Calcium dynamics and signaling in vascular regulation: computational models

Abstract

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.

FIGURE 1

Models of calcium dynamics in ECs. (a) A model introduced by Wong and Klassen. The model contained an internal calcium store that was divided into a superficial (sc) and deep (dc) compartments and four transmembrane currents: a sodium current (I_Na), a potassium current (I_K), a Ca²⁺-activated potassium current (I_K,Ca), and a stretch-activated calcium current (I_SA). (b) A calcium dynamics model in human umbilical vein endothelial cells (HUVECS) introduced by Wiesner and coworkers. The model includes descriptions for CICR, CCE, kinetics for IP₃ formations following thrombin receptor activation, and Ca²⁺ buffering. Formulations for the NCX and the PMCA pump were also included. (c) A model of endothelial cell electrophysiology presented by Schuster and coworkers (reprinted with permission from Ref . Copyright 2003 Springer). Model focuses on K⁺ currents from bradykinin-sensitive channels (i.e., the large-conductance BKCa channel, the apamin-insensitive small-conductance SKCa channel, and an NSC channel) and the apamin-sensitive SKCa channel that is not gated by bradykinin. (d) An endothelial cell electrophysiology/ionic dynamics model introduced by Silva et al. Model contains several transmembrane currents from: store-operated Ca²⁺ channels (SOC); nonselective cation (NSC) channels; voltage-regulated anion channel (VRAC); Ca²⁺-activated Cl^– channels (CACC); inward rectifier K⁺ channels; Ca²⁺-activated K⁺ channels (small-conductance SKCa, intermediate-conductance IKCa); Na⁺-K⁺-ATPase (NaK) pumps; plasma membrane Ca²⁺-ATPase (PMCA) pumps; Na⁺/Ca²⁺ exchanger (NCX). The model accounts for changes in membrane potential (V_m) and in the intracellular concentrations of the four main ionic species following agonist induced IP₃ release.

FIGURE 2

Models of calcium dynamics in SMCs. (a) The Wong and Klassen model of Ca²⁺ dynamics and membrane's electrical behavior in vascular smooth muscle (reproduced with permission from Ref . Copyright 1993 Elsevier). The model includes two intracellular stores. A-store is an IP₃-sensitive store. C-store is replenished by extracellular Ca²⁺. J_eff and J_D are the active and passive transmembrane Ca²⁺ efflux; J_p is the Ca²⁺ uptake rate of the A-store; I_Ca is the Ca²⁺ current through the voltage-operated channel; I_in is the inward current through the receptor-operated channel; I_K,Ca and I_K are the calcium and voltage-dependent K⁺ channels. (b) Minimal model reproduced from Parthimos and coworkers. Model contains two independent internal stores [a ryanodine-sensitive (CICR) and inositol 1,4,5-trisphosphate (InsP3) sensitive]. Membrane component contains descriptions for: transmembrane currents for Cl^–, K⁺ channels; voltage- and receptor-operated Ca²⁺ channels (VOCCs and ROCCs, respectively); Na⁺-K⁺-ATPase, and Ca²⁺-ATPase pumps; NCX. (c) Detailed model reproduced with permission from Ref . Copyright 2003 Elsevier. Membrane model (upper panel) describing ionic membrane currents and transmembrane potential. Black and white indicates resistance with voltage-dependent nonlinearity, and the all white resistor indicates linear element. Fluid compartment model (lower panel) describes ionic dynamics, Ca²⁺ buffering, and Ca²⁺ handling by sarcoplasmic reticulum (SR). The SR is divided into a ryonadine-sensitive release compartment and an uptake compartment containing SERCA pumps that communicate. (d) Diagram of the cell model presented and reproduced from Ref . Copyright 2007 American Physiological Society. Model contains three intracellular compartments: bulk cytosol (cyt), microdomains (md), and sarcoplasmic reticulum (SR). J_i,diff is the electrodiffusive flux of ion i between the microdomains and the bulk cytosol.

FIGURE 3

Multicellular models of the vascular wall. (a) The Koenigsberger model integrates on a two-dimensional grid model equations for SMCs superposed on a two-dimensional grid of ECs. ECs are arranged parallel and SMCs perpendicular to the vessel axis (reproduced with permission from Ref . Copyright 2005 Elsevier). Cell geometry is approximated by a rectangle. Each cell is connected with its nearest neighbors on the same layer (homocellular connection) and with the cells on the other layer directly superposed on it (hetero-cellular connection). (b) A model of the SMC layer introduced by Jacobsen and coworkers (Reproduced with permission from Ref . Copyright 2007 American Physiological Society). Each SMC (upper panel) contains: Na⁺/K⁺-ATPase (1), NCX (2), plasma membrane Ca²⁺-ATPase (3), sarco(endo)plasmic reticulum Ca²⁺-ATPase (4), SR calcium release channel (5), cytoplasmic calcium buffer (6), SR calcium buffer (7), cGMP-sensitive calcium-dependent chloride channel (8), calcium-activated potassium channels (9), voltage-sensitive calcium channel (L-type calcium channel; 10), and gap junction (11). The SMCs are arranged into a single-layered cell plate (lower panel). Each spindle-shaped cell couples to neighboring cells through gap junctions (black double-barrel structures). (c) Multicellular model of a vessel segment presented by Kapela and coworkers.^, ECs and SMCs are placed in appropriate arrangement. Cells are coupled by nitric oxide (NO) and myoendothelial gap junctions permeable to Ca²⁺, Na⁺, K⁺, and Cl^– ions, and IP₃. K_ir—inward rectifier K⁺ channel; VRAC—volume-regulated anion channel; SK_Ca, IK_Ca and BK_Ca—small-, intermediate-, and large-conductance Ca²⁺-activated K⁺ channels; SOC—store-operated channel; NSC—nonselective cation channel, CaCC and Cl_Ca—Ca²⁺-activated chloride channel; NaK—Na⁺-K⁺-ATPase; PMCA—plasma membrane Ca²⁺-ATPase; NCX—Na⁺/Ca²⁺ exchanger; NaKCl—Na⁺-K⁺-Cl^– cotransport; K_v—voltage-dependent K⁺ channel; K_leak—unspecified K⁺ leak current; VOCC—voltage-operated Ca²⁺ channels; SR/ER—sarco/endoplasmic reticulum; IP₃R—IP₃ receptor; RyR—ryanodine receptor; SERCA—SR/ER Ca²⁺-ATPase; CSQN—calsequestrin; CM—calmodulin; R—receptor; G—G-protein; DAG—diacylglycerol; PLC—phospholipase C; sGC—soluble guanylate cyclase; cGMP—cyclic guanosine monophosphate.

FIGURE 4

Strategy for integrative modeling of the vasculature. (a) Genomic, proteomic and in vitro data such as electrophysiological recordings can be utilized to provide mathematical formulations for subcellular components and signaling pathways. Representative example shows the three-dimensional folding of KIR_2.1 protein and the current–voltage behavior of a K_IR channel in an EC. (b) An EC model that integrates such formulations and incorporates membrane channels, pumps and exchangers, intracellular compartments, and signaling mechanisms. The model can simulate membrane electrophysiology, dynamic behavior of Ca²⁺ and other ions, and the generation of second messengers and signaling factors. (c) EC and SMCs can be coupled through gap junctions and the diffusion of species like NO and IP₃. (d) Multicellular models of the vascular wall can be constructed by placing EC and SMCs cells in an appropriate arrangement. (e) Example of a biotransport model investigating the diffusion of species (i.e., NO, O₂) in and around a single arteriole and incorporates RBCs and nearby capillaries. (f) A biomechanics model of a single arteriole presents the constriction of the vessel at the site of norepinephrine application in the absence of longitudinal signal conduction. (g) Detailed computational model investigating blood flow and O₂ distribution in three-dimensional vascular networks and mesoscale tissue volumes. (h) Reconstructed whole-organ vessel network and blood flow calculations.(Reprinted with permission from Ref . Copyright 2002 Society for Industrial and Applied Mathematics).