Clusters of calcium release channels harness the Ising phase transition to confine their elementary intracellular signals.

Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal-to-noise ratio is achieved by a cooperative action of Ca release channels such as IP3 receptors or ryanodine receptors arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced Ca release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. Despite the positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. Here we show that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field in an Ising ferromagnet. Our simple quantitative criterion closely predicts the Ca store depletion level required for spark termination for each cluster size. We further formulate exact requirements that a cluster of release channels should satisfy in any cell type for our mapping to the Ising model and the associated formula to remain valid. Thus, we describe deterministically the behavior of a system on a coarser scale (release unit) that is random on a finer scale (release channels), bridging the gap between scales. Our results provide exact mapping of a nanoscale biological signaling model to an interacting particle system in statistical physics, making the extensive mathematical apparatus available to quantitative biology

Mechanisms of Calcium Leak from Cardiac Sarcoplasmic Reticulum Revealed by Statistical Mechanics

Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR), a major intracellular Ca store. However, under abnormal conditions Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of Ca release unit (CRU) reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogues of magnetic field ($h$) and the inverse temperature ($\beta$) in the Ising model, for which we have explicit formulas in terms of SR Ca and release channel opening/closing rates. The classification of leak regimes takes the shape of a phase $\beta$-$h$ diagram, with the regime boundaries occurring at $h$=0 and a critical value of $\beta$ ($\beta*$) which we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when $h$>0 and $\beta>\beta*$ and a disordered leak occurs when $\beta<\beta*$ and $h$ is not too negative. The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths, an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective RyR parameters and SR Ca load that describe and classify phase transitions and leak emergence.Comment: 20 pages, 6 figures, supplemental materia

On Agmon metrics and exponential localization for quantum graphs

We investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using Agmon's method to prove $L^2$ and $L^\infty$ bounds on the product of an eigenfunction with the exponential of a certain metric. A generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line, which are of classical Liouville-Green (WKB) form. Examples reveal that this estimate can be the best possible, but that a more rapid rate of decay is typical when the graph has additional structure. In order to understand this fact, we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay. One of these depends on how the eigenfunction is distributed along a particular chosen path, while the other applies to an average of the eigenfunction over edges at a given distance from the root point

Localization and landscape functions on quantum graphs

6 figures6 figures6 figuresWe discuss explicit landscape functions for quantum graphs. By a "landscape function" $\Upsilon(x)$ we mean a function that controls the localization properties of normalized eigenfunctions $\psi(x)$ through a pointwise inequality of the form $$|\psi(x)| \le \Upsilon(x).$$ The ideal $\Upsilon$ is a function that a) responds to the potential energy $V(x)$ and to the structure of the graph in some formulaic way; b) is small in examples where eigenfunctions are suppressed by the tunneling effect, and c) relatively large in regions where eigenfunctions may - or may not - be concentrated, as observed in specific examples. It turns out that the connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy r\'egime, as we show with simple examples. We therefore apply different methods in different r\'egimes determined by the values of the potential energy $V(x)$ and the eigenvalue parameter $E$

Simulation and Mechanistic Investigation of the Arrhythmogenic Role of the Late Sodium Current in Human Heart Failure

Heart failure constitutes a major public health problem worldwide. The electrophysiological remodeling of failing hearts sets the stage for malignant arrhythmias, in which the role of the late Na+ current (INaL) is relevant and is currently under investigation. In this study we examined the role of INaL in the electrophysiological phenotype of ventricular myocytes, and its proarrhythmic effects in the failing heart. A model for cellular heart failure was proposed using a modified version of Grandi et al. model for human ventricular action potential that incorporates the formulation of INaL. A sensitivity analysis of the model was performed and simulations of the pathological electrical activity of the cell were conducted. The proposed model for the human INaL and the electrophysiological remodeling of myocytes from failing hearts accurately reproduce experimental observations. The sensitivity analysis of the modulation of electrophysiological parameters of myocytes from failing hearts due to ion channels remodeling, revealed a role for INaL in the prolongation of action potential duration (APD), triangulation of the shape of the AP, and changes in Ca2+ transient. A mechanistic investigation of intracellular Na+ accumulation and APD shortening with increasing frequency of stimulation of failing myocytes revealed a role for the Na+/K+ pump, the Na+/Ca2+ exchanger and INaL. The results of the simulations also showed that in failing myocytes, the enhancement of INaL increased the reverse rate-dependent APD prolongation and the probability of initiating early afterdepolarizations. The electrophysiological remodeling of failing hearts and especially the enhancement of the INaL prolong APD and alter Ca2+ transient facilitating the development of early afterdepolarizations. An enhanced INaL appears to be an important contributor to the electrophysiological phenotype and to the dysregulation of [Ca2+]i homeostasis of failing myocytes

Search for beautiful tetraquarks in the <i>ϒ</i>(1<i>S</i>)μ⁺μ⁻ invariant-mass spectrum

International audienceThe ϒ(1S)μ$^{+}$μ$^{−}$ invariant-mass distribution is investigated for a possible exotic meson state composed of two b quarks and two $\overline{b}$ quarks, ${X}_{b\overline{b}b\overline{b}}$ . The analysis is based on a data sample of pp collisions recorded with the LHCb detector at centre-of-mass energies $\sqrt{s}=7$ , 8 and 13 TeV, corresponding to an integrated luminosity of 6.3 fb$^{−1}$. No significant excess is found, and upper limits are set on the product of the production cross-section and the branching fraction as functions of the mass of the ${X}_{b\overline{b}b\overline{b}}$ state. The limits are set in the fiducial volume where all muons have pseudorapidity in the range [2.0, 5.0], and the ${X}_{b\overline{b}b\overline{b}}$ state has rapidity in the range [2.0, 4.5] and transverse momentum less than 15 GeV/c

Electrophysiological and Structural Remodeling in Heart Failure Modulate Arrhythmogenesis. 1D Simulation Study

Background: Heart failure is a final common pathway or descriptor for various cardiac pathologies. It is associated with sudden cardiac death, which is frequently caused by ventricular arrhythmias. Electrophysiological remodeling, intercellular uncoupling, fibrosis and autonomic imbalance have been identified as major arrhythmogenic factors in heart failure etiology and progression. Objective: In this study we investigate in silico the role of electrophysiological and structural heart failure remodeling on the modulation of key elements of the arrhythmogenic substrate, i.e., electrophysiological gradients and abnormal impulse propagation. Methods: Two different mathematical models of the human ventricular action potential were used to formulate models of the failing ventricular myocyte. This provided the basis for simulations of the electrical activity within a transmural ventricular strand. Our main goal was to elucidate the roles of electrophysiological and structural remodeling in setting the stage for malignant life-threatening arrhythmias. Results: Simulation results illustrate how the presence of M cells and heterogeneous electrophysiological remodeling in the human failing ventricle modulate the dispersion of action potential duration and repolarization time. Specifically, selective heterogeneous remodeling of expression levels for the Na+ /Ca2+ exchanger and SERCA pump decrease these heterogeneities. In contrast, fibroblast proliferation and cellular uncoupling both strongly increase repolarization heterogeneities. Conduction velocity and the safety factor for conduction are also reduced by the progressive structural remodeling during heart failure. Conclusion: An extensive literature now establishes that in human ventricle, as heart failure progresses, gradients for repolarization are changed significantly by protein specific electrophysiological remodeling (either homogeneous or heterogeneous). Our simulations illustrate and provide new insights into this. Furthermore, enhanced fibrosis in failing hearts, as well as reduced intercellular coupling, combine to increase electrophysiological gradients and reduce electrical propagation. In combination these changes set the stage for arrhythmias.This work was partially supported by (i) the "VI Plan Nacional de Investigacion Cientifica, Desarrollo e Innovacion Tecnologica" from the Ministerio de Economia y Competitividad of Spain (grant number TIN2012-37546-C03-01) and the European Commission (European Regional Development Funds - ERDF - FEDER), (ii) the Direccion General de Politica Cientifica de la Generalitat Valenciana (grant number GV/2013/119), and (iii) Programa Prometeo (PROMETEO/2012/030) de la Conselleria d'Educacio Formacio I Ocupacio, Generalitat Valenciana. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Gómez García, JF.; Cardona, K.; Romero Pérez, L.; Ferrero De Loma-Osorio, JM.; Trénor Gomis, BA. (2014). Electrophysiological and Structural Remodeling in Heart Failure Modulate Arrhythmogenesis. 1D Simulation Study. PLoS ONE. 9(9). https://doi.org/10.1371/journal.pone.0106602S9

A model model: a commentary on DiFrancesco and Noble (1985) ‘A model of cardiac electrical activity incorporating ionic pumps and concentration changes’

This paper summarizes the advances made by the DiFrancesco and Noble (DFN) model of cardiac cellular electrophysiology, which was published in Philosophical Transactions B in 1985. This model was developed at a time when the introduction of new techniques and provision of experimental data had resulted in an explosion of knowledge about the cellular and biophysical properties of the heart. It advanced the cardiac modelling field from a period when computer models considered only the voltage-dependent channels in the surface membrane. In particular, it included a consideration of changes of both intra- and extracellular ionic concentrations. In this paper, we summarize the most important contributions of the DiFrancesco and Noble paper. We also describe how computer modelling has developed subsequently with the extension from the single cell to the whole heart as well as its use in understanding disease and predicting the effects of pharmaceutical interventions. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society