The ups and downs of modeling the cell cycle

We discuss the impact of mathematical modeling on our understanding of the cell cycle. Although existing, detailed models confirm that the known interactions in the cell cycle can produce oscillations and predict behaviors such as hysteresis, they contain many parameters and are poorly constrained by data which are almost all qualitative. Questions about the basic architecture of the oscillator may be more amenable to modeling approaches that ignore molecular details. These include asking how the various elaborations of the basic oscillator affect the robustness of the system and how cells monitor their size and use this information to control the cell cycle.     

A statistical model of the human core-temperature circadian rhythm

We formulate a statistical model of the human core-temperature circadian rhythm in which the circadian signal is modeled as a van der Pol oscillator, the thermoregulatory response is represented as a first-order autoregressive process, and the evoked effect of activity is modeled with a function specific for each circadian protocol. The new model directly links differential equation-based simulation models and harmonic regression analysis methods and permits statistical analysis of both static and dynamical properties of the circadian pacemaker from experimental data. We estimate the model parameters by using numerically efficient maximum likelihood algorithms and analyze human core-temperature data from forced desynchrony, free-run, and constant-routine protocols. By representing explicitly the dynamical effects of ambient light input to the human circadian pacemaker, the new model can estimate with high precision the correct intrinsic period of this oscillator ( approximately 24 h) from both free-run and forced desynchrony studies. Although the van der Pol model approximates well the dynamical features of the circadian pacemaker, the optimal dynamical model of the human biological clock may have a harmonic structure different from that of the van der Pol oscillator.     

Oscillatory network with self-organized dynamical connections for synchronization-based image segmentation

An oscillatory network of columnar architecture located in 3D spatial lattice was recently designed by the authors as oscillatory model of the brain visual cortex. Single network oscillator is a relaxational neural oscillator with internal dynamics tunable by visual image characteristics - local brightness and elementary bar orientation. It is able to demonstrate either activity state (stable undamped oscillations) or "silence" (quickly damped oscillations). Self-organized nonlocal dynamical connections of oscillators depend on oscillator activity levels and orientations of cortical receptive fields. Network performance consists in transfer into a state of clusterized synchronization. At current stage grey-level image segmentation tasks are carried out by 2D oscillatory network, obtained as a limit version of the source model. Due to supplemented network coupling strength control the 2D reduced network provides synchronization-based image segmentation. New results on segmentation of brightness and texture images presented in the paper demonstrate accurate network performance and informative visualization of segmentation results, inherent in the model.     

Optical Properties of DNA in Aqueous Solution

In the study of DNA electric birefringence, it is usual to use theories that consider that molecules in solution are small in relation to the light wavelength. In this work, we study the DNA electric birefringence using a broken-rod macroion (BRM) model composed of two cylindrical arms which does not restrict the size of the molecules. To achieve this, we include the inhomogeneity effect of the light electric field through the molecule and the interaction between its different parts. To analyze the interaction between a molecule and the incident beam of light, we apply the discrete dipole approximation (DDA), according to which each molecule is described as a finite array of electronic coupled oscillators. The electric birefringence is calculated from the oscillator polarizability. This is obtained from experimental data of electric birefringence saturation and from the increment of the solution refraction index in relation to that of the solvent. Furthermore, the oscillator polarizability is also estimated from DNA absorption spectrum using the Kronig–Kramers relations. This allows us to analyze the contributions of the different absorption bands of DNA to the electric birefringence. We analyze the influence of the inhomogeneity of the light electric field and of the intramolecular interactions in the characterization of DNA optical properties using electric birefringence measurements.     

心肌振荡模型的非线性动力学研究

通过计算机对心肌振荡模型的研究,得到它在单刺激下的相角转移曲线.用相角转移曲线研究了振子在不同周期刺激下表现出来的锁相、分叉及混沌现象,进而对某些常见的心律失常产生的机制进行了讨论.     

Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine

The gait of current two-legged walking machines differs from that of humans, although the kinematic structures of these machines' legs frequently imitate human limbs. This paper presents a method of generating the trajectories of hip and knee joint angles resulting in a gait pattern similar to that of a human. For this purpose the solutions of coupled van der Pol oscillator equations are utilised. There is much evidence that these equations can be treated as a good model of the central pattern generator generating functional (also locomotional) rhythms in living creatures. The oscillator equations are solved by numerical integration. The method of changing the type of gait by changing appropriate parameter values in the oscillator equations is presented (change of velocity and trajectory of leg-ends). The results obtained enable enhanced control of twolegged walking systems by including gait pattern generators which will assume a similar role to that of biological generators.     

The circadian timekeeping system of Drosophila

Daily rhythms in behavior, physiology and metabolism are controlled by endogenous circadian clocks. At the heart of these clocks is a circadian oscillator that keeps circadian time, is entrained by environmental cues such as light and activates rhythmic outputs at the appropriate time of day. Genetic and molecular analyses in Drosophila have revealed important insights into the molecules and mechanisms underlying circadian oscillator function in all organisms. In this review I will describe the intracellular feedback loops that form the core of the Drosophila circadian oscillator and consider how they are entrained by environmental light cycles, where they operate within the fly and how they are thought to control overt rhythms in physiology and behavior. I will also discuss where work remains to be done to give a comprehensive picture of the circadian clock in Drosophila and likely many other organisms.     

Model of the sino-atrial and atrio-ventricular nodes of the conduction system of the human heart.

The coupled sino-atrial and atrio-ventricular nodes of the heart are discussed using a dedicated non-linear oscillator model. Several modes by which the oscillations cease in the system are obtained (asystole models). The oscillations of the model are compared with heart rate variability in heart block patients.     

Synchronization in a neural network of phase oscillators with the central element

A neural network model is considered which is designed as a system of phase oscillators and contains the central oscillator and peripheral oscillators which interact via the central oscillator. The regime of partial synchronization was studied when current frequencies of the central oscillator and one group of peripheral oscillators are near to each other while current frequencies of other peripheral oscillators are far from being synchronized with the central oscillator. Approximation formulas for the average frequency of the central oscillator in the regime of partial synchronization are derived, and results of computation experiments are presented which characterize the accuracy of the approximation.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

Use of current-voltage diagrams in locating peak energy barriers in cell membranes

Summary The current-voltage relations obtained by integrating the Nernst-Planck equations for a variety of energy profiles are obtained. A simple and approximate method for comparing these relations is described. The method is based on using a linearized transform of current-voltage relations for an Eyring single barrier model. A parameter, , related to the location of the single barrier in the Eyring model, and to the shape of the barrier in other models, is readily obtained from the slopes of the linearized relations. It is then a simple matter to determine whether a given current-voltage relation allows discrimination between any particular energy profiles. The results show that the equivalent Eyring model does not always place the peak energy barrier in the same position as other models and that quite large errors in the assignment of position may be made if such a model is used. The results are also used to test the ability of some experimental current-voltage diagrams to discriminate between various energy profiles.     

Quantum-mechanical kinetic study of the primary reaction of the photochemical cycle of Halobacterium halobium

A two-frequency oscillator model for the primary photochemical reaction bacteriorhodopsin batho-bacteriorhodopsin (K₆₁₀) is proposed. According to this model two conformational changes in the reaction are considered to take place: the first one is a distortion of the retinal in the bacteriorhodopsin active site and the second one is a conformational transition of the bacterioopsin, affecting the native structure hydrogen bonds. On the basis of this model the temperature dependences of the rate constants for normal and deuterated reactants are calculated in good agreement with the available experimental data. The relations of the reaction considered to the primary photochemical reaction of vision are discussed.     

Recycling the cell cycle: cyclins revisited

I discuss advances in the cell cycle in the 21 years since cyclin was discovered. The surprising redundancy amongst the classical cyclins (A, B, and E) and cyclin-dependent kinases (Cdk1 and Cdk2) show that the important differences between these proteins are when and where they are expressed rather than the proteins they phosphorylate. Although the broad principles of the cell cycle oscillator are widely accepted, we are surprisingly ignorant of its detailed mechanism. This is especially true of the anaphase promoting complex (APC), the machine that triggers chromosome segregation and the exit of mitosis by targeting securin and mitotic cyclins for destruction. I discuss how a cyclin/Cdk-based engine could have evolved to assume control of the cell cycle from other, older protein kinases.     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Role of the Pineal in the Circadian System of Birds

Pinealectomy of sparrows abolishes the free running rhythm oflocomotor activity in constant darkness. Without their pineals,sparrows still entrain to light-dark cycles and show other signsthat they retain part, but not all, of the circadian system.Interruption of either the neural input to the pineal or itsneural output, or both, does not abolish the free running rhythm.Rhythmicity can be restored to a pinealectomized bird by implantingthe pineal of a donor into the anterior chamber of its eye.A model of the circadian system controlling locomotor behavioris proposed to account for these facts. We suggest that theavian pineal contains a self-sustained oscillator, and as aconsequence, produces a rhythmic hormonal output. Circadianfluctuations of this hormone entrain a damped oscillator locatedelsewhere, which in turn drives the locomotor activity. Eachoscillator has separate access to environmental light cycles.The behavior of a pinealectomized bird is determined exclusivelyby its damped oscillator. It does not free run because it lacksthe self-sustained oscillator which, however, can be restoredby transplantation. The transplanted pineal continues its self-sustainedcircadian hormonal output which entrains the damped oscillatorand restores the system to its normal state.