Nephelometric determination of turgor pressure in growing gram-negative bacteria.

Gas vesicles were used as probes to measure turgor pressure in Ancylobacter aquaticus. The externally applied pressure required to collapse the vesicles in turgid cells was compared with that in cells whose turgor had been partially or totally removed by adding an impermeable solute to the external medium. Since gram-negative bacteria do not have rigid cell walls, plasmolysis is not expected to occur in the same way as it does in the cells of higher plants. Bacterial cells shrink considerably before plasmolysis occurs in hyperosmotic media. The increase in pressure required to collapse 50% of the vesicles as external osmotic pressure increases is less than predicted from the degree of osmotically inducible shrinkage seen with this organism or with another gram-negative bacterium. This feature complicates the calculation of the turgor pressure as the difference between the collapse pressure of vesicles with and without sucrose present in the medium. We propose a new model of the relationship between turgor pressure and the cell wall stress in gram-negative bacteria based on the behavior of an ideal elastic container when the pressure differential across its surface is decreased. We developed a new curve-fitting technique for evaluating bacterial turgor pressure measurements.     

Fluctuation-driven rhythmogenesis in an excitatory neuronal network with slow adaptation

We study an excitatory all-to-all coupled network of N spiking neurons with synaptically filtered background noise and slow activity-dependent hyperpolarization currents. Such a system exhibits noise-induced burst oscillations over a range of values of the noise strength (variance) and level of cell excitability. Since both of these quantities depend on the rate of background synaptic inputs, we show how noise can provide a mechanism for increasing the robustness of rhythmic bursting and the range of burst frequencies. By exploiting a separation of time scales we also show how the system dynamics can be reduced to low-dimensional mean field equations in the limit N → ∞. Analysis of the bifurcation structure of the mean field equations provides insights into the dynamical mechanisms for initiating and terminating the bursts.     

Temporal variability in a system of coupled mitotic timers

 Cell proliferation is considered a periodic process governed by a relaxation timer. The collective behavior of a system composed of three identical relaxation oscillators in numerically studied under the condition that diffusion of the slow mode dominates. We demonstrate: (1) the existence of three periodic regimes with different periods and phase relations and an unsymmetrical, stable steady-state (USSS); (2) the coexistence of in-phase oscillations and USSS; (3) the coexistence of periodic attractors; and (4) the emergence of a two-loop limit cycle coexisting with both in-phase oscillations and a stable steady-state. The qualitative reasons for such a diversitiy and its possible role in the generation of cell cycle variability are discussed. Received: 18 March 1992/Accepted in revised form: 16 April 1994     

A Phantom Bursting Mechanism for Episodic Bursting

We describe a novel dynamic mechanism for episodic or compound bursting oscillations, in which bursts of electrical impulses are clustered together into episodes, separated by long silent phases. We demonstrate the mechanism for episodic bursting using a minimal mathematical model for “phantom bursting.” Depending on the location in parameter space, this model can produce fast, medium, or slow bursting, or in the present case, fast, slow, and episodic bursting. The episodic bursting is modestly robust to noise and to parameter variation, and the effect that noise has on the episodic bursting pattern is quite different from that of an alternate episodic burst mechanism in which the slow envelope is produced by metabolic oscillations. This mechanism could account for episodic bursting produced in endocrine cells or neurons, such as pancreatic islets or gonadotropin releasing neurons of the hypothalamus.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

A theory of cell hydration governed by adsorption of water on cell proteins rather than by osmotic pressure

It is postulated that cell hydration is governed by adsorption of water on cell proteins in accord with the Bradley adsorption isotherm, and that the action of a solute in the surrounding solution is to lower the vapor pressure of the solution so that cell water adsorption is decreased by moving down the Bradley isotherm. From these concepts, it is derived that cell volume (V) should be related to solute concentration (x) by the equationV=−E log₁₀ x+F whereE andF are constants which are independent of type of solute. For a non-adsorbed solute this agrees well with experimental data. For solutes which are adsorbed by cell proteins, a correction in the above equation may be necessary at higher solute concentrations, which is shown to be compatible with various experimental data. The types of experiments which are generally used to support the osmotic pressure theory of cell hydration agree equally well with the adsorption theory. The virtue of the adsorption theory is that, unlike the osmotic pressure theory of cell swelling, it is compatible with permeability of the cell membrane to solutes, which has been experimentally observed for various solutes. The opinions and conclusions contained in this report are those of the author. They are not to be construed as necessarily reflecting the views or the endorsement of the Navy Department.     

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.     

Activity-dependent modulation of adaptation produces a constant burst proportion in a model of the lamprey spinal locomotor generator

The neuronal network underlying lamprey swimming has stimulated extensive modelling on different levels of abstraction. The lamprey swims with a burst frequency ranging from 0.3 to 8–10 Hz with a rostro-caudal lag between bursts in each segment along the spinal cord. The swimming motor pattern is characterized by a burst proportion that is independent of burst frequency and lasts around 30%–40% of the cycle duration. This also applies in preparations in which the reciprocal inhibition in the spinal cord between the left and right side is blocked. A network of coupled excitatory neurons producing hemisegmental oscillations may form the basis of the lamprey central pattern generator (CPG). Here we explored how such networks, in principle, could produce a large frequency range with a constant burst proportion. The computer simulations of the lamprey CPG use simplified, graded output units that could represent populations of neurons and that exhibit adaptation. We investigated the effect of an active modulation of the degree of adaptation of the CPG units to accomplish a constant burst proportion over the whole frequency range when, in addition, each hemisegment is assumed to be self-oscillatory. The degree of adaptation is increased with the degree of stimulation of the network. This will make the bursts terminate earlier at higher burst rates, allowing for a constant burst proportion. Without modulated adaptation the network operates in a limited range of swimming frequencies due to a progressive increase of burst duration with increasing background stimulation. By introducing a modulation of the adaptation, a broad burst frequency range can be produced. The reciprocal inhibition is thus not the primary burst terminating factor, as in many CPG models, and it is mainly responsible for producing alternation between the left and right sides. The results are compared with the Morris-Lecar oscillator model with parameters set to produce a type A and type B oscillator, in which the burst durations stay constant or increase, respectively, when the background stimulation is increased. Here as well, burst duration can be controlled by modulation of the slow variable in a similar way as above. When oscillatory hemisegmental networks are coupled together in a chain a phase lag is produced. The production of a phase lag in chains of such oscillators is compared with chains of Morris-Lecar relaxation oscillators. Models relating to the intact versus isolated spinal cord preparation are discussed, as well as the role of descending inhibition. Received: 1 April 1997 / Accepted in revised form: 20 March 1998     

Mixed mode oscillations as a mechanism for pseudo-plateau bursting

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called “pseudo-plateau bursting”. Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic β-cells.     

Control of leaf cell elongation in barley. Generation rates of osmotic pressure and turgor, and growth-associated water potential gradients

In a previous study on the effects of N-supply on leaf cell elongation, the spatial distribution of relative cell elongation rates (RCER), epidermal cell turgor, osmotic pressure (OP) and water potential (Ψ) along the elongation zone of the third leaf of barley was determined (W. Fricke et al. 1997, Planta 202: 522–530). The results suggested that in plants receiving N at fixed relative addition rates (N-supply limitation of growth), cell elongation was rate-limited by the rate of solute provision, whereas in plants growing on complete nutrient solution containing excessive amounts of N (N-demand limitation), cell elongation was rate-limited by the rate of water supply or wall yielding. In the present paper, these suggestions were tested further. The generation rates of cell OP, turgor and Ψ along the elongation zone were calculated by applying the continuity equation of fluid dynamics to the previous data. To allow a more conclusive interpretation of results, anatomical data were collected and bulk solute concentrations determined. The rate of OP generation generally exceeded the rate of turgor generation. As a result, negative values of cell Ψ were created, particularly in demand-limited plants. These plants showed highest RCER along the elongation zone and a Ψ gradient of at least −0.15 MPa between water source (xylem) and expanding epidermal cells. The latter was similar to a theoretically predicted value (−0.18 MPa). Highest rates of OP generation were observed in demand-limited plants, with a maximum rate of 0.112 MPa · h⁻¹ at 16–20 mm from the leaf base. This was almost twice the rate in N-supply-limited plants and implied that the cells in the leaf elongation zone were capable of importing (or synthesising) every minute almost 1 mM of osmolytes. Potassium, Cl⁻ and NO₃ ⁻ were the main inorganic osmolytes (only determined for demand-limited plants). Their concentrations suggest that, unlike the situation in fully expanded epidermal cells, sugars are used to generate OP and turgor. Anatomical data revealed that the zone of lateral cell expansion extended distally beyond the zone of cell elongation. It is concluded that leaf cell expansion in barley relies on high rates of water and solute supply, rates that may not be sustainable during periods of sufficient N-supply (limitation by water supply: Ψ gradients) or limiting N-supply (limitation by solute provision: reduced OP-generation rates). To minimise the possibility of growth limitation by water and osmolyte provision, longitudinal and lateral cell expansion peak at different locations along the growth zone. Received: 15 October 1997 / Accepted: 12 March 1998     

The Interrelations of Cell Turgor Pressure, Gas-vacuolation, and Buoyancy in a Blue-green Alga

The increase in pressure required to collapse gas vacuoles onsuspending the cells of the blue-green alga Anabaena flos-aquaein hypertonic sucrose solutions shows the turgor pressure tovary over the range of 265 to 459 KN m^–2 under differentculture conditions. The cell turgor increased at a rate of upto KN m^–2 h^–1 on transferring the alga from lowto high light intensity. This rise appears to be a result ofthe accumulation of photosynthate, as it is dependent on thepresence of carbon dioxide in the gas phase and is inhibitedby DCMU. Experiments using ¹⁴CO² indicate that the increasedrate of photosynthesis during the high light exposure is easilysufficient to account for the observed turgor rise. The rise in turgor can bring about collapse of sufficient ofthe alga's gas vacuoles to destroy its buoyancy. Higher turgorpressures, and consequently a lower degree of gas vacuolationand buoyancy, were maintained when the alga was kept at highlight intensitives for a week and more. The significance ofthis behaviour is discussed in relation to stratification ofplanktonic blue-green algae in natural habitats.     

Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans.

Electrical bursting oscillations of mammalian pancreatic beta-cells are synchronous among cells within an islet. While electrical coupling among cells via gap junctions has been demonstrated, its extent and topology are unclear. The beta-cells also share an extracellular compartment in which oscillations of K+ concentration have been measured (Perez-Armendariz and Atwater, 1985). These oscillations (1-2 mM) are synchronous with the burst pattern, and apparently are caused by the oscillating voltage-dependent membrane currents: Extracellular K+ concentration (Ke) rises during the depolarized active (spiking) phase and falls during the hyperpolarized silent phase. Because raising Ke depolarizes the cell membrane by increasing the potassium reversal potential (VK), any cell in the active phase should recruit nonspiking cells into the active phase. The opposite is predicted for the silent phase. This positive feedback system might couple the cells' electrical activity and synchronize bursting. We have explored this possibility using a theoretical model for bursting of beta-cells (Sherman et al., 1988) and K+ diffusion in the extracellular space of an islet. Computer simulations demonstrate that the bursts synchronize very quickly (within one burst) without gap junctional coupling among the cells. The shape and amplitude of computed Ke oscillations resemble those seen in experiments for certain parameter ranges. The model cells synchronize with exterior cells leading, though incorporating heterogeneous cell properties can allow interior cells to lead. The model islet can also be forced to oscillate at both faster and slower frequencies using periodic pulses of higher K+ in the medium surrounding the islet. Phase plane analysis was used to understand the synchronization mechanism. The results of our model suggest that diffusion of extracellular K+ may contribute to coupling and synchronization of electrical oscillations in beta-cells within an islet.     

A study of the bifurcation behaviour of a model of flow through a collapsible tube

Most of the elastic tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in collapsible tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.     

Simulation of gait and gait initiation associated with body oscillating behavior in the gravity environment on the Moon, Mars and Phobos

 A double-inverted pendulum model of body oscillations in the frontal plane during stepping [Brenière and Ribreau (1998) Biol Cybern 79: 337–345] proposed an equivalent model for studying the body oscillating behavior induced by step frequency in the form of: (1) a kinetic body parameter, the natural body frequency (NBF), which contains gravity and which is invariable for humans, (2) a parametric function of frequency, whose parameter is the NBF, which explicates the amplitude ratio of center of mass to center of foot pressure oscillation, and (3) a function of frequency which simulates the equivalent torque necessary for the control of the head-arms-trunk segment oscillations. Here, this equivalent model is used to simulate the duration of gait initiation, i.e., the duration necessary to initiate and execute the first step of gait in subgravity, as well as to calculate the step frequencies that would impose the same minimum and maximum amplitudes of the oscillating responses of the body center of mass, whatever the gravity value. In particular, this simulation is tested under the subgravity conditions of the Moon, Mars, and Phobos, where gravity is 1/6, 3/8, and 1/1600 times that on the Earth, respectively. More generally, the simulation allows us to establish and discuss the conditions for gait adaptability that result from the biomechanical constraints particular to each gravity system. Received: 15 February 1999 / Accepted in revised form: 9 October 2000     

The pressure relationships of halophilic and non-halophilic prokaryotic cells determined by using gas vesicles as pressure probes

Abstract Gas vesicles can be used to measure the hydrostatic pressure (turgor pressure) in prokaryotic cells. Halophilic cyanobacteria have turgor pressures that are substantially less than those of cyanobacteria from fresh water. Turgor pressure acts so as to tend to burst cell walls and collapse hollow gas vesicles. The halophiles take advantage of their lower turgor pressures by producing cell walls that are relatively thinner and gas vesicles that are relatively wider than in the mesophilic cyanobacteria. In this way the halophilic structure encounters the same stress and saves on material. Extreme halophiles, with negligible turgor, have been able to adopt various shapes and to produce the weakest and widest gas vesicles.     

A Model of Oscillatory Blood Cell Counts in Chronic Myelogenous Leukaemia

In certain blood diseases, oscillations are found in blood cell counts. Particularly, such oscillations are sometimes found in chronic myelogenous leukaemia, and then occur in all the derived blood cell types: red blood cells, white blood cells, and platelets. It has been suggested that such oscillations arise because of an instability in the pluri-potential stem cell population, associated with its regulatory control system. In this paper, we consider how such oscillations can arise in a model of competition between normal (S) and genetically altered abnormal (A) stem cells, as the latter population grows at the expense of the former. We use an analytic model of long period oscillations to describe regions of oscillatory behaviour in the S–A phase plane, and give parametric criteria to describe when such oscillations will occur. We also describe a mechanism which can explain dynamically how the transformation from chronic phase to acute phase and blast crisis can occur.     

Electrocoupling of Ion Transporters in Plants: Interaction with Internal Ion Concentrations

There are five major electroenzymes in the plasmalemma of plant cells: a driving electrogenic pump, inward and outward rectifying K⁺ channels, a Cl⁻-2H⁺ symporter, and Cl⁻-channels. It has been demonstrated previously (Gradmann, Blatt & Thiel 1993, J. Membrane Biol. 136:327–332) how voltage-gating of these electroenzymes causes oscillations of the transmembrane voltage (V) at constant substrate concentrations. The purpose of this study is to examine the interaction of the same transporter ensemble with cytoplasmic concentrations of K⁺ and Cl⁻. The former model system has been extended to account for changing internal concentrations. Constant-field theory has been applied to describe the influence of ion concentrations on current-voltage relationships of the active channels. The extended model is investigated using a reference set of model parameters. In this configuration, the system converges to stable slow oscillations with intrinsic changes in cytoplasmic K⁺ and Cl⁻ concentrations. These slow oscillations reflect alternation between a state of salt uptake at steady negative values of V and a state of net salt loss at rapidly oscillating V, the latter being analogous to the previously reported oscillations. By switching off either concentration changes or gating, it is demonstrated that the fast oscillations are mostly due to the gating properties of the Cl⁻ channel, whereas the slow oscillations are controlled by the effect of the Cl⁻ concentration on the current. The sensitivity of output results y (e.g., frequency of oscillations) to changes of the model parameters x (e.g., maximum Cl⁻ conductance) has been investigated for the reference system. Further examples are presented where some larger changes of specific model parameters cause fundamentally different behavior, e.g., convergence towards a stable state of only the fast oscillations without intrinsic concentration changes, or to a steady-state without any oscillations. The main and general result of this study is that the osmotic status of a plant cell is stabilized by the ensemble of familiar electroenzymes through oscillatory interactions with the internal concentrations of the most abundant ions. This convergent behavior of the stand-alone system is an important prerequisite for osmotic regulation by means of other physiological mechanisms, like second messengers and gating modifiers. Received: 23 February/Revised: 16 July 1998     

Dancing between the devil and deep blue sea: the stabilizing effect of enemy-free and victimless sinks

Theoretical and empirical studies have shown that enemy–victim interactions in spatially homogenous environments can exhibit diverging oscillations which result in the extinction of one or both species. For enemy–victim models with overlapping generations, we investigate the dynamical implications of spatial heterogeneity created by enemy-free sinks or victimless sinks. An enemy-free sink is a behavioral, physiological or ecological state that reduces or eliminates the victim's vulnerability to the enemy but cannot sustain the victim population. For victims that move in an ideal-free manner, we prove that the inclusion of an enemy-free sink shifts the population dynamics from diverging oscillations to stable oscillations. During these stable oscillations, the victim disperses in an oscillatory manner between the enemy-free sink and the enemy-occupied patch. Enemy-free sinks with lower mortality rates exhibit oscillations with smaller amplitudes and longer periods. A victimless sink, on the other hand, is a behavioral, physiological or ecological state in which the enemy has limited (or no) access to its victims. For enemies that move in an ideal-free manner, we prove that victimless sinks also stabilize diverging oscillations. Simulations suggest that suboptimal behavior due to information gathering or learning limitations amplify oscillations for systems with enemy-free sinks and dampen oscillations for systems with victimless sinks. These results illustrate that the coupling of a sink created by unstable enemy–victim interactions and a sink created by unsuitable environmental conditions can result in population persistence at the landscape level.     

Context-Specific Bioturbation Mediates Changes to Ecosystem Functioning

Species are often grouped according to their biological or functional traits to better understand their contribution to ecosystem functioning. However, it is becoming clear that a single species can perform different roles in different habitats. Austrohelice crassa, a burrow-building mud crab shifts its primary bioturbational role to that of a vertical mixer in non-cohesive sediments as frequent burrow collapse greatly enhances sediment reworking. We conducted in situ crab density manipulations in two sediment environments (a non-cohesive sand and a cohesive muddy-sand) to examine if the context-specific functional roles were linked to changes in solute fluxes across the sediment–water interface. Across both habitats, we show that A. crassa regulated nutrient cycling, creating strong density driven effects on solute exchanges. Increasing crab density increased sediment O₂ demand and the flux of NH₄ ⁺ from the sediment, indicating much of the response was physiologically driven. Clear interactions between A. crassa and microphytobenthos were also detected in both habitats. Despite lowering microphyte standing stock through deposit feeding, A. crassa increased benthic primary production per unit of chlorophyll a. Our experiment also revealed important context-specific differences, most notably for NH₄ ⁺ fluxes, which were higher where burrows and their associated microbial communities were most stable (muddy-sand). This study highlights the need to integrate interactions between organism behavior and habitat type into functional group studies to broaden conceptual frameworks and avoid oversimplification of highly complex organism–sediment interactions.