Analysis of biochemical phase shift oscillators by a harmonic balancing technique

The use of harmonic balancing techniques for theoretically investigating a large class of biochemical phase shift oscillators is outlined and the accuracy of this approximate technique for large dimension nonlinear chemical systems is considered. It is concluded that for the equations under study these techniques can be successfully employed to both find periodic solutions and to indicate those cases which can not oscillate. The technique is a general one and it is possible to state a step by step procedure for its application. It has a substantial advantage in producing results which are immediately valid for arbitrary dimension. As the accuracy of the method increases with dimension, it complements classical small dimension methods. The results obtained by harmonic balancing analysis are compared with those obtained by studying the local stability properties of the singular points of the differential equation. A general theorem is derived which identifies those special cases where the results of first order harmonic balancing are identical to those of local stability analysis, and a necessary condition for this equivalence is derived. As a concrete example, the n-dimensional Goodwin oscillator is considered where p, the Hill coefficient of the feedback metabolite, is equal to three and four. It is shown that for p = 3 or 4 and n less than or equal to 4 the approximation indicates that it is impossible to construct a set of physically permissible reaction constants such that the system possesses a periodic solution. However for n greater than or equal to 5 it is always possible to find a large domain in the reaction constant space giving stable oscillations. A means of constructing such a parameter set is given. The results obtained here are compared with previously derived results for p = 1 and p = 2.     

Metastable epialleles in mammals

There are some mammalian alleles that display the unusual characteristic of variable expressivity in the absence of genetic heterogeneity. It has recently become evident that this is because the activity of these alleles is dependent on their epigenetic state. Interestingly, the epigenetic state is somewhat labile, resulting in phenotypic mosaicism between cells (variegation) and also between individuals (variable expressivity). The establishment of the epigenetic state occurs during early embryogenesis and is a probabilistic event that is influenced by whether the allele is carried on the paternal or maternal alleles. In addition, the epigenetic state determines whether these alleles are dominant. We propose that mammalian alleles with such characteristics should be termed metastable epialleles to distinguish them from traditional alleles. At this stage, it is unclear how common these alleles are, but an appreciation of their existence will aid in their identification.     

A balancing act

Mary E. Martin, DVM, MPH, DACLAM, Chief of Veterinary and Educational Services, Cornell University, Center for Animal Resources and Education, Ithaca, NY.     

Cytosolic calcium oscillators

Many cells display oscillations in intracellular calcium resulting from the periodic release of calcium from intracellular reservoirs. Frequencies are varied, but most oscillations have periods ranging from 5 to 60 s. For any given cell, frequency can vary depending on external conditions, particularly the concentration of natural stimuli or calcium. This cytosolic calcium oscillator is particularly sensitive to those stimuli (neurotransmitters, hormones, growth factors) that hydrolyze phosphoinositides to give diacylglycerol and inositol 1,4,5-trisphosphate (Ins1,4,5P3). The ability of Ins1,4,5P3 to mobilize intracellular calcium is a significant feature of many of the proposed models that are used to explain oscillatory activity. Receptor-controlled oscillator models propose that there are complex feedback mechanisms that generate oscillations in the level of Ins1,4,5P3. Second messenger-controlled oscillator models demonstrate that the oscillator is a component of the calcium reservoir, which is induced to release calcium by a constant input of either Ins1,4,5P3 or calcium itself. In the latter case, the process of calcium-induced calcium release might be the basis of oscillatory activity in many cell types. The function of calcium oscillations is still unknown. Because oscillator frequency can vary with agonist concentration, calcium transients might be part of a frequency-encoded signaling system. When an external stimulus arrives at the cell surface the information is translated into a train of calcium spikes, i.e., the signal is digitized. Certain cells may then convey information by varying the frequency of this digital signal.     

Metastable Oxygen Molecules in the Troposphere

The sources and steady-state concentration of singlet oxygen in the atmosphere are assessed in view of potential effects on the biosphere. Collision-induced absorption of sunlight by molecular oxygen in 1 atm of air produces O₂(a'δ_g) at a rate P = 1.6 × 10'cm's^?1 in bright sunlight. Less than 10% are added to this purely natural source by the photolysis of ozone, and by anthropogenic sensitizers (SO₂, NO₂, volatile aromatics). Collisional quenching of O₂(a'δ_g) by ground state oxygen establishes a steady-state concentration of ca. 1.7 × 10⁸cm^?1'. Reactions of singlet oxygen with other atmospheric pollutants are entirely negligible when compared with the concurrent reactions of ambient OH and 0₃. Potential effects of atmospheric singlet oxygen on the biosphere are limited by the deposition rate F< 0.051 P which depends on the production rate P of O₂(a' δ_g) in the air layer immediately above the flat surface.     

Population Dynamics of Metastable Growth-Rate Phenotypes

Neo-Darwinian evolution has presented a paradigm for population dynamics built on random mutations and selection with a clear separation of time-scales between single-cell mutation rates and the rate of reproduction. Laboratory experiments on evolving populations until now have concentrated on the fixation of beneficial mutations. Following the Darwinian paradigm, these experiments probed populations at low temporal resolution dictated by the rate of rare mutations, ignoring the intermediate evolving phenotypes. Selection however, works on phenotypes rather than genotypes. Research in recent years has uncovered the complexity of genotype-to-phenotype transformation and a wealth of intracellular processes including epigenetic inheritance, which operate on a wide range of time-scales. Here, by studying the adaptation dynamics of genetically rewired yeast cells, we show a novel type of population dynamics in which the intracellular processes intervene in shaping the population structure. Under constant environmental conditions, we measure a wide distribution of growth rates that coexist in the population for very long durations (>100 generations). Remarkably, the fastest growing cells do not take over the population on the time-scale dictated by the width of the growth-rate distributions and simple selection. Additionally, we measure significant fluctuations in the population distribution of various phenotypes: the fraction of exponentially-growing cells, the distributions of single-cell growth-rates and protein content. The observed fluctuations relax on time-scales of many generations and thus do not reflect noisy processes. Rather, our data show that the phenotypic state of the cells, including the growth-rate, for large populations in a constant environment is metastable and varies on time-scales that reflect the importance of long-term intracellular processes in shaping the population structure. This lack of time-scale separation between the intracellular and population processes calls for a new framework for population dynamics which is likely to be significant in a wide range of biological contexts, from evolution to cancer.     

Modeling synthetic gene oscillators

Genetic oscillators have long held the fascination of experimental and theoretical synthetic biologists alike. From an experimental standpoint, the creation of synthetic gene oscillators represents a yardstick by which our ability to engineer synthetic gene circuits can be measured. For theorists, synthetic gene oscillators are a playground in which to test mathematical models for the dynamics of gene regulation. Historically, mathematical models of synthetic gene circuits have varied greatly. Often, the differences are determined by the level of biological detail included within each model, or which approximation scheme is used. In this review, we examine, in detail, how mathematical models of synthetic gene oscillators are derived and the biological processes that affect the dynamics of gene regulation.     

Metastable polymerization of sickle hemoglobin in droplets

Sickle cell disease arises from a genetic mutation of one amino acid in each of the two hemoglobin beta chains, leading to the polymerization of hemoglobin in the red cell upon deoxygenation, and is characterized by vascular crises and tissue damage due to the obstruction of small vessels by sickled cells. It has been an untested assumption that, in red cells that sickle, the growing polymer mass would consume monomers until the thermodynamically well-described monomer solubility was reached. By photolysing droplets of sickle hemoglobin suspended in oil we find that polymerization does not exhaust the available store of monomers, but stops prematurely, leaving the solutions in a supersaturated, metastable state typically 20% above solubility at 37 degrees C, though the particular values depend on the details of the experiment. We propose that polymer growth stops because the growing ends reach the droplet edge, whereas new polymer formation is thwarted by long nucleation times, since the concentration of hemoglobin is lowered by depletion of monomers into the polymers that have formed. This finding suggests a new aspect to the pathophysiology of sickle cell disease; namely, that cells deoxygenated in the microcirculation are not merely undeformable, but will actively wedge themselves tightly against the walls of the microvasculature by a ratchet-like mechanism driven by the supersaturated solution.     

Stuck in a balancing act

Comment on: Lee, et al. Cancer Res 2011; 71:3921-31.