from a discrete unitary potential basis

Spontaneously rhythmic pacemaker activity produced by interstitial cells of Cajal (ICC) is the result of the entrainment of unitary potential depolarizations generated at intracellular sites termed pacemaker units. In this study, we present a mathematical modeling framework that quantitatively represents the transmembrane ion flows and intracellular Ca2+ dynamics from a single ICC operating over the physiological membrane potential range. The mathematical model presented here extends our recently developed biophysically based pacemaker unit modeling framework by including mechanisms necessary for coordinating unitary potential events, such as a T-Type Ca2+ current, Vm-dependent K+ currents, and global Ca2+ diffusion. Model simulations produce spontaneously rhythmic slow wave depolarizations with an amplitude of 65 mV at a frequency of 17.4 cpm. Our model predicts that activity at the spatial scale of the pacemaker unit is fundamental for ICC slow wave generation, and Ca2+ influx from activation of the T-Type Ca2+ current is required for unitary potential entrainment. These results suggest that intracellular Ca2+ levels, particularly in the region local to the mitochondria and endoplasmic reticulum, significantly influence pacing frequency and synchronization of pacemaker unit discharge. Moreover, numerical investigations show that our ICC model is capable of qualitatively replicating a wide range of experimental observations.

A Ca(2+)-activated Cl(-) conductance in interstitial cells of Cajal linked to slow wave currents and pacemaker activity

Interstitial cells of Cajal (ICC) are unique cells that generate electrical pacemaker activity in gastrointestinal (GI) muscles. Many previous studies have attempted to characterize the conductances responsible for pacemaker current and slow waves in the GI tract, but the precise mechanism of electrical rhythmicity is still debated. We used a new transgenic mouse with a bright green fluorescent protein (copGFP) constitutively expressed in ICC to facilitate study of these cells in mixed cell dispersions. We found that ICC express a specialized 'slow wave' current. Reversal of tail current analysis showed this current was due to a Cl(-) selective conductance. ICC express ANO1, a Ca(2+)-activated Cl(-) channel. Slow wave currents are not voltage dependent, but a secondary voltage-dependent process underlies activation of these currents. Removal of extracellular Ca(2+), replacement of Ca(2+) with Ba(2+), or extracellular Ni(2+) (30 microm) blocked the slow wave current. Single Ca(2+)-activated Cl() channels with a unitary conductance of 7.8 pS were resolved in excised patches of ICC. These are similar in conductance to ANO1 channels (8 pS) expressed in HEK293 cells. Slow wave current was blocked in a concentration-dependent manner by niflumic acid (IC(50) = 4.8 microm). Slow wave currents are associated with transient depolarizations of ICC in current clamp, and these events were blocked by niflumic acid. These findings demonstrate a role for a Ca(2+)-activated Cl(-) conductance in slow wave current in ICC and are consistent with the idea that ANO1 participates in pacemaker activity.

A Ca(2+)-activated Cl(-) conductance in interstitial cells of Cajal linked to slow wave currents and pacemaker activity

Interstitial cells of Cajal (ICC) are unique cells that generate electrical pacemaker activity in gastrointestinal (GI) muscles. Many previous studies have attempted to characterize the conductances responsible for pacemaker current and slow waves in the GI tract, but the precise mechanism of electrical rhythmicity is still debated. We used a new transgenic mouse with a bright green fluorescent protein (copGFP) constitutively expressed in ICC to facilitate study of these cells in mixed cell dispersions. We found that ICC express a specialized 'slow wave' current. Reversal of tail current analysis showed this current was due to a Cl(-) selective conductance. ICC express ANO1, a Ca(2+)-activated Cl(-) channel. Slow wave currents are not voltage dependent, but a secondary voltage-dependent process underlies activation of these currents. Removal of extracellular Ca(2+), replacement of Ca(2+) with Ba(2+), or extracellular Ni(2+) (30 microm) blocked the slow wave current. Single Ca(2+)-activated Cl() channels with a unitary conductance of 7.8 pS were resolved in excised patches of ICC. These are similar in conductance to ANO1 channels (8 pS) expressed in HEK293 cells. Slow wave current was blocked in a concentration-dependent manner by niflumic acid (IC(50) = 4.8 microm). Slow wave currents are associated with transient depolarizations of ICC in current clamp, and these events were blocked by niflumic acid. These findings demonstrate a role for a Ca(2+)-activated Cl(-) conductance in slow wave current in ICC and are consistent with the idea that ANO1 participates in pacemaker activity.

Quantitative cellular description of gastric slow wave activity

Interstitial cells of Cajal (ICC) are responsible for the spontaneous and omnipresent electrical activity in the stomach. A quantitative description of the intracellular processes whose coordinated activity is believed to generate electrical slow waves has been developed and is presented here. In line with recent experimental evidence, the model describes how the interplay between the mitochondria and the endoplasmic reticulum in cycling intracellular Ca(2+) provides the primary regulatory signal for the initiation of the slow wave. The major ion channels that have been identified as influencing slow wave activity have been modeled according to data obtained from isolated ICC. The model has been validated by comparing the simulated profile of the slow waves with experimental recordings and shows good correspondence in terms of frequency, amplitude, and shape in both control and pharmacologically altered conditions.

A biophysically based mathematical model of unitary potential activity in interstitial cells of Cajal

Unitary potential (UP) depolarizations are the basic intracellular events responsible for pacemaker activity in interstitial cells of Cajal (ICCs), and are generated at intracellular sites termed "pacemaker units". In this study, we present a mathematical model of the transmembrane ion flows and intracellular Ca(2+) dynamics from a single ICC pacemaker unit acting at near-resting membrane potential. This model quantitatively formalizes the framework of a novel ICC pacemaking mechanism that has recently been proposed. Model simulations produce spontaneously rhythmic UP depolarizations with an amplitude of approximately 3 mV at a frequency of 0.05 Hz. The model predicts that the main inward currents, carried by a Ca(2+)-inhibited nonselective cation conductance, are activated by depletion of sub-plasma-membrane [Ca(2+)] caused by sarcoendoplasmic reticulum calcium ATPase Ca(2+) sequestration. Furthermore, pacemaker activity predicted by our model persists under simulated voltage clamp and is independent of [IP(3)] oscillations. The model presented here provides a basis to quantitatively analyze UP depolarizations and the biophysical mechanisms underlying their production.

Interstitial cells of Cajal: a new perspective on smooth muscle function

Interstitial cells of Cajal (ICC) were described more than 100 years ago by Ramon y Cajal. For many years these cells were identified only by non-specific histological stains and later, more reliably, by electron microscopy. Ultrastructural features and the anatomical locations of ICC suggested important physiological roles for these cells. A breakthrough occurred in our ability to study ICC when it was recognized that antibodies for Kit could be used to identify ICC, even in living tissues. Signalling via Kit, a receptor tyrosine kinase, is also necessary for ICC development and maintenance of phenotype. Thus, blocking Kit, by a variety of techniques, caused loss of ICC in experimental animals and demonstrated the critical physiological functions of these cells in gastrointestinal motility. Loss of ICC in human gastrointestinal diseases may contribute to the motor pathologies observed. Unrestrained Kit signalling leads to the transformation of ICC and the development of gastrointestinal stromal tumours. Now ICC-like cells have been identified in a variety of smooth muscle tissues, and the race is on to discover whether these cells have equivalent or even novel functions in organs outside the gastrointestinal tract. This perspectives article gives a short overview of the history of ICC research and directions for future investigation.

Mathematical models in physiology

Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term (modelling OR modeling) yields over 58000 entries in PubMed, with more than 34000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5-6 years, compared to only about one-third in the preceding 5-6 decades. The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function-the Physiome. The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations.

Simulation of Ca2+-activated Cl- current of cardiomyocytes in rabbit pulmonary vein: implications of subsarcolemmal Ca2+ dynamics

In recent studies, we recorded transiently activated outward currents by the application of three-step voltage pulses to induce a reverse mode of Na+-Ca2+ exchange (NCX). We found that these currents were mediated by a Ca2+-activated Cl- current. Based on the recent reports describing the atrial Ca2+ transients, the Ca2+ transient at the subsarcolemmal space was initiated and then diffused into the cytosolic space. Because the myocardium in the pulmonary vein is an extension of the atrium, the Ca2+-activated Cl- current may reflect the subsarcolemmal Ca2+ dynamics. We tried to predict the subsarcolemmal Ca2+ dynamics by simulating these current traces. According to recent reports on the geometry of atrial myocytes, we assumed that there were three compartments of sarcoplasmic reticulum (SR): a network SR, a junctional SR and a central SR. Based on these structures, we also divided the cytosolic space into three compartments: the junctional, subsarcolemmal and cytosolic spaces. Geometry information and cellular capacitance suggested that there were essentially no T-tubules in these cells. The basic physical data, such as the compartmental volumes, the diffusion coefficients and the stability coefficients of the Ca2+ buffers, were obtained from the literature. In the simulation, we incorporated the NCX, the L-type Ca2+ channel, the rapid activating outward rectifier K+ channel, the Na+-K+ pump, the SR Ca2+-pump, the ryanodine receptor, the Ca2+-activated Cl- channel and the dynamics of Na+, K+, Ca2+ and Cl-. In these conditions, we could successfully reconstruct the Ca2+-activated Cl- currents. The simulation allowed estimation of the Ca2+ dynamics of each compartment and the distribution of the Ca2+-activated Cl- channel and the NCX in the sarcolemma on the junctional or subsarcolemmal space.

Modelling Cl- homeostasis and volume regulation of the cardiac cell

We aim at introducing a Cl- homeostasis to the cardiac ventricular cell model (Kyoto model), which includes the sarcomere shortening and the mitochondria oxidative phosphorylation. First, we examined mechanisms underlying the cell volume regulation in a simple model consisting of Na+/K+ pump, Na+-K+-2Cl- cotransporter 1 (NKCC1), cystic fibrosis transmembrane conductance regulator, volume-regulated Cl- channel and background Na+, K+ and Cl- currents. The high intracellular Cl- concentration of approximately 30 mM was achieved by the balance between the secondary active transport via NKCC1 and passive currents. Simulating responses to Na+/K+ pump inhibition revealed the essential role of Na+/K+ pump in maintaining the cellular osmolarity through creating the negative membrane potential, which extrudes Cl- from a cell, confirming the previous model study in the skeletal muscle. In addition, this model well reproduced the experimental data such as the responses to hypotonic shock in the presence or absence of beta-adrenergic stimulation. Finally, the volume regulation via Cl- homeostasis was successfully incorporated to the Kyoto model. The steady state was well established in the comprehensive cell model in respect to both the intracellular ion concentrations and the shape of the action potential, which are all in the physiological range. The source code of the model, which can reproduce every result, is available from http://www.sim-bio.org/.