By D’Artagnan Greene and Yohannes Shiferaw
Approximately one in five human deaths occur when the periodic beating of the heart suddenly transitions into a chaotic rhythm [7]. This transition—called a cardiac arrhythmia—prevents the heart from effectively pumping blood and can have deadly consequences. To identify the cause of arrhythmias, one must understand the basic physiology of heart cells and tissue [4]. Heart cells regulate voltage across their membranes via the complex interplay of millions of nanometer-scale proteins called ion channels. Ion channels are sensitive to the voltage gradient; their nonlinear response to this gradient endows the cells with the properties of an excitable system. Heart cells are excitable in the sense that injection of a small initial current can induce a much larger secondary current flow that generates a rapid increase in the voltage across the cell membrane.
Cells are connected to each other via ion channels, which means that an excitation of one cell can rapidly excite its nearest neighbors. As with falling dominoes, this process can rapidly spread the signal outward across many cells to eventually excite all cardiac tissue. This rapid propagation of electrical excitation orchestrates the heart’s rhythmic beating. In an unhealthy heart, however, the typically organized electrical excitation can break down into a swirl of turbulent electrical activity that overruns the heart’s intrinsic rhythm and causes an arrhythmia.
Figure 1. A paced cardiac cell can exhibit a sustained alternating pattern in which a quiescent beat (Q) follows a calcium (Ca) wave (W). A large amount of Ca is released into the cell during the wave, which generates an alternating Ca and voltage signal. Figure courtesy of the authors.
In recent decades, biologists have worked closely with applied mathematicians and physicists to decipher the mechanisms that drive cardiac arrhythmias. The mathematical perspective has yielded important insights about the structure of coherent excitations in tissue—such as spiral and scroll waves—that practitioners have experimentally observed in the heart [2]. Additionally, molecular biologists have discovered that arrhythmias can arise due to defects at the level of individual amino acids in proteins that regulate the voltage across cell membranes. However, the mechanism through which nanometer-scale molecular defects influence the spatiotemporal activity of an entire heart remains unknown. This is because sub-microsecond time scale fluctuations between stable conformational states—such as an ion channel’s open and closed state—dictate the function of proteins at the nanometer scale. Furthermore, an electrical activation in tissue involves the excitation of millions of electrically coupled cells. Understanding the cause-and-effect relationships across these vast length and time scales is a daunting yet necessary challenge that will allow researchers to fully comprehend the mechanisms of cardiac arrhythmias.
In order for a cardiac cell to function properly, a vast array of proteins and ions must work together like a finely tuned instrument. Cells contribute to this partnership by utilizing signal transduction — a process that transports and delivers information between different parts of the cell with precise timing. The most ubiquitous signaling messenger is the calcium (Ca) ion, which regulates a broad range of cellular processes. Cardiac cells have subcellular compartments that store Ca ions at concentration levels that are roughly four orders of magnitude higher than those in the cell. To induce a cellular response, a small initial increase in Ca ions triggers the release of a large amount of Ca from these compartments into the cell interior, which in turn diffuses and activates an array of Ca-sensitive proteins. This architecture also endows the cell interior with the properties of an excitable system, since a signal can spread in the interior much like electrical activity spreads between cells in tissue.
As a result, waves of released Ca—which occur randomly in cardiac cells—can propagate within a cell under appropriate conditions. These Ca waves are dangerous because Ca wave propagation is a highly nonlinear function of the Ca concentration in intracellular stores, and the waves disrupt the rhythmic responses of cardiac cells. Nucleation of these waves is also highly sensitive to local fluctuations, meaning that their timing and extent is stochastic. Consequently, there is now a consensus in the cardiac community that Ca waves can cause a wide range of cardiac arrhythmias [8].
Figure 2. The dynamics of calcium (Ca) waves and voltage in cardiac tissue. 2a. At steady state, the order parameter \(s_{ij}\) that measures the phase of the alternating waves can synchronize across large patches of cardiac tissue. Here, black regions denote cells with \(s_{ij}=1\) and white regions denote cells with \(s_{ij}=-1\). Orange points denote cells wherein no Ca wave occurs. This pattern then drives spatial patterns of voltage that are quantified by the action potential amplitude \(\Delta a_{ij}\), which measures beat-to-beat differences in the voltage time course. 2b. When the same tissue is paced from the bottom edge, planar waves undergo wave break that is caused by the spatial patterns that form due to the synchronization transition. Here we simulate a piece of cardiac tissue that consists of a \(150 \times 150\) cell grid. The wave break occurs after 30 beats when the spatial patterns have developed. Figure courtesy of [3].
Although biologists have clearly demonstrated that a disruption in Ca signaling can trigger cardiac arrhythmia, researchers still do not understand how a molecular-scale defect can prompt a breakdown of electrical activity for the whole heart. Our group has recently found that Ca waves can synchronize across millions of cells in cardiac tissue. Ca waves fire in unison when this synchronization occurs, which dramatically amplifies their effect on the tissue scale. To discern the impetus behind such synchronization, we must apply concepts from the theory of phase transitions — an elaborate mathematical framework that explains sudden changes in material properties with temperature [1].
The concept of symmetry plays a fundamental role in phase transitions. This symmetry arises in the heart because periodically driven cells can acquire temporal patterns in which Ca waves occur in the cells on alternate beats (see Figure 1). A sequence of Ca waves (W) followed by quiescence (Q) produces this alternating pattern, so that a given cell can alternate with a sequence
\[...Q \enspace W \enspace Q \enspace W...\]
However, simply shifting the sequence by one beat yields
\[...W \enspace Q \enspace W \enspace Q...,\]
which is dynamically equivalent. Paced cardiac cells thus possess a subtle symmetry—first described in a different context [6]—that has important consequences at the tissue scale. To explore these consequences, we introduce an order parameter that measures the phase of the alternating sequence. On a two-dimensional (2D) lattice, we can therefore describe the cell at site \(ij\) with an order parameter
\[s_{ij}= \left\{\begin{array} 1+1 \qquad \rightarrow \qquad ... Q \enspace W \enspace Q \enspace W... \\ -1 \qquad \rightarrow \qquad ... W \enspace Q \enspace W \enspace Q...\end{array}\right\}.\]
On a lattice of coupled cells, the probability of observing sequence \(\{s_{ij}\}\) is the same as observing \(-\{s_{ij}\}\). This phenomenon—known as Ising symmetry—is shared by the interacting spin-\(1/2\) systems that characterize ferromagnetic materials. The “spins” \(s_{ij}\) interact in a manner that is governed by the interplay between the voltage and Ca signals in cardiac tissue. By accounting for these interactions, we realize that we can map the organization of Ca waves in tissue to an equivalent statistical mechanics problem with a Hamiltonian, given by
\[H = \frac{\epsilon}{N}\Bigg(\sum_{ij}s_{ij}\Bigg)^2.\]
\(N\) refers to the total number of cells and \(\epsilon\) is a parameter whose sign is determined by the way in which voltage couples to Ca according to the multitude of ion channels that regulate the cell membrane. This model is analogous to the classic Curie-Weiss model for ferromagnetism, which exhibits an order-disorder phase transition at a critical temperature \(T_C\) [5]. In the context of the heart, the transition maps directly to a synchronization transition wherein Ca waves self-organize across millions of cells in cardiac tissue. Here, the role of thermal fluctuation corresponds to the stochasticity of Ca wave formation and a critical pacing rate replaces the critical temperature. Below the critical pacing rate, the voltage nudges the stochastic subcellular Ca signal to synchronize millions of cells so that they fire Ca waves in unison. Numerical simulations of simplified 2D systems reveal that when this transition commences, a large piece of cardiac tissue coarsens into regions of synchronized waves (see Figure 2a). These regions of synchronized waves can induce large voltage perturbations in that tissue, potentially leading to wave break and reentry (see Figure 2b). In systems with disrupted Ca signaling, we find that arrhythmias only occur at parameter regimes where the synchronization transition has transpired.
Our results highlight new relationships between biological tissue and material science systems. Under certain conditions, biological tissue can share symmetries that also exist in unrelated systems (like ferromagnetic materials). Furthermore, the phase transition that we identify here has direct physiological relevance — it provides a precise mechanism that relates subcellular defects in Ca cycling to a tissue-scale phenomenon that involves millions of cells. In the future, researchers might be able to utilize this novel perspective to develop new therapeutics that prevent the synchronization transition from occurring.
This article is based on Yohannes Shiferaw’s invited talk at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually this May.
References
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D’Artagnan Greene is a computational biochemist who applies molecular dynamics to understand the molecular basis of cardiac arrhythmias. He is currently a postdoctoral research scientist in the Department of Physics and Astronomy at California State University, Northridge. Yohannes Shiferaw is a physicist who works on the application of computational methods to understand cardiac arrhythmias. He is a professor in the Department of Physics and Astronomy at California State University, Northridge.