Population dynamics and competition in chemostat models with adaptive nutrient uptake

 The standard Monod model for microbial population dynamics in the chemostat is modified to take into consideration that cells can adapt to the change of nutrient concentration in the chemostat by switching between fast and slow nutrient uptake and growing modes with asymmetric thresholds for transition from one mode to another. This is a generalization of a modified Monod model which considers adaptation by transition between active growing and quiescent cells. Global analysis of the model equations is obtained using the theory of asymptotically autonomous systems. Transient oscillatory population density and hysteresis growth pattern observed experimentally, which do not occur for the standard Monod model, can be explained by such adaptive mechanism of the cells. Competition between two species that can switch between fast and slow nutrient uptake and growing modes is also considered. It is shown that generically there is no coexistence steady state, and only one steady state, corresponding to the survival of at most one species in the chemostat, is a local attractor. Numerical simulations reproduce the qualitative feature of some experimental data which show that the population density of the winning species approaches a positive steady state via transient oscillations while that of the losing species approaches the zero steady state monotonically. Received 4 August 1995; received in revised form 15 December 1995     

Irreversible transitions in the 6-phosphofructokinase/fructose 1,6-bisphosphatase cycle.

The dynamics of the fructose 6-phosphate fructose-1,6-bisphosphate cycle operating in an open and homogeneous system reconstituted from purified enzymes was extensively studied. In addition to 6-phosphofructokinase and fructose-1,6-bisphosphatase, pyruvate kinase, adenylate kinae and glucose-6-phosphate isomerase were involved. In that multi-enzyme system, the main source of non-linearity is the reciprocal effect of AMP on the activities of 6-phosphofructokinase and fructose-1,6-bisphosphatase. Depending upon the experimental parameter values, stable attractors, various types of multiple states and sustained oscillations were shown to occur. In the present report we show that irreversible transitions are also likely to occur for realistic operating conditions. Two parameters of the system, that is the adenylate energy charge of the influx and the fructose-1,6-bisphosphatase maximal activity, are potential candidates to provoke such irreversible transitions from one steady state to the other: (a) when varying the maximal activity of fructose-1,6-bisphosphatase, the system can jump irreversibly from a low to a high stable steady state, and (b) when the adenylate energy charge of the influx is the changing parameter, irreversible transitions occur from a high stable steady state to a stable oscillatory state (limit cycle motion). This behavior can be predicted by constructing the loci of limit points and Hopf bifurcation points.     

Modeling the interrelations between the calcium oscillations and ER membrane potential oscillations

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.     

Rhythmical processes in the cerebral cortex bioelectrical activity in arousal reaction: qualitative nonlinear analysis involving refractoriness

An integral equation that takes into account the absolute and relative refractory periods of cortical cells was added to the earlier obtained integral equations describing the cooperation of excitatory and inhibitory neurons of the cerebral cortex, and its substantiation was suggested. The steady states and the stable region of oscillations of the mean membrane potential of excitatory neurons were investigated depending on the mean afferent influx value on the assumption of the normal threshold distribution of cells. The existence of a parameter multitude in the region of stability was shown when a suppression of the delta-rhythm and an enhancement of the theta-rhythm up to its dominance take place increasing nonspecific afferent influx, which shows up a white noise, and decrease of the theta-rhythm during further increase in nonspecific activation. Simultaneously the frequencies of the major cerebral rhythms increase. The conformity of the theoretically calculated dependences of changes of the spectrum upon increasing nonspecific afferent influx to the experimental data obtained during the electrical stimulation of the rabbit midbrain reticular formation was demonstrated.     

Active transport of potassium by the Cecropia midgut; tracer kinetic theory and transport pool size.

1. Tracer influx kinetics have been analysed theoretically to determine the size of the transport pool with no assumptions regarding the transport pathway. 2. For a calculation of the size of the transport pool to be made, the following six conditions are required by the theory: tracer steady state attained, tissue steady state attained, Isc measures next flux, small magnitude and constant time-course of efflux, and correction for decay in pumping rate. 3. The size of the pool, SI, is given by the steady state influx, Finfinity, divided by the mixing-time constant, alpha. 4. Some experimental results are analysed by three different graphical methods, and it is shown that these three methods are equivalent. Specifically, alpha is equal to the reciprocal of the 75% mixing time, t75, divided by 1n 4 and is equal to the reciprocal of the lag time, X. 5. The tracer kinetic theory is applied to active potassium transport across the isolated short-circuited midgut: the transport meets the six conditions required by the theory. 6. The size of the transport pool of potassium in one midgut is calculated to be 80.5 muequiv./g wet weight under high-K steady-state conditions. A value as high as this suggests that the pool is intracellular.     

A model of intracellular Ca2+ oscillations based on the activity of the intermediate-conductance Ca2+-activated K+ channels

Intracellular Ca2+ oscillations are observed in a large number of non-excitable cells. While most appear to reflect an intermittent Ca2+ release from intracellular stores, in some instances intracellular Ca2+ oscillations strongly depend on Ca2+ influx, and are coupled to oscillations of the membrane potential, suggesting that a plasma membrane-based mechanism may be involved. We have developed a theoretical model for the latter type of intracellular Ca2+ oscillations based on the Ca2+-dependent modulation of the intermediate-conductance, Ca2+-activated K+ (IKCa) channel. The functioning of this model relies on the Ca2+-dependent activation, and the much slower Ca2+-dependent rundown of this channel. We have shown that Ca2+-dependent activation of the IKCa channels, the consequent membrane hyperpolarization and the resulting increase in Ca2+ influx may confer the positive feedback mechanism required for the ascending phase of the oscillation. The much slower Ca2+-dependent rundown process will conversely halt this positive loop, and establish the descending phase of the intracellular Ca2+ oscillation. We found that this simple model gives rise to intracellular Ca2+ oscillations when using physiologically reasonable parameters, suggesting that IKCa channels could participate in the generation of intracellular Ca2+ oscillations.     

Analysis of Mn(2+)/Ca(2+) influx and release during Ca(2+) oscillations in mouse eggs injected with sperm extract

Repetitive Ca(2+) release from the endoplasmic reticulum (ER) is necessary for activation of mammalian eggs. Influx and release of Mn(2+) and Ca(2+) during Ca(2+) oscillations induced by injection of sperm extract (SE) into mouse eggs were investigated by Mn(2+)-quenching of intracellular Fura-2 after adding Mn(2+) to external medium. Mn(2+)/Ca(2+) influx was detected at the resting state. A marked Mn(2+)/Ca(2+) influx occurred during the first Ca(2+) release upon SE injection, and persistently facilitated Mn(2+)/Ca(2+) influx was observed during steady Ca(2+) oscillations. As intracellular Mn(2+) concentration ([Mn(2+)](i)) increased progressively, periodic [Mn(2+)](i) rises appeared, corresponding to each Ca(2+)transient but taking a slower time course. A numerical simulation based on continuous Mn(2+)/Ca(2+) influx-extrusion across the plasma membrane and release-uptake across the ER membrane in a competitive manner mimicked well the Mn(2+) oscillations calculated from experimental data, strongly suggesting that repetitive Mn(2+) release develops after Mn(2+) entry and uptake into the ER. In other experiments, a marked Mn(2+) influx occurred upon Mn(2+) addition to Ca(2+)-free medium after depletion of the ER using an ER Ca(2+) pump inhibitor plus repeated injection of inositol 1,4,5-trisphosphate (InsP(3)). No significant increase in Mn(2+) influx was induced by injection of SE, InsP(3), or Ca(2+), when Ca(2+) release was prevented by pre-injection of an antibody against the InsP(3) receptor. We concluded that Ca(2+) influx is activated during the initial large Ca(2+)release possibly by a capacitative mechanism and kept facilitated during steady Ca(2+) oscillations. The finding that repetitive Mn(2+) release is caused by continuous Mn(2+) entry suggests that continuous Ca(2+) influx may play a critical role in refilling the ER and, thereby, maintaining Ca(2+)oscillations in mammalian fertilization.     

Ryanodine receptor adaptation and Ca2+(-)induced Ca2+ release-dependent Ca2+ oscillations.

A simplified mechanism that mimics "adaptation" of the ryanodine receptor (RyR) has been developed and its significance for Ca2+(-)induced Ca2+ release and Ca2+ oscillations investigated. For parameters that reproduce experimental data for the RyR from cardiac cells, adaptation of the RyR in combination with sarco/endoplasmic reticulum Ca2+ ATPase Ca2+ pumps in the internal stores can give rise to either low [Cai2+] steady states or Ca2+ oscillations coexisting with unphysiologically high [Cai2+] steady states. In this closed-cell-type model rapid, adaptation-dependent Ca2+ oscillations occur only in limited ranges of parameters. In the presence of Ca2+ influx and efflux from outside the cell (open-cell model) Ca2+ oscillations occur for a wide range of physiological parameter values and have a period that is determined by the rate of Ca2+ refilling of the stores. Although the rate of adaptation of the RyR has a role in determining the shape and the period of the Ca2+ spike, it is not essential for their existence. This is in marked contrast with what is observed for the inositol 1,4,5-trisphosphate receptor for which the biphasic activation and inhibition of its activity by Ca2+ are sufficient to produce oscillations. Results for this model are compared with those based on Ca2+(-)induced Ca2+ release alone in the bullfrog sympathetic neuron. This kinetic model should be suitable for analyzing phenomena associated with "Ca2+ sparks," including their merger into Ca2+ waves in cardiac myocytes.     

Transition state of the glycolytic pathway under FDP saturating conditions: experimental studies and a theoretical model

1. The transition state of the glycolytic pathway, under FDP saturating conditions, from no ADP to ADP-saturating levels, is studied in a metabolic model in vitro obtained from rat skeletal muscle. 2. When ADP is absent from the reaction mixture a steady state for NADH concentration is observed. After ADP addition, a new steady state is reached. The transition state from the first steady state to the second one shows a pulse of NADH. Both the profile and the size of this pulse depend on the enzyme concentration. 3. A kinetic model of the lower part of glycolysis (after PFK reaction) is proposed, and this is described by a set of first order coupled nonlinear differential equations. The results obtained through stability analysis and numerical integration of these equations agree with the experimental ones. 4. The possible role of the above mentioned transition state on the transmitter mechanism of glycolytic oscillations from PFK to the lower part of the glycolysis is discussed.     

Oscillations of the p53-Akt Network: Implications on Cell Survival and Death

Intracellular protein levels of p53 and MDM2 have been shown to oscillate in response to ionizing radiation (IR), but the physiological significance of these oscillations remains unclear. The p53-MDM2 negative feedback loop – the putative cause of the oscillations – is embedded in a network involving a mutual antagonism (or positive feedback loop) between p53 and AKT. We have shown earlier that this p53-AKT network predicts an all-or-none switching behavior between a pro-survival cellular state (low p53 and high AKT levels) and a pro-apoptotic state (high p53 and low AKT levels). Here, we show that upon exposure to IR, the p53-AKT network can also reproduce the experimentally observed p53 and MDM2 oscillations. The present work is based on the hypothesis that the physiological significance of the experimentally observed oscillations could be found in their role in regulating the switching behavior of the p53-AKT network between pro-survival and pro-apoptotic states. It is shown here that these oscillations are associated with a significant decrease in the threshold level of IR at which switching from a pro-survival to a pro-apoptotic state occurs. Moreover, oscillations in p53 protein levels induce higher levels of expression of p53-target genes compared to non-oscillatory p53, and thus influence cell-fate decisions between cell cycle arrest/DNA damage repair versus apoptosis.     

Kinetics of Mg2+ flux into rat liver mitochondria.

Unidirectional fluxes of Mg2+ across the limiting membranes of rat liver mitochondria have been measured in the presence of the respiratory substrate succinate by means of the radioisotope 28Mg. Rates of both influx and efflux of Mg2+ are decreased when respiration is inhibited. A linear dependence of the reciprocal of the Mg2+ influx rate on the reciprocal of the Mg2+ concentration is observed. The apparent Km for Mg2+ averages about 0.7 mM. N-Ethyl-maleimide, an inhibitor of transmembrane phosphate-hydroxyl exchanges, enhances the observed pH dependence of Mg2+, influx. In the presence of MalNEt, the apparent Vmax of Mg2+ influx is greater at pH 8 than at pH 7, and there is a linear dependence of the Mg2+ influx rate on the external OH- concentration. The K+ analogue Tl+ inhibits Mg2+ influx, while La3+, an inhibitor of mitochondrial Ca2+ transport, has no effect on Mg2+ influx. Mg2+ competitively inhibits the flux of K+ into rat liver mitochondria. The mechanism(s) mediating mitochondrial Mg2+ and K+ fluxes appear to be similar in their energy dependence, pH dependence, sensitivity to Tl+, and insensitivity to La3+.     

Dependence of stability on inter-enzyme distance in cooperative systems

A formulation in terms of quasilinear parabolic equations is given for a two-component cooperative system, with components free to diffuse in one dimension and cooperative reactive sites fixed at the boundaries of the interval. Localization of enzymes introduces Dirac delta functions into the equations. A unique, space-independent steady state is found, which is unstable with respect to diffusion and inter-enzyme distance. The biological implications of the results are formulated as a mechanism for “off-on” switching, or coupling of structure and function, in mitochondria.     

Bifurcation analysis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations

The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Dynamics of chemostat in which one microbial population grows on multiple complementary nutrients

The equations of a chemostat in which one microbial population grows on multiple rate-limiting nutrients are formulated. The dynamics of a chemostat involving growth on complementary nutrients is studied through stability analysis of the system of equations. Some conditions are derived that relate the dynamic behavior of the chemostat to its operating conditions and can be applied to any model for the specific growth rate of the population. It is shown that, if maintenance of the population is neglected, the system exhibits no sustained or damped oscillations. If maintenance of the population is considered, damped oscillations are observed for some operating conditions.     

Orthophosphate influx and efflux rates of Chlorella fusca measured in a continuous turbidostat culture with 32P under various conditions

The main objective of the experiments with Chlorella fusca strain 211-8b was to measure, with adequate time resolution, the unidirectional influx rates of phosphate into non-phosphate-starved algae under different steady state conditions (light, temperature, 3-phosphoglycerate influence) or following the addition of several photosynthesis and phosphate transport inhibitors (phenylmercuric acetate, p-chloromercuribenzoate, arsenate). the algae were cultivated in a phosphate rich medium in a continuous turbidostat culture. The phosphate exchange experiments with carrier-free ³²PO ₄ ^3- were performed directly in the continuous culture. The sampling intervals after the tracer addition were 15 s.For a continuous steady state culture grown in the light (25° C) the unidirectional influx rate measured with ³²P is 260 times higher than the net uptake rate (=influx minus efflux rate) calculated from the mass balance using the data of this culture. In all experiments, except the control experiment with trichloroacetic acid killed cells, the specific activity of the intracellular inorganic orthophosphate compartment oscillates around a constant mean value which never reaches the specific activity of the nutrient medium within the duration of the short-term experiments (7.5 min). The inhibitors strongly affect the characteristics of the oscillations. The unidirectional influx rates are constant. Oscillating flushing rates with unlabelled phosphate from a storage compartment have been postulates to explain the oscillations. Oscillating rates from the individual cells are apparently synchronized by an unknown mechanism.     

The influence of the potential dependence of the amplitude of postsynaptic potentials on rhythmic processes of bioelectric activity of the cerebral cortex

The dependence of the postsynaptic potential amplitude on the membrane potential was entered into the earlier derived integral equations describing the interactions between excitatory and inhibitory populations of neocortical neurons. The influence of the potential dependence on steady states and the stable region of oscillations of the mean membrane potential of neurons were investigated. Encephalograms of humans and animals in different functional states were numerically simulated. The real form of a power spectrum of electroencephalogram was obtained. The occurrence of the nonregular spindle-shaped activity was revealed, which expands the frequency of basic oscillations and widens the spectral peak. In the unsteady region, the existence of a limiting cycle and the possibility of arising of the pathological activity observed upon abnormal brain functioning were shown with the help of the numerical nonlinear analysis.     

Stochastic Hierarchical Systems: Excitable Dynamics

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.