Synchronization of non-linear biochemical oscillators coupled by diffusion

The behaviour of similar coupled non-linear oscillators of the type \(\dot x\) =f(x, y, µ \(\dot y\) =g(x, y, µ is to be investigated. The oscillators are assumed to be coupled by diffusion gradients. If some conditions on the magnitude of the diffusion coefficients are satisfied, it is proved that: 1) if the oscillators have the same period (identical value of the parameter μ) and different phases before coupling, after coupling they tend to synchronize the phases; 2) if the periods of the oscillators are not too different (in terms of the values of the parameter μ) before coupling, after coupling they tend to oscillate with the same period. It is suggested the possible role of diffusion as a synchronizing mechanism in some biological phenomena.     

Design principles of biochemical oscillators

Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions. Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism.     

Analysis of entrainment of circadian oscillators by skeleton photoperiods using phase transition curves

A skeleton photoperiod consists of two short pulses which are applied on the circadian oscillator at times corresponding to the beginning and to the end of a continuous light stimulus. To study several problems in entrainment of circadian rhythms by skeleton photoperiods, we develop a simple diagrammatic solution of the steady state entrainment making use of phase transition curves which are directly gotten from phase response curves. The graphical method is simple and systematic to study entrainment by light cycles with various day lengths. As the method is also intuitive, we can easily examine three problems. (1) In Drosophila the phase relation (ψ) between rhythm and light cycle is a continuous function of day length of skeleton photoperiods up to about 12 h, but a marked discontinuity (ψ-jump) sets in between 13 and 14h. By the diagrammatic method we find that ψ-jump is mathematically a bifurcation phenomenon. (2) The action of photoperiods up to about 12 h is fully simulated by two 15-min skeleton pulses. Do 3-min skeleton pulses imitate the complete photoperiods? We find that pulse width is arbitrary to some extent. (3) Why skeleton photoperiods up to about 12 h are good models of complete photoperiods? The reason is the small amplitude and the nearly symmetrical form of phase response curves in the subjective day.     

Populations of biochemical oscillators as circadian clocks

Various types of populations of interacting oscillators were analyzed and their synchronization states were determined. One of the systems involving biochemical oscillators was simulated on the computer and the occurence of rhythm splitting was observed. A comparison of its attributes with experimental results on circadian ryhythms showed good agreement. This allows us to distinguish between types of mechanisms held responsible for the splitting phenomenon in the past. The present model also offers a new explanation about the differences of light action on diurnal and nocturnal organisms.     

Homeostasis of the period of post-translational biochemical oscillators

Generally, circadian clocks or biological oscillations are resistant to external conditions such as temperature and nutrient concentration. We propose that enzyme-limited competition provides a general mechanism of homeostasis of the period of post-translational oscillators based on protein modifications, and demonstrate it by nutrient compensation in a theoretical model of cyanobacterial circadian clock. The rate change by nutrient concentration is counterbalanced by the amount of available free enzyme, which occurs because of the competition among the various substrates for the limited enzyme. The temperature and nutrient compensation are determined by the postulate that the catalytic modification reactions are rate limiting.     

Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

Selective visual attention in a neurocomputational model of phase oscillators

In order to understand the dynamic property of covert selective visual attention, which is different from the proposed mechanism of the spotlight metaphor, a two-layered network of phase oscillators was developed. The first layer is related to the hippocampus and controls attention focus formation. The second layer is related to the visual cortex, and each cortical oscillator in it simulates an assembly of cells coding for a particular stimulus in the sense of feature binding. Selective visual attention is interpreted as the result of the emergent synchronization of hippocampus oscillators and a part of cortical oscillators. Numerical experiments are presented to illustrate attention focus formation and attention shifting from one set of stimuli to another. From a neurocomputational point of view, our results demonstrate that attention is an emergent property of the dynamical cell assemblies responding to the whole visual field. Received: 2 January 1998 / Accepted in revised form: 10 November 1998     

Cluster associative memory formation in a three-layer network of phase oscillators

A three-layer network model of oscillatory associative memory is proposed. The network is capable of storing binary images, which can be retrieved upon presenting an appropriate stimulus. Binary images are encoded in the form of the spatial distribution of oscillatory phase clusters in-phase and anti-phase relative to a reference periodic signal. The information is loaded into the network using a set of interlayer connection weights. A condition for error-free pattern retrieval is formulated, delimiting the maximal number of patterns to be stored in the memory (storage capacity). It is shown that the capacity can be significantly increased by generating an optimal alphabet (basis pattern set). The number of stored patterns can reach values of the network size (the number of oscillators in each layer), which is significantly higher than the capacity of conventional oscillatory memory models. The dynamical and information characteristics of the retrieval process based on the optimal alphabet, including the size of “attraction basins“ and the input pattern distortion admissible for error-free retrieval, are investigated.     

An unusual form of phase walk in a system of coupled oscillators

The authors describe an unusual form of phase walk (i.e., a progressive change in phase angle between coupled oscillators) using the 10-Hz rhythmic discharges of the inferior cardiac and vertebral postganglionic sympathetic nerves (CN and VN, respectively) in hypercapnic, baroreceptor-denervated, and vagotomized cats anesthetized with urethane. Unlike phase walk ascribable to weakened coupling (desynchronization of oscillators), the phase walk of VN 10- Hz activity relative to CN10-Hz activity 1) recurred on the time scale of the respiratory cycle, 2) was bidirectional with CN-VN phase angle increasing during expiration and decreasing during inspiration, and 3) occurred over a range equivalent to one-half the period of the 10-Hz rhythm rather than a full cycle. Moreover, this form of phase walk occurred during strong coupling of the 10-Hz oscillators, as reflected by CN-VN coherence values approaching 1.0. The authors propose that the bidirectional phase walk reflects a state of strong coupling of the 10-Hz oscillators controlling the CN and VN, the angle of which is reset from cycle to cycle by the continuously changing level of activity in their respiratory inputs. In addition, the data demonstrate that frequency and amplitude modulation of sympathetic nerve discharge can be independently regulated by respiratory inputs.     

Analysis of biological wastewater treatment processes using multicomponent gas phase mass balancing.

A reactor system using off-gas analysis was developed for analyzing wastewater treatment process reactions. Using a mass spectrometer for the gas analysis provides the ability to simultaneously measure several gas components (such as oxygen, nitrogen, carbon dioxide, and argon). One of the benefits of the reactor design was the precise control of the dissolved oxygen concentration, uncoupled from the system turbulence, which was controlled via a gas recycle loop. This feature allowed control of the turbulence within the reactor without any need for mechanical stirring. Using oxygen as the test gas, the reactor was shown to perform well in the measurement of oxygen uptake rate of nitrifying activated sludge. The oxygen uptake rate calculations were made using a simple calibration method developed for the reactor system. The reactor was able to provide precise and accurate results for this test case. Furthermore, the system was capable of measuring under dynamic process conditions, as well as when the process rates were constant (steady state).     

Synchronization in a neural network of phase oscillators with the central element

A neural network model is considered which is designed as a system of phase oscillators and contains the central oscillator and peripheral oscillators which interact via the central oscillator. The regime of partial synchronization was studied when current frequencies of the central oscillator and one group of peripheral oscillators are near to each other while current frequencies of other peripheral oscillators are far from being synchronized with the central oscillator. Approximation formulas for the average frequency of the central oscillator in the regime of partial synchronization are derived, and results of computation experiments are presented which characterize the accuracy of the approximation.     

Quantification of DNA and protein adsorption by optical phase shift

A primary advantage of label-free detection methods over fluorescent measurements is its quantitative detection capability, since an absolute measure of adsorbed material facilitates kinetic characterization of biomolecular interactions. Interferometric techniques relate the optical phase to biomolecular layer density on the surface, but the conversion factor has not previously been accurately determined. We present a calibration method for phase shift measurements and apply it to surface-bound bovine serum albumin, immunoglobulin G, and single-stranded DNA.Biomolecules with known concentrations dissolved in salt-free water were spotted with precise volumes on the array surface and upon evaporation of the water, left a readily calculated mass. Using our label-free technique, the calculated mass of the biolayer was compared with the measured thickness, and we observed a linear dependence over 4 orders of magnitude. We determined that the widely accepted conversion of 1 nm of thickness corresponds to 1 ng/mm² surface density held reasonably well for these substances and through our experiments can now be further specified for different types of biomolecules. Through accurate calibration of the dependence of thickness on surface density, we have established a relation allowing precise determination of the absolute number of molecules for single-stranded DNA and two different proteins.     

Deterministic characterization of phase noise in biomolecular oscillators

On top of the many external perturbations, cellular oscillators also face intrinsic perturbations due the randomness of chemical kinetics. Biomolecular oscillators, distinct in their parameter sets or distinct in their architecture, show different resilience with respect to such intrinsic perturbations. Assessing this resilience can be done by ensemble stochastic simulations. These are computationally costly and do not permit further insights into the mechanistic cause of the observed resilience. For reaction systems operating at a steady state, the linear noise approximation (LNA) can be used to determine the effect of molecular noise. Here we show that methods based on LNA fail for oscillatory systems and we propose an alternative ansatz. It yields an asymptotic expression for the phase diffusion coefficient of stochastic oscillators. Moreover, it allows us to single out the noise contribution of every reaction in an oscillatory system. We test the approach on the one-loop model of the Drosophila circadian clock. Our results are consistent with those obtained through stochastic simulations with a gain in computational efficiency of about three orders of magnitude.     

The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division

Intercellular communication of molecules between normal cells by tight junctions, and lack of this in some cancer cells (Loewenstein), can explain contact inhibition of cellular division in tissues. A general theory has been based on assuming the continual rise and fall (intrinsic oscillation) of a key substance x in each cell, with the period of the cell cycle. Periods are asynchronous in different cells, and x is exchanged between cells in contact by diffusion. A reduction in the resultant amplitude of fluctuation of x results, so that it does not reach the threshold x_t required for division to ensue; hence contact inhibition.The mathematical model is defined in its simplest form, and the sets of differential equations for arrays of cells are solved, from the isolated cell to the cell in an infinite sheet. The relative probability of division, P, is computed by numerical analysis from the area of resultant curves of x that lies above the threshold x_t. P depends on four dimensionless parameters, the order of coupling n (the number of cells directly communicating with a given cell), the total number of cells N in the aggregate, the communication constant K, and x_t, as a fraction of the amplitude of the intrinsic oscillation. The degree of synchrony, measured by the coefficient of variation σ of the periods, is important. If σ < ± 4%, contact inhibition is much reduced. The theory predicts that a paradoxical “contact-facilitation” is possible for very small aggregates of cells. For a cell in an infinite sheet, the amplitude of oscillation of x is reduced approximately by the factor $\text{1}\text{nK}$. For normal cells K is probably > 1, for cancer cells that lack communication, K is probably «< 1. However, two other basic causes for lack of regulation of tissue growth (cancer) could be excessive intrinsic oscillation of x, cf. x_t, and partial or complete synchronization of groups of cells by some unknown mechanism.     

Feedback quenching as a means of effectively increasing the period of biochemical and biological oscillators

A cellular oscillator can be expected to modify the levels of other constituents. In turn, some of these are likely to modulate the behaviour of the oscillating system. Under appropriate conditions this feedback can temporarily quench the periodicity. By such means the frequency of the oscillation can be effectively reduced by a factor of ten or more.     

Biological oscillators can be stopped—Topological study of a phase response curve

Many biological oscillators are stable against noise and perturbation (e.g. circadian rhythms, biochemical oscillators, pacemaker neurons, bursting neurons and neural networks with periodic outputs). The experiment of phase shifts resulting from discrete perturbation of stable biological rhythms was developed by Perkel and coworkers (Perkel et al., 1964). By these methods, they could get important insights into the entrainment behaviors of biological rhythms. Phase response curves, which are measured in these experiments, can be classified into two types. The one is the curve with one mapping degree (Type 1), and the other is that with zero mapping degree (Type 0) (Winfree, 1970). We define the phase response curve mathematically, and explain the difference between these two types by the homotopy theory. Moreover, we prove that, if a Type 0 curve is obtained at a certain magnitude of perturbation, there exists at least one lower magnitude for which the phase response curve cannot be measured. Some applications of these theoretical results are presented.     

Application of a metabolic balancing technique to the analysis of microbial fermentation data

A general method for the development of fermentation models, based on elemental and metabolic balances, is illustrated with three examples from the literature. Physiological parameters such as the (maximal) yield on ATP, the energetic maintenance coefficient, the P/O ratio and others are estimated by fitting model equations to experimental data. Further, phenomenological relations concerning kinetics of product formation and limiting enzyme activities are assessed. The results are compared with the conclusions of the original articles, and differences due to the application of improved models are discussed.     

A simple model for phase locking of biological oscillators

Summary A mathematical model is presented for phase locking of a biological oscillator to a sinusoidal stimulus. Analytical, numerical and topological considerations are used to discuss the patterns of phase locking as a function of the amplitude of the sinusoidal stimulus and the relative frequencies of the oscillator and the sinusoidal stimulus. The sorts of experimental data which are needed to make comparisons between theory and experiment are discussed.