Global dynamics of biological systems from time-resolved omics experiments

The emergent properties of biological systems, organized around complex networks of irregularly connected elements, limit the applications of the direct scientific method to their study. The current lack of knowledge opens new perspectives to the inverse scientific paradigm where observations are accumulated and analysed by advanced data-mining techniques to enable a better understanding and the formulation of testable hypotheses about the structure and functioning of these systems. The current technology allows for the wide application of omics analytical methods in the determination of time-resolved molecular profiles of biological samples. Here it is proposed that the theory of dynamical systems could be the natural framework for the proper analysis and interpretation of such experiments. A new method is described, based on the techniques of non-linear time series analysis, which is providing a global view on the dynamics of biological systems probed with time-resolved omics experiments.     

Mutual Information Rate and Bounds for It

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.     

Identification of nonstationary dynamics in physiological recordings

We present a novel framework for the analysis of time series from dynamical systems that alternate between different operating modes. The method simultaneously segments and identifies the dynamical modes by using predictive models. In extension to previous approaches, it allows an identification of smooth transition between successive modes. The method can be used for analysis, diagnosis, prediction, and control. In an application to EEG and respiratory data recorded from humans during afternoon naps, the obtained segmentations of the data agree with the sleep stage segmentation of a medical expert to a large extent. However, in contrast to the manual segmentation, our method does not require a priori knowledge about physiology. Moreover, it has a high temporal resolution and reveals previously unclassified details of the transitions. In particular, a parameter is found that is potentially helpful for vigilance monitoring. We expect that the method will generally be useful for the analysis of nonstationary dynamical systems, which are abundant in medicine, chemistry, biology and engineering. Received: 5 May 1999 / Accepted in revised form: 28 December 1999     

Prospects for advancing the understanding of complex biochemical systems

The application of mathematical theories to understanding the behaviour of complex biochemical systems is reviewed. Key aspects of behaviour are identified as the flux through particular pathways in a steady state, the nature and stability of dynamical states, and the thermodynamic properties of systems. The first of these is dealt primarily in theories of metabolic control, and metabolic control analysis (MCA) is an important example. The valid application of this theory is limited to steady-state systems, and the cases where the essential features of control can be derived from calibration experiments which perturb the state of the system by a sufficiently small amount from its operating point. In practice, time-dependent systems exist, it is not always possible to know a priori whether applied perturbations are sufficiently small, and important features of control may lie farther from the operating point than the application of the theory permits. The nature and stability of dynamical and thermodynamical states is beyond the scope of MCA. To understand the significance of these limitations fully, and to address the dynamical and thermodynamical properties, more complete theories are required. Non-linear systems theory offers the possibility of studying important questions regarding control of steady and dynamical states. It can also link to thermodynamic properties of the system including the energetic efficiency of particular pathways. However, its application requires a more detailed characterisation of the system under study. This extra detail may be an essential feature of the study of non-equilibrium states in general, and non-ideal pathways in particular. Progress requires considerably more widespread integration of theoretical and experimental approaches than currently exists.     

基于估计熵确定心率变异信号中动力噪声强度

目前普遍的观点认为心率变异信号是确定性低维混沌系统产生的,同时该系统受到大量生理噪声的影响。研究了观测噪声和动力噪声对低维混沌系统的影响,提出了一种基于复杂性度量动力噪声强度刻划方法,表明随着动力噪声强度的增加,系统的复杂程度增加。该方法不需要对动力学系统完全重建,并对观测噪声具有一定鲁棒性。最后利用３类共５３只狗的心电图,每只狗的心电数据为１０００个Ｒ－Ｒ间期的微分序列检验了该算法。分析表明,估计出的动力噪声强度与不同的心脏功能状态相关联     

A computational modeling and analysis in cell biological dynamics using electric cell-substrate impedance sensing (ECIS)

In this paper, a study of computational modeling and multi-scale analysis in cell dynamics is presented. Our study aims at: (1) deriving and validating a mathematical model for cell growth, and (2) quantitatively detecting and analyzing the biological interdependencies across multiple observational scales with a variety of time and frequency resolutions. This research was conducted using the time series data practically measured from a novel on-line cell monitoring technique, referred to as electric cell-substrate impedance sensing (ECIS), which allows continuously tracking the cellular behavior such as adhesion, proliferation, spreading and micromotion. First, comparing our ECIS-based cellular growth modeling analysis results with those determined by hematocytometer measurement using different time intervals, we found that the results obtained from both experimental methods consistently agreed. However, our study demonstrated that it is much easier and more convenient to operate with the ECIS system for on-line cellular growth monitoring. Secondly, for multi-scale analysis our results showed that the proposed wavelet-based methodology can effectively quantify the fluctuations associated with cell micromotions and quantitatively capture the biological interdependencies across multiple observational scales. Note that although the wavelet method is well known, its application into the ECIS time series analysis is novel and unprecedented in computational cell biology. Our analyses indicated that the proposed study on ECIS time series could provide a hopeful start and great potentials in both modeling and elucidating the complex mechanisms of cell biological systems.     

Size Effects on Correlation Measures

The detection and quantification of long-range correlations in time series is a fundamental tool to characterize the properties of different dynamical systems, and is applied in many different fields, including physics, biology or engineering. Due to the diversity of applications, many techniques for measuring correlations have been designed. Here, we study systematically the influence of the length of a time series on the results obtained from several techniques commonly used to detect and quantify long-range correlations: the autocorrelation analysis, Hursts analysis, and detrended fluctuation analysis (DFA). Using the Fourier filtering method, we generate artificial time series with known and controlled long-range correlations and with a broad range of lengths, and apply on them the different correlation measures we have studied. Our results indicate that while the DFA method is practically unaffected by the length of the time series, and almost always provides accurate results, the results from Hursts analysis and the autocorrelation analysis strongly depend on the length of the time series.     

Nonlinear analysis of gait kinematics to track changes in oxygen consumption in prolonged load carriage walking: A pilot study

Linking human mechanical work to physiological work for the purpose of developing a model of physical fatigue is a complex problem that cannot be solved easily by conventional biomechanical analysis. The purpose of the study was to determine if two nonlinear analysis methods can address the fundamental issue of utilizing kinematic data to track oxygen consumption from a prolonged walking trial: we evaluated the effectiveness of dynamical systems and fractal analysis in this study. Further, we selected, oxygen consumption as a measure to represent the underlying physiological measure of fatigue. Three male US Army Soldier volunteers (means: 23.3 yr; 1.80 m; 77.3 kg) walked for 120 min at 1.34 m/s with a 40-kg load on a level treadmill. Gait kinematic data and oxygen consumption (VO₂) data were collected over the 120-min period. For the fractal analysis, utilizing stride interval data, we calculated fractal dimension. For the dynamical systems analysis, kinematic angle time series were used to estimate phase space warping based features at uniform time intervals: smooth orthogonal decomposition (SOD) was used to extract slowly time-varying trends from these features. Estimated fractal dimensions showed no apparent trend or correlation with independently measured VO₂. While inter-individual difference did exist in the VO₂ data, dominant SOD time trends tracked and correlated with the VO₂ for all volunteers. Thus, dynamical systems analysis using gait kinematics may be suitable to develop a model to predict physiologic fatigue based on biomechanical work.     

Detecting determinism in short time series, with an application to the analysis of a stationary EEG recording

We have developed a new method for detecting determinism in a short time series and used this method to examine whether a stationary EEG is deterministic or stochastic. The method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded phase space. The angles between two successive directional vectors in the trajectory reconstructed from a time series at a minimum embedding dimension were calculated as a function of time. We measured the irregularity of the angle variations obtained from the time series using second-order difference plots and central tendency measures, and compared these values with those from surrogate data. The ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated time series, including data from Lorenz, R?ssler, and Van der Pol attractors, high-dimensional equations, and 1/f noise. We then applied this method to the analysis of stationary segments of EEG recordings consisting of 750 data points (6-s segments) from five normal subjects. The stationary EEG segments were not found to exhibit deterministic components. This method can be used to analyze determinism in short time series, such as those from physiological recordings, that can be modeled using differentiable dynamical processes.     

Periodic orbits: a new language for neuronal dynamics.

A new nonlinear dynamical analysis is applied to complex behavior from neuronal systems. The conceptual foundation of this analysis is the abstraction of observed neuronal activities into a dynamical landscape characterized by a hierarchy of "unstable periodic orbits" (UPOs). UPOs are rigorously identified in data sets representative of three different levels of organization in mammalian brain. An analysis based on UPOs affords a novel alternative method of decoding, predicting, and controlling these neuronal systems.     

Congruent qualitative behavior of complete and reconstructed phase space trajectories from biomolecular dynamics simulation

Central to the study of a complex dynamical system is knowledge of its phase space behavior. Experimentally, it is rarely possible to record a system's (multidimensional) phase space variables. Rather, the system is observed via one (or few) scalar-valued signal(s) of emission or response. In dynamical systems analysis, the multidimensional phase space of a system can be reconstructed by manipulation of a one-dimensional signal. The trick is in the construction of a (higher-dimensional) space through the use of a time lag (or delay) on the signal time series. The trajectory in this embedding space can then be examined using phase portraits generated in selected subspaces. By contrast, in computer simulation, one has an embarrassment of riches: direct access to the complete multidimensional phase space variables, at arbitrary time resolution and precision. Here, the problem is one of reducing the dimensionality to make analysis tractable. This can be achieved through linear or nonlinear projection of the trajectory into subspaces containing high information content. This study considers trajectories of the small protein crambin from molecular dynamics simulations. The phase space behavior is examined using principal component analysis on the Cartesian coordinate covariance matrix of 138 dimensions. In addition, the phase space is reconstructed from a one dimensional signal, representing the radius of gyration of the structure along the trajectory. Comparison of low-dimensional phase portraits obtained from the two methods shows that the complete phase space distribution is well represented by the reconstruction. The study suggests that it may be possible to develop a deeper connection between the experimental and simulated dynamics of biomolecules via phase space reconstruction using data emerging from recent advances in single-molecule time-resolved biophysical techniques.     

Dimensionality Reduction of Bistable Biological Systems

Time hierarchies, arising as a result of interactions between system’s components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems.     

Automated Design of Complex Dynamic Systems

Several fields of study are concerned with uniting the concept of computation with that of the design of physical systems. For example, a recent trend in robotics is to design robots in such a way that they require a minimal control effort. Another example is found in the domain of photonics, where recent efforts try to benefit directly from the complex nonlinear dynamics to achieve more efficient signal processing. The underlying goal of these and similar research efforts is to internalize a large part of the necessary computations within the physical system itself by exploiting its inherent non-linear dynamics. This, however, often requires the optimization of large numbers of system parameters, related to both the system''s structure as well as its material properties. In addition, many of these parameters are subject to fabrication variability or to variations through time. In this paper we apply a machine learning algorithm to optimize physical dynamic systems. We show that such algorithms, which are normally applied on abstract computational entities, can be extended to the field of differential equations and used to optimize an associated set of parameters which determine their behavior. We show that machine learning training methodologies are highly useful in designing robust systems, and we provide a set of both simple and complex examples using models of physical dynamical systems. Interestingly, the derived optimization method is intimately related to direct collocation a method known in the field of optimal control. Our work suggests that the application domains of both machine learning and optimal control have a largely unexplored overlapping area which envelopes a novel design methodology of smart and highly complex physical systems.     

A Method for Comparing Multivariate Time Series with Different Dimensions

In many situations it is desirable to compare dynamical systems based on their behavior. Similarity of behavior often implies similarity of internal mechanisms or dependency on common extrinsic factors. While there are widely used methods for comparing univariate time series, most dynamical systems are characterized by multivariate time series. Yet, comparison of multivariate time series has been limited to cases where they share a common dimensionality. A semi-metric is a distance function that has the properties of non-negativity, symmetry and reflexivity, but not sub-additivity. Here we develop a semi-metric – SMETS – that can be used for comparing groups of time series that may have different dimensions. To demonstrate its utility, the method is applied to dynamic models of biochemical networks and to portfolios of shares. The former is an example of a case where the dependencies between system variables are known, while in the latter the system is treated (and behaves) as a black box.     

Automated refinement and inference of analytical models for metabolic networks

The reverse engineering of metabolic networks from experimental data is traditionally a labor-intensive task requiring a priori systems knowledge. Using a proven model as a test system, we demonstrate an automated method to simplify this process by modifying an existing or related model--suggesting nonlinear terms and structural modifications--or even constructing a new model that agrees with the system's time series observations. In certain cases, this method can identify the full dynamical model from scratch without prior knowledge or structural assumptions. The algorithm selects between multiple candidate models by designing experiments to make their predictions disagree. We performed computational experiments to analyze a nonlinear seven-dimensional model of yeast glycolytic oscillations. This approach corrected mistakes reliably in both approximated and overspecified models. The method performed well to high levels of noise for most states, could identify the correct model de novo, and make better predictions than ordinary parametric regression and neural network models. We identified an invariant quantity in the model, which accurately derived kinetics and the numerical sensitivity coefficients of the system. Finally, we compared the system to dynamic flux estimation and discussed the scaling and application of this methodology to automated experiment design and control in biological systems in real time.     

Nonlinear biological oscillators: Action-angle variable approach with an application to Cowan's model of neuroelectric activity

We obtain within the action-angle variable approach new expressions, involving the Dirac delta function, for time periods and time averages of dynamical variables which are useful for nonlinear biological oscillator problems. We combine these with Laplace transformation techniques for evaluating the required perturbation expansions. The radii of convergence of these series are determined through a complex variable approach. The method is powerful enough to yield explicit results for such systems as the two species Volterra model, Goodwin's model of protein synthesis etc. and as an illustration, is applied here to Cowan's model of neuroelectric activity. We also point out the usefulness of the action integral in the case where parameters occurring in dynamics have slow time variations.     

Integrating heterogeneous gene expression data for gene regulatory network modelling

Gene regulatory networks (GRNs) are complex biological systems that have a large impact on protein levels, so that discovering network interactions is a major objective of systems biology. Quantitative GRN models have been inferred, to date, from time series measurements of gene expression, but at small scale, and with limited application to real data. Time series experiments are typically short (number of time points of the order of ten), whereas regulatory networks can be very large (containing hundreds of genes). This creates an under-determination problem, which negatively influences the results of any inferential algorithm. Presented here is an integrative approach to model inference, which has not been previously discussed to the authors' knowledge. Multiple heterogeneous expression time series are used to infer the same model, and results are shown to be more robust to noise and parameter perturbation. Additionally, a wavelet analysis shows that these models display limited noise over-fitting within the individual datasets.     

Dynamical Analysis and Visualization of Tornadoes Time Series

In this paper we analyze the behavior of tornado time-series in the U.S. from the perspective of dynamical systems. A tornado is a violently rotating column of air extending from a cumulonimbus cloud down to the ground. Such phenomena reveal features that are well described by power law functions and unveil characteristics found in systems with long range memory effects. Tornado time series are viewed as the output of a complex system and are interpreted as a manifestation of its dynamics. Tornadoes are modeled as sequences of Dirac impulses with amplitude proportional to the events size. First, a collection of time series involving 64 years is analyzed in the frequency domain by means of the Fourier transform. The amplitude spectra are approximated by power law functions and their parameters are read as an underlying signature of the system dynamics. Second, it is adopted the concept of circular time and the collective behavior of tornadoes analyzed. Clustering techniques are then adopted to identify and visualize the emerging patterns.     

Epilepsy and Nonlinear Dynamics

This overview summarizes findings obtained from analyzing electroencephalographic (EEG) recordings from epilepsy patients with methods from the theory of nonlinear dynamical systems. The last two decades have shown that nonlinear time series analysis techniques allow an improved characterization of epileptic brain states and help to gain deeper insights into the spatial and temporal dynamics of the epileptic process. Nonlinear EEG analyses can help to improve the evaluation of patients prior to neurosurgery, and with an unequivocal identification of precursors of seizures, they can be of great value in the development of seizure warning and prevention techniques.