John J. Tyson

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

The physiological responses of cells to external and internal stimuli are governed by genes and proteins interacting in complex networks whose dynamical properties are impossible to understand by intuitive reasoning alone. Recent advances by theoretical biologists have demonstrated that molecular regulatory networks can be accurately modeled in mathematical terms. These models shed light on the design principles of biological control systems and make predictions that have been verified experimentally.

Singular perturbation theory of traveling waves in excitable media (a review)

Abstract Waves of chemical or electrical activity, traveling through space, have been observed in several contexts: chemical reaction mixtures in non-convecting liquid phase, cell suspensions, nerve axons, and neuromuscular tissues. Typically, wave-supporting preparations are excitable; that is, they respond and sensitivity to perturbations are rapidly damped out, but suprathreshold disturbances trigger an abrupt and substantial response. The abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media. Singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious, but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways. We compare and contrast the approaches taken by Greenberg, Zykov, Fife, Krinskii and his collaborators, and ourselves, with particular emphasis on the case of rotating spiral waves. Our intention is to bring some order to the important but difficult theory of propagating waves in two-dimensional excitable media. In conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three-dimensional space.

Network dynamics and cell physiology

Complex assemblies of interacting proteins carry out most of the interesting jobs in a cell, such as metabolism, DNA synthesis, movement and information processing. These physiological properties play out as a subtle molecular dance, choreographed by underlying regulatory networks. To understand this dance, a new breed of theoretical molecular biologists reproduces these networks in computers and in the mathematical language of dynamical systems.

Computational cell biology

Preface.-Introductory Course.-Dynamic Phenomena in Cells.-Voltage Gated Ionic Currents.-Transporters and Pumps.-Fast and Slow time Scales.-Whole Cell Models.-Intercellular Communication.-Advanced Material.-Spatial Modeling.-Modeling Intracellular Calcium Waves and Sparks.-Biochemical Oscillations.-Cell Cycle Controls.- Modeling the Stochastic Gating of Ion Channels.-Molecular Motors: Theory.-Molecular Motors: Examples.-Numerical Algorithms.-References.-Index.

Hysteresis drives cell-cycle transitions in Xenopus laevis egg extracts

Cells progressing through the cell cycle must commit irreversibly to mitosis without slipping back to interphase before properly segregating their chromosomes. A mathematical model of cell-cycle progression in cell-free egg extracts from frog predicts that irreversible transitions into and out of mitosis are driven by hysteresis in the molecular control system. Hysteresis refers to toggle-like switching behavior in a dynamical system. In the mathematical model, the toggle switch is created by positive feedback in the phosphorylation reactions controlling the activity of Cdc2, a protein kinase bound to its regulatory subunit, cyclin B. To determine whether hysteresis underlies entry into and exit from mitosis in cell-free egg extracts, we tested three predictions of the Novak–Tyson model. (i) The minimal concentration of cyclin B necessary to drive an interphase extract into mitosis is distinctly higher than the minimal concentration necessary to hold a mitotic extract in mitosis, evidence for hysteresis. (ii) Unreplicated DNA elevates the cyclin threshold for Cdc2 activation, indication that checkpoints operate by enlarging the hysteresis loop. (iii) A dramatic “slowing down” in the rate of Cdc2 activation is detected at concentrations of cyclin B marginally above the activation threshold. All three predictions were validated. These observations confirm hysteresis as the driving force for cell-cycle transitions into and out of mitosis.

Target patterns in a realistic model of the Belousov–Zhabotinskii reaction

Periodic expanding target patterns of chemical activity are observed in thin layers of solution containing bromate, malonic acid and ferroin in dilute sulfuric acid. Commonly these patterns appear as thin blue (oxidized) rings propagating out from a central point into red (reduced) bulk medium. Recently, the opposite pattern has been observed: red waves of reduction propagating through an oxidized bulk medium. We discuss both of these patterns under the assumption that there is a heterogeneity at the center of the pattern—most likely a dust particle or a scratch on the glass—which changes the kinetics locally from a stable excitable steady state to a stable periodic oscillatory state. The temporal oscillation at the origin triggers waves of chemical activity which propagate radially into the excitable medium. Our approach is to combine recent advances in the mathematical description of traveling wave front solutions of reaction–diffusion equations with a realistic model of the kinetics of the reaction med...

Spiral waves in the Belousov-Zhabotinskii reaction

Abstract The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media. It has long been known that these spirals are similar to involutes of circles, at least some distance from the center, but until now, no way has been known to determine the correct wavelength and frequency. In this paper, we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution. The critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two-dimensional media, and the dispersion of plane waves Our analytical results are shown to be in good agreement with experimental data for the Belousov-Zhabotinskii reagent.

The Belousov-Zhabotinskii reaction

I. Preliminaries.- Chemical kinetics.- Ordinary differential equations.- Reaction-diffusion equations.- II. Chemistry of the Belousov-Zhabotinskii reaction.- Overall reaction.- The FKN mechanism.- III. The Oregonator.- The model, steady states and stability.- Existence of periodic solutions.- Limit cycles in the relaxation-oscillator regime.- Hard self-excitation.- IV. Chemical waves.- Kinematic waves.- a) Phase gradients.- b) Frequency gradients.- Trigger waves.- Velocity of propagation of trigger waves.- Scroll waves.- Plane wave and spiral wave solutions of reactiondiffusion equations.- Appendix. The Zhabotinskii-Zaikin-Korzukhin-Kreitser model.- References.

Functional Motifs in Biochemical Reaction Networks

The signal-response characteristics of a living cell are determined by complex networks of interacting genes, proteins, and metabolites. Understanding how cells respond to specific challenges, how these responses are contravened in diseased cells, and how to intervene pharmacologically in the decision-making processes of cells requires an accurate theory of the information-processing capabilities of macromolecular regulatory networks. Adopting an engineers approach to control systems, we ask whether realistic cellular control networks can be decomposed into simple regulatory motifs that carry out specific functions in a cell. We show that such functional motifs exist and review the experimental evidence that they control cellular responses as expected.

Steady States and Oscillations in the p53/Mdm2 Network

p53 is activated in response to events compromising the genetic integrity of a cell. Recent data show that p53 activity does not increase steadily with genetic damage but rather fluctuates in an oscillatory fashion (Lahav et al., Nature Genetics, 36, 147-150, 2004). Theoretical studies suggest that oscillations can arise from a combination of positive and negative feedbacks or from a long negative feedback loop alone. Both negative and positive feedbacks are present in the p53/Mdm2 network, but it is not known what roles they play in the oscillatory response to DNA damage. We developed a mathematical model of p53 oscillations based on positive and negative feedbacks in the p53/Mdm2 network. According to the model, the system reacts to DNA damage by moving from a stable steady state into a region of stable limit cycles. Oscillations in the model are born with large amplitude, which guarantees an all-or-none response to damage. As p53 oscillates, damage is repaired and the system moves back to a stable steady state with low p53 activity. The model reproduces experimental data in quantitative detail. We suggest new experiments for dissecting the contributions of negative and positive feedbacks to the generation of oscillations.