Range Of Bistability Research Articles

Bifurcation Analysis for Phage Lambda with Binding Energy Uncertainty

In a phageλgenetic switch model, bistable dynamical behavior can be destroyed due to the bifurcation caused by inappropriately chosen model parameters. Since the values of many parameters with biological significance often cannot be accurately acquired, it is thus of fundamental importance to analyze how and to which extent the system dynamics is influenced by model parameters, especially those parameters pertaining to binding energies. In this paper, we apply a Jacobian method to investigate the relation between bifurcation and parameter uncertainties on a phageλOR model. By introducing bistable range as a measure of system robustness, we find that RNA polymerase binding energies have the minimum bistable ranges among all the binding energies, which implies that the uncertainties on these parameters tend to demolish the bistability more easily. Moreover, parameters describing mutual prohibition between proteins Cro and CI have finite bistable ranges, whereas those describing self-prohibition have infinity bistable ranges. Hence, the former are more sensitive to parameter uncertainties than the latter. These results help to understand the influence of parameter uncertainties on the bifurcation and thus bistability.

Tuning the range and stability of multiple phenotypic states with coupled positive–negative feedback loops

Positive feedback loops can produce multistability, resulting in different phenotypic states. However, many transcription networks contain counteracting positive and negative feedbacks. Here we explore the dynamics of an interlinked positive and negative feedback motif based on the galactose-uptake control system of Saccharomyces cerevisiae modified to make the strength of each feedback externally controllable. Our results show that although the positive feedback loop determines the range of bistability and the width of the regions where intermediate activation is possible, the transition rates between states are mostly sensitive to the negative feedback strength. Thus, our results suggest that the function of the negative loop in this motif is to allow separate tuning of the range and transition rates between phenotypic states. This could enhance fitness by allowing improved matching of the stochastic switching to the frequency of environmental changes.

Optical Bistability in a ${\rm Nd}{:}{\rm Lu}_{0.14}{\rm Gd}_{0.86}{\rm VO}_{4}/{\rm Cr{:}YAG}$ Microchip Laser

Optical bistability was realized in a passively Q-switched Nd:Lu <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.14</sub> Gd <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.86</sub> VO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> microchip laser. The bistable operation range for the incident pump power was 5.9-7.3 W, and the jump output power at the turning point was 0.24 W. By using a four-level laser model with a saturable absorber and Fabry-Perot cavity, the calculated results agreed well with the experimental data.

Bistability of silence and seizure-like bursting

Neuronal circuits exhibiting seizure episodes have been shown to be prone to multistability. The coexistence of normal and pathological regimes could explain why seizures suddenly start and stop. Methods developed in dynamical systems theory are powerful tools for determining the cellular mechanisms that underlie multistable seizure dynamics. Here, we present two different approaches to assess multistability in a model neuron. In this model, we identified a bursting regime and a silent regime. First, we investigated properties of a square pulse of injected current which produced a switch from seizure-like bursting into silence. By systematically varying the phase and amplitude of the pulse, we found contiguous pulse parameter sets, so-called windows, that satisfied this criterion, and we determined the dependence of these windows on the parameter gleak. As gleak increased, the size of each window scaled according to the same law as the amplitude of the saddle orbit. Second, we examined the role of each current in supporting bistability of bursting and silence. We defined the index of propensity for multistability as the range of gleak for which bursting and silence coexisted. We computed this quantity while iteratively varying the maximal conductance of each voltage-gated current one at a time. Increasing the maximal conductance of the slow potassium current or the hyperpolarization-activated current increased the range of bistability. In contrast, decreasing the maximal conductance of the persistent sodium current increased the range of bistability.

Optical bistability in the oscillation of an inhomogeneously broadened quasi-three-level laser

A theoretical modeling analysis is presented to study the optical bistability exhibited in the oscillation of an inhomogeneously broadened quasi-three-level laser. All the major characteristics of optical bistability depend on two normalized parameters, f and xa, which are defined by f = Isat,a/Isat,m and xa = 2αa0pa/δ and are related to measurable properties of the laser medium. In comparison with the case of a homogeneously broadened laser, the essential condition for the occurrence of such bistability, f < xa/(xa + 1), turns out to be the same, whereas the intensities at the up- and down-thresholds are substantially increased and the bistability range is reduced.

DNA looping increases the range of bistability in a stochastic model of the lac genetic switch

Conditions and parameters affecting the range of bistability of the lac genetic switch in Escherichia coli are examined for a model which includes DNA looping interactions with the lac repressor and a lactose analogue. This stochastic gene–mRNA–protein model of the lac switch describes DNA looping using a third transcriptional state. We exploit the fast bursting dynamics of mRNA by combining a novel geometric burst extension with the finite state projection method. This limits the number of protein/mRNA states, allowing for an accelerated search of the model's parameter space. We evaluate how the addition of the third state changes the bistability properties of the model and find a critical region of parameter space where the phenotypic switching occurs in a range seen in single molecule fluorescence studies. Stochastic simulations show induction in the looping model is preceded by a rare complete dissociation of the loop followed by an immediate burst of mRNA rather than a slower build up of mRNA as in the two-state model. The overall effect of the looped state is to allow for faster switching times while at the same time further differentiating the uninduced and induced phenotypes. Furthermore, the kinetic parameters are consistent with free energies derived from thermodynamic studies suggesting that this minimal model of DNA looping could have a broader range of application.

Generalization of the Modeling Analysis of Optical Bistability in Quasi-Three-Level Lasers

In this paper, we generalize a previously developed model describing the behavior of optical bistability in the operation of a quasi-three-level laser by taking into account a Gaussian transverse distribution for the laser mode and including inhomogeneous line broadening for the laser medium. The results show that apart from some quantitative modifications, the primary characteristics of bistability predicted by the simplified model remain correct; in particular, the fundamental condition for the existence of bistability, f <; xa/(xa+1), turns out to hold universally. It is also revealed that in comparison with the results obtained in the simplified model, the intensities at the up- and down-threshold will be increased, whereas the bistability range will be reduced.

Coupling between feedback loops in autoregulatory networks affects bistability range, open-loop gain and switching times

Biochemical regulatory networks governing diverse cellular processes such as stress-response, differentiation and cell cycle often contain coupled feedback loops. We aim at understanding how features of feedback architecture, such as the number of loops, the sign of the loops and the type of their coupling, affect network dynamical performance. Specifically, we investigate how bistability range, maximum open-loop gain and switching times of a network with transcriptional positive feedback are affected by additive or multiplicative coupling with another positive- or negative-feedback loop. We show that a network's bistability range is positively correlated with its maximum open-loop gain and that both quantities depend on the sign of the feedback loops and the type of feedback coupling. Moreover, we find that the addition of positive feedback could decrease the bistability range if we control the basal level in the signal-response curves of the two systems. Furthermore, the addition of negative feedback has the capacity to increase the bistability range if its dissociation constant is much lower than that of the positive feedback. We also find that the addition of a positive feedback to a bistable network increases the robustness of its bistability range, whereas the addition of a negative feedback decreases it. Finally, we show that the switching time for a transition from a high to a low steady state increases with the effective fold change in gene regulation. In summary, we show that the effect of coupled feedback loops on the bistability range and switching times depends on the underlying mechanistic details.

Exploring Neuronal Bistability at the Depolarization Block

Many neurons display bistability–coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na+ current. Our study suggests a connection between the amount of the Na+ window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.

Bistability of seizure-like bursting and silence

Heart interneurons (HNs) in the third and fourth ganglia of the medicinal leech are units of the central pattern generator driving the heartbeat. When placed in bath with Co2+ and 4-aminopyridine, cells in a ganglion exhibit seizure-like activity with slow plateau oscillations [1]. In this scenario, the synaptic currents, the Ca2+ currents, and most of the K+ currents are blocked. We consider this bursting activity to be a model for chemically induced seizures. We present a novel Hodgkin-Huxley type model of the pharmacologically reduced HN. The model contains a leak current and four voltage-gated ionic currents: a fast Na+ current, a persistent Na+ current (IP), a non-inactivating K+ current (IK2), and a hyperpolarization-activated current (Ih). This model produced slow plateau-like bursting at an initial set of parameters. We systematically varied two parameters of the leak current (gLeak and ELeak) and recorded a variety of silent and oscillatory regimes. For a given value of ELeak, we established the range of values of gLeak for which the system supported bursting activity. For large values of gLeak, the model exhibited quiescence. As gLeak was decreased, the equilibrium lost stability in an Andronov-Hopf bifurcation, and a saddle orbit was born. The ranges of values in gLeak for which the system exhibited bursting and quiescence overlapped defining a range of values in gLeak supporting coexistence of the two regimes. Coexistence means that an external perturbation can switch the activity from one of the stable regimes to another. We explored bistability of bursting and silence by injecting a square pulse of current to perturb bursting activity. Pulses of current were characterized by two parameters: the amplitude of the pulse and the phase in the cycle period at which the pulse was applied. We varied the phase and amplitude through a range of values and determined whether a given pulse switched the activity from bursting to silence. We found contiguous parameter sets that satisfied this criterion. We examined the role of each current in supporting bistability of bursting and silence. The propensity of the model to bistability of bursting and silence was defined as the range of gLeak for which bursting and silence coexisted. We computed this value while iteratively varying the maximal conductance of each voltage-gated current. Increasing the maximal conductance of IK2 increased the range of bistability, and there was a similar trend for Ih. In contrast, the range of bistability increased as the maximal conductance of IP decreased. We tested whether these effects were complimentary. We selected values of the maximal conductances of IK2, Ih, and IP that each supported the largest range of bistability found. We performed a two-parameter (gLeak, ELeak) bifurcation analysis in both the initial model and this new model. The range of bistability in the new model increased substantially over that exhibited in the initial model. We suggest that this study could lead to the development of novel methods for controlling seizures.

PageRank model of opinion formation on social networks

We propose the PageRank model of opinion formation and investigate its rich properties on real directed networks of the Universities of Cambridge and Oxford, LiveJournal, and Twitter. In this model, the opinion formation of linked electors is weighted with their PageRank probability. Such a probability is used by the Google search engine for ranking of web pages. We find that the society elite, corresponding to the top PageRank nodes, can impose its opinion on a significant fraction of the society. However, for a homogeneous distribution of two opinions, there exists a bistability range of opinions which depends on a conformist parameter characterizing the opinion formation. We find that the LiveJournal and Twitter networks have a stronger tendency to a totalitarian opinion formation than the university networks. We also analyze the Sznajd model generalized for scale-free networks with the weighted PageRank vote of electors.

Direct measurements of IPTG enable analysis of the induction behavior of E. coli in high cell density cultures

BackgroundThe E. coli lac operon and its components have been studied for decades, and lac-derived systems are widely used for recombinant protein production. However, lac operon dynamics and induction behavior remain the paradigm of gene regulation. Recently, an HPLC-MS-based method to quantify IPTG in the medium and inside the biomass has been established, and this tool may be useful to uncover the lack of knowledge and allow optimization of biotechnological processes.ResultsThe results obtained from the study of IPTG distribution profiles in fed-batch, high cell density cultures allowed discrimination between two different depletion patterns of an inducer from the medium to the biomass in E. coli-expressing rhamnulose-1-phosphate aldolase (RhuA). Moreover, we could demonstrate that active transport mediates the uptake of this gratuitous inducer. Additionally, we could study the induction behaviors of this expression system by taking into account the biomass concentration at the induction time.ConclusionsIn the bistable range, partial induction occurred, which led to intermediate levels of RhuA activity. There was a direct relationship between the initial inducer concentrations and the initial inducer transport rate together with the specific activity. A majority of the inducer remains in the medium to reach equilibrium with the intracellular level. The intracellular inducer accumulation was a further evidence of bistability of the lac operon.

Modeling the effects of climate change on eelgrass stability and resilience: future scenarios and leading indicators of collapse

Seagrass meadows influence local hydrodynamics in coastal bays, resulting in a decrease in the shear stress acting on the underlying bed sediment. The reduced sediment suspension and water column turbidity creates a more favorable light environment for further seagrass growth. This positive feedback is strong enough to induce depth-dependent bistable dynamics with 2 possible stable states, an extant meadow and a bare sediment surface. A coupled vegetation-growth hydrodynamic model was used to investigate eelgrass stability and leading indicators of ecosystem shift under the effects of sea-level rise and increases in water temperature associated with climate change. The model was applied to Hog Island Bay, a shallow coastal bay within the Virginia Coast Reserve, USA, where eelgrass restoration efforts are ongoing. The results indicate that while extant eelgrass meadows are likely to tolerate sea-level rise, an increase in the frequency of days when summer water temperature exceeds 30°C will cause more frequent summer die-offs. This increase in the number of higher temperature disturbance events is likely to push a dense meadow initially located within the bistable depth range (1.6 to 1.8 m mean sea level) toward and eventually past a critical bifurcation point, from which recovery is not possible. We identified 2 leading indicators of a meadow nearing this bifurcation point, both associated with the number of leaves per shoot: ‘flickering,’ which reflects conspicuous fluctuations from one attractor to the other across the threshold, and ‘slowing down,’ which is the decreased recovery from perturbations as a system gets close to a threshold. Our model indicates that the eelgrass in these coastal bays has limited resilience to increases in water temperatures predicted from current climate change models.

Stability and resilience of seagrass meadows to seasonal and interannual dynamics and environmental stress

Shallow coastal bays provide habitat for diverse fish and invertebrate populations and are an important source of sediment for surrounding marshes. The sediment dynamics of these bays are strongly affected by seagrass meadows, which limit sediment resuspension, thereby providing a more favorable light environment for their own survival and growth. Due to this positive feedback between seagrass and light conditions, it has been suggested that bare sediment and seagrass meadows are potential alternate stable states of the benthos in shallow coastal bays. To investigate the stability and resilience of seagrass meadows subjected to variation in environmental conditions (e.g., light, temperature), a coupled model of vegetation–sediment–water flow interactions and vegetation growth was developed. The model was used to examine the effect of dynamically varying seasonal and interannual seagrass density on sediment resuspension, water column turbidity, and the subsequent light environment on hourly time steps and then run over decadal time scales. A daily growth model was designed to capture both belowground biomass and the growth and senescence of aboveground biomass structural components (e.g., leaves and stems). This allowed us to investigate how the annual and seasonal variability in shoot and leaf density within a meadow affects the strength of positive feedbacks between seagrass and their light environment. The model demonstrates both the emergence of bistable behavior from 1.6 to 1.8 m mean sea level due to the strength of the positive feedback, as well as the limited resilience of seagrass meadows within this bistable range.

Optical Bistability in an Acousto-Optic Tunable Filter (AOTF) Operating with Short Optical Pulses

In this paper we report for the first time the presence of bistability in an acoustic-optic tunable filter (AOTF) operating with ultrashort (2 ps) optical light pulses. The results for the study of bistability has shown the dependence of the hysteresis curve with the product of the coupling constant (κ) by the length of the device (ξL) and the conversion power-coupling constant factor (G). The range of bistability varies significantly with both G and with κξL parameters. The variation of κξL directly increases the size of the range of bistability hysteresis while the increase in G causes the bistability to occur at low powers. The phenomenon of optical bistability (OB) is the object of increasing interest due to its possibilities for important device applications. A bistable device is a device with a capability to generate two different outputs for a given input and the physical requirements for this are an intensity-dependence refractive index and an optical feedback mechanism.

Bi-stable flow around tandem cylinders of different diameters at low Reynolds number

The flow around two circular cylinders in tandem arrangement with different diameters has been investigated numerically. The upstream to downstream cylinder diameter ratio was kept constant at a value of d/D=2/3 and the centre-to-centre distance was varied from 1.2D to 5D. The Reynolds number based on upstream cylinder diameter and free-stream velocity used in this study is equal to 200. Force coefficients and flow frequencies have been obtained from 2D fluid flow finite-element simulations. Single-wake shedding, reattachment and co-shedding regimes have been observed. The results show that, depending on initial perturbations, reattachment and co-shedding can be observed on a wide range of centre-to-centre distances (L/D∊[1.8, 3.8]), which defines the bi-stable flow range. The lower part of the bi-stable range is characterized by a lock-in phenomenon for which vortex-shedding from the upstream and downstream cylinders are synchronized when there are co-shedding flow conditions.

Investigation on the bistability of linearly chirped nonlinear Bragg gratings

Based on the nonlinear coupled mode theory, by using the reversely recursive transmission matrix method, the switching characteristics of linearly chirped nonlinear Bragg gratings (LC-NLBG) have been investigated numerically. The results show that, introducing spatial chirp in fiber gratings is helpful to widen the bistable wavelength range, and the influence of the chirp on the switching-on threshold and the on-off ratio of the bistable switching are obvious, the chirp-dependent bistability is due to the difference of the axial distribution of forward wave intensity in the gratings.

Stability and bistability of seagrass ecosystems in shallow coastal lagoons: Role of feedbacks with sediment resuspension and light attenuation

Shallow coastal lagoons are environments where a dynamic equilibrium exists between water quality and seagrass cover. Dense seagrass canopies limit the resuspension of bed sediments thereby creating a clearer water column and a positive feedback for seagrass growth. Positive feedbacks are often associated with the existence of bistable dynamics in ecosystems. For example, a bare and a seagrass covered sediment bed could both be stable states of the system. This study describes a one‐dimensional hydrodynamic model of vegetation‐sediment‐water flow interactions and uses it to investigate the strengths of positive feedbacks between seagrass cover, stabilization of bed sediments, turbidity of the water column, and the existence of a favorable light environment for seagrasses. The model is applied to Hog Island Bay, a shallow coastal lagoon on the eastern shore of Virginia. The effects of temperature, eutrophication, and bed grain size on bistability of seagrass ecosystems in the lagoon are explored. The results indicate that under typical conditions, seagrass is stable in water depths &lt; 2.2 m (51% of the bay bottom deep enough for seagrass growth) and bistable conditions exist for depths of 2.2–3.6 m (23% of bay) where the preferred state depends on initial seagrass cover. The remaining 26% of the bay is too deep to sustain seagrass. Decreases in sediment size and increases in water temperature and degree of eutrophication shift the bistable range to shallower depths, with more of the bay bottom unable to sustain seagrass.

Spatial pattern formation and polarization dynamics of a nonequilibrium spinor polariton condensate

Quasiparticles in semiconductors---such as microcavity polaritons---can form condensates in which the steady-state density profile is set by the balance of pumping and decay. By taking account of the polarization degree of freedom for a polariton condensate, and considering the effects of an applied magnetic field, we theoretically discuss the interplay between polarization dynamics, and the spatial structure of the pumped decaying condensate. If spatial structure is neglected, this dynamics has attractors that are linearly polarized condensates (fixed points), and desynchronized solutions (limit cycles), with a range of bistability. Considering spatial fluctuations about the fixed point, the collective spin modes can either be diffusive, linearly dispersing, or gapped. Including spatial structure, interactions between the spin components can influence the dynamics of vortices; produce stable complexes of vortices and rarefaction pulses with both co- and counter-rotating polarizations; and increase the range of possible limit cycles for the polarization dynamics, with different attractors displaying different spatial structures.

Multi-phase-field analysis of short-range forces between diffuse interfaces

We characterize both analytically and numerically short-range forces between spatially diffuse interfaces in multi-phase-field models of polycrystalline materials. During late-stage solidification, crystal-melt interfaces may attract or repel each other depending on the degree of misorientation between impinging grains, temperature, composition, and stress. To characterize this interaction, we map the multiphase-field equations for stationary interfaces to a multidimensional classical mechanical scattering problem. From the solution of this problem, we derive asymptotic forms for short-range forces between interfaces for distances larger than the interface thickness. The results show that forces are always attractive for traditional models where each phase-field represents the phase fraction of a given grain. Those predictions are validated by numerical computations of forces for all distances. Based on insights from the scattering problem, we propose a multi-phase-field formulation that can describe both attractive and repulsive forces in real systems. This model is then used to investigate the influence of solute addition and a uniaxial stress perpendicular to the interface. Solute addition leads to bistability of different interfacial equilibrium states, with the temperature range of bistability increasing with strength of partitioning. Stress in turn, is shown to be equivalent to a temperature change through a standard Clausius-Clapeyron relation. The implications of those results for understanding grain boundary premelting are discussed.