A mathematical model of plasma membrane electrophysiology and calcium dynamics in vascular endothelial cells

Simulation of Conducted Responses in Microvascular Networks: Role of Gap Junction Current Rectification

OBJECTIVE

Local control of blood flow depends on signaling to arterioles via upstream conducted responses. Here, the objective is to examine how electrical properties of gap junctions between endothelial cells (EC) affect the spread of conducted responses in microvascular networks of the brain cortex, using a theoretical model based on EC electrophysiology.

METHODS

Modeled EC currents are an inward-rectifying potassium current, a non-voltage-dependent potassium current, a leak current, and a gap junction current between adjacent ECs. Effects of varying gap junction conductance are considered, including asymmetric conductance, with higher conductance for forward currents (positive currents from upstream to downstream, based on blood flow direction). The response is initiated by a local increase in extracellular potassium concentration. The model is applied to a 45-segment synthetic network and a 4881-segment network from mouse brain cortex.

RESULTS

The conducted response propagates preferentially to upstream arterioles when the conductance for forward currents is at least 20 times that for backward currents. The response depends strongly on the site of stimulation. With symmetric gap junction conductance, the network acts as a syncytium and the conducted response is dissipated.

CONCLUSIONS

Upstream propagation of conducted responses may depend on the asymmetric conductance of EC gap junctions.

Mathematical Modeling of PI3K/Akt Pathway in Microglia

The motility of microglia involves intracellular signaling pathways that are predominantly controlled by changes in cytosolic Ca2+ and activation of PI3K/Akt (phosphoinositide-3-kinase/protein kinase B). In this letter, we develop a novel biophysical model for cytosolic Ca2+ activation of the PI3K/Akt pathway in microglia where Ca2+ influx is mediated by both P2Y purinergic receptors (P2YR) and P2X purinergic receptors (P2XR). The model parameters are estimated by employing optimization techniques to fit the model to phosphorylated Akt (pAkt) experimental modeling/in vitro data. The integrated model supports the hypothesis that Ca2+ influx via P2YR and P2XR can explain the experimentally reported biphasic transient responses in measuring pAkt levels. Our predictions reveal new quantitative insights into P2Rs on how they regulate Ca2+ and Akt in terms of physiological interactions and transient responses. It is shown that the upregulation of P2X receptors through a repetitive application of agonist results in a continual increase in the baseline [Ca2+], which causes the biphasic response to become a monophasic response which prolongs elevated levels of pAkt.

The conducted vascular response as a mediator of hypercapnic cerebrovascular reactivity: A modelling study

It is well established that the cerebral blood flow (CBF) shows exquisite sensitivity to changes in the arterial blood partial pressure of CO2 ( [Formula: see text] ), which is reflected by an index termed cerebrovascular reactivity. In response to elevations in [Formula: see text] (hypercapnia), the vessels of the cerebral microvasculature dilate, thereby decreasing the vascular resistance and increasing CBF. Due to the challenges of access, scale and complexity encountered when studying the microvasculature, however, the mechanisms behind cerebrovascular reactivity are not fully understood. Experiments have previously established that the cholinergic release of the Acetylcholine (ACh) neurotransmitter in the cortex is a prerequisite for the hypercapnic response. It is also known that ACh functions as an endothelial-dependent agonist, in which the local administration of ACh elicits local hyperpolarization in the vascular wall; this hyperpolarization signal is then propagated upstream the vascular network through the endothelial layer and is coupled to a vasodilatory response in the vascular smooth muscle (VSM) layer in what is known as the conducted vascular response (CVR). Finally, experimental data indicate that the hypercapnic response is more strongly correlated with the CO2 levels in the tissue than in the arterioles. Accordingly, we hypothesize that the CVR, evoked by increases in local tissue CO2 levels and a subsequent local release of ACh, is responsible for the CBF increase observed in response to elevations in [Formula: see text] . By constructing physiologically grounded dynamic models of CBF and control in the cerebral vasculature, ones that integrate the available knowledge and experimental data, we build a new model of the series of signalling events and pathways underpinning the hypercapnic response, and use the model to provide compelling evidence that corroborates the aforementioned hypothesis. If the CVR indeed acts as a mediator of the hypercapnic response, the proposed mechanism would provide an important addition to our understanding of the repertoire of metabolic feedback mechanisms possessed by the brain and would motivate further in-vivo investigation. We also model the interaction of the hypercapnic response with dynamic cerebral autoregulation (dCA), the collection of mechanisms that the brain possesses to maintain near constant CBF despite perturbations in pressure, and show how the dCA mechanisms, which otherwise tend to be overlooked when analysing experimental results of cerebrovascular reactivity, could play a significant role in shaping the CBF response to elevations in [Formula: see text] . Such in-silico models can be used in tandem with in-vivo experiments to expand our understanding of cerebrovascular diseases, which continue to be among the leading causes of morbidity and mortality in humans.

Computational modeling of the variation of the transmembrane potential of the endothelial cells of the blood-brain-barrier subject to an external electric field

The electromechanical properties of the membrane of endothelial cells forming the blood-brain barrier play a vital role in the function of this barrier. The mechanical effect exerted by external electric fields on the membrane could change its electrical properties. In this study the effect of extremely low frequency (ELF) external electric fields on the electrical activity of these cells has been studied by considering the mechanical effect of these fields on the capacitance of the membrane. The effect of time-dependent capacitance of the membrane is incorporated in the current components of the parallel conductance model for the electrical activity of the cells. The results show that the application of ELF electric fields induces hyperpolarization, having an indirect effect on the release of nitric oxide from the endothelial cell and the polymerization of actin filaments. Accordingly, this could play an important role in the permeability of the barrier. Our finding can have possible consequences in the field of drug delivery into the central nervous system.

On the Ca2＋ elevation in vascular endothelial cells due to inositol trisphosphate-sensitive store receptors activation: A data-driven modeling approach

Agonist-induced Ca2+ signaling is essential for the regulation of many vital functions in endothelial cells (ECs). A broad range of stimuli elevate the cytosolic Ca2＋ concentration by promoting a pathway mediated by inositol 1,4,5 trisphosphate (IP3) which causes Ca2＋ release from intracellular stores. Despite its importance, there are very few studies focusing on the quantification of such dynamics in the vascular endothelium. Here, by using data from isolated ECs, we established a minimalistic modeling framework able to quantitatively capture the main features (averaged over a cell population) of the cytosolic Ca2＋ response to different IP3 stimulation levels. A suitable description of Ca2＋-regulatory function of inositol 1,4,5 trisphosphate receptors (IP3Rs) and corresponding parameter space are identified by comparing the different model variants against experimental mean population data. The same approach is used to numerically assess the relevance of cytosolic Ca2＋ buffering, as well as Ca2＋ store IP3-sensitivity in the overall cell dynamics. The variability in the dynamics' features observed across the population can be explained (at least in part) through variation of certain model parameters (such as buffering capacity or Ca2＋ store sensitivity to IP3). The results, in terms of experimental fitting and validation, support the proposed minimalistic model as a reference framework for the quantification of the EC Ca2＋ dynamics induced by IP3Rs activation.

Diversity of cells and signals in the cardiovascular system

This white paper is the outcome of the seventh UC Davis Cardiovascular Research Symposium on Systems Approach to Understanding Cardiovascular Disease and Arrhythmia. This biannual meeting aims to bring together leading experts in subfields of cardiovascular biomedicine to focus on topics of importance to the field. The theme of the 2022 Symposium was 'Cell Diversity in the Cardiovascular System, cell-autonomous and cell-cell signalling'. Experts in the field contributed their experimental and mathematical modelling perspectives and discussed emerging questions, controversies, and challenges in examining cell and signal diversity, co-ordination and interrelationships involved in cardiovascular function. This paper originates from the topics of formal presentations and informal discussions from the Symposium, which aimed to develop a holistic view of how the multiple cell types in the cardiovascular system integrate to influence cardiovascular function, disease progression and therapeutic strategies. The first section describes the major cell types (e.g. cardiomyocytes, vascular smooth muscle and endothelial cells, fibroblasts, neurons, immune cells, etc.) and the signals involved in cardiovascular function. The second section emphasizes the complexity at the subcellular, cellular and system levels in the context of cardiovascular development, ageing and disease. Finally, the third section surveys the technological innovations that allow the interrogation of this diversity and advancing our understanding of the integrated cardiovascular function and dysfunction.

Modeling Reactive Hyperemia to Better Understand and Assess Microvascular Function: A Review of Techniques

Reactive hyperemia is a well-established technique for the non-invasive evaluation of the peripheral microcirculatory function, measured as the magnitude of limb re-perfusion after a brief period of ischemia. Despite widespread adoption by researchers and clinicians alike, many uncertainties remain surrounding interpretation, compounded by patient-specific confounding factors (such as blood pressure or the metabolic rate of the ischemic limb). Mathematical modeling can accelerate our understanding of the physiology underlying the reactive hyperemia response and guide in the estimation of quantities which are difficult to measure experimentally. In this work, we aim to provide a comprehensive guide for mathematical modeling techniques that can be used for describing the key phenomena involved in the reactive hyperemia response, alongside their limitations and advantages. The reported methodologies can be used for investigating specific reactive hyperemia aspects alone, or can be combined into a computational framework to be used in (pre-)clinical settings.

Mathematical modeling of intracellular calcium in presence of receptor: a homeostatic model for endothelial cell

Calcium is a ubiquitous molecule and second messenger that regulates many cellular functions ranging from exocytosis to cell proliferation at different time scales. In the vasculature, a constant adenosine triphosphate (ATP) concentration is maintained because of ATP released by red blood cells (RBCs). These ATP molecules continuously react with purinergic receptors on the surface of endothelial cells (ECs). Consequently, a cascade of chemical reactions are triggered that result in a transient cytoplasmic calcium (Ca[Formula: see text]), followed by return to its basal concentration. The mathematical models proposed in the literature are able to reproduce the transient peak. However, the trailing concentration is always higher than the basal cytoplasmic Ca[Formula: see text] concentrations, and the Ca[Formula: see text] concentration in endoplasmic reticulum (ER) remains lower than its initial concentration. This means that the intracellular homeostasis is not recovered. We propose, herein, a minimal model of calcium kinetics. We find that the desensitization of EC surface receptors due to phosphorylation and recycling plays a vital role in maintaining calcium homeostasis in the presence of a constant stimulus (ATP). The model is able to capture several experimental observations such as refilling of Ca[Formula: see text] in the ER, variation of cytoplasmic Ca[Formula: see text] transient peak in ECs, the resting cytoplasmic Ca[Formula: see text] concentration, the effect of removing ATP from the plasma on Ca[Formula: see text] homeostasis, and the saturation of cytoplasmic Ca[Formula: see text] transient peak with increase in ATP concentration. Direct confrontation with several experimental results is conducted. This work paves the way for systematic studies on coupling between blood flow and chemical signaling, and should contribute to a better understanding of the relation between (patho)physiological conditions and Ca[Formula: see text] kinetics.

Mathematical modelling of human P2X-mediated plasma membrane electrophysiology and calcium dynamics in microglia

Regulation of cytosolic calcium (Ca²⁺) dynamics is fundamental to microglial function. Temporal and spatial Ca²⁺ fluxes are induced from a complicated signal transduction pathway linked to brain ionic homeostasis. In this paper, we develop a novel biophysical model of Ca²⁺ and sodium (Na⁺) dynamics in human microglia and evaluate the contribution of purinergic receptors (P2XRs) to both intracellular Ca²⁺ and Na⁺ levels in response to agonist/ATP binding. This is the first comprehensive model that integrates P2XRs to predict intricate Ca²⁺ and Na⁺ transient responses in microglia. Specifically, a novel compact biophysical model is proposed for the capture of whole-cell patch-clamp currents associated with P2X₄ and P2X₇ receptors, which is composed of only four state variables. The entire model shows that intricate intracellular ion dynamics arise from the coupled interaction between P2X₄ and P2X₇ receptors, the Na⁺/Ca²⁺ exchanger (NCX), Ca²⁺ extrusion by the plasma membrane Ca²⁺ ATPase (PMCA), and Ca²⁺ and Na⁺ leak channels. Both P2XRs are modelled as two separate adenosine triphosphate (ATP) gated Ca²⁺ and Na⁺ conductance channels, where the stoichiometry is the removal of one Ca²⁺ for the hydrolysis of one ATP molecule. Two unique sets of model parameters were determined using an evolutionary algorithm to optimise fitting to experimental data for each of the receptors. This allows the proposed model to capture both human P2X₇ and P2X₄ data (hP2X₇ and hP2X₄). The model architecture enables a high degree of simplicity, accuracy and predictability of Ca²⁺ and Na⁺ dynamics thus providing quantitative insights into different behaviours of intracellular Na⁺ and Ca²⁺ which will guide future experimental research. Understanding the interactions between these receptors and other membrane-bound transporters provides a step forward in resolving the qualitative link between purinergic receptors and microglial physiology and their contribution to brain pathology.

Mathematical modelling and computer simulation are powerful tools by which we can analyse complex biological systems, particularly, neural phenomena involved in brain dysfunction. In this research, we develop a theoretical foundation for studying P2X-mediated calcium and sodium signalling in human microglial cells. Microglia, which are brain-resident macrophages, restructure their intracellular actin cytoskeleton to enable motility; this restructuring requires a complex molecular cascade involving a set of ionic channels, membrane-coupled receptors and cytosolic components. Recent studies highlight the importance for increasing our understanding of microglia physiology, since their functions play critical roles in both normal physiological and pathological dynamics of the brain. There is a need to develop reliable human cellular models to investigate the biology of microglia aimed at understanding the influence of purinergic signalling in brain dysfunction to provide novel drug discovery targets. In this work, a detailed mathematical model is built for the dynamics of human P2XRs in microglia. Subsequently, experimental whole-cell currents are used to derive P2X-mediated electrophysiology of human microglia (i.e. sodium and calcium dynamics, and membrane potential). Our predictions reveal new quantitative insights into P2XRs on how they regulate ionic concentrations in terms of physiological interactions and transient responses.

Local IP3 receptor-mediated Ca2+ signals compound to direct blood flow in brain capillaries

Healthy brain function depends on the finely tuned spatial and temporal delivery of blood-borne nutrients to active neurons via the vast, dense capillary network. Here, using in vivo imaging in anesthetized mice, we reveal that brain capillary endothelial cells control blood flow through a hierarchy of IP3 receptor-mediated Ca2+ events, ranging from small, subsecond protoevents, reflecting Ca2+ release through a small number of channels, to high-amplitude, sustained (up to ~1 min) compound events mediated by large clusters of channels. These frequent (~5000 events/s per microliter of cortex) Ca2+ signals are driven by neuronal activity, which engages Gq protein-coupled receptor signaling, and are enhanced by Ca2+ entry through TRPV4 channels. The resulting Ca2+-dependent synthesis of nitric oxide increases local blood flow selectively through affected capillary branches, providing a mechanism for high-resolution control of blood flow to small clusters of neurons.

Electrical Propagation of Vasodilatory Signals in Capillary Networks

We have developed a computational model to study electrical propagation of vasodilatory signals and arteriolar regulation of blood flow depending on the oxygen tension and agonist distribution in the capillary network. The involving key parameters of endothelial cell-to-cell electrical conductivity and plasma membrane area per unit volume were calibrated with the experimental data on an isolated endothelial tube of mouse skeletal feeding arteries. We have estimated the oxygen saturation parameters in terms of erythrocyte ATP release from the data of a left anterior descending coronary blood perfusion of dog. Regarding the acetylcholine-induced upstream conduction, our model shows that spatially uniform superfusion of acetylcholine attenuates the electrical signal propagation, and blocking calcium-activated potassium channels suppresses that attenuation. On the other hand, a local infusion of acetylcholine induces enhanced electrical propagation that corresponds to physiological relevance. Integrating the electrophysiology of endothelial tube and the electrophysiology/mechanics of a lumped arteriole, we show mechanistically that endothelial purinergic oxygen sensing of ATP released from erythrocytes and local infusion of acetylcholine are individually effective to induce vasodilatory signals to regulate blood flow in arterioles. We have recapitulated the upstream vasomotion in arterioles from the elevated oxygen tension in the downstream capillary domain. This study is a foundation for characterizing effective pharmaceutical strategies for ascending vasodilation and oxygenation.

Connexins: Key Players in the Control of Vascular Plasticity and Function

Of the 21 members of the connexin family, 4 (Cx37, Cx40, Cx43, and Cx45) are expressed in the endothelium and/or smooth muscle of intact blood vessels to a variable and dynamically regulated degree. Full-length connexins oligomerize and form channel structures connecting the cytosol of adjacent cells (gap junctions) or the cytosol with the extracellular space (hemichannels). The different connexins vary mainly with regard to length and sequence of their cytosolic COOH-terminal tails. These COOH-terminal parts, which in the case of Cx43 are also translated as independent short isoforms, are involved in various cellular signaling cascades and regulate cell functions. This review focuses on channel-dependent and -independent effects of connexins in vascular cells. Channels play an essential role in coordinating and synchronizing endothelial and smooth muscle activity and in their interplay, in the control of vasomotor actions of blood vessels including endothelial cell reactivity to agonist stimulation, nitric oxide-dependent dilation, and endothelial-derived hyperpolarizing factor-type responses. Further channel-dependent and -independent roles of connexins in blood vessel function range from basic processes of vascular remodeling and angiogenesis to vascular permeability and interactions with leukocytes with the vessel wall. Together, these connexin functions constitute an often underestimated basis for the enormous plasticity of vascular morphology and function enabling the required dynamic adaptation of the vascular system to varying tissue demands.

The capillary Kir channel as sensor and amplifier of neuronal signals: Modeling insights on K+-mediated neurovascular communication

Neuronal activity leads to an increase in local cerebral blood flow (CBF) to allow adequate supply of oxygen and nutrients to active neurons, a process termed neurovascular coupling (NVC). We have previously shown that capillary endothelial cell (cEC) inwardly rectifying K⁺ (Kir) channels can sense neuronally evoked increases in interstitial K⁺ and induce rapid and robust dilations of upstream parenchymal arterioles, suggesting a key role of cECs in NVC. The requirements of this signal conduction remain elusive. Here, we utilize mathematical modeling to investigate how small outward currents in stimulated cECs can elicit physiologically relevant spread of vasodilatory signals within the highly interconnected brain microvascular network to increase local CBF. Our model shows that the Kir channel can act as an "on-off" switch in cECs to hyperpolarize the cell membrane as extracellular K⁺ increases. A local hyperpolarization can be amplified by the voltage-dependent activation of Kir in neighboring cECs. Sufficient Kir density enables robust amplification of the hyperpolarizing stimulus and produces responses that resemble action potentials in excitable cells. This Kir-mediated excitability can remain localized in the stimulated region or regeneratively propagate over significant distances in the microvascular network, thus dramatically increasing the efficacy of K⁺ for eliciting local hyperemia. Modeling results show how changes in cEC transmembrane current densities and gap junctional resistances can affect K⁺-mediated NVC and suggest a key role for Kir as a sensor of neuronal activity and an amplifier of retrograde electrical signaling in the cerebral vasculature.

Mathematical Model Predicts Effective Strategies to Inhibit VEGF-eNOS Signaling

The endothelial nitric oxide synthase (eNOS) signaling pathway in endothelial cells has multiple physiological significances. It produces nitric oxide (NO), an important vasodilator, and enables a long-term proliferative response, contributing to angiogenesis. This signaling pathway is mediated by vascular endothelial growth factor (VEGF), a pro-angiogenic species that is often targeted to inhibit tumor angiogenesis. However, inhibiting VEGF-mediated eNOS signaling can lead to complications such as hypertension. Therefore, it is important to understand the dynamics of eNOS signaling in the context of angiogenesis inhibitors. Thrombospondin-1 (TSP1) is an important angiogenic inhibitor that, through interaction with its receptor CD47, has been shown to redundantly inhibit eNOS signaling. However, the exact mechanisms of TSP1's inhibitory effects on this pathway remain unclear. To address this knowledge gap, we established a molecular-detailed mechanistic model to describe VEGF-mediated eNOS signaling, and we used the model to identify the potential intracellular targets of TSP1. In addition, we applied the predictive model to investigate the effects of several approaches to selectively target eNOS signaling in cells experiencing high VEGF levels present in the tumor microenvironment. This work generates insights for pharmacologic targets and therapeutic strategies to inhibit tumor angiogenesis signaling while avoiding potential side effects in normal vasoregulation.

Overview of mathematical modeling of myocardial blood flow regulation

The oxygen consumption by the heart and its extraction from the coronary arterial blood are the highest among all organs. Any increase in oxygen demand due to a change in heart metabolic activity requires an increase in coronary blood flow. This functional requirement of adjustment of coronary blood flow is mediated by coronary flow regulation to meet the oxygen demand without any discomfort, even under strenuous exercise conditions. The goal of this article is to provide an overview of the theoretical and computational models of coronary flow regulation and to reveal insights into the functioning of a complex physiological system that affects the perfusion requirements of the myocardium. Models for three major control mechanisms of myogenic, flow, and metabolic control are presented. These explain how the flow regulation mechanisms operating over multiple spatial scales from the precapillaries to the large coronary arteries yield the myocardial perfusion characteristics of flow reserve, autoregulation, flow dispersion, and self-similarity. The review not only introduces concepts of coronary blood flow regulation but also presents state-of-the-art advances and their potential to impact the assessment of coronary microvascular dysfunction (CMD), cardiac-coronary coupling in metabolic diseases, and therapies for angina and heart failure. Experimentalists and modelers not trained in these models will have exposure through this review such that the nonintuitive and highly nonlinear behavior of coronary physiology can be understood from a different perspective. This survey highlights knowledge gaps, key challenges, future research directions, and novel paradigms in the modeling of coronary flow regulation.

Biophysical properties of microvascular endothelium: Requirements for initiating and conducting electrical signals

OBJECTIVE

Electrical signaling along the endothelium underlies spreading vasodilation and blood flow control. We use mathematical modeling to determine the electrical properties of the endothelium and gain insight into the biophysical determinants of electrical conduction.

METHODS

Electrical conduction data along endothelial tubes (40 μm wide, 2.5 mm long) isolated from mouse skeletal muscle resistance arteries were analyzed using cable equations and a multicellular computational model.

RESULTS

Responses to intracellular current injection attenuate with an axial length constant (λ) of 1.2-1.4 mm. Data were fitted to estimate the axial (r_a ; 10.7 MΩ/mm) and membrane (r_m ; 14.5 MΩ∙mm) resistivities, EC membrane resistance (R_m ; 12 GΩ), and EC-EC coupling resistance (R_gj ; 4.5 MΩ) and predict that stimulation of ≥30 neighboring ECs is required to elicit 1 mV of hyperpolarization at distance = 2.5 mm. Opening Ca²⁺ -activated K⁺ channels (K_Ca ) along the endothelium reduced λ by up to 55%.

CONCLUSIONS

High R_m makes the endothelium sensitive to electrical stimuli and able to conduct these signals effectively. Whereas the activation of a group of ECs is required to initiate physiologically relevant hyperpolarization, this requirement is increased by myoendothelial coupling and K_Ca activation along the endothelium inhibits conduction by dissipating electrical signals.

Computer Simulation of TSP1 Inhibition of VEGF-Akt-eNOS: An Angiogenesis Triple Threat

The matricellular protein thrombospondin-1 (TSP1) is a potent inhibitor of angiogenesis. Specifically, TSP1 has been experimentally shown to inhibit signaling downstream of vascular endothelial growth factor (VEGF). The molecular mechanism of this inhibition is not entirely clear. We developed a detailed computational model of VEGF signaling to Akt-endothelial nitric oxide synthase (eNOS) to investigate the quantitative molecular mechanism of TSP1 inhibition. The model demonstrated that TSP1 acceleration of VEGFR2 degradation is sufficient to explain the inhibition of VEGFR2 and eNOS phosphorylation. However, Akt inhibition requires TSP1-induced phosphatase recruitment to VEGFR2. The model was then utilized to test various strategies for the rescue of VEGF signaling to Akt and eNOS. Inhibiting TSP1 was predicted to be not as effective as CD47 depletion in rescuing signaling to Akt. The model further predicts that combination strategy involving depletion of CD47 and inhibition of TSP1 binding to CD47 is necessary for effective recovery of signaling to eNOS. In all, computational modeling offers insight to molecular mechanisms involving TSP1 interaction with VEGF signaling and provides strategies for rescuing angiogenesis by targeting TSP1-CD47 axis.

Calcium and electrical dynamics in lymphatic endothelium

KEY POINTS

Endothelial cell function in resistance arteries integrates Ca2+ signalling with hyperpolarization to promote relaxation of smooth muscle cells and increase tissue blood flow. Whether complementary signalling occurs in lymphatic endothelium is unknown. Intracellular calcium and membrane potential were evaluated in endothelial cell tubes freshly isolated from mouse collecting lymphatic vessels of the popliteal fossa. Resting membrane potential measured using intracellular microelectrodes averaged ∼-70 mV. Stimulation of lymphatic endothelium by acetylcholine or a TRPV4 channel agonist increased intracellular Ca2+ with robust depolarization. Findings from Trpv4-/- mice and with computational modelling suggest that the initial mobilization of intracellular Ca2+ leads to influx of Ca2+ and Na+ through TRPV4 channels to evoke depolarization. Lymphatic endothelial cells lack the Ca2+ -activated K+ channels present in arterial endothelium to generate endothelium-derived hyperpolarization. Absence of this signalling pathway with effective depolarization may promote rapid conduction of contraction along lymphatic muscle during lymph propulsion.

ABSTRACT

Subsequent to a rise in intracellular Ca2+ ([Ca2+ ]i ), hyperpolarization of the endothelium coordinates vascular smooth muscle relaxation along resistance arteries during blood flow control. In the lymphatic vasculature, collecting vessels generate rapid contractions coordinated along lymphangions to propel lymph, but the underlying signalling pathways are unknown. We tested the hypothesis that lymphatic endothelial cells (LECs) exhibit Ca2+ and electrical signalling properties that facilitate lymph propulsion. To study electrical and intracellular Ca2+ signalling dynamics in lymphatic endothelium, we excised collecting lymphatic vessels from the popliteal fossa of mice and removed their muscle cells to isolate intact LEC tubes (LECTs). Intracellular recording revealed a resting membrane potential of ∼-70 mV. Acetylcholine (ACh) increased [Ca2+ ]i with a time course similar to that observed in endothelium of resistance arteries (i.e. rapid initial peak with a sustained 'plateau'). In striking contrast to the endothelium-derived hyperpolarization (EDH) characteristic of arteries, LECs depolarized (>15 mV) to either ACh or TRPV4 channel activation. This depolarization was facilitated by the absence of Ca2+ -activated K+ (KCa ) channels as confirmed with PCR, persisted in the absence of extracellular Ca2+ , was abolished by LaCl3 and was attenuated ∼70% in LECTs from Trpv4-/- mice. Computational modelling of ion fluxes in LECs indicated that omitting K+ channels supports our experimental results. These findings reveal novel signalling events in LECs, which are devoid of the KCa activity abundant in arterial endothelium. Absence of EDH with effective depolarization of LECs may promote the rapid conduction of contraction waves along lymphatic muscle during lymph propulsion.

Inhibition of VEGFR2 Activation and Its Downstream Signaling to ERK1/2 and Calcium by Thrombospondin-1 (TSP1): In silico Investigation

VEGF signaling through VEGFR2 is a central regulator of the angiogenic response. Inhibition of VEGF signaling by the stress-induced matricellular protein TSP1 plays a role in modulating the angiogenic response to VEGF in both health and disease. TSP1 binding to CD47 inhibits VEGFR2 activation. The full implications of this inhibitory interaction are unknown. We developed a detailed rule-based computational model to inquire if TSP1-CD47 signaling through VEGF had downstream effects upon ERK1/2 and calcium. Our Simulations suggest that enhanced degradation of VEGFR2 initiated by the binding of TSP1 to CD47 is sufficient to explain the inhibition of VEGFR2 phosphorylation, calcium elevation, and ERK1/2 activation downstream of VEGF. A complementary mechanism involving the recruitment of phosphatases to the VEGFR2 complex with consequent increase in the rate of receptor dephosphorylation may augment the inhibition of the VEGF signal. The model was then utilized to simulate the effect of inhibiting external TSP1 or the depletion of CD47 as potential therapeutic strategies in restoring VEGF signaling. Results suggest that depleting CD47 is a more efficient strategy in inhibiting the effects of TSP1/CD47 on VEGF signaling. Our results highlight the utility of in silico investigations in elucidating and clarifying molecular mechanisms at the intersection of TSP1 and VEGF biology and in differentiating between competing pro-angiogenic therapeutic strategies relevant to peripheral arterial disease (PAD) and wound healing.

Can endothelial hemoglobin-α regulate nitric oxide vasodilatory signaling?

We used mathematical modeling to investigate nitric oxide (NO)-dependent vasodilatory signaling in the arteriolar wall. Detailed continuum cellular models of calcium (Ca²⁺) dynamics and membrane electrophysiology in smooth muscle and endothelial cells (EC) were coupled with models of NO signaling and biotransport in an arteriole. We used this theoretical approach to examine the role of endothelial hemoglobin-α (Hbα) as a modulator of NO-mediated myoendothelial feedback, as previously suggested in Straub et al. (Nature 491: 473-477, 2012). The model considers enriched expression of inositol 1,4,5-triphosphate receptors (IP₃Rs), endothelial nitric oxide synthase (eNOS) enzyme, Ca²⁺-activated potassium (K_Ca) channels and Hbα in myoendothelial projections (MPs) between the two cell layers. The model suggests that NO-mediated myoendothelial feedback is plausible if a significant percentage of eNOS is localized within or near the myoendothelial projection. Model results show that the ability of Hbα to regulate the myoendothelial feedback is conditional to its colocalization with eNOS near MPs at concentrations in the high nanomolar range (>0.2 μM or 24,000 molecules). Simulations also show that the effect of Hbα observed in in vitro experimental studies may overestimate its contribution in vivo, in the presence of blood perfusion. Thus, additional experimentation is required to quantify the presence and spatial distribution of Hbα in the EC, as well as to test that the strong effect of Hbα on NO signaling seen in vitro, translates also into a physiologically relevant response in vivo.NEW & NOTEWORTHY Mathematical modeling shows that although regulation of nitric oxide signaling by hemoglobin-α (Hbα) is plausible, it is conditional to its presence in significant concentrations colocalized with endothelial nitric oxide synthase in myoendothelial projections. Additional experimentation is required to test that the strong effect of Hbα seen in vitro translates into a physiologically relevant response in vivo.

A mechanistic model of a PDGFRα(+) cell

A novel platelet-derived growth factor receptor alpha-positive cell (PDGFRα(+)) has recently been identified as part of the purinergic inhibitory neural control mechanism in the gastrointestinal (GI) tract. The mechanism through which PDGFRα(+) cells mediate GI muscle relaxation has been found to be associated with the purine receptors P2Y1 and apamin-sensitive SK3 channels that are highly expressed in these cells. This study aims to develop a mechanistic model elucidating a proposed mechanism through which PDGFRα(+) cells contribute to purinergic inhibitory neuromuscular transmission. In accordance with recent experimental findings, the model describes how the binding of neurotransmitters, released from enteric neurons, triggers the release of Ca(2+) from the endoplasmic reticulum in the PDGFRα(+) cells, and how this subsequently leads to large amplitude transient outward currents, which in turn hyperpolarize the cell. The model has been validated against experimental recordings and good agreement was found under normal and pharmacologically-altered conditions. This model demonstrates the feasibility of the proposed mechanism and provides a basis for understanding the mechanism underlying purinergic control of colonic motility.

Stochastic model of endothelial TRPV4 calcium sparklets: effect of bursting and cooperativity on EDH

We examined the endothelial transient receptor vanilloid 4 (TRPV4) channel's vasodilatory signaling using mathematical modeling. The model analyzes experimental data by Sonkusare and coworkers on TRPV4-induced endothelial Ca(2+) events (sparklets). A previously developed continuum model of an endothelial and a smooth muscle cell coupled through microprojections was extended to account for the activity of a TRPV4 channel cluster. Different stochastic descriptions for the TRPV4 channel flux were examined using finite-state Markov chains. The model also took into consideration recent evidence for the colocalization of intermediate-conductance calcium-activated potassium channels (IKCa) and TRPV4 channels near the microprojections. A single TRPV4 channel opening resulted in a stochastic localized Ca(2+) increase in a small region (i.e., few μm(2) area) close to the channel. We predict micromolar Ca(2+) increases lasting for the open duration of the channel sufficient for the activation of low-affinity endothelial KCa channels. Simulations of a cluster of four TRPV4 channels incorporating burst and cooperative gating kinetics provided quantal Ca(2+) increases (i.e., steps of fixed amplitude), similar to the experimentally observed Ca(2+) sparklets. These localized Ca(2+) events result in endothelium-derived hyperpolarization (and SMC relaxation), with magnitude that depends on event frequency. The gating characteristics (bursting, cooperativity) of the TRPV4 cluster enhance Ca(2+) spread and the distance of KCa channel activation. This may amplify the EDH response by the additional recruitment of distant KCa channels.

On the roles of vascular smooth muscle contraction in cerebral blood flow autoregulation - a modeling perspective

We here review existing models of vascular smooth muscle cell, endothelial cell and cell-cell communication, which have been developed to better understand vascular tone and blood flow autoregulation. In particular, we discuss models that intended to explain modulation of myogenic tone by intraluminal pressure in resistance arterioles. Modeling efforts in the recent past have witnessed a shift from empirical models to models with mechanistic details that underscore different physical aspects of vascular regulation. Future models should synthesize mechanistic interactions in a hierarchy, from molecular regulation of ion channels to whole organ blood flow control.

Polytrichum commune L.ex Hedw ethyl acetate extract-triggered perturbations in intracellular Ca²⁺ homeostasis regulates mitochondrial-dependent apoptosis

ETHNOPHARMACOLOGICAL RELEVANCE

Polytrichum commune L.ex Hedw (PCLH), a moss of Bryopsida, has been used as a traditional Chinese medicine and shown to possess anticancer activities. Previous studies have indicated its anti-leukemia effect but the potential mechanisms have not been fully explained.

AIM OF THE STUDY

The present study aimed to further investigate the efficacy of PCLH ethyl acetate fraction (PC-EEF) and the associated mechanisms in human leukemia cells.

MATERIALS AND METHODS

Phytochemical analysis of PC-EEF was performed by spectrophotometry and HPLC. MTT analysis and trypan blue exclusion assay were adopted to examine its cytotoxicity on a panel of leukemia cells (K562, U937, HL-60 and K562/DOX cells) and non-cancerous cells (human PBMCs). Anti-proliferative effect was monitored by colony formation assay and EdU incorporation assay. Ultrastructural alterations on K562 cell membrane surface were observed by scanning electron microscopy. Changes on plasma membrane integrity, cell membrane potential, mitochondrial membrane potential and apoptosis were analyzed by flow cytometry. Fluorescence microscope was performed to assess [Ca(2+)]i level, mitochondrial injury and cytochrome c release. Apoptosis-associated protein expression was analyzed by western blot. The role of Ca(2+) in PC-EEF-induced cell death was investigated by Ca(2+) chelating reagent BAPTA-AM.

RESULTS

PC-EEF possessed relatively high flavonoid content (about 88.84 ± 0.89%) and showed significant cytotoxicity to human leukemia cells. PC-EEF could cause obvious cell morphological deformation, membrane integrity loss and membrane depolarization. Meanwhile, PC-EEF treatment could dramatically potentiate perturbations in cellular Ca(2+) homeostasis. Subsequently, mitochondrial membrane potential (MMP) collapse, cytochrome c release and Bcl-2/Bax down-regulation were all observed. Consistent with these results, PC-EEF treatment resulted in significant activation of caspase 3, poly (ADP-ribose) polymerase (PARP) degradation and apoptosis. Moreover, PC-EEF-caused cytotoxicity, membrane damage, mitochondrial injury and apoptosis were remarkably reversed by BAPTA-AM.

CONCLUSIONS

PC-EEF damaged the membrane system and triggered Ca(2+)-dependent mitochondrial apoptosis, which may provide some new insights into its efficacy against human leukemia cells.

Gap junctions suppress electrical but not [Ca(2+)] heterogeneity in resistance arteries

Despite stochastic variation in the molecular composition and morphology of individual smooth muscle and endothelial cells, the membrane potential along intact microvessels is remarkably uniform. This is crucial for coordinated vasomotor responses. To investigate how this electrical homogeneity arises, a virtual arteriole was developed that introduces variation in the activities of ion-transport proteins between cells. By varying the level of heterogeneity and subpopulations of gap junctions (GJs), the resulting simulations shows that GJs suppress electrical variation but can only reduce cytosolic [Ca(2+)] variation. The process of electrical smoothing, however, introduces an energetic cost due to permanent currents, one which is proportional to the level of heterogeneity. This cost is particularly large when electrochemically different endothelial-cell and smooth-muscle-cell layers are coupled. Collectively, we show that homocellular GJs in a passively open state are crucial for electrical uniformity within the given cell layer, but homogenization may be limited by biophysical or energetic constraints. Owing to the ubiquitous presence of ion transport-proteins and cell-cell heterogeneity in biological tissues, these findings generalize across most biological fields.

Measurement of shear stress-mediated intracellular calcium dynamics in human dermal lymphatic endothelial cells

The shear stress applied to lymphatic endothelial cells (LEC) by lymph flow changes dramatically under normal conditions as well as in response to disease conditions and immune reactions. In general, LEC are known to regulate the contraction frequency and strength of lymphatic pumping in response to shear stress. Intracellular calcium concentration ([Ca(2+)]i) is an important factor that regulates lymphatic contraction characteristics. In this study, we measured changes in the [Ca(2+)]i under different shear stress levels and determined the source of this calcium signal. Briefly, human dermal LEC were cultured in custom-made microchannels for 3 days before loading with 2 µM fura-2 AM, a ratiometric calcium dye to measure [Ca(2+)]i. Step changes in shear stress resulted in a rapid increase in [Ca(2+)]i followed by a gradual return to the basal level and sometimes below the initial baseline (45.2 ± 2.2 nM). The [Ca(2+)]i reached a peak at 126.2 ± 5.6 nM for 10 dyn/cm(2) stimulus, whereas the peak was only 71.8 ± 5.4 nM for 1 dyn/cm(2) stimulus, indicating that the calcium signal depends on the magnitude of shear stress. Removal of the extracellular calcium from the buffer or pharmocological blockade of calcium release-activated calcium (CRAC) channels significantly reduced the peak [Ca(2+)]i, demonstrating a role of extracellular calcium entry. Inhibition of endoplasmic reticulum (ER) calcium pumps showed the importance of intracellular calcium stores in the initiation of this signal. In conclusion, we demonstrated that the shear-mediated calcium signal is dependent on the magnitude of the shear and involves ER store calcium release and extracellular calcium entry.

Conducted vasoreactivity: the dynamical point of view

Conducted vasodilation is part of the physiological response to increasing metabolic demand of the tissue. Similar responses can be elicited by focal electrical or chemical stimulation. Some evidence suggests an endothelial pathway for nondecremental transmission of hyperpolarizing pulses. However, the underlying mechanisms are debated. Here, we focus on dynamical aspects of the problem hypothesizing the existence of a bistability-powered mechanism for regenerative pulse transmission along the endothelium. Bistability implies that the cell can have two different stable resting potentials and can switch between those states following an appropriate stimulus. Bistability is possible if the current–voltage curve is N shaped instead of monotonically increasing. Specifically, the presence of an inwardly rectifying potassium current may provide the endothelial cell with such properties. We provide a theoretical analysis as well as numerical simulations of both single- and multiunit bistable systems mimicking endothelial cells to investigate the self-consistence and stability of the proposed mechanism. We find that the individual cell may switch readily between two stable potentials. An array of coupled cells, however, as found in the vascular wall, requires a certain adaptation of the membrane currents after a switch, in order to switch back. Although the formulation is generic, we suggest a combination of specific membrane currents that could underlie the phenomenon.

Origins of variation in conducted vasomotor responses

Regulation of blood flow in the microcirculation depends on synchronized vasomotor responses. The vascular conducted response is a synchronous dilatation or constriction, elicited by a local electrical event that spreads along the vessel wall. Despite the underlying electrical nature, however, the efficacy of conducted responses varies significantly between different initiating stimuli within the same vascular bed as well as between different vascular beds following the same stimulus. The differences have stimulated proposals of different mechanisms to account for the experimentally observed variation. Using a computational approach that allows for introduction of structural and electrophysiological heterogeneity, we systematically tested variations in both arteriolar electrophysiology and modes of stimuli. Within the same vessel, our simulations show that conduction efficacy is influenced by the type of cell being stimulated and, in case of depolarization, by the stimulation strength. Particularly, simultaneous stimulation of both endothelial and vascular smooth muscle cells augments conduction. Between vessels, the specific electrophysiology determines membrane resistance and conduction efficiency-notably depolarization or radial currents reduce electrical spread. Random cell-cell variation, ubiquitous in biological systems, only cause small or no reduction in conduction efficiency. Collectively, our simulations can explain why CVRs from hyperpolarizing stimuli tend to conduct longer than CVRs from depolarizing stimuli and why agonists like acetylcholine induce CVRs that tend to conduct longer than electrical injections. The findings demonstrate that although substantial heterogeneity is observed in conducted responses, it can be largely ascribed to the origin of electrical stimulus combined with the specific electrophysiological properties of the arteriole. We conclude by outlining a set of "principles of electrical conduction" in the microcirculation.

Less is more: minimal expression of myoendothelial gap junctions optimizes cell-cell communication in virtual arterioles

Dysfunctional electrical signalling within the arteriolar wall is a major cause of cardiovascular disease. The endothelial cell layer constitutes the primary electrical pathway, co-ordinating contraction of the overlying smooth muscle cell (SMC) layer. As myoendothelial gap junctions (MEGJs) provide direct contact between the cell layers, proper vasomotor responses are thought to depend on a high, uniform MEGJ density. However, MEGJs are observed to be expressed heterogeneously within and among vascular beds. This discrepancy is addressed in the present study. As no direct measures of MEGJ conductance exist, we employed a computational modelling approach to vary the number, conductance and distribution of MEGJs. Our simulations demonstrate that a minimal number of randomly distributed MEGJs augment arteriolar cell-cell communication by increasing conduction efficiency and ensuring appropriate membrane potential responses in SMCs. We show that electrical coupling between SMCs must be tailored to the particular MEGJ distribution. Finally, observation of non-decaying mechanical conduction in arterioles without regeneration has been a long-standing controversy in the microvascular field. As heterogeneous MEGJ distributions provide for different conduction profiles along the cell layers, we demonstrate that a non-decaying conduction profile is possible in the SMC layer of a vessel with passive electrical properties. These intriguing findings redefine the concept of efficient electrical communication in the microcirculation, illustrating how heterogeneous properties, ubiquitous in biological systems, may have a profound impact on system behaviour and how acute local and global flow control is explained from the biophysical foundations.

Role of microprojections in myoendothelial feedback--a theoretical study

We investigated the role of myoendothelial projections (MPs) in endothelial cell (EC) feedback response to smooth muscle cell (SMC) stimulation using mathematical modelling. A previously developed compartmental EC-SMC model is modified to include MPs as subcellular compartments in the EC. The model is further extended into a 2D continuum model using a finite element method (FEM) approach and electron microscopy images to account for MP geometry. The EC and SMC are coupled via non-selective myoendothelial gap junctions (MEGJs) which are located on MPs and allow exchange of Ca(2+), K(+), Na(+) and Cl(-) ions and inositol 1,4,5-triphosphate (IP3). Models take into consideration recent evidence for co-localization of intermediate-conductance calcium-activated potassium channels (IKCa) and IP3 receptors (IP3Rs) in the MPs. SMC stimulation causes an IP3-mediated Ca(2+) transient in the MPs with limited global spread in the bulk EC. A hyperpolarizing feedback generated by the localized IKCa channels is transmitted to the SMC via MEGJs. MEGJ resistance (Rgj) and the density of IKCa and IP3R in the projection influence the extent of EC response to SMC stimulation. The predicted Ca(2+) transients depend also on the volume and geometry of the MP. We conclude that in the myoendothelial feedback response to SMC stimulation, MPs are required to amplify the SMC initiated signal. Simulations suggest that the signal is mediated by IP3 rather than Ca(2+) diffusion and that a localized rather than a global EC Ca(2+) mobilization is more likely following SMC stimulation.

Intercellular communication in the vascular wall: a modeling perspective

Movement of ions (Ca(2+) , K(+) , Na(+) , and Cl(-) ) and second messenger molecules like inositol 1, 4, 5-trisphosphate inside and in between different cells is the basis of many signaling mechanisms in the microcirculation. In spite of the vast experimental efforts directed toward evaluation of these fluxes, it has been a challenge to establish their roles in many essential microcirculatory phenomena. Recently, detailed theoretical models of calcium dynamics and plasma membrane electrophysiology have emerged to assist in the quantification of these intra and intercellular fluxes and enhance understanding of their physiological importance. This perspective reviews selected models relevant to estimation of such intra and intercellular ionic and second messenger fluxes and prediction of their relative significance to a variety of vascular phenomena, such as myoendothelial feedback, conducted responses, and vasomotion.

Long distance conduction of vasodilation: a passive or regenerative process?

The mechanism enabling coordination of the resistance of feed arteries with microcirculatory arterioles to rapidly regulate tissue blood flow in line with changes in metabolic demand has preoccupied scientists for a quarter of a century. As experiments uncovered the underlying electrical events, it was frequently questioned how vasodilation could conduct over long distances without appreciable attenuation. This perspective reviews the data pertinent to this phenomenon and provides evidence that this remarkable response could be made possible by a simple mechanism based on the steep relationship between membrane potential and calcium entry demonstrated by the voltage-dependent calcium channels which mediate the control of vascular tone in vivo.

Multiscale modeling and data integration in the virtual physiological rat project

It has become increasingly evident that the descriptions of many complex diseases are only possible by taking into account multiple influences at different physiological scales. To do this with computational models often requires the integration of several models that have overlapping scales (genes to molecules, molecules to cells, cells to tissues). The Virtual Physiological Rat (VPR) Project, a National Institute of General Medical Sciences (NIGMS) funded National Center of Systems Biology, is tasked with mechanistically describing several complex diseases and is therefore identifying methods to facilitate the process of model integration across physiological scales. In addition, the VPR has a considerable experimental component and the resultant data must be integrated into these composite multiscale models and made available to the research community. A perspective of the current state of the art in model integration and sharing along with archiving of experimental data will be presented here in the context of multiscale physiological models. It was found that current ontological, model and data repository resources and integrative software tools are sufficient to create composite models from separate existing models and the example composite model developed here exhibits emergent behavior not predicted by the separate models.

Multiple factors influence calcium synchronization in arterial vasomotion

The intercellular synchronization of spontaneous calcium (Ca(2+)) oscillations in individual smooth muscle cells is a prerequisite for vasomotion. A detailed mathematical model of Ca(2+) dynamics in rat mesenteric arteries shows that a number of synchronizing and desynchronizing pathways may be involved. In particular, Ca(2+)-dependent phospholipase C, the intercellular diffusion of inositol trisphosphate (IP(3), and to a lesser extent Ca(2+)), IP(3) receptors, diacylglycerol-activated nonselective cation channels, and Ca(2+)-activated chloride channels can contribute to synchronization, whereas large-conductance Ca(2+)-activated potassium channels have a desynchronizing effect. Depending on the contractile state and agonist concentrations, different pathways become predominant, and can be revealed by carefully inhibiting the oscillatory component of their total activity. The phase shift between the Ca(2+) and membrane potential oscillations can change, and thus electrical coupling through gap junctions can mediate either synchronization or desynchronization. The effect of the endothelium is highly variable because it can simultaneously enhance the intercellular coupling and affect multiple smooth muscle cell components. Here, we outline a system of increased complexity and propose potential synchronization mechanisms that need to be experimentally tested.

The nanostructure of myoendothelial junctions contributes to signal rectification between endothelial and vascular smooth muscle cells

Micro-anatomical structures in tissues have potential physiological effects. In arteries and arterioles smooth muscle cells and endothelial cells are separated by the internal elastic lamina, but the two cell layers often make contact through micro protrusions called myoendothelial junctions. Cross talk between the two cell layers is important in regulating blood pressure and flow. We have used a spatiotemporal mathematical model to investigate how the myoendothelial junctions affect the information flow between the two cell layers. The geometry of the model mimics the structure of the two cell types and the myoendothelial junction. The model is implemented as a 2D axi-symmetrical model and solved using the finite element method. We have simulated diffusion of Ca²⁺ and IP₃ between the two cell types and we show that the micro-anatomical structure of the myoendothelial junction in itself may rectify a signal between the two cell layers. The rectification is caused by the asymmetrical structure of the myoendothelial junction. Because the head of the myoendothelial junction is separated from the cell it is attached to by a narrow neck region, a signal generated in the neighboring cell can easily drive a concentration change in the head of the myoendothelial protrusion. Subsequently the signal can be amplified in the head, and activate the entire cell. In contrast, a signal in the cell from which the myoendothelial junction originates will be attenuated and delayed in the neck region as it travels into the head of the myoendothelial junction and the neighboring cell.

Modeling Ca2+ signaling in the microcirculation: intercellular communication and vasoreactivity

A network of intracellular signaling pathways and complex intercellular interactions regulate calcium mobilization in vascular cells, arteriolar tone, and blood flow. Different endothelium-derived vasoreactive factors have been identified and the importance of myoendothelial communication in vasoreactivity is now well appreciated. The ability of many vascular networks to conduct signals upstream also is established. This phenomenon is critical for both short-term changes in blood perfusion as well as long-term adaptations of a vascular network. In addition, in a phenomenon termed vasomotion, arterioles often exhibit spontaneous oscillations in diameter. This is thought to improve tissue oxygenation and enhance blood flow. Experimentation has begun to reveal important aspects of the regulatory machinery and the significance of these phenomena for the regulation of local perfusion and oxygenation. Mathematical modeling can assist in elucidating the complex signaling mechanisms that participate in these phenomena. This review highlights some of the important experimental studies and relevant mathematical models that provide the current understanding of these mechanisms in vasoreactivity.

Nitric oxide signaling in the microcirculation

Several apparent paradoxes are evident when one compares mathematical predictions from models of nitric oxide (NO) diffusion and convection in vasculature structures with experimental measurements of NO (or related metabolites) in animal and human studies. Values for NO predicted from mathematical models are generally much lower than in vivo NO values reported in the literature for experiments, specifically with NO microelectrodes positioned at perivascular locations next to different sizes of blood vessels in the microcirculation and NO electrodes inserted into a wide range of tissues supplied by the microcirculation of each specific organ system under investigation. There continues to be uncertainty about the roles of NO scavenging by hemoglobin versus a storage function that may conserve NO, and other signaling targets for NO need to be considered. This review describes model predictions and relevant experimental data with respect to several signaling pathways in the microcirculation that involve NO.

Applicability of cable theory to vascular conducted responses

Conduction processes in the vasculature have traditionally been described using cable theory, i.e., locally induced signals decaying passively along the arteriolar wall. The decay is typically quantified using the steady-state length-constant, λ, derived from cable theory. However, the applicability of cable theory to blood vessels depends on assumptions that are not necessarily fulfilled in small arteries and arterioles. We have employed a morphologically and electrophysiologically detailed mathematical model of a rat mesenteric arteriole to investigate if the assumptions hold and whether λ adequately describes simulated conduction profiles. We find that several important cable theory assumptions are violated when applied to small blood vessels. However, the phenomenological use of a length-constant from a single exponential function is a good measure of conduction length. Hence, λ should be interpreted as a descriptive measure and not in light of cable theory. Determination of λ using cable theory assumes steady-state conditions. In contrast, using the model it is possible to probe how conduction behaves before steady state is achieved. As ion channels have time-dependent activation and inactivation, the conduction profile changes considerably during this dynamic period with an initially longer spread of current. This may have implications in relation to explaining why different agonists have different conduction properties. Also, it illustrates the necessity of using and developing models that handle the nonlinearity of ion channels.

BKCa and KV channels limit conducted vasomotor responses in rat mesenteric terminal arterioles

Intracellular Ca(2+) signals underlying conducted vasoconstriction to local application of a brief depolarizing KCl stimulus was investigated in rat mesenteric terminal arterioles (<40 μm). Using a computer model of an arteriole segment comprised of coupled endothelial cells (EC) and vascular smooth muscle cells (VSMC) simulations of both membrane potential and intracellular [Ca(2+)] were performed. The "characteristic" length constant, λ, was approximated using a modified cable equation in both experiments and simulations. We hypothesized that K(+) conductance in the arteriolar wall limit the electrotonic spread of a local depolarization along arterioles by current dissipation across the VSMC plasma membrane. Thus, we anticipated an increased λ by inhibition of voltage-activated K(+) channels. Application of the BK(Ca) channel blocker iberiotoxin (100 nM) onto mesenteric arterioles in vitro and inhibition of BK(Ca) channel current in silico increased λ by 34% and 32%, respectively. Similarly, inhibition of K(V) channels in vitro (4-aminopyridine, 1 mM) or in silico increased λ by 41% and 21%, respectively. Immunofluorescence microscopy demonstrated expression of BK(Ca), Kv1.5, Kv2.1, but not Kv1.2, in VSMCs of rat mesenteric terminal arterioles. Our results demonstrate that inhibition of voltage-activated K(+) channels enhance vascular-conducted responses to local depolarization in terminal arterioles by increasing the membrane resistance of VSMCs. These data contribute to our understanding of how differential expression patterns of voltage-activated K(+) channels may influence conducted vasoconstriction in small arteriolar networks. This finding is potentially relevant to understanding the compromised microcirculatory blood flow in systemic vascular diseases such as diabetes mellitus and hypertension.

Multiscale FEM modeling of vascular tone: from membrane currents to vessel mechanics

Regulation of vascular tone is a complex process that remains poorly understood. Here, we present our recent efforts for the development of physiologically realistic models of arterial segments for the analysis of vasoreactivity in health and disease. Multiscale modeling integrates intracellular and cell membrane components into whole-cell models of calcium and membrane potential dynamics. Single-cell models of vascular cells are combined into a multicellular model of the vascular wall, and vessel wall biomechanics are integrated with calcium dynamics in the smooth muscle layer. At each scale, continuum models using finite element method can account for spatial heterogeneity in calcium signaling and for nonuniform deformations of a vessel segment. The outlined approach can be used to investigate cellular mechanisms underlying altered vasoreactivity in hypertension.

Fluid flow induced calcium response in osteoblasts: mathematical modeling

Fluid flow in the bone lacuno-canalicular network can induce dynamic fluctuation of intracellular calcium concentration ([Ca(2+)](i)) in osteoblasts, which plays an important role in bone remodeling. There has been limited progress in the mathematical modeling of this process probably due to its complexity, which is controlled by various factors such as Ca(2+) channels and extracellular messengers. In this study we developed a mathematical model to describe [Ca(2+)](i) response induced by fluid shear stress (SS) by integrating the major factors involved and analyzed the effects of different experimental setups (e.g. [Ca(2+)](i) baseline, pretreatment with ATP). In this model we considered the ATP release process and the activities of multiple ion channels and purinergic receptors. The model was further verified quantitatively by comparing the simulation results with experimental data reported in literature. The results showed that: (i) extracellular ATP concentration has more significant effect on [Ca(2+)](i) baseline (73% increase in [Ca(2+)](i) with extracellular ATP concentration varying between 0 and 10 μM), as compared to that induced by SS (25% variation in [Ca(2+)](i) with SS varying from 0 to 3.5 Pa); (ii) Pretreatment with ATP-medium results in different [Ca(2+)](i) response as compared to the control group (ATP-free medium) under SS; (iii) Relative [Ca(2+)](i) fluctuation over baseline is more reliable to show the [Ca(2+)](i) response process than the absolute [Ca(2+)](i) response peak. The developed model may improve the experimental design and facilitate our understanding of the mechanotransduction process in osteoblasts.

Calcium dynamics and signaling in vascular regulation: computational models

Calcium is a universal signaling molecule with a central role in a number of vascular functions including in the regulation of tone and blood flow. Experimentation has provided insights into signaling pathways that lead to or affected by Ca(2+) mobilization in the vasculature. Mathematical modeling offers a systematic approach to the analysis of these mechanisms and can serve as a tool for data interpretation and for guiding new experimental studies. Comprehensive models of calcium dynamics are well advanced for some systems such as the heart. This review summarizes the progress that has been made in modeling Ca(2+) dynamics and signaling in vascular cells. Model simulations show how Ca(2+) signaling emerges as a result of complex, nonlinear interactions that cannot be properly analyzed using only a reductionist's approach. A strategy of integrative modeling in the vasculature is outlined that will allow linking macroscale pathophysiological responses to the underlying cellular mechanisms.

A mathematical model of vasoreactivity in rat mesenteric arterioles: I. Myoendothelial communication

To study the effect of myoendothelial communication on vascular reactivity, we integrated detailed mathematical models of Ca(2+) dynamics and membrane electrophysiology in arteriolar smooth muscle (SMC) and endothelial (EC) cells. Cells are coupled through the exchange of Ca(2+), Cl(-), K(+), and Na(+) ions, inositol 1,4,5-triphosphate (IP(3)), and the paracrine diffusion of nitric oxide (NO). EC stimulation reduces intracellular Ca(2+) ([Ca(2+)](i)) in the SMC by transmitting a hyperpolarizing current carried primarily by K(+). The NO-independent endothelium-derived hyperpolarization was abolished in a synergistic-like manner by inhibition of EC SK(Ca) and IK(Ca) channels. During NE stimulation, IP(3) diffusing from the SMC induces EC Ca(2+) release, which, in turn, moderates SMC depolarization and [Ca(2+)](i) elevation. On the contrary, SMC [Ca(2+)](i) was not affected by EC-derived IP(3). Myoendothelial Ca(2+) fluxes had no effect in either cell. The EC exerts a stabilizing effect on calcium-induced calcium release-dependent SMC Ca(2+) oscillations by increasing the norepinephrine concentration window for oscillations. We conclude that a model based on independent data for subcellular components can capture major features of the integrated vessel behavior. This study provides a tissue-specific approach for analyzing complex signaling mechanisms in the vasculature.

A mathematical model of vasoreactivity in rat mesenteric arterioles. II. Conducted vasoreactivity

This study presents a multicellular computational model of a rat mesenteric arteriole to investigate the signal transduction mechanisms involved in the generation of conducted vasoreactivity. The model comprises detailed descriptions of endothelial (ECs) and smooth muscle (SM) cells (SMCs), coupled by nonselective gap junctions. With strong myoendothelial coupling, local agonist stimulation of the EC or SM layer causes local changes in membrane potential (V(m)) that are conducted electrotonically, primarily through the endothelium. When myoendothelial coupling is weak, signals initiated in the SM conduct poorly, but the sensitivity of the SMCs to current injection and agonist stimulation increases. Thus physiological transmembrane currents can induce different levels of local V(m) change, depending on cell's gap junction connectivity. The physiological relevance of current and voltage clamp stimulations in intact vessels is discussed. Focal agonist stimulation of the endothelium reduces cytosolic calcium (intracellular Ca(2+) concentration) in the prestimulated SM layer. This SMC Ca(2+) reduction is attributed to a spread of EC hyperpolarization via gap junctions. Inositol (1,4,5)-trisphosphate, but not Ca(2+), diffusion through homocellular gap junctions can increase intracellular Ca(2+) concentration in neighboring ECs. The small endothelial Ca(2+) spread can amplify the total current generated at the local site by the ECs and through the nitric oxide pathway, by the SMCs, and thus reduces the number of stimulated cells required to induce distant responses. The distance of the electrotonic and Ca(2+) spread depends on the magnitude of SM prestimulation and the number of SM layers. Model results are consistent with experimental data for vasoreactivity in rat mesenteric resistance arteries.

A mathematical model of Ca2+ dynamics in rat mesenteric smooth muscle cell: agonist and NO stimulation

A mathematical model of calcium dynamics in vascular smooth muscle cell (SMC) was developed based on data mostly from rat mesenteric arterioles. The model focuses on (a) the plasma membrane electrophysiology; (b) Ca2+ uptake and release from the sarcoplasmic reticulum (SR); (c) cytosolic balance of Ca2+, Na+, K+, and Cl ions; and (d) IP3 and cGMP formation in response to norepinephrine(NE) and nitric oxide (NO) stimulation. Stimulation with NE induced membrane depolarization and an intracellular Ca2+ ([Ca2+]i) transient followed by a plateau. The plateau concentrations were mostly determined by the activation of voltage-operated Ca2+ channels. NE causes a greater increase in [Ca2+]i than stimulation with KCl to equivalent depolarization. Model simulations suggest that the effect of[Na+]i accumulation on the Na+/Ca2+ exchanger (NCX) can potentially account for this difference.Elevation of [Ca2+]i within a concentration window (150-300 nM) by NE or KCl initiated [Ca2+]i oscillations with a concentration-dependent period. The oscillations were generated by the nonlinear dynamics of Ca2+ release and refilling in the SR. NO repolarized the NE-stimulated SMC and restored low [Ca2+]i mainly through its effect on Ca2+-activated K+ channels. Under certain conditions, Na+-K+-ATPase inhibition can result in the elevation of [Na+]i and the reversal of NCX, increasing resting cytosolic and SR Ca2+ content, as well as reactivity to NE. Blockade of the NCX's reverse mode could eliminate these effects. We conclude that the integration of the selected cellular components yields a mathematical model that reproduces, satisfactorily, some of the established features of SMC physiology. Simulations suggest a potential role of intracellular Na+ in modulating Ca2+ dynamics and provide insights into the mechanisms of SMC constriction, relaxation, and the phenomenon of vasomotion. The model will provide the basis for the development of multi-cellular mathematical models that will investigate microcirculatory function in health and disease.

Nitric oxide in the vasculature: where does it come from and where does it go? A quantitative perspective

Nitric oxide (NO) affects two key aspects of O2 supply and demand: It regulates vascular tone and blood flow by activating soluble guanylate cyclase (sGC) in the vascular smooth muscle, and it controls mitochondrial O2 consumption by inhibiting cytochrome c oxidase. However, significant gaps exist in our quantitative understanding of the regulation of NO production in the vascular region. Large apparent discrepancies exist among the published reports that have analyzed the various pathways in terms of the perivascular NO concentration, the efficacy of NO in causing vasodilation (EC50), its efficacy in tissue respiration (IC50), and the paracrine and endocrine NO release. In this study, we review the NO literature, analyzing NO levels on various scales, identifying and analyzing the discrepancies in the reported data, and proposing hypotheses that can potentially reconcile these discrepancies. Resolving these issues is highly relevant to improving our understanding of vascular biology and to developing pharmaceutical agents that target NO pathways, such as vasodilating drugs.