A Dynamic Gene Regulatory Network Model That Recovers the Cyclic Behavior of Arabidopsis thaliana Cell Cycle

Cell cycle control is fundamental in eukaryotic development. Several modeling efforts have been used to integrate the complex network of interacting molecular components involved in cell cycle dynamics. In this paper, we aimed at recovering the regulatory logic upstream of previously known components of cell cycle control, with the aim of understanding the mechanisms underlying the emergence of the cyclic behavior of such components. We focus on Arabidopsis thaliana, but given that many components of cell cycle regulation are conserved among eukaryotes, when experimental data for this system was not available, we considered experimental results from yeast and animal systems. We are proposing a Boolean gene regulatory network (GRN) that converges into only one robust limit cycle attractor that closely resembles the cyclic behavior of the key cell-cycle molecular components and other regulators considered here. We validate the model by comparing our in silico configurations with data from loss- and gain-of-function mutants, where the endocyclic behavior also was recovered. Additionally, we approximate a continuous model and recovered the temporal periodic expression profiles of the cell-cycle molecular components involved, thus suggesting that the single limit cycle attractor recovered with the Boolean model is not an artifact of its discrete and synchronous nature, but rather an emergent consequence of the inherent characteristics of the regulatory logic proposed here. This dynamical model, hence provides a novel theoretical framework to address cell cycle regulation in plants, and it can also be used to propose novel predictions regarding cell cycle regulation in other eukaryotes.     

Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

The nonlinear behavior of the Holling-Tanner predatory-prey differential equation system, employed by R.M. May to illustrate the apparent robustness of Kolmogorov’s Theorem when applied to such exploitation systems, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature-dependent mite interaction on fruit trees. The most significant result of this analysis is that there exists a temperature range wherein multiple stable states can occur, in direct violation of May’s interpretation of this system’s satisfaction of Kolmogorov’s Theorem: namely, that linear stability predictions have global consequences. In particular these stable states consist of a focus (spiral point) and a limit cycle separated from each other in the phase plane by an unstable limit cycle, all of which are associated with the single community equilibrium point of the system. The ecological implications of such metastability, hysteresis, and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system, and the biological control of mite populations are discussed.     

一类非线性微分动力系统的定性分析

本文研究了一类非线性微分动力系统０,ｂ＞０,Ｐ＞０）的定性行为,完整地解决了系统的极限环的不存在性、存在性和唯一性问题。得到系统有唯一极限环当且仅当（Ｐ一１）ａ－ｂ＞（ａ＋ｂ）~（Ｐ＋１）     

Some conditions for sustained oscillations in chain processes with feedback inhibition and saturable removal

A biochemical chain with feedback inhibition, enzymatic removal and catalyzed input is considered. Some conditions on the parameters are given for which the result is a simple limit cycle behavior or a limit cycle surrounding an unstable limit cycle and a stable point. In the latter case, the system in the steady state can either be at rest or oscillate around the point at rest, depending on the initial conditions. Presented at the May 23–24, 1975 meeting of the Society for Mathematical Biology, Bowling Green State University, Bowling Green, Ohio.     

Co-operative components, spatial localization and oscillatory cellular dynamics

The consequences of spatial localization on the dynamic behavior of catalytic components whose outputs control one another's activities but are not the substrates of those activities are explored. For some simple model systems we naturally find limit cycle oscillations, multiple asymptotic states, dependence of the dynamic states upon spatial configuration of the components, and complex switching between dynamic states as a function of external perturbation to the system.     

Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees

The nonlinear behavior of a particular Kolmogorov-type exploitation differential equation system assembled by May (1973,Stability and Complexity in Model Ecosystems, Princeton University Press) from predator and prey components developed by Leslie (1948,Biometrica 35, 213–245) and Holling (1973,Mem. Entomol. Soc. Can. 45, 1–60), respectively, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature dependent mite interaction on fruit trees. The most significant result of this analysis is that, in addition to the temperature ranges over which the single community equilibrium point of the system iseither globally stableor gives rise to a globally stable limit cycle, there can also exist a range wherein multiple stable states occur. These stable states consist of a focus (spiral point) and a limit cycle, separated from each other in the phase plane by an unstable limit cycle. The ecological implications of such metastability, hysteresis and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system and the biological control of mite populations are discussed. It is further suggested that a model of this sort which possesses a single community equilibrium point may be more useful for representing outbreak phenomena, especially in the presence of oscillations, than the non-Kolmogorov predator-prey systems possessing three community equilibrium points, two of which are stable and the other a saddle point, traditionally employed for this purpose.     

具非线性饱和功能反应的捕食者-食饵系统的定性分析

研究了一类具有非线性饱和功能反应的捕食者—食饵系统的定性行为.结果表明:当正平衡点稳定时,系统为全局渐近稳定的;当正平衡点不稳定时,系统存在唯一稳定的极限环.     

Bounds on the populations in the Dreitlein-Smoes model of oscillatory kinetic systems

The model recently proposed by Dreitlein and Smoes for oscillatory kinetic systems is studied. Diffusion of the oscillating species is taken into account, and bounds on the total number of individuals of each species are determined for both two- and three-dimensional finite regions with various boundary conditons applied. It is found that in general the effect of diffusion on the system behavior is to reduce the maximum possible radius of limit cycles. In particular, in some cases global limit cycle behavior is precluded.     

Possible functional roles of phase resetting during walking

The walking rhythm is known to show phase shift or "reset" in response to external impulsive perturbations. We tried to elucidate functional roles of the phase reset possibly used for the neural control of locomotion. To this end, a system with a double pendulum as a simplified model of the locomotor control and a model of bipedal locomotion were employed and analyzed in detail. In these models, a movement corresponding to the normal steady-state walking was realized as a stable limit cycle solution of the system. Unexpected external perturbations applied to the system can push the state point of the system away from its limit cycle, either outside or inside the basin of attraction of the limit cycle. Our mathematical analyses of the models suggested functional roles of the phase reset during walking as follows. Function 1: an appropriate amount of the phase reset for a given perturbation can contribute to relocating the system's state point outside the basin of attraction of the limit cycle back to the inside. Function 2: it can also be useful to reduce the convergence time (the time necessary for the state point to return to the limit cycle). In experimental studies during walking of animals and humans, the reset of walking rhythm induced by perturbations was investigated using the phase transition curve (PTC) or the phase resetting curve (PRC) representing phase-dependent responses of the walking. We showed, for the simple double-pendulum model, the existence of the optimal phase control and the corresponding PTC that could optimally realize the aforementioned functions in response to impulsive force perturbations. Moreover, possible forms of PRC that can avoid falling against the force perturbations were predicted by the biped model, and they were compared with the experimentally observed PRC during human walking. Finally, physiological implications of the results were discussed.     

Multisite phosphorylation and network dynamics of cyclin-dependent kinase signaling in the eukaryotic cell cycle

Multisite phosphorylation of regulatory proteins has been proposed to underlie ultrasensitive responses required to generate nontrivial dynamics in complex biological signaling networks. We used a random search strategy to analyze the role of multisite phosphorylation of key proteins regulating cyclin-dependent kinase (CDK) activity in a model of the eukaryotic cell cycle. We show that multisite phosphorylation of either CDK, CDC25, wee1, or CDK-activating kinase is sufficient to generate dynamical behaviors including bistability and limit cycles. Moreover, combining multiple feedback loops based on multisite phosphorylation do not destabilize the cell cycle network by inducing complex behavior, but rather increase the overall robustness of the network. In this model we find that bistability is the major dynamical behavior of the CDK signaling network, and that negative feedback converts bistability into limit cycle behavior. We also compare the dynamical behavior of several simplified models of CDK regulation to the fully detailed model. In summary, our findings suggest that multisite phosphorylation of proteins is a critical biological mechanism in generating the essential dynamics and ensuring robust behavior of the cell cycle.     

Instability and oscillatory behavior of membrane-chemical reaction systems

The stability problem of a chemical system consisting of reversible reactions is analyzed with the aid of computer calculations. The system is based on the model proposed by Edelstein and exhibits oscillations of chemical species. The analysis shows that the oscillatory character is of limit cycle type. The results are applied to the construction of a membrane-chemical reaction system, which shows characteristic instability behavior. This is useful as a model of cell division.     

A prediction of mRNA hybridization kinetics based on polypeptide abundances

An oscillating chemical reaction dealt with in this paper is the Brusselator, which is capable of generating a limit cycle oscillation beyond instability point. The rate equations of this reaction scheme are solved by two-time scale method. An evolution criterion is derived for the limit cycle to first order approximation. The time average of entropy change over the period of the limit cycle takes a definite negative value, which is dependent not on the initial conditions but on the external parameters bringing the kinetic behaviour of the system under control. It can be expressed as the product of two quantities. The first, which is kinetic in character, is the areal velocity of the limit cycle. The second, which is thermodynamic in character, is the rotation of anti-symmetric flow with respect to thermodynamic force. It is shown that the latter is equivalent to the irreversible circulation of fluctuation.     

Bifurcation analysis of a neural network model

This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory populations of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.     

具常数投放率功能反应为／x的食饵-捕食系统的定性分析

本文研究一类具有常数投放率的食饵-捕食系统的定性行为,得到了正平衡点全局渐近稳定以及在正平衡点周围存在唯一极限环的充分条件．利用数值模拟检验了结论．     

Two-neuron networks

The behavior of two pacemaker neurons simulated by leaky integrators and connected reciprocally by synapses was studied. In every case the firing of both neurons phase-locks. The resulting limit cycle may or may not show simultaneous firing of both neurons. When both synapses are excitatory, phase-locking with simultaneous neuronal firing is always present. When one synapse is excitatory and the other inhibitory, phase-locking is also present always, while the neurons may or may not fire simultaneously. For a restricted set of parameters, bistability appears; the initial conditions determine whether or not the limit cycle presents simultaneous firing. When both synapses are inhibitory, the system phase-locks without simultaneous firing for almost every set of parameters.     

Transient and steady state phase response curves of limit cycle oscillators

The experiment of phase shifts resulting from discrete perturbations of stable biological rhythms has been carried out to study entrainment behavior of oscillators. There are two kinds of phase response curves, which are measured in experiments, according to as one measures the phase shifts immediately or long after the perturbation. The former is the first transient phase response curve and the latter is the steady state phase response curve. We redefine both curves within the framework of dynamical system theory and homotopy theory. Several topological properties of both curves are clarified. Consequently, it is shown that we must compare the shapes of both two phase response curves to investigate the inner structures of biological oscillators. Moreover, we prove that a single limit cycle oscillator involving only two variables cannot simulate transient resetting behavior reported by Pittendrigh and Minis (1964). In other words, the circadian oscillator of Drosophila pseudoobscura does not consist of a single oscillator of two variables. Finally we show that a model which consists of two limit cycle oscillators is able to simulate qualitatively the phase response curves of Drosophila.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Dynamical behavior of two predators competing over a single prey

Dynamical behavior of a food web comprising two predators competing over a single prey has been investigated. The analysis of the food web model shows that the persistence is not possible for two competing predators sharing a single prey species in the cases when any one of the boundary prey–predator planes has a stable equilibrium point. The principle of competitive exclusion holds in such cases. However, numerical simulations exhibit persistence in the presence of periodic solutions in the boundary planes. The system exhibits quasi-periodic behavior in the positive octant. The co-existence in the form of a limit cycle is also possible in some cases.     

Dynamical behavior of two predators competing over a single prey

Dynamical behavior of a food web comprising two predators competing over a single prey has been investigated. The analysis of the food web model shows that the persistence is not possible for two competing predators sharing a single prey species in the cases when any one of the boundary prey–predator planes has a stable equilibrium point. The principle of competitive exclusion holds in such cases. However, numerical simulations exhibit persistence in the presence of periodic solutions in the boundary planes. The system exhibits quasi-periodic behavior in the positive octant. The co-existence in the form of a limit cycle is also possible in some cases.     

A nonlinear,second-order population model

Delay differential, difference, and partial differential equation models are being used more extensively to explain single-species population oscillations and limit cycle behavior. Ordinary differential equation (ODE) models have been largely ignored. This is because first-order ODE models are inherently monotonic. Certainly this is not usual population behavior in the real world. If it is assumed that the per capita growth rate of a population changes over time as a result of regulating factors impinging on it, then a more realistic biological model results. The model translates into a second-order nonlinear ODE. Such a model can exhibit oscillatory and limit cycle as well as monotonic solutions, i.e., behavior for which non-ODE models have been used to explain. Although first-order ODE models are gross simplifications of real phenomena, ODE models in general should not be disregarded as important analytical tools.