Dynamics of karyotype evolution.

In the evolution of species, the karyotype changes with a timescale of tens to hundreds of thousand years. In the development of cancer, the karyotype often is modified in cancerous cells over the lifetime of an individual. Characterizing these changes and understanding the mechanisms leading to them has been of interest in a broad range of disciplines including evolution, cytogenetics, and cancer genetics. A central issue relates to the relative roles of random vs deterministic mechanisms in shaping the changes. Although it is possible that all changes result from random events followed by selection, many results point to other non-random factors that play a role in karyotype evolution. In cancer, chromosomal instability leads to characteristic changes in the karyotype, in which different individuals with a specific type of cancer display similar changes in karyotype structure over time. Statistical analyses of chromosome lengths in different species indicate that the length distribution of chromosomes is not consistent with models in which the lengths of chromosomes are random or evolve solely by simple random processes. A better understanding of the mechanisms underlying karyotype evolution should enable the development of quantitative theoretical models that combine the random and deterministic processes that can be compared to experimental determinations of the karyotype in diverse settings.

Rhythms from Two Competing Periodic Sources Embedded in an Excitable Medium.

In an excitable medium, a stimulus generates a wave that propagates in space until it reaches the boundary or collides with another wave and annihilates. We study the dynamics generated by two periodic sources with different frequencies in excitable cardiac tissue culture using optogenetic techniques. The observed rhythms, which can be modeled using cellular automata and studied analytically, show unexpected regularities related to classic results in number theory. We apply the results to identify cardiac arrhythmias in people that are due to a putative mechanism of two competing pacemakers.

Some elements for a history of the dynamical systems theory.

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Using mathematics to diagnose, cure, and predict cardiac arrhythmia.

Mathematics can be used to analyze and model cardiac arrhythmia. I discuss three different problems. (1) Diagnosis of atrial fibrillation based on the time intervals between subsequent beats. The probability density histograms of the differences of the intervals between consecutive beats have characteristic shapes for atrial fibrillation. (2) Curing atrial fibrillation by ablation of the core of rotors. Recent clinical studies have proposed that ablating the core of rotors in atrial tissue can cure atrial fibrillation. However, the claims are controversial. One problem that arises relates to difficulties associated with developing algorithms to identify the core of rotors. In model tissue culture systems, heterogeneity in the structure makes it difficult to unambiguously locate the core of rotors. (3) Risk stratification for sudden cardiac death (SCD). Despite numerous clinical studies, there is still a need for improved criteria to assess the risk of SCD. I discuss the possibility of using the dynamics of premature ventricular complexes to help make predictions. The development of wearable devices to record and analyze cardiac rhythms offers new prospects for the diagnosis and treatment of cardiac arrhythmia.

Universal mechanisms for self-termination of rapid cardiac rhythm.

Excitable media sustain circulating waves. In the heart, sustained circulating waves can lead to serious impairment or even death. To investigate factors affecting the stability of such waves, we have used optogenetic techniques to stimulate a region at the apex of a mouse heart at a fixed delay after the detection of excitation at the base of the heart. For long delays, rapid circulating rhythms can be sustained, whereas for shorter delays, there are paroxysmal bursts of activity that start and stop spontaneously. By considering the dependence of the action potential and conduction velocity on the preceding recovery time using restitution curves, as well as the reduced excitability (fatigue) due to the rapid excitation, we model prominent features of the dynamics including alternation of the duration of the excited phases and conduction times, as well as termination of the bursts for short delays. We propose that this illustrates universal mechanisms that exist in biological systems for the self-termination of such activities.

Double-wave reentry in excitable media.

A monolayer of chick embryo cardiac cells grown in an annular geometry supports two simultaneous reentrant excitation waves that circulate as a doublet. We propose a mechanism that can lead to such behavior. The velocity restitution gives the instantaneous velocity of a wave as a function of the time since the passage of the previous wave at a given point in space. Nonmonotonic restitution relationships will lead to situations in which various spacings between circulating waves are possible. In cardiology, the situation in which two waves travel in an anatomically defined circuit is referred to as double-wave reentry. Since double-wave reentry may arise as a consequence of pacing during cardiac arrhythmias, understanding the dynamic features of double-wave reentry may be helpful in understanding the physiological properties of cardiac tissue and in the design of therapy.

Hybrid models of genetic networks: Mathematical challenges and biological relevance.

We review results concerning dynamics in a class of hybrid ordinary differential equations which incorporates logical control to yield piecewise linear equations. These equations relate qualitative features of the structure of networks to qualitative properties of the dynamics. Because of their simple structure, they have been studied using techniques from discrete mathematics and nonlinear dynamics. Initially developed as a qualitataive description of gene regulatory networks, many generalizations of the basic approach have been developed. In particular, we show how this qualitative approach may be adapted to switching biochemical systems without degradation, illustrated by an example of a motif in which two branches of a pathway may be regulated differently when the thresholds for the two pathways are separated.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems.

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

Defining the pattern of initiation of monomorphic ventricular tachycardia using the beat-to-beat intervals recorded on implantable cardioverter defibrillators from the RAFT study: A computer-based algorithm.

Arrhythmia onset pattern may have important implications on morbidity, recurrent implantable cardioverter defibrillator (ICD) shocks, and mortality, given the proposed correlation between initiation pattern and arrhythmia mechanism. Therefore, we developed and tested a computer-based algorithm to differentiate the pattern of initiation based on the beat-to-beat intervals of the ventricular tachycardia (VT) episodes in ICD recordings from the Resynchronization-Defibrillation for Ambulatory Heart Failure Trial (RAFT). Intervals on intracardiac electrograms from ICDs were analyzed backwards starting from the marker of VT detection, comparing each interval with the average tachycardia cycle length. If the morphology of the beat initiating the VT was similar to the morphology of the VT itself, the episode was considered sudden. If the morphology of the beat initiating the VT was not similar to the morphology of the VT itself, the episode was considered non-sudden. The capability of the algorithm to classify the pattern of initiation based only on the beat-to-beat intervals allows for the classification and analysis of large datasets to further investigate the clinical importance of classifying VT initiation. If analysis of the VT initiation proves to be of clinical value, this algorithm could potentially be integrated into ICD software, which would make it easily accessible and potentially helpful in clinical decision-making.

Varieties of reentrant dynamics.

Experiments were carried out in monolayer tissue cultures of embryonic chick heart cells imaged using a calcium sensitive fluorescent dye. The cells were grown in annular geometries and in annular geometries with an isthmus connecting antipodal region of the annulus. We observed a large number of spatially different patterns of propagation consisting of one or more circulating waves. As well, we also observed rhythms in which rotors embedded in the annuli generated propagating pulses. These results demonstrate that many different patterns of excitation can be present in cardiac tissue with simple geometries.

Demonstration of cardiac rotor and source mapping techniques in embryonic chick monolayers.

Excitable media, such as the heart, display propagating waves with different geometries including target patterns and rotors (spiral waves). Collision of two waves leads to annihilation of both. We present algorithms for data processing and analysis to identify the core of rotors. In this work, we show that as the spatial sampling resolution decreases it becomes increasingly difficult to identify rotors-there are instances of false negatives and false positives. These observations are relevant to current controversies concerning the role of rotors in the initiation, maintenance, and treatment of cardiac arrhythmias, especially atrial fibrillation. Currently some practitioners target the core of rotors for ablation, but the effectiveness of this procedure has been questioned. In view of the difficulties inherent in the identification of rotors, we conclude that methods to identify rotors need to first be validated prior to assessing the efficacy of ablation.

Predicting the risk of sudden cardiac death.

Sudden cardiac death (SCD) is the result of a change of cardiac activity from normal (typically sinus) rhythm to a rhythm that does not pump adequate blood to the brain. The most common rhythms leading to SCD are ventricular tachycardia (VT) or ventricular fibrillation (VF). These result from an accelerated ventricular pacemaker or ventricular reentrant waves. Despite significant efforts to develop accurate predictors for the risk of SCD, current methods for risk stratification still need to be improved. In this article we briefly review current approaches to risk stratification. Then we discuss the mathematical basis for dynamical transitions (called bifurcations) that may lead to VT and VF. One mechanism for transition to VT or VF involves a perturbation by a premature ventricular complex (PVC) during sinus rhythm. We describe the main mechanisms of PVCs (reentry, independent pacemakers and abnormal depolarizations). An emerging approach to risk stratification for SCD involves the development of individualized dynamical models of a patient based on measured anatomy and physiology. Careful analysis and modelling of dynamics of ventricular arrhythmia on an individual basis will be essential in order to improve risk stratification for SCD and to lay a foundation for personalized (precision) medicine in cardiology.

Slowing Down of Recovery as Generic Risk Marker for Acute Severity Transitions in Chronic Diseases.

OBJECTIVE: We propose a novel paradigm to predict acute attacks and exacerbations in chronic episodic disorders such as asthma, cardiac arrhythmias, migraine, epilepsy, and depression. A better generic understanding of acute transitions in chronic dynamic diseases is increasingly important in critical care medicine because of the higher prevalence and incidence of these chronic diseases in our aging societies. DATA SOURCES: PubMed, Medline, and Web of Science. STUDY SELECTION: We selected studies from biology and medicine providing evidence of slowing down after a perturbation as a warning signal for critical transitions. DATA EXTRACTION: Recent work in ecology, climate, and systems biology has shown that slowing down of recovery upon perturbations can indicate loss of resilience across complex, nonlinear biologic systems that are approaching a tipping point. This observation is supported by the empiric studies in pathophysiology and controlled laboratory experiments with other living systems, which can flip from one state of clinical balance to a contrasting one. We discuss examples of such evidence in bodily functions such as blood pressure, heart rate, mood, and respiratory regulation when a tipping point for a transition is near. CONCLUSIONS: We hypothesize that in a range of chronic episodic diseases, indicators of critical slowing down, such as rising variance and temporal correlation, may be used to assess the risk of attacks, exacerbations, and even mortality. Identification of such early warning signals over a range of diseases will enhance the understanding of why, how, and when attacks and exacerbations will strike and may thus improve disease management in critical care medicine.

Functional characterization of oscillatory and excitable media.

Cardiac and neural systems share common features of intrinsic oscillation in some cells as well as the ability to propagate excitation. One theoretical approach to study such systems is to develop realistic models for the tissue. This involves first developing detailed ionic "Hodgkin-Huxley"-type models of individual cells and then connecting the individual cells via synaptic and gap junctions in realistic geometries. An alternative approach is to characterize tissue in terms of functional properties such as phase resetting curves and restitution curves. Using simple models based on one-dimensional difference equations, the measured functional curves can be used to predict, analyze, and interpret nonlinear dynamical phenomena. This approach offers the prospects of providing simplified descriptions that offer insight into the experimental and clinical cardiac dynamics.

Vitamin D receptor as a master regulator of the c-MYC/MXD1 network.

Vitamin D signaling regulates cell proliferation and differentiation, and epidemiological data suggest that it functions as a cancer chemopreventive agent, although the underlying mechanisms are poorly understood. Vitamin D signaling can suppress expression of genes regulated by c-MYC, a transcription factor that controls epidermal differentiation and cell proliferation and whose activity is frequently elevated in cancer. We show through cell- and animal-based studies and mathematical modeling that hormonal 1,25-dihydroxyvitamin D (1,25D) and the vitamin D receptor (VDR) profoundly alter, through multiple mechanisms, the balance in function of c-MYC and its antagonist the transcriptional repressor MAD1/MXD1. 1,25D inhibited transcription of c-MYC-regulated genes in vitro, and topical 1,25D suppressed expression of c-MYC and its target setd8 in mouse skin, whereas MXD1 levels increased. 1,25D inhibited MYC gene expression and accelerated its protein turnover. In contrast, it enhanced MXD1 expression and stability, dramatically altering ratios of DNA-bound c-MYC and MXD1. Remarkably, F-box protein FBW7, an E3-ubiquitin ligase, controlled stability of both arms of the c-MYC/MXD1 push-pull network, and FBW7 ablation attenuated 1,25D regulation of c-MYC and MXD1 turnover. Additionally, c-MYC expression increased upon VDR knockdown, an effect abrogated by ablation of MYC regulator ß-catenin. c-MYC levels were widely elevated in vdr(-/-) mice, including in intestinal epithelium, where hyperproliferation has been reported, and in skin epithelia, where phenotypes of VDR-deficient mice and those overexpressing epidermal c-MYC are similar. Thus, 1,25D and the VDR regulate the c-MYC/MXD1 network to suppress c-MYC function, providing a molecular basis for cancer preventive actions of vitamin D.

Dynamical disease: Challenges for nonlinear dynamics and medicine.

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of some bodily function. There is a large literature in mathematics and physics which proposes mathematical models for the physiological systems and carries out analyses of the properties of these models using nonlinear dynamics concepts involving analyses of the stability and bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.

Chaotic dynamics and diffusion in a piecewise linear equation.

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

Spatial symmetry breaking determines spiral wave chirality.

Chirality represents a fundamental property of spiral waves. Introducing obstacles into cardiac monolayers leads to the initiation of clockwise-rotating, counterclockwise-rotating, and pairs of spiral waves. Simulations show that the precise location of the obstacle and the pacing frequency determine spiral wave chirality. Instabilities predicted by curves relating the action potential duration and the pacing frequency at different spatial locations predict sites of wave break initiation and, hence, spiral wave chirality.

Patterns of ectopy leading to increased risk of fatal or near-fatal cardiac arrhythmia in patients with depressed left ventricular function after an acute myocardial infarction.

AIMS: To identify potential new markers for assessing the risk of sudden arrhythmic events based on a method that captures features of premature ventricular complexes (PVCs) in relation to sinus RR intervals in Holter recordings (heartprint). METHODS AND RESULTS: Holter recordings obtained 6 weeks after acute myocardial infarction from 227 patients with reduced ventricular function (left ventricular ejection fraction ≤ 40%) were used to produce heartprints. Measured indices were: PVCs per hour, standard deviation of coupling interval (SDCI), and the number of occurrences of the most prevalent form of PVCs (SNIB). Predictive values, survival analysis, and Cox regression with adjustment for clinical variables were performed based on primary endpoint, defined as an electrocardiogram-documented fatal or near-fatal arrhythmic event, death from any cause, and cardiac death. High ectopy (PVCs per hour ≥10) was a predictor of all endpoints. Repeating forms of PVCs (SNIB ≥ 83) was a predictor of primary endpoint, hazard ratio = 3.5 (1.3-9.5), and all-cause death, hazard ratio = 2.8 (1.1-7.3), but not cardiac death. SDCI ≤ 80 ms was a predictor of all-cause death and cardiac death, but not of primary endpoint. CONCLUSION: High ectopy, prevalence of repeating forms of PVCs, and low coupling interval variability are potentially useful risk markers of fatal or near-fatal arrhythmias after myocardial infarction.

Introduction to focus issue: quantitative approaches to genetic networks.

All cells of living organisms contain similar genetic instructions encoded in the organism's DNA. In any particular cell, the control of the expression of each different gene is regulated, in part, by binding of molecular complexes to specific regions of the DNA. The molecular complexes are composed of protein molecules, called transcription factors, combined with various other molecules such as hormones and drugs. Since transcription factors are coded by genes, cellular function is partially determined by genetic networks. Recent research is making large strides to understand both the structure and the function of these networks. Further, the emerging discipline of synthetic biology is engineering novel gene circuits with specific dynamic properties to advance both basic science and potential practical applications. Although there is not yet a universally accepted mathematical framework for studying the properties of genetic networks, the strong analogies between the activation and inhibition of gene expression and electric circuits suggest frameworks based on logical switching circuits. This focus issue provides a selection of papers reflecting current research directions in the quantitative analysis of genetic networks. The work extends from molecular models for the binding of proteins, to realistic detailed models of cellular metabolism. Between these extremes are simplified models in which genetic dynamics are modeled using classical methods of systems engineering, Boolean switching networks, differential equations that are continuous analogues of Boolean switching networks, and differential equations in which control is based on power law functions. The mathematical techniques are applied to study: (i) naturally occurring gene networks in living organisms including: cyanobacteria, Mycoplasma genitalium, fruit flies, immune cells in mammals; (ii) synthetic gene circuits in Escherichia coli and yeast; and (iii) electronic circuits modeling genetic networks using field-programmable gate arrays. Mathematical analyses will be essential for understanding naturally occurring genetic networks in diverse organisms and for providing a foundation for the improved development of synthetic genetic networks.