Long-range power-law correlations have been reported recently for DNA sequences containing noncoding regions. We address the question of whether such correlations may be a trivial consequence of the known mosaic structure ("patchiness") of DNA. We analyze two classes of controls consisting of patchy nucleotide sequences generated by different algorithms--one without and one with long-range power-law correlations. Although both types of sequences are highly heterogenous, they are quantitatively distinguishable by an alternative fluctuation analysis method that differentiates local patchiness from long-range correlations. Application of this analysis to selected DNA sequences demonstrates that patchiness is not sufficient to account for long-range correlation properties.

Abstract-The

The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium-like state [Physiol. Rev. 9, 399-431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343-1346 (1993); Fractals in Biology and Medicine (Birkhauser-Verlag, Basel, 1994), pp. 55-65] reveal that under normal conditions, beat-to-beat fluctuations in heart rate display the kind of long-range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381-384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior. We describe a new method--detrended fluctuation analysis (DFA)--for quantifying this correlation property in non-stationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long-range scaling exponents. This method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties.

Abstract

Traditional approaches to measuring the complexity of biological signals fail to account for the multiple time scales inherent in such time series. These algorithms have yielded contradictory findings when applied to real-world datasets obtained in health and disease states. We describe in detail the basis and implementation of the multiscale entropy (MSE) method. We extend and elaborate previous findings showing its applicability to the fluctuations of the human heartbeat under physiologic and pathologic conditions. The method consistently indicates a loss of complexity with aging, with an erratic cardiac arrhythmia (atrial fibrillation), and with a life-threatening syndrome (congestive heart failure). Further, these different conditions have distinct MSE curve profiles, suggesting diagnostic uses. The results support a general "complexity-loss" theory of aging and disease. We also apply the method to the analysis of coding and noncoding DNA sequences and find that the latter have higher multiscale entropy, consistent with the emerging view that so-called "junk DNA" sequences contain important biological information.

According to classical concepts of physiologic control, healthy systems are self-regulated to reduce variability and maintain physiologic constancy. Contrary to the predictions of homeostasis, however, the output of a wide variety of systems, such as the normal human heartbeat, fluctuates in a complex manner, even under resting conditions. Scaling techniques adapted from statistical physics reveal the presence of long-range, power-law correlations, as part of multifractal cascades operating over a wide range of time scales. These scaling properties suggest that the nonlinear regulatory systems are operating far from equilibrium, and that maintaining constancy is not the goal of physiologic control. In contrast, for subjects at high risk of sudden death (including those with heart failure), fractal organization, along with certain nonlinear interactions, breaks down. Application of fractal analysis may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as to monitoring the aging process. Similar approaches show promise in assessing other regulatory systems, such as human gait control in health and disease. Elucidating the fractal and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the postgenomic era.A hallmark of physiologic systems is their extraordinary complexity. The nonstationarity and nonlinearity of signals ( Fig. 1) generated by living organisms defy traditional mechanistic approaches based on homeostasis and conventional biostatistical methodologies. Recognition that physiologic time series contain ''hidden information'' has fueled growing interest in applying concepts and techniques from statistical physics, including chaos theory, to a wide range of biomedical problems from molecular to organismic levels (1, 2).This presentation describes one area of investigation that has engaged our collaborative attention, namely, fractal analysis of physiologic time series in health and disease. The discussion will focus primarily on certain features of the human heartbeat, one important example of complex physiologic fluctuations. The dynamics of another physiologic control system-human gait-is also briefly discussed. Recognizing that this topic represents only one selected aspect of the broad and rapidly expanding applications of complexity theory to biomedicine (Table 1), readers are referred to a number of useful reviews and references therein (1, 3-10).A motivating problem for our work is depicted in Fig. 1, which presents a dynamical self-test. Shown are 30-min heart rate time series from four subjects. Only one is from a healthy individual; the other three are from patients with life-threatening forms of heart disease. The problem is to identify the normal record. The (perhaps nonintuitive) answer to this ''test'' is given in the figure caption. Beyond its obvious diagnostic import, the problem of classifying temporal assays of integrated cardiac physiology has implications for understanding a...

DNA sequences have been analysed using models, such as an n-step Markov chain, that incorporate the possibility of short-range nucleotide correlations. We propose here a method for studying the stochastic properties of nucleotide sequences by constructing a 1:1 map of the nucleotide sequence onto a walk, which we term a 'DNA walk'. We then use the mapping to provide a quantitative measure of the correlation between nucleotides over long distances along the DNA chain. Thus we uncover in the nucleotide sequence a remarkably long-range power law correlation that implies a new scale-invariant property of DNA. We find such long-range correlations in intron-containing genes and in nontranscribed regulatory DNA sequences, but not in complementary DNA sequences or intron-less genes.

There is evidence that physiological signals under healthy conditions may have a fractal temporal structure. Here we investigate the possibility that time series generated by certain physiological control systems may be members of a special class of complex processes, termed multifractal, which require a large number of exponents to characterize their scaling properties. We report on evidence for multifractality in a biological dynamical system, the healthy human heartbeat, and show that the multifractal character and nonlinear properties of the healthy heart rate are encoded in the Fourier phases. We uncover a loss of multifractality for a life-threatening condition, congestive heart failure.