Modeling Temperature Compensation in Chemical and Biological Oscillators

All physicochemical and biological oscillators maintain a balance between destabilizing reactions (as, for example, intrinsic autocatalytic or amplifying reactions) and stabilizing processes. These two groups of processes tend to influence the period in opposite directions and may lead to temperature compensation whenever their overall influence balances. This principle of “antagonistic balance” has been tested for several chemical and biological oscillators. The Goodwin negative feedback oscillator appears of particular interest for modeling the circadian clocks in Neurospora and Drosophila and their temperature compensation. Remarkably, the Goodwin oscillator not only gives qualitative, correct phase response curves for temperature steps and temperature pulses, but also simulates the temperature behavior of Neurospora frq and Drosophila per mutants almost quantitatively. The Goodwin oscillator predicts that circadian periods are strongly dependent on the turnover of the clock mRNA or clock protein. A more rapid turnover of clock mRNA or clock protein results, in short, a slower turnover in longer period lengths. (Chronobiology International, 14(5), 499–510, 1997)     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

Weakly Circadian Cells Improve Resynchrony

The mammalian suprachiasmatic nuclei (SCN) contain thousands of neurons capable of generating near 24-h rhythms. When isolated from their network, SCN neurons exhibit a range of oscillatory phenotypes: sustained or damping oscillations, or arrhythmic patterns. The implications of this variability are unknown. Experimentally, we found that cells within SCN explants recover from pharmacologically-induced desynchrony by re-establishing rhythmicity and synchrony in waves, independent of their intrinsic circadian period We therefore hypothesized that a cell''s location within the network may also critically determine its resynchronization. To test this, we employed a deterministic, mechanistic model of circadian oscillators where we could independently control cell-intrinsic and network-connectivity parameters. We found that small changes in key parameters produced the full range of oscillatory phenotypes seen in biological cells, including similar distributions of period, amplitude and ability to cycle. The model also predicted that weaker oscillators could adjust their phase more readily than stronger oscillators. Using these model cells we explored potential biological consequences of their number and placement within the network. We found that the population synchronized to a higher degree when weak oscillators were at highly connected nodes within the network. A mathematically independent phase-amplitude model reproduced these findings. Thus, small differences in cell-intrinsic parameters contribute to large changes in the oscillatory ability of a cell, but the location of weak oscillators within the network also critically shapes the degree of synchronization for the population.     

Positive Feedback Promotes Oscillations in Negative Feedback Loops

A simple three-component negative feedback loop is a recurring motif in biochemical oscillators. This motif oscillates as it has the three necessary ingredients for oscillations: a three-step delay, negative feedback, and nonlinearity in the loop. However, to oscillate, this motif under the common Goodwin formulation requires a high degree of cooperativity (a measure of nonlinearity) in the feedback that is biologically “unlikely.” Moreover, this recurring negative feedback motif is commonly observed augmented by positive feedback interactions. Here we show that these positive feedback interactions promote oscillation at lower degrees of cooperativity, and we can thus unify several common kinetic mechanisms that facilitate oscillations, such as self-activation and Michaelis-Menten degradation. The positive feedback loops are most beneficial when acting on the shortest lived component, where they function by balancing the lifetimes of the different components. The benefits of multiple positive feedback interactions are cumulative for a majority of situations considered, when benefits are measured by the reduction in the cooperativity required to oscillate. These positive feedback motifs also allow oscillations with longer periods than that determined by the lifetimes of the components alone. We can therefore conjecture that these positive feedback loops have evolved to facilitate oscillations at lower, kinetically achievable, degrees of cooperativity. Finally, we discuss the implications of our conclusions on the mammalian molecular clock, a system modeled extensively based on the three-component negative feedback loop.     

Unfolding an electronic integrate-and-fire circuit

Many physical and biological phenomena involve accumulation and discharge processes that can occur on significantly different time scales. Models of these processes have contributed to understand excitability self-sustained oscillations and synchronization in arrays of oscillators. Integrate-and-fire (I+F) models are popular minimal fill-and-flush mathematical models. They are used in neuroscience to study spiking and phase locking in single neuron membranes, large scale neural networks, and in a variety of applications in physics and electrical engineering. We show here how the classical first-order I+F model fits into the theory of nonlinear oscillators of van der Pol type by demonstrating that a particular second-order oscillator having small parameters converges in a singular perturbation limit to the I+F model. In this sense, our study provides a novel unfolding of such models and it identifies a constructible electronic circuit that is closely related to I+F.     

The complex circadian system of Gonyaulax polyedra

Circadian oscillations are a fundamental biological property from bacteria to humans. The molecular mechanisms which produce a ca 24-h rhythmicity are still unknown but it has become clear that they are part of the biochemical machinery of the single cell. The cellular circadian system can be favorably studied in single-cell organisms such as the dinoflagellate Gonyaulax polyedra . The complexity of this circadian model system, which consists of at least two circadian oscillators, receives light via two input systems with different spectral sensitivities, and has several feed–back loops between the central oscillator(s) and the environment, is described here.     

The Impact of Time Delays on the Robustness of Biological Oscillators and the Effect of Bifurcations on the Inverse Problem

Differential equation models for biological oscillators are often not robust with respect to parameter variations. They are based on chemical reaction kinetics, and solutions typically converge to a fixed point. This behavior is in contrast to real biological oscillators, which work reliably under varying conditions. Moreover, it complicates network inference from time series data. This paper investigates differential equation models for biological oscillators from two perspectives. First, we investigate the effect of time delays on the robustness of these oscillator models. In particular, we provide sufficient conditions for a time delay to cause oscillations by destabilizing a fixed point in two-dimensional systems. Moreover, we show that the inclusion of a time delay also stabilizes oscillating behavior in this way in larger networks. The second part focuses on the inverse problem of estimating model parameters from time series data. Bifurcations are related to nonsmoothness and multiple local minima of the objective function.     

The temporal ‘structure’ and function of the insect photoperiodic clock: a tribute to Colin S. Pittendrigh

In 1936, Erwin Bünning suggested that photoperiodic time measurement was a function of the circadian system. Colin Pittendrigh became an ardent supporter of Bünning's hypothesis, drawing parallels between photoperiodism and his own group's investigations of adult eclosion rhythmicity in the fruit fly Drosophila pseudoobscura. They developed several more modern versions of Bünning's general hypothesis based on the entrainment of circadian oscillations to the light cycle, including ‘external coincidence’, which is a derivation of Bünning's original model, and ‘internal coincidence’, which relied upon seasonal changes in the mutual phase relationship of oscillators within a multi‐oscillator circadian system. This review considers the experimental evidence for the central role of the circadian system in photoperiodic timing and, in some species, for both external and internal coincidence. Pittendrigh, however, pursued the idea of internal coincidence further with his analysis of the pacemaker–slave organization of eclosion rhythmicity in D. pseudoobscura and proposed a similar theoretical model for photoperiodism comprising a group of slave oscillators driven by a light‐sensitive pacemaker. In this model, the phase relationships of the slaves to the pacemaker were affected by (i) the relative periods of the pacemaker and slave(s); (ii) the strength(s) of the coupling between the two; and (iii) the dampening coefficients of the various slaves. Manipulation of these variables showed that the slaves adopted different internal phase relationships (both to each other and to the pacemaker) under the influence of changes in daily photophase, the period of the Zeitgeber and phase shifts of the entraining light cycle.     

Mode coupling in living systems: implications for biology and medicine

Complex systems, and in particular biological ones, are characterized by large numbers of oscillations of widely differing frequencies. Various prejudices tend to lead to the assumption that such oscillators should generically be very weakly interacting. This paper reviews the basic ideas of linearity and nonlinearity as seen by a physicist, but with a view to biological systems. In particular, it is argued that large couplings between different oscillators of disparate frequencies are common, being present even in rather simple systems which are well-known in physics, although this issue is often glossed over. This suggests new experiments and investigations, as well as new approaches to therapies and human-environment interactions which, without the concepts described here, may otherwise seem unlikely to be interesting. The style of the paper is conversational with a minimum of mathematics, and no attempt at a complete list of references.     

A computational model of the hypothalamic - pituitary - gonadal axis in female fathead minnows (Pimephales promelas) exposed to 17α-ethynylestradiol and 17β-trenbolone

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.     

Entrainment of two coupled van der pol oscillators by an external oscillation

A system composed of two coupled internal oscillators and an external oscillation is studied as a model of biological control systems. The type of interaction between the internal oscillators is a mutual and dissipative one. Three macroscopic states of the internal oscillators are demonstrated in the absence of the external oscillation. Strict and loose entrainment regions of the internal oscillators by the external oscillation are shown in respect to the intensity of the mutual interaction, the intrinsic frequency difference of the internal oscillations, and the magnitude and frequency of the external oscillation. On the other hand, an idea of holonic system is introduced and the fundamental properties of the model as a holonic system are elucidated.     

From lamprey to salamander: an exploratory modeling study on the architecture of the spinal locomotor networks in the salamander

The evolutionary transition from water to land required new locomotor modes and corresponding adjustments of the spinal “central pattern generators” for locomotion. Salamanders resemble the first terrestrial tetrapods and represent a key animal for the study of these changes. Based on recent physiological data from salamanders, and previous work on the swimming, limbless lamprey, we present a model of the basic oscillatory network in the salamander spinal cord, the spinal segment. Model neurons are of the Hodgkin–Huxley type. Spinal hemisegments contain sparsely connected excitatory and inhibitory neuron populations, and are coupled to a contralateral hemisegment. The model yields a large range of experimental findings, especially the NMDA-induced oscillations observed in isolated axial hemisegments and segments of the salamander Pleurodeles waltlii. The model reproduces most of the effects of the blockade of AMPA synapses, glycinergic synapses, calcium-activated potassium current, persistent sodium current, and $h$ -current. Driving segments with a population of brainstem neurons yields fast oscillations in the in vivo swimming frequency range. A minimal modification to the conductances involved in burst-termination yields the slower stepping frequency range. Slow oscillators can impose their frequency on fast oscillators, as is likely the case during gait transitions from swimming to stepping. Our study shows that a lamprey-like network can potentially serve as a building block of axial and limb oscillators for swimming and stepping in salamanders.     

A Model of Activity Generation in the Epileptic Focus

A model of activity generation in the epileptic focus was proposed based on the ideas that synchronization occurs between the oscillations that arise to regulate the resting potential level and that neuronal discharges are synchronized with slow oscillations. Four phases were identified in seizure development: an increase in slow-wave activity, high-frequency spike activity, synchronization in the presence of sharp peaks, and extinction of the seizure. The model was compared with an intraoperative electrocorticography record.

     

Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions

Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, and we demonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.     

Inter- and intraspecific diversity of ontogeny and fecundity patterns in relation to reproductive strategy choice in Myomorpha (Rodentia: Calomyscidae,Cricetidae, Muridae)

Studying the life history of various organisms is essential for ecological and evolutionary inferences. However, publications regarding models of evolutionary shifts to either r- or K-reproductive strategies are scarce. Here, the combined original and previously published data on reproduction and development of Goodwin’s brush-tailed mouse (Calomyscus elburzensis), golden hamster (Mesocricetus auratus), and house mouse (Mus musculus) were compared. An adult male and female in estrous state (from each species) were coupled with each other for a night and embryos were harvested at embryonic days (E) 10–17 from pregnant females. The mean gestation period in Goodwin’s brush-tailed mouse was 31.5?±?2.1 days; growth was rapid at the first half of this period and after that it was reduced by half. During the postnatal period, growth was rapid up to weaning, but after day 35, the rate of growth decreased abruptly. Growth rate of head and body length was very rapid prior to weaning (PN35) in comparison with weight but the rates of the two characters were inversed following weaning. Totally, weight gain was more dynamic in rate, as compared with head and body length, up to adultness. For golden hamster and house mouse, gestation period was 15.6?±?0.8 and 20.8?±?0.8 days, respectively. Growth was slow during approximately the first half of the gestation period in both species but increased during the second half. During postnatal period in golden hamster, growth rate of head and body length was rapid both before and after weaning as compared with that of weight. However, head and body length showed a stable rate of growth before and after weaning (around PN15), but for body weight, the growth rate was slightly faster after weaning. Similarly, changes in head and body length in house mouse were rapid both before and after weaning (PN16) as compared with that of weight. Since approximately two-fold increases were observed in the growth rates of weight and head and body length in the second stage of postnatal development (PN16-adult) compared with the rates during the first stage (PN0-PN16), both showed dynamic changes during the postnatal development in this species. Typically, r-strategic house mouse appeared to be more similar in early morphological development to comparatively K-strategic Goodwin’s brush-tailed mouse, than to comparatively r-strategic golden hamster. Nevertheless, growth rate and pattern were different in the three species after birth. Hence, results showed that peculiarities of association between an ontogenetic pattern and a tendency in reproductive strategy can differ in diverse groups of myomorph rodents.

     

The manipulation of calcium oscillations by harnessing self-organisation

This paper investigates how self-organisation might be harnessed for the manipulation and control of calcium oscillations. Calcium signalling mechanisms are responsible for a number of important functions within biological systems, such as fertilization, secretion, contraction, neuronal signalling and learning. In this paper, calcium oscillations are investigated as a biological periodic process. Within biological systems such periodic behaviour is one of the outcomes from self-organisation. The understanding of periodic processes in living systems can enable more accurate diagnosis and physiologically suitable clinical therapies to be proposed, for diseases such as cancer, epilepsy, cardiac diseases and other dynamic diseases. In this paper these ideas are investigated by means of the calcium-induced calcium release (CICR) model and a number of representative simulations of intra and inter-cellular calcium oscillations are used to illustrate the manipulation and control of these oscillations in normal and pathological situations.