The motion of untwisted untorted scroll waves in belousov-zhabotinsky reagent.

Rotating waves of activity are seen in various biological phenomena and in chemical mixtures. In thin layers of these media, the waves often appear as spirals spinning around a pivot point, but actually they are scroll-shaped waves rotating around curved filament in three-space. The filament about which the scroll rotates is not stationary, but rather moves through space until it achieves a stable configuration or disappears altogether. Some features of the temporal evolution of a planar scroll wave filament can be understood in terms of the simple equation N = Dkappa, where N is the velocity of the filament in the direction of its principal normal, kappa is the curvature of the filament, and D is the diffusion coefficient of the active chemical species. This equation of motion implies that a scroll ring shrinks in size and collapses in finite time, that an elongated spiral evolves into a symmetric spiral, and that an elongated target pattern becomes more symmetrical and vanishes in finite time. Characteristic times for these processes are estimated. In each case, good quantitative agreement is found between implications of the model and observations of scroll-wave evolution in shallow layers of the Belousov-Zhabotinsky reagent.

A cellular automation model of excitable media including curvature and dispersion.

Excitable media are spatially distributed systems characterized by their ability to propagate signals undamped over long distances. Wave propagation in excitable media has been modeled extensively both by continuous partial differential equations and by discrete cellular automata. Cellular automata are desirable because of their intuitive appeal and efficient digital implementation, but until now they have not served as reliable models because they have lacked two essential properties of excitable media. First, traveling waves show dispersion, that is, the speed of wave propagation into a recovering region depends on the time elapsed since the preceding wave passed through that region. Second, wave speed depends on wave front curvature: curved waves travel with normal velocities noticeably different from the plane-wave velocity. These deficiencies of cellular automation models are remedied by revising the classical rules of the excitation and recovery processes. The revised model shows curvature and dispersion effects comparable to those of continuous models, it predicts rotating spiral wave solutions in quantitative accord with the theory of continuous excitable media, and it is parameterized so that the spatial step size of the automation can be adjusted for finer resolution of traveling waves.

Chemical kinetic theory: understanding cell-cycle regulation.

Progress of a cell through its reproductive cycle of DNA synthesis and division is governed by a complex network of biochemical reactions controlling the activities of both M-phase- and S-phase-promoting factors. Standard chemical kinetic theory provides a disciplined method for expressing the molecular biologists' diagrams and intuition in precise mathematical form, so that qualitative and quantitative implications of our 'working models' can be derived and compared with experiment.

Kinetic analysis of a molecular model of the budding yeast cell cycle.

The molecular machinery of cell cycle control is known in more detail for budding yeast, Saccharomyces cerevisiae, than for any other eukaryotic organism. In recent years, many elegant experiments on budding yeast have dissected the roles of cyclin molecules (Cln1-3 and Clb1-6) in coordinating the events of DNA synthesis, bud emergence, spindle formation, nuclear division, and cell separation. These experimental clues suggest a mechanism for the principal molecular interactions controlling cyclin synthesis and degradation. Using standard techniques of biochemical kinetics, we convert the mechanism into a set of differential equations, which describe the time courses of three major classes of cyclin-dependent kinase activities. Model in hand, we examine the molecular events controlling "Start" (the commitment step to a new round of chromosome replication, bud formation, and mitosis) and "Finish" (the transition from metaphase to anaphase, when sister chromatids are pulled apart and the bud separates from the mother cell) in wild-type cells and 50 mutants. The model accounts for many details of the physiology, biochemistry, and genetics of cell cycle control in budding yeast.

A kinetic hairpin transfer model for parvoviral DNA replication.

The DNAs encapsidated by parvoviruses show distinctly different patterns with respect to the ratio of plus-to-minus strands and sequence heterogeneity at the ends. A kinetic model, based on differential rates of hairpin transfer at 3' termini, is described and shown to account for all known parvoviral DNA distributions.

Dynamic Modeling of the Interaction Between Autophagy and Apoptosis in Mammalian Cells.

Autophagy is a conserved biological stress response in mammalian cells that is responsible for clearing damaged proteins and organelles from the cytoplasm and recycling their contents via the lysosomal pathway. In cases of mild stress, autophagy acts as a survival mechanism, while in cases of severe stress cells may switch to programmed cell death. Understanding the decision process that moves a cell from autophagy to apoptosis is important since abnormal regulation of autophagy occurs in many diseases, including cancer. To integrate existing knowledge about this decision process into a rigorous, analytical framework, we built a mathematical model of cell fate decisions mediated by autophagy. Our dynamical model is consistent with existing quantitative measurements of autophagy and apoptosis in rat kidney proximal tubular cells responding to cisplatin-induced stress.

A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM.

Many organisms display rhythms of physiology and behavior that are entrained to the 24-h cycle of light and darkness prevailing on Earth. Under constant conditions of illumination and temperature, these internal biological rhythms persist with a period close to 1 day ("circadian"), but it is usually not exactly 24h. Recent discoveries have uncovered stunning similarities among the molecular circuitries of circadian clocks in mice, fruit flies, and bread molds. A consensus picture is coming into focus around two proteins (called PER and TIM in fruit flies), which dimerize and then inhibit transcription of their own genes. Although this picture seems to confirm a venerable model of circadian rhythms based on time-delayed negative feedback, we suggest that just as crucial to the circadian oscillator is a positive feedback loop based on stabilization of PER upon dimerization. These ideas can be expressed in simple mathematical form (phase plane portraits), and the model accounts naturally for several hallmarks of circadian rhythms, including temperature compensation and the per(L) mutant phenotype. In addition, the model suggests how an endogenous circadian oscillator could have evolved from a more primitive, light-activated switch.

Models of cell cycle control in eukaryotes.

The molecular mechanisms of cell cycle control are now known in enough detail to warrant mathematical modeling by kinetic equations. Despite the repetitive nature of the cell division cycle, the most appropriate models emphasize steady-state solutions rather than limit cycle oscillations, because cells progress toward division by passing a series of checkpoints (steady states).

A stochastic, molecular model of the fission yeast cell cycle: role of the nucleocytoplasmic ratio in cycle time regulation.

We propose a stochastic version of a recently published, deterministic model of the molecular mechanism regulating the mitotic cell cycle of fission yeast, Schizosaccharomyces pombe. Stochasticity is introduced in two ways: (i) by considering the known asymmetry of cell division, which produces daughter cells of slightly different sizes; and (ii) by assuming that the nuclear volumes of the two newborn cells may also differ. In this model, the accumulation of cyclins in the nucleus is proportional to the ratio of cytoplasmic to nuclear volumes. We have simulated the cell-cycle statistics of populations of wild-type cells and of wee1(-) mutant cells. Our results are consistent with well known experimental observations.

Modeling M-phase control in Xenopus oocyte extracts: the surveillance mechanism for unreplicated DNA.

Alternating phases of DNA synthesis and mitosis, during the first 12 cell divisions of frog embryos, are driven by autonomous cytoplasmic oscillations of M-phase promoting factor (MPF). Cell-free extracts of frog eggs provide a convenient preparation for studying the molecular machinery that generates MPF oscillations and the surveillance mechanism that normally prevents entry into mitosis until chromosomal DNA is fully replicated. Early experiments suggested that unreplicated DNA blocks MPF activity by inducing phosphorylation of a crucial tyrosine residue, but recent evidence implicates a stoichiometric inhibitor (an MPF binding protein) as the 'braking' agent. Using a realistic mathematical model of the mitotic control system in frog egg extracts, we suggest that both tyrosine phosphorylation and a stoichiometric inhibitors are involved in the block of MPF by unreplicated DNA. Both pathways operate by raising the cyclin threshold for MPF activation. As a bonus, in the process of analyzing these experiments, we obtain more direct and reliable estimates of the rate constants in the model.

Mathematical model of the fission yeast cell cycle with checkpoint controls at the G1/S, G2/M and metaphase/anaphase transitions.

All events of the fission yeast cell cycle can be orchestrated by fluctuations of a single cyclin-dependent protein kinase, the Cdc13/Cdc2 heterodimer. The G1/S transition is controlled by interactions of Cdc13/Cdc2 and its stoichiometric inhibitor, Rum1. The G2/M transition is regulated by a kinase-phosphatase pair, Wee1 and Cdc25, which determine the phosphorylation state of the Tyr-15 residue of Cdc2. The meta/anaphase transition is controlled by interactions between Cdc13/Cdc2 and the anaphase promoting complex, which labels Cdc13 subunits for proteolysis. We construct a mathematical model of fission yeast growth and division that encompasses all three crucial checkpoint controls. By numerical simulations we show that the model is consistent with a broad selection of cell cycle mutants, and we predict the phenotypes of several multiple-mutant strains that have not yet been constructed.

A proposal for temperature compensation of the circadian rhythm in Drosophila based on dimerization of the per protein.

Recently Goldbeter suggested an interesting model of circadian rhythms based on feedback inhibition by the PER protein on its own rate of transcription. In his model, the long delay necessary for generating 24 h periodicity is associated with slow phosphorylations of PER protein in the cytoplasm, assuming that only highly phosphorylated forms of PER are able to enter the nucleus and there interfere with transcription of the per gene. By casting this molecular mechanism in mathematical form, Goldbeter showed that it is consistent with many known features of circadian oscillations in PER abundance. However, he did not address one of the most important characteristics of the circadian rhythm: the near constancy of the 24 h period over a broad temperature range. Huang, Curtin, and Rosbash have recently suggested that dimerization of the PER protein is involved in temperature compensation of the circadian rhythm in Drosophila, because in mutant flies lacking the PER dimerization domain, the period is strongly dependent on temperature. We incorporate this idea into Goldbeter's model by introducing parallel pathways of phosphorylation of PER monomers and dimers. We assume that both monomers and dimers can be transported into the nucleus as long as at least one PER subunit is multiply phosphorylated. Temperature compensation in our model arises from opposing effects of temperature (T) on the rate of association of PER monomers and the rate of nuclear import of PER protein. In mutant flies, when PER subunits cannot dimerize, the period of the oscillation increases with T, so we assume that the rate constant for nuclear import is a decreasing function of T. To compensate for this effect in wild-type flies, we assume that the rate of association of PER subunits is an increasing function of T. The mathematical model reveals the relationship between these opposing tendencies that must be satisfied to achieve effective temperature compensation.

Effects of asymmetric division on a stochastic model of the cell division cycle.

The stochastic model of cell division formulated by Alt and Tyson is generalized to the case of imprecise binary fission. Closed-form expressions are derived for the generation-time distribution, the birth-size and division-size distributions, the beta curve, and the correlation coefficient of generation times of sister cells. The theoretical results are compared to observations of cell division statistics in a culture of fission yeast.

An improved data analysis method for interleukin 2 microassay.

Development of the interleukin 2(IL 2) microassay, coupled with the use of highly purified or recombinant factors has allowed a detailed examination of the mechanism of action of this important biological response modifier. However, probit analysis of the microassay data does not allow inherent error of the system to be approximated nor can units of activity be assessed for significance. A computer program was developed to analyze the validity of each regression line and to generate 95% confidence intervals around each line. This program employs analysis of variance, linear regression analysis and the parallel line assay to fix confidence intervals for each IL 2 unit value. The use of recombinant IL 2 as an immunomodulator in clinical settings warrants a more precise statistical method to evaluate normal fluctuations of this factor than currently in use. The development of such a method is presented here.

Computer evaluation of network dynamics models with application to cell cycle control in budding yeast.

Cellular processes are governed by complex networks of interacting genes and proteins. Theoretical molecular biologists attempt to describe these processes via mathematical models by writing biochemical reaction equations. Modellers are building increasingly larger and complex mathematical models to describe these cellular processes, making model evaluation a time consuming and difficult task. The authors describe an automatable process for model evaluation and a software system that implements this process. The software is adaptable to many types of models and is freely available along with all needed data files. The cell cycle control system for budding yeast is known in fine detail and constrained by more than 100 phenotypic observations in mutant strains. As an example, the authors apply their process to a model of cell cycle control in budding yeast containing dozens of regulatory equations and explaining nearly all of the known mutant phenotypes.

Globally optimised parameters for a model of mitotic control in frog egg extracts.

DNA synthesis and nuclear division in the developing frog egg are controlled by fluctuations in the activity of M-phase promoting factor (MPF). The biochemical mechanism of MPF regulation is most easily studied in cytoplasmic extracts of frog eggs, for which careful experimental studies of the kinetics of phosphorylation and dephosphorylation of MPF and its regulators have been made. In 1998 Marlovits et al. used these data sets to estimate the kinetic rate constants in a mathematical model of the control system originally proposed by Novak & Tyson. In a recent publication, we showed that a gradient-based optimisation algorithm finds a locally optimal parameter set quite close to the 'Marlovits' estimates. In this paper, we combine global and local optimisation strategies to show that the 'refined Marlovits' parameter set, with one minor but significant modification to the Novak & Tyson equations, is the unique, best-fitting solution to the parameter estimation problem.

Modeling the cell division cycle: cdc2 and cyclin interactions.

The proteins cdc2 and cyclin form a heterodimer (maturation promoting factor) that controls the major events of the cell cycle. A mathematical model for the interactions of cdc2 and cyclin is constructed. Simulation and analysis of the model show that the control system can operate in three modes: as a steady state with high maturation promoting factor activity, as a spontaneous oscillator, or as an excitable switch. We associate the steady state with metaphase arrest in unfertilized eggs, the spontaneous oscillations with rapid division cycles in early embryos, and the excitable switch with growth-controlled division cycles typical of nonembryonic cells.

Periodic enzyme synthesis and oscillatory repression: why is the period of oscillation close to the cell cycle time?

During exponential growth of a cell culture, some enzymes are synthesized periodically. In a synchronous culture, in which all cells undergo DNA synthesis and division more-or-less synchronously, the burst of enzyme synthesis also occurs synchronously in each cell once per division cycle. However, there are a number of interesting cases in which periodic enzyme synthesis continues in the absence of synchronous DNA replication or cell division. In all cases of periodic enzyme synthesis in asynchronous cultures, the time between bursts of enzyme synthesis, though no longer identical to the cell cycle time, is still close to the interdivision time of the growing, replicating cells. The theory of oscillatory repression looks for an explanation of this phenomenon in the periodic repression of gene transcription caused by periodic fluctuations in the concentration of the endproduct of the metabolic pathway of which the enzyme is a part. A major difficulty with this theory is that there is no obvious relationship between the periodicity of the negative feedback loop, which is determined by the kinetics of synthesis and degradation of the individual components of the feedback loop, and the periodicity of the cell cycle, which is determined by overall net synthetic rates of cellular macromolecules. Why should the period of oscillation of a repressible gene transcription system be close to the interdivision time of a population of growing cells? In this paper, I show that the relationship may be coincidental: the two fundamental periods are close to each other because they are both close to the mass-doubling time of the cell culture. That the mean interdivision time must be close to the mass-doubling time is a consequence of "balanced" growth: there is a stable size distribution of cells in a growing culture. That the period of oscillation of the negative feedback loop is also close to the mass-doubling time is shown to be a consequence of the large, nearly constant demand for endproduct and the assumed stability of the enzyme. The period of oscillation is largely attributable to the slow dilution of the stable enzyme by cell growth. For reasonable values of the parameters describing the gene-control system, I show that the enzyme must be diluted by a factor of two (approximately), that is, by the growth accomplished by one mass-doubling (nearly).

Unstable activator models for size control of the cell cycle.

The unstable activator model of Wheals & Silverman (1982) is extended to account for the delay of nuclear division in the acellular slime mold, Physarum polycephalum, that is caused by pulse treatments with inhibitors of protein synthesis. The model is solved exactly to predict the delay as a function of the half-life of the activator. The Wheals-Silverman model is found to give results comparable, but not superior, to other unstable activator models of the cell cycle.

Cyclic AMP waves during aggregation of Dictyostelium amoebae.

During the aggregation phase of their life cycle, Dictyostelium discoideum amoebae communicate with each other by traveling waves of cyclic AMP. These waves are generated by an interplay between random diffusion of cyclic AMP in the extracellular milieu and the signal-reception/signal/relaying capabilities of individual amoebae. Kinetic properties of the enzymes, transport proteins and cell-surface receptor proteins involved in the cyclic AMP signaling system have been painstakingly worked out over the past fifteen years in many laboratories. Recently Martiel & Goldbeter (1987) incorporated this biochemical information into a unified mathematical model of communication among Dictyostelium amoebae. Numerical simulations of the mathematical model, carried out by Tyson et al. (1989), agree in quantitative detail with experimental observations of cyclic AMP traveling waves in Dictyostelium cultures. Such mathematical modeling and numerical experimentation provide a necessary link between detailed studies of the molecular control mechanism and experimental observations of the intact developmental system.