Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales ( e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Quantized cell cycle times: interaction between a relaxation oscillator and ultradian clock pulses

Control of the timing of cell division is considered to result from a relaxation cell cycle oscillator: this has one slow and one rapid component and obeys a system of two ordinary differential equations. Interactions of the slow component with an ultradian oscillator leads to quantization of cell cycle times when the free parameters of the cell cycle oscillator are chosen close to its bifurcation point. This model fits the experimental results previously reported.     

Interaction of mitotic oscillators as a source of variability of cell cycle duration

Interaction between membrane mitotic oscillators at the expense of exchange with the molecules of lipids (slow variable) and antioxidants (fast variable) was considered. Parameters of all the oscillators are equal, excluding a small noise added to the equation for lipids. These parameters are chosen in such a way that the oscillators are not far from the transition to the stable stationary state. The numerical modeling has shown that the exchange with lipids brings about the appearance of an additional limit cycle whose period is significantly greater than that of an autonomous oscillator. The addition of noise averages the behaviour of oscillators, and distribution according to cycle duration becomes broad and bimodal. Thus the exchange of the slow variable increases the dispersion of distribution of cell generation times. This conclusion seems to be true for any oscillator with similar dynamic properties.     

Dependency of size of Saccharomyces cerevisiae cells on growth rate.

The mean size and percentage of budded cells of a wild-type haploid strain of Saccharomyces cerevisiae grown in batch culture over a wide range of doubling times (tau) have been measured using microscopic measurements and a particle size analyzer. Mean size increased over a 2.5-fold range with increasing growth rate (from tau = 450 min to tau = 75 min). Mean size is principally a function of growth rate and not of a particular carbon source. The duration of the budded phase increased at slow growth rates according to the empirical equation, budded phase = 0.5 tau + 27 (all in minutes). Using a recent model of the cell cycle in which division is thought to be asymmetric, equations have been derived for mean cell age and mean cell volume. The data are consistent with the notion that initiation of the cell cycle occurs at "start" after attainment of a critical cell size, and this size is dependent on growth rate, being, at slow growth rates, 63% of the volume of fast growth rates. Previous reports are reanalyzed in the light of the unequal division model and associated population equations.     

Morphological analysis of nuclear separation and cell division during the life cycle of Escherichia coli.

Quantitative electron microscope observations were performed on Escherichia coli B/r after balanced growth with doubling times (tau) of 32 and 60 min. The experimental approach allowed the timing of morphological events during the cell cycle by classifying serially sectioned cells according to length. Visible separation of the nucleoplasm was found to coincide with the time of termination of chromosome replication as predicted by the Cooper-Helmstetter model. The duration of the process of constrictive cell division (10 min) appeared to be independent of the growth rate for tau equals 60 min or less but to increase with increase doubling time in more slowly growing cells. Physiological division, i.e., compartmentalization prior to physical separation of the cells, was only observed to occur in the last minute of the cell cycle. The morphological results indicate that cell elongation continues during the division process in cells with tau equals 32 min, but fails to continue in cells with tau equals 60 min.     

Alteration of period and amplitude of circadian rhythms in shift workers. With special reference to temperature, right and left hand grip strength

48 male shift workers in various industries volunteered to document circadian rhythms in sleeping and working, oral temperature, grip strength of both hands, peak expiratory flow and heart rate. All physiological variables were self-measured 4 to 5 times a day for 2 to 4 weeks. Individual time series were analyzed according to several statistical methods (power spectrum, cosinor, chi squares, ANOVA, correlation, etc.) in order to estimate rhythm parameters such as circadian period (tau) and amplitude (A), and to evaluate subgroup differences with regard to tolerance to shift work, age, duration of shift work, speed of rotation and type of industry. The present study confirms for oral temperature and extends to other variables (grip strength of both hands, heart rate) that intolerance to shift work is frequently associated with both internal desynchronization and small circadian amplitude. The internal desynchronization among several circadian rhythms supports the hypothesis that these latter are driven by several oscillators. Many differences were observed between circadian rhythms in right and left hand grip strength: circadian tau in oral temperature was correlated with that in the grip strength of the dominant hand but not with that of the other hand; changes in tau s of the non-dominant hand were age-related but did not correlate with temperature tau; only the circadian A of the non-dominant hand was associated with a desynchronization. Thus, circadian rhythms in oral temperature and dominant hand grip strength may be driven by the same oscillator while that of the non-dominant hand may be governed by a different one. Internal desynchronization between both hand grip rhythms as well as desynchronization of performance rhythms reported by others provide indirect evidence that circadian oscillator(s) may be located in the human cerebral cortex.     

Neurogenesis and cell cycle-reactivated neuronal death during pathogenic tau aggregation

The aim of the present study was to investigate the relation between neurogenesis, cell cycle reactivation and neuronal death during tau pathology in a novel tau transgenic mouse line THY-Tau22 with two frontotemporal dementia with parkinsonism linked to chromosome-17 mutations in a human tau isoform. This mouse displays all Alzheimer disease features of neurodegeneration and a broad timely resolution of tau pathology with hyperphosphorylation of tau at younger age (up to 6 months) and abnormal tau phosphorylation and tau aggregation in aged mice (by 10 months). Here, we present a follow-up of cell cycle markers with aging in control and transgenic mice from different ages. We show that there is an increased neurogenesis during tau hyperphosphorylation and cell cycle events during abnormal tau phosphorylation and tau aggregation preceding neuronal death and neurodegeneration. However, besides phosphorylation, other mechanisms including tau mutations and changes in tau expression and/or splicing may be also involved in these mechanisms of cell cycle reactivation. Altogether, these data suggest that cell cycle events in THY-Tau22 are resulting from neurogenesis in young animals and cell death in older ones. It suggests that neuronal cell death in such models is much more complex than believed.     

Properties of mutual coupling between the two circadian pacemakers in the eyes of the mollusc Bulla gouldiana.

We examined, in vitro, the effects of changing the free-running period (tau) of one oscillator on the phase relationship between the circadian rhythms of impulse activity in the optic nerves that are driven by the bilaterally paired ocular pacemakers in Bulla gouldiana. One eye of the coupled pair was treated either with lithium artificial seawater (to lengthen tau) or with low-chloride artificial seawater (to shorten tau). The results suggested that the coupling is relatively weak, since the majority (9 to 16) of eyes were unable to maintain a stable phase relationship when tau differences between the eyes were only about 1 hr. When stable phase differences were achieved, the tau of the coupled system was intermediate between the tau's of the individual oscillators, and the eye with the shorter intrinsic tau would invariably phase-lead the pair. Interestingly, in a few instances, pairs of eyes that had desynchronized by 9.5-10.5 hr resynchronized within a single cycle via a massive phase advance in the rhythm from the phase-lagging eye. The result suggests the existence of a novel phase-shifting mechanism that is part of the mutual coupling pathway. We found evidence that connection of the eye with the cerebral ganglion increases the tau of the ocular pacemaker, suggesting that efferent signals from the central nervous system influence tau. These signals may also modulate the phase-shifting response.     

Simulating psoriasis by altering transit amplifying cells

Computational models of tissue homeostasis will facilitate a deeper understanding of many diseases. They link molecular networks, cellular differentiation and the spatial and temporal organization of tissues. Here we show an approach which is able to computationally turn a healthy in silico epidermis into one with four central properties of psoriatic epidermis. We achieve this by altering a single simulation parameter in the cellular differentiation program of the simulated epidermal keratinocytes: the fractional time period during which transit amplifying cells proliferate (tau). Prolonging tau results in the four main pathological characteristics of psoriatic skin: (1) an absolute increase of the germinative compartment, (2) an absolute increase of the differentiated compartment, (3) a higher proportion of germinative cells and (4) a marked reduction in turnover time. The prolongation of tau is able to increase the proliferation capacity of the epidermal tissue without altering the cell cycle frequency.     

The ups and downs of biological timers

Background  

The need to execute a sequence of events in an orderly and timely manner is central to many biological processes, including cell cycle progression and cell differentiation. For self-perpetuating systems, such as the cell cycle oscillator, delay times between events are defined by the network of interacting proteins that propagates the system. However, protein levels inside cells are subject to genetic and environmental fluctuations, raising the question of how reliable timing is maintained.     

Chlamydomonas reinhardtii: duration of its cell cycle and phases at growth rates affected by temperature

Synchronized cultures of the green alga Chlamydomonas reinhardtii were grown photoautotrophically under a wide range of environmental conditions including temperature (15–37°C), different mean light intensities (132, 150, 264 μmol m⁻² s⁻¹), different illumination regimes (continuous illumination or alternation of light/dark periods of different durations), and culture methods (batch or continuous culture regimes). These variable experimental approaches were chosen in order to assess the role of temperature in the timing of cell division, the length of the cell cycle and its pre- and post-commitment phases. Analysis of the effect of temperature, from 15 to 37°C, on synchronized cultures showed that the length of the cell cycle varied markedly from times as short as 14 h to as long as 36 h. We have shown that the length of the cell cycle was proportional to growth rate under any given combination of growth conditions. These findings were supported by the determination of the temperature coefficient (Q ₁₀), whose values were above the level expected for temperature-compensated processes. The data presented here show that cell cycle duration in C. reinhardtii is a function of growth rate and is not controlled by a temperature independent endogenous timer or oscillator, including a circadian one.     

Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data

We consider an age-maturity structured model arising from a blood cell proliferation problem. This model is "hybrid", i.e., continuous in time and age but the maturity variable is discrete. This is due to the fact that we include the cell division marker carboxyfluorescein diacetate succinimidyl ester. We use our mathematical analysis in conjunction with experimental data taken from the division analysis of primitive murine bone marrow cells to characterize the maturation/proliferation process. Cell cycle parameters such as proliferative rate beta, cell cycle duration tau, apoptosis rate gamma, and loss rate micro can be evaluated from CarboxyFluorescein diacetate Succinimidyl Ester + cell tracking experiments. Our results indicate that after three days in vitro, primitive murine bone marrow cells have parameters beta = 2.2 day(-1), tau = 0.3 day, gamma = 0.3 day(-1), and micro = 0.05 day(-1).     

Length growth of two Escherichia coli B/r substrains.

Length growth of synchronized Escherichia coli B/r substrain A (ATCC 12407) and B/r substrain F26 (Thy his) was followed with an electron microscope. Cells were grown with doubling times (tau) of 60 min (B/rA) and of 82 and 165 min (B/rF26). Different length growth patterns were found for the two substrains. In B/rF, the length growth rate increased about midway in the cell cycle. For tau = 165 min, the rate increase was preceded by a short period of slow growth. For B/r A (r = 60 min), this period seemed to occur at the beginning of the cell cycle. The possibility is raised that the different length growth patterns are related to different deoxyribonucleic acid replication patterns of the respective strains.     

Determination of erythrocyte transit times through micropores. I--Basic operational principles

To study the transit times of each red blood cell passing through cylindrical micropores and in order to evaluate sub-population of cells with regard to their deformability, we have developed a new system called the cell transit time analyser (CTTA). By using an AC voltage (100 KHz) across a special filter, we measure the electrical conductance change produced by the cells passing through the pores under a known driving pressure. This computer based device provides the distribution of transit times tau for 2000 cells in 1 minute and as a result the mean transit time [tau]. Experiments with red cells were designed to evaluate the flow behavior of both normal cells and cells whose mechanical properties were artificially altered. Cell volume was changed by use of non-isotonic media. Cell shape and cell volume were modified by varying the pH of the suspending buffer. Results of these experiments are: 1) a skew distribution of transit times towards high tau values for both control cells and artificially altered cells is observed: 2) [tau] is minimum for isotonic conditions and increases sharply for either hypotonic or hypertonic media: 3) [tau] is minimum at physiological pH and increases for either acid or alcaline changes of pH.     

An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasi-linear viscoelastic theory

The quasi-linear viscoelastic (QLV) theory proposed by Fung (1972) has been frequently used to model the nonlinear time- and history-dependent viscoelastic behavior of many soft tissues. It is common to use five constants to describe the instantaneous elastic response (constants A and B) and reduced relaxation function (constants C, tau 1, and tau 2) on experiments with finite ramp times followed by stress relaxation to equilibrium. However, a limitation is that the theory is based on a step change in strain which is not possible to perform experimentally. Accounting for this limitation may result in regression algorithms that converge poorly and yield nonunique solutions with highly variable constants, especially for long ramp times (Kwan et al. 1993). The goal of the present study was to introduce an improved approach to obtain the constants for QLV theory that converges to a unique solution with minimal variability. Six goat femur-medial collateral ligament-tibia complexes were subjected to a uniaxial tension test (ramp time of 18.4 s) followed by one hour of stress relaxation. The convoluted QLV constitutive equation was simultaneously curve-fit to the ramping and relaxation portions of the data (r2 > 0.99). Confidence intervals of the constants were generated from a bootstrapping analysis and revealed that constants were distributed within 1% of their median values. For validation, the determined constants were used to predict peak stresses from a separate cyclic stress relaxation test with averaged errors across all specimens measuring less than 6.3 +/- 6.0% of the experimental values. For comparison, an analysis that assumed an instantaneous ramp time was also performed and the constants obtained for the two approaches were compared. Significant differences were observed for constants B, C, tau 1, and tau 2, with tau 1 differing by an order of magnitude. By taking into account the ramping phase of the experiment, the approach allows for viscoelastic properties to be determined independent of the strain rate applied. Thus, the results obtained from different laboratories and from different tissues may be compared.     

Changes in Oscillatory Dynamics in the Cell Cycle of Early Xenopus laevis Embryos

During the early development of Xenopus laevis embryos, the first mitotic cell cycle is long (∼85 min) and the subsequent 11 cycles are short (∼30 min) and clock-like. Here we address the question of how the Cdk1 cell cycle oscillator changes between these two modes of operation. We found that the change can be attributed to an alteration in the balance between Wee1/Myt1 and Cdc25. The change in balance converts a circuit that acts like a positive-plus-negative feedback oscillator, with spikes of Cdk1 activation, to one that acts like a negative-feedback-only oscillator, with a shorter period and smoothly varying Cdk1 activity. Shortening the first cycle, by treating embryos with the Wee1A/Myt1 inhibitor PD0166285, resulted in a dramatic reduction in embryo viability, and restoring the length of the first cycle in inhibitor-treated embryos with low doses of cycloheximide partially rescued viability. Computations with an experimentally parameterized mathematical model show that modest changes in the Wee1/Cdc25 ratio can account for the observed qualitative changes in the cell cycle. The high ratio in the first cycle allows the period to be long and tunable, and decreasing the ratio in the subsequent cycles allows the oscillator to run at a maximal speed. Thus, the embryo rewires its feedback regulation to meet two different developmental requirements during early development.     

Timing the start of division in E. coli: a single-cell study

We monitor the shape dynamics of individual E. coli cells using time-lapse microscopy together with accurate image analysis. This allows measuring the dynamics of single-cell parameters throughout the cell cycle. In previous work, we have used this approach to characterize the main features of single-cell morphogenesis between successive divisions. Here, we focus on the behavior of the parameters that are related to cell division and study their variation over a population of 30 cells. In particular, we show that the single-cell data for the constriction width dynamics collapse onto a unique curve following appropriate rescaling of the corresponding variables. This suggests the presence of an underlying time scale that determines the rate at which the cell cycle advances in each individual cell. For the case of cell length dynamics a similar rescaling of variables emphasizes the presence of a breakpoint in the growth rate at the time when division starts, tau(c). We also find that the tau(c) of individual cells is correlated with their generation time, tau(g), and inversely correlated with the corresponding length at birth, L(0). Moreover, the extent of the T-period, tau(g) - tau(c), is apparently independent of tau(g). The relations between tau(c), tau(g) and L(0) indicate possible compensation mechanisms that maintain cell length variability at about 10%. Similar behavior was observed for both fast-growing cells in a rich medium (LB) and for slower growth in a minimal medium (M9-glucose). To reveal the molecular mechanisms that lead to the observed organization of the cell cycle, we should further extend our approach to monitor the formation of the divisome.     

The ups and downs of modeling the cell cycle

We discuss the impact of mathematical modeling on our understanding of the cell cycle. Although existing, detailed models confirm that the known interactions in the cell cycle can produce oscillations and predict behaviors such as hysteresis, they contain many parameters and are poorly constrained by data which are almost all qualitative. Questions about the basic architecture of the oscillator may be more amenable to modeling approaches that ignore molecular details. These include asking how the various elaborations of the basic oscillator affect the robustness of the system and how cells monitor their size and use this information to control the cell cycle.