Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions.

Cyclic four-state models are frequently used in biology to represent a variety of molecular behaviors. A common experimental strategy to test such models is to follow the behavior of the real system after some of the rate constants are changed in a stepwise manner. We analyze the mathematical behavior of a simple example of such a model applicable to the regulation of contraction of smooth muscle, but our results apply in general to any linear, cyclic four-state model. We discuss detailed balance and requirements for linearity. We find that the only way to have sustained oscillations is for the rate constants of the model to be themselves oscillatory. We state conditions for decaying oscillations and find that in models that do not follow strictly first-order kinetics and do not satisfy detailed balance, these conditions can hold. We show analytically that the response of any state to step changes in the rate constants is the sum of three weighted exponentials plus a constant term, the steady-state value. We provide explicit expressions for the time dependence of all state variables. We discuss a simple way to use these results to obtain numerical solutions in cases where the rate constants change in an arbitrary way.     

An expanded latch-bridge model of protein kinase C-mediated smooth muscle contraction.

A thin-filament-regulated latch-bridge model of smooth muscle contraction is proposed to integrate thin-filament-based inhibition of actomyosin ATPase activity with myosin phosphorylation in the regulation of smooth muscle mechanics. The model included two latch-bridge cycles, one of which was identical to the four-state model as proposed by Hai and Murphy (Am J Physiol Cell Physiol 255: C86-C94, 1988), whereas the ultraslow cross-bridge cycle has lower cross-bridge cycling rates. The model-fitted phorbol ester induced slow contractions at constant myosin phosphorylation and predicted steeper dependence of force on myosin phosphorylation in phorbol ester-stimulated smooth muscle. By shifting cross bridges between the two latch-bridge cycles, the model predicts that a smooth muscle cell can either maintain force at extremely low-energy cost or change its contractile state rapidly, if necessary. Depending on the fraction of cross bridges engaged in the ultraslow latch-bridge cycle, the model predicted biphasic kinetics of smooth muscle mechanics and variable steady-state dependencies of force and shortening velocity on myosin phosphorylation. These results suggest that thin-filament-based regulatory proteins may function as tuners of actomyosin ATPase activity, thus allowing a smooth muscle cell to have two discrete cross-bridge cycles with different cross-bridge cycling rates.     

Smooth muscle contraction: mechanochemical formulation for homogeneous finite strains

Chemical kinetics of smooth muscle contraction affect mechanical properties of organs that function under finite strains. In an effort to gain further insight into organ physiology, we formulate a mechanochemical finite strain model by considering the interaction between mechanical and biochemical components of cell function during activation. We propose a new constitutive framework and use a mechanochemical device that consists of two parallel elements: (i) spring for the cell stiffness; (ii) contractile element for the sarcomere. We use a multiplicative decomposition of cell elongation into filament contraction and cross-bridge deformation, and suggest that the free energy be a function of stretches, four variables (free unphosphorylated myosin, phosphorylated cross-bridges, phosphorylated and dephosphorylated cross-bridges attached to actin), chemical state variable driven by Ca2+-concentration, and temperature. The derived constitutive laws are thermodynamically consistent. Assuming isothermal conditions, we specialize the mechanical phase such that we recover the linear model of Yang et al. [2003a. The myogenic response in isolated rat cerebrovascular arteries: smooth muscle cell. Med. Eng. Phys. 25, 691-709]. The chemical phase is also specialized so that the linearized chemical evolution law leads to the four-state model of Hai and Murphy [1988. Cross-bridge phosphorylation and regulation of latch state in smooth muscle. Am. J. Physiol. 254, C99-C106]. One numerical example shows typical mechanochemical effects and the efficiency of the proposed approach. We discuss related parameter identification, and illustrate the dependence of muscle contraction (Ca2+-concentration) on active stress and related stretch. Mechanochemical models of this kind serve the mathematical basis for analyzing coupled processes such as the dependency of tissue properties on the chemical kinetics of smooth muscle.     

A multistate model for bivariate interval-censored failure time data

SUMMARY: Interval-censored life-history data arise when the events of interest are only detectable at periodic assessments. When interest lies in the occurrence of two such events, bivariate-interval censored event time data are obtained. We describe how to fit a four-state Markov model useful for characterizing the association between two interval-censored event times when the assessment times for the two events may be generated by different inspection processes. The approach treats the two events symmetrically and enables one to fit multiplicative intensity models that give estimates of covariate effects as well as relative risks characterizing the association between the two events. An expectation-maximization (EM) algorithm is described for estimation in which the maximization step can be carried out with standard software. The method is illustrated by application to data from a trial of HIV patients where the events are the onset of viral shedding in the blood and urine among individuals infected with cytomegalovirus.     

Role of the adrenal cortex and sodium in the pathogenesis of human hypertension.

After 30 years of continuous research into the mechanisms of human hypertension, we summarize the results obtained by the members of the multidisciplinary research group on hypertension of the Clinical Research Institute of Montreal on the disturbances of minerlocorticoid activity in a rigorously selected group of patients with early, mild essential hypertension. We attempt to integrate these findings with those of many other groups working on other aspects of hypertensive cardiovascular diseases. On the assumption that the increased peripheral resistance responsible for hypertension results from an imbalance or a disturbance of the equilibrium between the sympathetic nervous system and norepinephrine on one hand, and the vascular tone, sensitivity and responsiveness of the arterial smooth muscle to norepinephrine and to angiotensin II on the other hand, three models that fit the experimental and clinical facts as known at present are described.     

Contribution of circulating vascular progenitors in lesion formation and vascular healing: lessons from animal models

PURPOSE OF REVIEW: It is a widely accepted view that vascular repair results from migration and proliferation of adjacent cells in animal models. On the contrary, accumulating evidence suggests that bone marrow can give rise to endothelial-like cells and smooth muscle like cells that potentially contribute to vascular healing, remodeling, and lesion formation under physiological and pathological conditions. The aim of this article is to review recent findings obtained from animal models of vascular diseases regarding bone marrow derived progenitor cells. RECENT FINDINGS: Studies using chimeric animals revealed that bone marrow derived cells exist at the sites of vascular healing and lesion formation after injury. High-resolution histological analyses revealed that those bone marrow derived cells do express some markers for endothelial cells or smooth muscle cells. Peripheral mononuclear cells could differentiate into endothelial-like cells or smooth muscle like cells in vitro according to the culture conditions. SUMMARY: Circulating progenitors significantly contribute to vascular repair and lesion formation. These findings provide the basis for the development of new therapeutic strategies that involve targeting the mobilization, homing, differentiation, and proliferation of bone marrow- derived vascular progenitor cells.     

Cloning of a gap junctional protein from vascular smooth muscle and expression in two-cell mouse embryos.

Gap junctional proteins (connexins) form aqueous channels that enable direct cell-cell transfer of ions and small molecules. The distribution and conductance of gap junction channels in cardiac muscle determine the pattern and synchrony of cellular activation. However, the capacity for smooth muscle to restrict contractile events temporally and spatially suggests that cell-cell coupling or its regulation may be decidedly different in this tissue. We isolated a cDNA from vascular smooth muscle which encodes a connexin (Mr 43,187) structurally homologous to cardiac connexin43. Vascular smooth muscle connexin43 mRNA was expressed prominently in smooth muscle tissues, cultured vascular myocytes, and arterial endothelial cells. A model for functional expression of connexins was developed in two-cell B6D2 mouse embryos. Microinjection of in vitro transcribed vascular smooth muscle connexin43 mRNA was shown to be sufficient to induce intercellular coupling in previously uncoupled blastomeres. Through the construction of two deletion mutants of connexin43, we also show that the formation of cell-to-cell connections does not depend upon a predicted cytoplasmic region within 98 residues of the carboxyl terminus. Finally, the identification of connexin43 in smooth muscle and endothelial cells provides supporting evidence for the existence of heterocellular coupling between cells of the vascular intima.     

Emergent properties of electrically coupled smooth muscle cells

Asynchronous and synchronous calcium oscillations occur in a variety of cells. A well-established pathway for intercellular communication is provided by gap junctions which connect adjacent cells and can mediate electrical and chemical coupling. Several experimental studies report that cells presenting only a transient increase when freshly dispersed may oscillate when they are coupled. Such observations suggest that the role of gap junctions is not only to coordinate calcium oscillations of adjacent cells. Gap junctions may also be important to generate oscillations. Here we illustrate the emergent properties of electrically coupled smooth muscle cells using a model that we recently proposed. A bifurcation analysis in the case of two cells reveals that synchronous and asynchronous calcium oscillations can be induced by electrical coupling. In a larger population of smooth muscle cells, electrical coupling may result in the creation of groups of cells presenting synchronous calcium oscillations. The elements of one group may be distant from each other. Moreover, our results highlight a general mechanism by which gap junctional electrical coupling can give rise to out of phase calcium oscillations in smooth muscle cells that are non-oscillating when uncoupled. All these observations remain true in the case of non-identical cells, except that the solution corresponding to synchronous calcium oscillations disappears and that the formation of groups is sensitive to the degree of heterogeneity. The first two authors contributed equally to this work.     

Somite-derived cells replace ventral aortic hemangioblasts and provide aortic smooth muscle cells of the trunk

We have previously shown that endothelial cells of the aortic floor give rise to hematopoietic cells, revealing the existence of an aortic hemangioblast. It has been proposed that the restriction of hematopoiesis to the aortic floor is based on the existence of two different and complementary endothelial lineages that form the vessel: one originating from the somite would contribute to the roof and sides, another from the splanchnopleura would contribute to the floor. Using quail/chick orthotopic transplantations of paraxial mesoderm, we have traced the distribution of somite-derived endothelial cells during aortic hematopoiesis. We show that the aortic endothelium undergoes two successive waves of remodeling by somitic cells: one when the aortae are still paired, during which the initial roof and sides of the vessels are renewed; and a second, associated to aortic hematopoiesis, in which the hemogenic floor is replaced by somite endothelial cells. This floor thus appears as a temporary structure, spent out and replaced. In addition, the somite contributes to smooth muscle cells of the aorta. In vivo lineage tracing experiments with non-replicative retroviral vectors showed that endothelial cells do not give rise to smooth muscle cells. However, in vitro, purified endothelial cells acquire smooth muscle cells characteristics. Taken together, these data point to the crucial role of the somite in shaping the aorta and also give an explanation for the short life of aortic hematopoiesis.     

Modeling Facilitative Sugar Transporters: Transitions Between Single and Double Ligand Occupancy of Multiconformational Channel Models Explain Anomalous Kinetics

The four-state simple carrier model (SCM) is employed to describe ligand translocation by diverse passive membrane transporters. However, its application to systems like facilitative sugar transporters (GLUTs) is controversial: unidirectional fluxes under zero-trans and equilibrium-exchange experimental conditions fit a SCM, but flux data from infinite-cis and infinite-trans experiments appear not to fit the same SCM. More complex kinetic models have been proposed to explain this ``anomalous' behavior of GLUTs, but none of them accounts for all the experimental findings. We propose an alternative model in which GLUTs are channels subject to conformational transitions, and further assume that the results from zero-trans and equilibrium-exchange experiments as well as trans-effects corresponds to a single-occupancy channel regime, whereas the results from the infinite-cis and infinite-trans experiments correspond to a regime including higher channel occupancies. We test the plausibility of this hypothesis by studying a kinetic model of a two-site channel with two conformational states. In each state, the channel can bind the ligand from only one of the compartments. Under single-occupancy, for conditions corresponding to zero-trans and equilibrium-exchange experiments, the model behaves as a SCM capable of exhibiting trans-stimulations. For a regime including higher degrees of occupancy and infinite-cis and infinite-trans conditions, the same channel model can exhibit a behavior qualitatively similar to a SCM, albeit with kinetic parameters different from those for the single-occupancy regime. Numerical results obtained with our model are consistent with available experimental data on facilitative glucose transport across erythrocyte membranes. Hence, if GLUTs are multiconformational channels, their particular kinetic properties can result from transitions between single and double channel occupancies. Received: 12 April 1995/Revised: 28 August 1995     

An automated decision-tree approach to predicting protein interaction hot spots

Protein-protein interactions can be altered by mutating one or more "hot spots," the subset of residues that account for most of the interface's binding free energy. The identification of hot spots requires a significant experimental effort, highlighting the practical value of hot spot predictions. We present two knowledge-based models that improve the ability to predict hot spots: K-FADE uses shape specificity features calculated by the Fast Atomic Density Evaluation (FADE) program, and K-CON uses biochemical contact features. The combined K-FADE/CON (KFC) model displays better overall predictive accuracy than computational alanine scanning (Robetta-Ala). In addition, because these methods predict different subsets of known hot spots, a large and significant increase in accuracy is achieved by combining KFC and Robetta-Ala. The KFC analysis is applied to the calmodulin (CaM)/smooth muscle myosin light chain kinase (smMLCK) interface, and to the bone morphogenetic protein-2 (BMP-2)/BMP receptor-type I (BMPR-IA) interface. The results indicate a strong correlation between KFC hot spot predictions and mutations that significantly reduce the binding affinity of the interface.     

Cardiovascular malformations with normal smooth muscle differentiation in neural crest-specific type II TGFbeta receptor (Tgfbr2) mutant mice

Previous studies have demonstrated that TGFbeta induces a smooth muscle fate in primary neural crest cells in culture. By crossing a conditional allele of the type II TGFbeta receptor with the neural crest-specific Wnt1cre transgene, we have addressed the in vivo requirement for TGFbeta signaling in smooth muscle specification and differentiation. We find that elimination of the TGFbeta receptor does not alter neural crest cell specification to a smooth muscle fate in the cranial or cardiac domains, and that a smooth muscle fate is not realized by trunk neural crest cells in either control or mutant embryos. Instead, mutant embryos exhibit with complete penetrance two very specific and mechanistically distinct cardiovascular malformations--persistent truncus arteriosus (PTA) and interrupted aortic arch (IAA-B). Pharyngeal organ defects such as those seen in models of DiGeorge syndrome were not observed, arguing against an early perturbation of the cardiac neural crest cell lineage. We infer that TGFbeta is an essential morphogenic signal for the neural crest cell lineage in specific aspects of cardiovascular development, although one that is not required for smooth muscle differentiation.     

Adenosine 5''-triphosphate consumption by smooth muscle as predicted by the coupled four-state crossbridge model.

We have proposed a four-state crossbridge model to explain contraction and the latch state in arterial smooth muscle. Ca(2+)-dependent crossbridge phosphorylation was the only postulated regulatory mechanism and the latchbridge (a dephosphorylated, attached crossbridge) was the only novel element in the model. In this study, we used the model to predict rates of ATP consumption by crossbridge phosphorylation (JPhos) and cycling (JCycle) during isometric and isotonic contractions in arterial smooth muscle; then we compared model predictions with experimental data. The model predicted that JPhos and JCycle were similar in magnitude in isometric contractions, and both increased almost linearly with myosin phosphorylation. The predicted relationship between isometric stress and ATP consumption was quasihyperbolic, but approximately linear when myosin phosphorylation was below 35%, in agreement with most of the available data. Muscle shortening increased the predicted values of JCycle up to 3.7-fold depending on shortening velocity and the level of myosin phosphorylation. The predicted maximum work output per ATP was 7.4-7.8 kJ/mol ATP and was relatively insensitive to changes in myosin phosphorylation. The predicted increase in JCycle with shortening was in agreement with available data, but the model prediction that work output per ATP was insensitive to changes in myosin phosphorylation was unexpected and remains to be tested in future experiments.     

Local control models of cardiac excitation-contraction coupling. A possible role for allosteric interactions between ryanodine receptors

In cardiac muscle, release of activator calcium from the sarcoplasmic reticulum occurs by calcium- induced calcium release through ryanodine receptors (RyRs), which are clustered in a dense, regular, two-dimensional lattice array at the diad junction. We simulated numerically the stochastic dynamics of RyRs and L-type sarcolemmal calcium channels interacting via calcium nano-domains in the junctional cleft. Four putative RyR gating schemes based on single-channel measurements in lipid bilayers all failed to give stable excitation-contraction coupling, due either to insufficiently strong inactivation to terminate locally regenerative calcium-induced calcium release or insufficient cooperativity to discriminate against RyR activation by background calcium. If the ryanodine receptor was represented, instead, by a phenomenological four-state gating scheme, with channel opening resulting from simultaneous binding of two Ca2+ ions, and either calcium-dependent or activation-linked inactivation, the simulations gave a good semiquantitative accounting for the macroscopic features of excitation-contraction coupling. It was possible to restore stability to a model based on a bilayer-derived gating scheme, by introducing allosteric interactions between nearest-neighbor RyRs so as to stabilize the inactivated state and produce cooperativity among calcium binding sites on different RyRs. Such allosteric coupling between RyRs may be a function of the foot process and lattice array, explaining their conservation during evolution.     

Contribution of nNOS- and eNOS-derived NO to microvascular smooth muscle NO exposure.

Nitric oxide (NO) plays an important role in autocrine and paracrine manner in numerous physiological processes, including regulation of blood pressure and blood flow, platelet aggregation, and leukocyte adhesion. In vascular wall, most of the bioavailable NO is believed to derive from endothelial cell NO synthase (eNOS). Recently, neuronal NOS (nNOS) has been identified as a source of NO in the vicinity of microvessels and has been shown to participate in vascular function. Thus NO can be produced and transported to the vascular smooth muscle cells from 1). endothelial cells and 2). perivascular nerve fibers, mast cells, and other nNOS-containing sources. In this study, a mathematical model of NO diffusion-reaction in a cylindrical arteriolar segment was formulated. The model quantifies the relative contribution of these NO sources and the smooth muscle availability of NO in a tissue containing an arteriolar blood vessel. The results indicate that a source of NO derived through nNOS in the perivascular region can be a significant contributor to smooth muscle NO. Predicted smooth muscle NO concentrations are as high as 430 nM, which is consistent with reported experimental measurements ( approximately 400 nM). In addition, we used the model to analyze the smooth muscle NO availability in 1). eNOS and nNOS knockout experiments, 2). the presence of myoglobin, and 3). the presence of cell-free Hb, e.g., Hb-based oxygen carriers. The results show that NO release by nNOS would significantly affect available smooth muscle NO. Further experimental and theoretical studies are required to account for distribution of NOS isoforms and determine NO availability in vasculatures of different tissues.     

Notch and bone morphogenetic protein differentially act on dermomyotome cells to generate endothelium, smooth, and striated muscle

We address the mechanisms underlying generation of skeletal muscle, smooth muscle, and endothelium from epithelial progenitors in the dermomyotome. Lineage analysis shows that of all epithelial domains, the lateral region is the most prolific producer of smooth muscle and endothelium. Importantly, individual labeled lateral somitic cells give rise to only endothelial or mural cells (not both), and endothelial and mural cell differentiation is driven by distinct signaling systems. Notch activity is necessary for smooth muscle production while inhibiting striated muscle differentiation, yet it does not affect initial development of endothelial cells. On the other hand, bone morphogenetic protein signaling is required for endothelial cell differentiation and/or migration but inhibits striated muscle differentiation and fails to impact smooth muscle cell production. Hence, although different mechanisms are responsible for smooth muscle and endothelium generation, the choice to become smooth versus striated muscle depends on a single signaling system. Altogether, these findings underscore the spatial and temporal complexity of lineage diversification in an apparently homogeneous epithelium.     

Arterial smooth muscle cells dysfunction in hyperglycaemia and hyperglycaemia associated with hyperlipidaemia: from causes to effects

Given the important role of smooth muscle cells in arterial wall dysfunction in diabetes, as well as in diabetes associated with accelerated atherosclerosis, we provide a brief review of the recent achievements in identification of signalling molecules underlying their altered cellular responses, and examine the consequences of these pathological insults on smooth muscle cells properties. The original results emerging from the Golden Syrian hamster model (rendered diabetic or simultaneously hyperlipidaemic-diabetic) and from human aortic smooth muscle cells cultured in 25 mM glucose (to mimic diabetic condition) or sera of obese type 2 diabetic patients (to mimic the metabolic syndrome condition) are presented in this context. We conclude this review with several open issues disclosed by the most recent literature that deserve essential attention for targeting the translational medicine.     

A micromechanical model of skeletal muscle to explore the effects of fiber and fascicle geometry

Computational models of muscle generally lump the material properties of connective tissue, muscle fibers, and muscle fascicles together into one constitutive relationship that assumes a transversely isotropic microstructure. These models do not take into account how variations in the microstructure of muscle affect its macroscopic material properties. The goal of this work was to develop micromechanical models of muscle to determine the effects of variations in muscle microstructure on the macroscopic constitutive behavior. We created micromechanical models at the fiber and fascicle levels based on histological cross-sections of two rabbit muscles, the rectus femoris (RF) and the soleus, to determine the effects of microstructure geometry (fiber and fascicle shapes) on the along-fiber shear modulus of muscle. The two fiber-level models predicted similar macroscopic shear moduli (within 13.5% difference); however, the two fascicle-level models predicted very different macroscopic shear moduli (up to 161% difference). We also used the micromechanical models to test the assumption that the macroscopic properties of muscle are transversely isotropic about the fiber (or fascicle) direction. The fiber-level models exhibited behavior consistent with the transverse isotropy assumption; however, the fascicle-level models exhibited transversely anisotropic behavior. Micromechanical models, combined with fiber and fiber bundle mechanical experiments, are needed to understand how normal or pathological variations in microstructure give rise to the observed macroscopic behavior of muscle.     

ACE-inhibitors and defence reflexes of the airways.

The group of angiotensin-converting enzyme (ACE) inhibitors is one of the drugs of choice for the treatment of hypertension and congestive heart disease. However, it has been reported that in some of patients ACE-inhibitors induce hyperreactivity of the airways with occurrence of a persistent dry cough, dyspnoe and wheezing. We supposed that the mechanism of these hyperreactivity is connected to accumulation of bradykinin, tachykinins and other inflammatory mediators in the airways. Increased local concentration of inflammatory neuropeptides stimulates bronchial C fibres and rapidly adapting receptors and provoke the cough reflex. Inflammatory processes in the airways could be followed by contraction of airway smooth muscle. In this study, our aim was to measure the changes of the number and intensity of mechanical induced cough in cats, which were treated for days with enalapril (5 mg/kg b.w.). After 15 days of treatment the reactivity of the lung and tracheal smooth muscles to the bronchoconstrictor mediator histamine was estimated. As to our finding 15 days of administration of enalapril results in significant increase of cough parameters measured with a more significant sensitivity of the laryngopharyngeal part. In the experimental animals we observed increased reactivity of bronchial smooth muscle to histamine after 15 days of enalapril treatment. The reactivity of the lung smooth muscle to the histamine was not significantly changed. These results confirmed the increased cough sensitivity and increased bronchial reactivity after enalapril treatment. These experimental animal model may be useful for the investigation of the pharmacological minimization of respiratory adverse effect of ACE-inhibitors.     

Melanoblast proliferation dynamics during mouse embryonic development. Modeling and validation

In this paper, we are looking for mathematical modeling of mouse embryonic melanoblast proliferation dynamics, taking into account, the expression level of β‐catenin. This protein plays an important role into the whole signal pathway process. Different assumptions on some unobservable features lead to different candidate models. From real data measured, from biological experiments and from a priori biological knowledge, it was able to validate or invalidate some of the candidate models. Data assimilation and parameter identification allowed us to derive a mathematical model that is in very good agreement with biological data. As a result, the produced model can give tracks for biologists into their biological investigations and experimental evidence. Another interest is the use of this model for robust hidden parameter identification like double times or number of founder melanoblasts.