A simple kinetic model for dynein oscillatory activity

A kinetic model for dynein, a molecular motor, complexed with microtubule fragments, is considered. The model explains the experimental observations of oscillatory movements in surprisingly simple axonemal fragments perfused by the ATP solution. The model explains at first time the oscillatory dynein activity as a phenomenon induced by two dynein heads cooperative interaction in the axoneme. The oscillation form, frequency, and amplitude, observed for the model, are close to these experimental characteristics. Kinetic parameters, used in the model, are close to the known experimental parameters.     

Modeling relapse in infectious diseases

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.     

Transitions and new dynamical states induced by noise in a multiply regulated biochemical system

Noise-induced transitions between coexisting states, and the emergence of a new oscillatory state, are examined in a model for a multiply regulated biochemical system. For the undisturbed system, three oscillatory states, I, II, and III, coexist. It is found that noise above a critical amplitude can cause a transition between states III and II and between states III or II and state I, whereas a transition from state I to either states II or III is never observed. This indicates that the relative stability under noise perturbations is greatest for state I, and progressively less for states II and III. In addition to this transition behaviour, a purely noise-induced state is found. Under noise perturbations, the average concentration of metabolites may depend on both the time duration and amplitude of the superimposed noise. The implications of these results for understanding the in vivo behaviour of complex biochemical systems are discussed.     

A simple kinetic model for dynein oscillatory activity

A kinetic model was proposed for dynein, a motor protein, complexed with microtubule fragments. The model explains the experimental observations of oscillatory movements in surprisingly simple axoneme fragments perfused with an ATP solution. This is the first model explaining the oscillatory activity of dynein as determined by a cooperative interaction of two dynein heads in the axoneme. The oscillation shape, frequency, and amplitude obtained for the model are close to the corresponding parameters determined experimentally.     

Regulator or Driving Force? The Role of Turgor Pressure in Oscillatory Plant Cell Growth

Turgor generates the stress that leads to the expansion of plant cell walls during cellular growth. This has been formalized by the Lockhart equation, which can be derived from the physical laws of the deformation of viscoelastic materials. However, the experimental evidence for such a direct correlation between growth rate and turgor is inconclusive. This has led to challenges of the Lockhart model. We model the oscillatory growth of pollen tubes to investigate this relationship. We couple the Lockhart equation to the dynamical equations for the change in material properties. We find that the correct implementation of the Lockhart equation within a feedback loop leading to low amplitude oscillatory growth predicts that in this system changes in the global turgor do not influence the average growth rate in a linear manner, consistent with experimental observations. An analytic analysis of our model demonstrates in which regime the average growth rate becomes uncorrelated from the turgor pressure.     

Hippocampal Theta Modulation of Neocortical Spike Times and Gamma Rhythm: A Biophysical Model Study

The hippocampal theta and neocortical gamma rhythms are two prominent examples of oscillatory neuronal activity. The hippocampus has often been hypothesized to influence neocortical networks by its theta rhythm, and, recently, evidence for such a direct influence has been found. We examined a possible mechanism for this influence by means of a biophysical model study using conductance-based model neurons. We found, in agreement with previous studies, that networks of fast-spiking GABA -ergic interneurons, coupled with shunting inhibition, synchronize their spike activity at a gamma frequency and are able to impose this rhythm on a network of pyramidal cells to which they are coupled. When our model was supplied with hippocampal theta-modulated input fibres, the theta rhythm biased the spike timings of both the fast-spiking and pyramidal cells. Furthermore, both the amplitude and frequency of local field potential gamma oscillations were influenced by the phase of the theta rhythm. We show that the fast-spiking cells, not pyramidal cells, are essential for this latter phenomenon, thus highlighting their crucial role in the interplay between hippocampus and neocortex.     

Boolean analysis of cell regulation networks

A comparison is made between the predictions of the Boolean and continuous analysis of a regulation model when the formation of two mediators interacting by cross-inhibition is stimulated by one or two specific signals. For such a system, the Boolean analysis reproduces the characteristics of behaviour previously predicted by continuous analysis (multiple stable states of opposite type, discontinuous transition, and associated hysteresis phenomenon). The qualitative agreement between the two methods allows a qualitative but rigorous treatment of regulation systems in which the Boolean analysis is applicable. From a general schematic representation of interaction in bidirectional control systems, we analyse by the Boolean method a large range of possible systems of increasing complexities which could theoretically apply. Previously unforeseen consequences of some systems are described. After that, we give a logical analysis of a well-known system (negative loop grafted with additional external controls) and discuss the application of such a system to explain certain oscillatory phenomena in the cell, showing the disrupting role of an additional control on the expected behaviour. Thus, when the analysis of a model including a negative loop does not indicate the possibility of experimentally suggested oscillations, we propose other simple logical structures which can predict this behaviour. Finally, we show a logical analysis of an opposite type of example of cell regulation where the biochemical observations can be accounted for simply by a negative loop grafted with one input variable.     

Self-excitation in a porous membrane doped with sorbitan monooleate (Span-80) induced by an Na+/K+ concentration gradient

The electrical potential across a fine-pore membrane doped with sorbitan monooleate (Span-80) imposed between aqueous solutions of NaCl and KCl was studied. It was found that this system showed rhythmic and sustained oscillations of electrical potential between the two aqueous solutions. These oscillations were attributed to the change of permeability of Na+ and K+ across the membrane, which originated from the phase transition of Span-80 molecules within the fine pores. Impedance measurement across the membrane also suggested a change in permeability. It was found that this membrane exhibited the property of differential negative resistance. In relation to this, it was shown that Na+ and K+ have different effects on the aggregation of Span-80 molecules. The mechanism of oscillation is discussed in relation to the ability of Span-80 molecules to behave as a dynamic channel through the membrane. This oscillatory phenomenon is interesting because in biological nervous membranes a difference between the concentrations of Na+ and K+ across the membranes is essential for excitability.     

The inflow of the substrate can regulate the persistence of memory of enzymes having the properties of hysteresis: theoretical analysis

A simple kinetic model of hysteretic enzymes with the influx of substrate or ion (transported by the enzyme) is considered. Two alternative steady activity levels are shown to arise in the system with a hysteretic enzyme. The transition between these levels can proceed in an oscillatory manner. The duration of the initial steady activity level is shown to be determined by the initial substrate (or ion) level, and the oscillatory transition between the activity levels is the property of hysteretic enzymes. It was shown for plasma membrane Ca2+-ATPase as an example that the level of the signal can be encoded into the time interval in which the enzyme retains the memory about this signal.     

A theoretical exploration of birhythmicity in the p53-Mdm2 network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities.     

How modeling can reconcile apparently discrepant experimental results: the case of pacemaking in dopaminergic neurons

Midbrain dopaminergic neurons are endowed with endogenous slow pacemaking properties. In recent years, many different groups have studied the basis for this phenomenon, often with conflicting conclusions. In particular, the role of a slowly-inactivating L-type calcium channel in the depolarizing phase between spikes is controversial, and the analysis of slow oscillatory potential (SOP) recordings during the blockade of sodium channels has led to conflicting conclusions. Based on a minimal model of a dopaminergic neuron, our analysis suggests that the same experimental protocol may lead to drastically different observations in almost identical neurons. For example, complete L-type calcium channel blockade eliminates spontaneous firing or has almost no effect in two neurons differing by less than 1% in their maximal sodium conductance. The same prediction can be reproduced in a state of the art detailed model of a dopaminergic neuron. Some of these predictions are confirmed experimentally using single-cell recordings in brain slices. Our minimal model exhibits SOPs when sodium channels are blocked, these SOPs being uncorrelated with the spiking activity, as has been shown experimentally. We also show that block of a specific conductance (in this case, the SK conductance) can have a different effect on these two oscillatory behaviors (pacemaking and SOPs), despite the fact that they have the same initiating mechanism. These results highlight the fact that computational approaches, besides their well known confirmatory and predictive interests in neurophysiology, may also be useful to resolve apparent discrepancies between experimental results.     

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.     

Unique biphasic band shape of the visible circular dichroism of bacteriorhodopsin in purple membrane: Excitons,multiple transitions or protein heterogeneity?

Over a decade and a half ago, when the first visible membrane suspension circular dichroic (CD) spectrum of the purple membrane (PM) was presented, two mechanisms were proposed to account for the observed biphasic shaped CD band: (a) excitonic interactions among the retinals of the sole protein bacteriorhodopsin (bR) in the crystalline structure of the PM, and (b) combination of CD bands with opposite rotational strengths due to a retinal-apoprotein heterogeneity of the bR molecules or due to two possible close-lying long-wavelength transitions of the retinal of the bR with opposite rotational strengths. Since that time, an impressive body of experimental and theoretical evidence has been accumulated, mostly consistent with an exciton model but many at serious odds with any heterogeneity or multiple transition model. Recently, a number of articles have appeared reporting analyses of new experimental observations which are proposed to cast serious doubts on the viability of the exciton model, and therefore, may revive the heterogeneity or multiple transition model as an explanation for the unique shape of the CD band of the PM. The intent of this article is to demonstrate that if all observations found in literature baring on this question are considered in toto and in a consistent manner, they can be interpreted without exception by excitons, and furthermore, that there is no plausible evidence available to warrant the revival of the heterogeneity or multiple transition model as an explanation for the unique shape of the biphasic CD band of the PM.     

Compliant bipedal model with the center of pressure excursion associated with oscillatory behavior of the center of mass reproduces the human gait dynamics

Although the compliant bipedal model could reproduce qualitative ground reaction force (GRF) of human walking, the model with a fixed pivot showed overestimations in stance leg rotation and the ratio of horizontal to vertical GRF. The human walking data showed a continuous forward progression of the center of pressure (CoP) during the stance phase and the suspension of the CoP near the forefoot before the onset of step transition. To better describe human gait dynamics with a minimal expense of model complexity, we proposed a compliant bipedal model with the accelerated pivot which associated the CoP excursion with the oscillatory behavior of the center of mass (CoM) with the existing simulation parameter and leg stiffness. Owing to the pivot acceleration defined to emulate human CoP profile, the arrival of the CoP at the limit of the stance foot over the single stance duration initiated the step-to-step transition. The proposed model showed an improved match of walking data. As the forward motion of CoM during single stance was partly accounted by forward pivot translation, the previously overestimated rotation of the stance leg was reduced and the corresponding horizontal GRF became closer to human data. The walking solutions of the model ranged over higher speed ranges (~1.7 m/s) than those of the fixed pivoted compliant bipedal model (~1.5 m/s) and exhibited other gait parameters, such as touchdown angle, step length and step frequency, comparable to the experimental observations. The good matches between the model and experimental GRF data imply that the continuous pivot acceleration associated with CoM oscillatory behavior could serve as a useful framework of bipedal model.     

Oscillations in plant membrane transport: model predictions, experimental validation, and physiological implications

Although oscillations in membrane-transport activity are ubiquitous in plants, the ionic mechanisms of ultradian oscillations in plant cells remain largely unknown, despite much phenomenological data. The physiological role of such oscillations is also the subject of much speculation. Over the last decade, much experimental evidence showing oscillations in net ion fluxes across the plasma membrane of plant cells has been accumulated using the non-invasive MIFE technique. In this study, a recently proposed feedback-controlled oscillatory model was used. The model adequately describes the observed ion flux oscillations within the minute range of periods and predicts: (i) strong dependence of the period of oscillations on the rate constants for the H+ pump; (ii) a substantial phase shift between oscillations in net H+ and K+ fluxes; (iii) cessation of oscillations when H+ pump activity is suppressed; (iv) the existence of some 'window' of external temperatures and ionic concentrations, where non-damped oscillations are observed: outside this range, even small changes in external parameters lead to progressive damping and aperiodic behaviour; (v) frequency encoding of environmental information by oscillatory patterns; and (vi) strong dependence of oscillatory characteristics on cell size. All these predictions were successfully confirmed by direct experimental observations, when net ion fluxes were measured from root and leaf tissues of various plant species, or from single cells. Because oscillatory behaviour is inherent in feedback control systems having phase shifts, it is argued from this model that suitable conditions will allow oscillations in any cell or tissue. The possible physiological role of such oscillations is discussed in the context of plant adaptive responses to salinity, temperature, osmotic, hypoxia, and pH stresses.     

Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

A model for the dynamics of human weight cycling

The resolution to lose weight by cognitive restraint of nutritional intake often leads to repeated bouts of weight loss and regain, a phenomenon known as weight cycling or “yo-yo dieting”. A simple mathematical model for weight cycling is presented. The model is based on a feedback of psychological nature by which a subject decides to reduce dietary intake once a threshold weight is exceeded. The analysis of the model indicates that sustained oscillations in body weight occur in a parameter range bounded by critical values. Only outside this range can body weight reach a stable steady state. The model provides a theoretical framework that captures key facets of weight cycling and suggests ways to control the phenomenon. The view that weight cycling represents self-sustained oscillations has indeed specific implications. In dynamical terms, to bring weight cycling to an end, parameter values should change in such a way as to induce the transition of body weight from sustained oscillations around an unstable steady state to a stable steady state. Maintaining weight under a critical value should prevent weight cycling and allow body weight to stabilize below the oscillatory range.     

Viscoplasticity of respiratory tissues

Low-frequency mechanical behavior of various respiratory tissues shows certain similarities. In this study we test the hypothesis that rate-independent plastic processes along with rate-dependent viscoelastic processes are responsible. We considered oscillatory responses of several respiratory tissues measured over prescribed ranges of frequency (up to 6 Hz) and amplitude of forcing. These included the excised cat lung, the human chest wall in vivo, and two components of the chest wall: the excised dog rib cage and the excised rabbit abdominal viscera; some data were previously reported and some are new. We analyzed these data using the viscoplastic model of Hildebrandt (J. Appl. Physiol. 28: 365-372, 1970). It consists of three compartments: a plastoelastic compartment mechanically in parallel with a viscoelastic compartment, both in series with a lumped inertia. We fitted oscillatory data of the above respiratory tissues to the model by a least-squares technique. The fit was qualitatively consistent with the observations and exhibited moderately good to very good quantitative correspondence. As an independent verification of this approach, we obtained the stress relaxation after a step-volume change. Based on the oscillatory response of cat lungs, the calculated stress relaxation function was found to be generally consistent with corresponding observations. This study indicates that both plasticity and viscoelasticity appear to be important determinants of mechanical behavior of respiratory tissues at low frequencies and that inertial effects are negligible.