The effects of time delays in a phosphorylation-dephosphorylation pathway.

Complex signaling cascades involve many interlocked positive and negative feedback loops which have inherent delays. Modeling these complex cascades often requires a large number of variables and parameters. Delay differential equation models have been helpful in describing inherent time lags and also in reducing the number of governing equations. However the consequences of model reduction via delay differential equations have not been fully explored. In this paper we systematically examine the effect of delays in a complex network of phosphorylation-dephosphorylation cycles (described by Gonze and Goldbeter, J. Theor. Biol., 210, (2001) 167-186), which commonly occur in many biochemical pathways. By introducing delays in the positive and negative regulatory interactions, we show that a delay differential model can indeed reduce the number of cycles actually required to describe the phosphorylation-dephosphorylation pathway. In addition, we find some of the unique properties of the network and a quantitative measure of the minimum number of delay variables required to model the network. These results can be extended for modeling complex signalling cascades.

On the evidence of deterministic chaos in ECG: Surrogate and predictability analysis.

The question whether the human cardiac system is chaotic or not has been an open one. Recent results in chaos theory have shown that the usual methods, such as saturation of correlation dimension D(2) or the existence of positive Lyapunov exponent, alone do not provide sufficient evidence to confirm the presence of deterministic chaos in an experimental system. The results of surrogate data analysis together with the short-term prediction analysis can be used to check whether a given time series is consistent with the hypothesis of deterministic chaos. In this work nonlinear dynamical tools such as surrogate data analysis, short-term prediction, saturation of D(2) and positive Lyapunov exponent have been applied to measured ECG data for several normal and pathological cases. The pathology presently studied are PVC (Premature Ventricular Contraction), VTA (Ventricular Tachy Arrhythmia), AV (Atrio-Ventricular) block and VF (Ventricular Fibrillation). While these results do not prove that ECG time series is definitely chaotic, they are found to be consistent with the hypothesis of chaotic dynamics. (c) 1998 American Institute of Physics.

A simple time delay model for eukaryotic cell cycle.

We propose a seven variable model with time delay in one of the variables for the cell cycle in higher eukaryotes. The model consists of four important phosphorylation-dephosphorylation (P-D) cycles that govern the cell cycle, namely Pre-MPF-MPF, Cdc25P-Cdc25, Wee1P-Wee1 and APCP-APC. Other variables are cyclin, free cyclin dependent kinase (Cdk) and mass. The mass acts as a G2/M checkpoint and the checkpoint is represented by a saddle node loop bifurcation. The key feature of the model is that a time lag has been introduced in the activation of anaphase promoting complex (APC) by maturation promoting factor (MPF). This is effected by treating MPF as a time-delayed variable in the activation step of APC. The time lag acts as a spindle checkpoint. Absence of time delay induces a bistability in our model. Time delay also brings about variability in G1 phase timings. The model also reproduces the mutant phenotype experiments on wee1 cells. Stochasticity has been introduced in the model to simulate the dependence of the cycle time on cell birth length. Mutant phenotypes in the stochastic model reproduce the experimental observations better than the deterministic model.

A two variable delay model for the circadian rhythm of Neurospora crassa.

A two variable model with delay in both the variables, is proposed for the circadian oscillations of protein concentrations in the fungal species Neurospora crassa. The dynamical variables chosen are the concentrations of FRQ and WC-1 proteins. Our model is a two variable simplification of the detailed model of Smolen et al. (J. Neurosci. 21 (2001) 6644) modeling circadian oscillations with interlocking positive and negative feedback loops, containing 23 variables. In our model, as in the case of Smolen's model, a sustained limit cycle oscillation takes place in both FRQ and WC-1 protein in continuous darkness, and WC-1 is anti-phase to FRQ protein, as observed in experiments. The model accounts for various characteristic features of circadian rhythms such as entrainment to light dark cycles, phase response curves and robustness to parameter variation and molecular fluctuations. Simulations are carried out to study the effect of periodic forcing of circadian oscillations by light-dark cycles. The periodic forcing resulted in a rich bifurcation diagram that includes quasiperiodicity and chaotic oscillations, depending on the magnitude of the periodic changes in the light controlled parameter. When positive feedback is eliminated, our model reduces to the generic one dimensional delay model of Lema et al. (J. Theor. Biol. 204 (2000) 565), delay model of the circadian pace maker with FRQ protein as the dynamical variable which represses its own production. This one-dimensional model also exhibits all characteristic features of circadian oscillations and gives rise to circadian oscillations which are reasonably robust to parameter variations and molecular noise.

Concentration of CO2 over melting ice oscillates.

We report that the concentration of CO2 over melting ice oscillates as long as water and ice coexist. A phenomenological model involving melting of CO2 containing ice leading to its release, readsorption of the vapor on ice, and dissolution in water is proposed. Thermokinetics of these processes lead to nonlinearity of the dynamics. This phenomenon is also observed over impure ice contaminated with salts or in the presence of nitrogen or air. Oscillations have been observed in several other solute or ice-water systems.

Nonlinear analysis of continuous ECG during sleep I. Reconstruction.

In recent years evidence has accumulated that ECG signals are of a nonlinear nature. It has been recognized that strictly periodic cardiac rhythms are not accompanied by healthy conditions but, on the contrary, by pathological states. Therefore, the application of methods from nonlinear system theory for the analysis of ECG signals has gained increasing interest. Crucial for the application of nonlinear methods is the reconstruction (embedding) of the time series in a phase space with appropriate dimension. In this study continuous ECG signals of 12 healthy subjects recorded during different sleep stages were analysed. Proper embedding dimension was determined by application of two techniques the false nearest neighbours method and the saturation of the correlation dimension. Results for the ECG signals were compared with findings for simulated data (quasiperiodic dynamics, Lorenz data, white noise) and for phase randomized surrogates. Findings obtained with the two approaches suggest that embedding dimensions from 6 to 8 may be regarded as suitable for the topologically proper reconstruction of ECG signals.

Nonlinear analysis of continuous ECG during sleep II. Dynamical measures.

The hypothesis that cardiac rhythms are associated with chaotic dynamics implicating a healthy flexibility has motivated the investigation of continuous ECG with methods of nonlinear system theory. Sleep is known to be associated with modulations of the sympathetic and parasympathetic control of cardiac dynamics. Thus, the differentiation of ECG signals recorded during different sleep stages can serve to determine the usefulness of nonlinear measures in discriminating ECG states in general. For this purpose the following six nonlinear measures were implemented: correlation dimension D2, Lyapunov exponent L1. Kolmogorov entropy K2, as well as three measures derived from the analysis of unstable periodic orbits. Results of this study show that continuous ECG signals can be differentiated from linear stochastic surrogates by each of the nonlinear measures. The most significant finding with respect to the sleep-related differentiation of ECG signals is an increase in dominant chaoticity assessed by L1 and a reduction in the degrees of freedom estimated by D2 during REM sleep compared to slow wave sleep. Our findings suggest that the increase in dominant chaoticity during REM sleep with regard to time-continuous nonlinear analysis is comparable to an increased heart rate variability. The reduction in the correlation dimension may be interpreted as an expression of the withdrawal of respiratory influences during REM sleep.