Revised limit cycle oscillator model of human circadian pacemaker

In 1990, Kronauer proposed a mathematical model of the effects of light on the human circadian pacemaker. This study presents several refinements to Kronauer's original model of the pacemaker that enable it to predict more accurately the experimental results from a number of different studies of the effects of the intensity, timing, and duration of light stimuli on the human circadian pacemaker. These refinements include the following: The van der Pol oscillator from Kronauer's model has been replaced with a higher order limit cycle oscillator so that the system's amplitude recovery is slower near the singularity and faster near the limit cycle; the phase and amplitude of the circadian rhythm in sensitivity to light from Kronauer's model has been refined so that the peak sensitivity to light on the limit cycle now occurs approximately 4 h before the core body temperature minimum (CBTmin) and is three times as great as the minimum sensitivity on the limit cycle; the critical phase (at which type 1 phase response curves [PRCs] can be distinguished from type 0 PRCs) that occurs at CBT,n now corresponds to 0.8 h after the minimum of x (x(min) in this refined model rather than to the exact timing of x(min) as in Kronauer's model; a direct effect of light on circadian period was incorporated into the model such that as light intensity increases, the period decreases, which is in accordance with Aschoff's rule.     

Quasi-exponential generation time distributions from a limit cycle oscillator

In spite of the apparently random behaviour and the often exponential distribution of generation times expressed in cell populations, there is evidence for rather precise timekeeping in the cell cycle. In experiments using time-lapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hr intervals. Phase shift responses following application of heat shock, ionizing radiation or serum pulses in each case show a pattern which is repeated twice in cells with an 8-9 hr modal generation time. We describe here a cell cycle model with an independent cellular clock controlling cell cycle events which accounts for the phase response data, while also reconciling the stochastic and periodic behaviour characteristic of animal cells.     

Comparison of amplitude recovery dynamics of two limit cycle oscillator models of the human circadian pacemaker

At an organism level, the mammalian circadian pacemaker is a two-dimensional system. For these two dimensions, phase (relative timing) and amplitude of the circadian pacemaker are commonly used. Both the phase and the amplitude (A) of the human circadian pacemaker can be observed within multiple physiological measures--including plasma cortisol, plasma melatonin, and core body temperature (CBT)--all of which are also used as markers of the circadian system. Although most previous work has concentrated on changes in phase of the circadian system, critically timed light exposure can significantly reduce the amplitude of the pacemaker. The rate at which the amplitude recovers to its equilibrium level after reduction can have physiological significance. Two mathematical models that describe the phase and amplitude dynamics of the pacemaker have been reported. These models are essentially equivalent in predictions of phase and in predictions of amplitude recovery for small changes from an equilibrium value (A = 1), but are markedly different in the prediction of recovery rates when A < 0.6. To determine which dynamic model best describes the amplitude recovery observed in experimental data; both models were fit to CBT data using a maximum likelihood procedure and compared using Akaike's Information Criterion (AIC). For all subjects, the model with the lower recovery rate provided a better fit to data in terms of AIC, supporting evidence that the amplitude recovery of the endogenous pacemaker is slow at low amplitudes. Experiments derived from model predictions are proposed to test the influence of low amplitude recovery on the physiological and neurobehavioral functions.     

The deterministic limit of a stochastic logistic model with individual variation

We consider a Markov chain model similar to the stochastic logistic model except that it allows for variation amongst individuals in the population. We prove that as the population size grows, the process can be approximated by a deterministic process. The equilibrium points of the limiting process and their stability are determined. Applications to modelling epidemics and metapopulations are discussed.     

A data-driven model of the generation of human EEG based on a spatially distributed stochastic wave equation

We discuss a model for the dynamics of the primary current density vector field within the grey matter of human brain. The model is based on a linear damped wave equation, driven by a stochastic term. By employing a realistically shaped average brain model and an estimate of the matrix which maps the primary currents distributed over grey matter to the electric potentials at the surface of the head, the model can be put into relation with recordings of the electroencephalogram (EEG). Through this step it becomes possible to employ EEG recordings for the purpose of estimating the primary current density vector field, i.e. finding a solution of the inverse problem of EEG generation. As a technique for inferring the unobserved high-dimensional primary current density field from EEG data of much lower dimension, a linear state space modelling approach is suggested, based on a generalisation of Kalman filtering, in combination with maximum-likelihood parameter estimation. The resulting algorithm for estimating dynamical solutions of the EEG inverse problem is applied to the task of localising the source of an epileptic spike from a clinical EEG data set; for comparison, we apply to the same task also a non-dynamical standard algorithm.     

A stochastic, molecular model of the fission yeast cell cycle: role of the nucleocytoplasmic ratio in cycle time regulation

We propose a stochastic version of a recently published, deterministic model of the molecular mechanism regulating the mitotic cell cycle of fission yeast, Schizosaccharomyces pombe. Stochasticity is introduced in two ways: (i) by considering the known asymmetry of cell division, which produces daughter cells of slightly different sizes; and (ii) by assuming that the nuclear volumes of the two newborn cells may also differ. In this model, the accumulation of cyclins in the nucleus is proportional to the ratio of cytoplasmic to nuclear volumes. We have simulated the cell-cycle statistics of populations of wild-type cells and of wee1(-) mutant cells. Our results are consistent with well known experimental observations.     

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.     

A reinvestigation of the Geman-Miller respiratory oscillator model

The model of Geman-Miller of the respiratory oscillator is reinvestigated for its interpretation of the parameters: W and T. It was found that the interpretation of Geman-Miller, that the parameters T and W represent the chemosensitive feedback, is incorrect. The extension to the model made by Engeman and Swanson is not necessary to produce afterdischarge. It is demonstrated that the afterdischarge can be predicted in the original Geman-Miller model from the Jacobian Matrix.This work was supported by grant 13-36-17 of the Netherlands Foundation for the Advancement of pure Research (Z.W.O.)     

Effects of asymmetric division on a stochastic model of the cell division cycle

The stochastic model of cell division formulated by Alt and Tyson is generalized to the case of imprecise binary fission. Closed-form expressions are derived for the generation-time distribution, the birth-size and division-size distributions, the beta curve, and the correlation coefficient of generation times of sister cells. The theoretical results are compared to observations of cell division statistics in a culture of fission yeast.     

Cell cycle variability: The oscillator model of the cell cycle yields transition probability alpha and beta type curves

The limit cycle concept of cell replication attributes cell cycle variability to continuous random modulation of the rates of reactions forming the intracellular control system believed to be responsible for replication processes. It is shown that this model can yield frequency histograms and both alpha and beta type accumulative distribution curves (with respect to generation times and also to cycle phases) which are of the various forms seen experimentally. The results thus provide additional support for this concept.     

A physiological model of a circannual oscillator

Recent evidence based on studies in hypothalamo-pituitary disconnected Soay sheep suggests that the generation of circannual rhythms may be local to specific tissues or physiological systems. Now, the authors present a physiological model of a circannual rhythm generator centered in the pituitary gland based on the interaction between melatonin-responsive cells in the pars tuberalis that act to decode photoperiod, and lactotroph cells of the adjacent pars distalis that secrete prolactin. The model produces a self-sustained, circannual rhythm in endocrine output that the authors explore by mathematical modeling. The circannual oscillation requires a delayed negative feedback mechanism. The authors highlight specific features of the pituitary dynamics as a guide to future research on circannual rhythms.     

An oscillator cell cycle model needs no first or second chance event

Theoretical data is presented to support the view that a limit cycle oscillator model of the cell cycle (as originally described) which accounts for random triggering of replication, can also accomodate the experimental results which have caused the supporters of the Transition Probability model to introduce the occurrence of yet a second chance event. In particular, it explains why the addition of a suboptimal amount of mitogen in a split dose alters the probability of triggering at each treatment but, generally, has little effect on the delay before the initiation of S-phase. It is also confirmed that, other than in some exceptional circumstances, lowering the probability of triggering in the oscillator model may have little effect on the minimum “intermitotic” time (as required by the evidence of the Transition Probability proponents) and, moreover, the oscillator concept provides reason why this should be so.     

A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.     

The "battle of the sexes": a genetic model with limit cycle behavior

A two-locus genetic model, based on Dawkins "sex war" game, with the fitness of the genotypes at each locus depending on the gene frequencies at the other, is shown to give rise to a stable limit cycle. The mathematical analysis involves averaging techniques and elliptic integrals.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

A stochastic model for cancer risk

We propose a simple stochastic model based on the two successive mutations hypothesis to compute cancer risks. Assume that only stem cells are susceptible to the first mutation and that there are a total of D stem cell divisions over the lifetime of the tissue with a first mutation probability mu(1) per division. Our model predicts that cancer risk will be low if m = mu(1)D is low even in the case of very advantageous mutations. Moreover, if mu(1)D is low the mutation probability of the second mutation is practically irrelevant to the cancer risk. These results are in contrast with existing models but in agreement with a conjecture of Cairns. In the case where m is large our model predicts that the cancer risk depends crucially on whether the first mutation is advantageous or not. A disadvantageous or neutral mutation makes the risk of cancer drop dramatically.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.