Chaos in biological systems

Fascinating rhythm: a primer on chaos theory and its application to cardiology

Nonlinear dynamics is an exciting new way of looking at peculiarities that in the past have been ignored or explained away. We have attempted to give a general introduction to the basics of the mathematics, applications to cardiology, and a brief review of the new tools needed to use the concepts of nonlinear mathematics. The careful mathematical approach to problems in cardiac electrical dynamics and blood flow is opening a window on behaviors and mechanisms previously inaccessible.

Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics

Whereas case rates for some childhood diseases (chickenpox) often vary according to an almost regular annual cycle, the incidence of more efficiently transmitted infections such as measles is more variable. Three hypotheses have been proposed to account for such fluctuations. (i) Irregular dynamics result from random shocks to systems with stable equilibria. (ii) The intrinsic dynamics correspond to biennial cycles that are subject to stochastic forcing. (iii) Aperiodic fluctuations are intrinsic to the epidemiology. Comparison of real world data and epidemiological models suggests that measles epidemics are inherently chaotic. Conversely, the extent to which chickenpox outbreaks approximate a yearly cycle depends inversely on the population size.

Regular oscillations in suspensions of a putatively chaotic mutant of Dictyostelium discoideum

We have tested the light-scattering properties of suspensions of the Dictyostelium discoideum mutant HH201 derived from the mutant Fr17. Previous studies indicated that HH201 and Fr17 possess highly irregular rhythmic properties which might represent aperiodic oscillations, i.e. chaos. We report that the former mutant can display regular oscillatory behavior. Possible explanations for this result are discussed, including that of a transition from chaotic to periodic behavior resulting from some parameter change or from strong intercellular coupling in cell suspensions.

Limit-cycle oscillations and chaos in reaction networks subject to conservation of mass

A cyclic network of autocatalytic reactions involving an unbuffered cofactor and a number of components subject to conservation of mass displays a surprising richness of dynamical behaviors. Limit-cycle oscillations are possible over a wide range of parameter values. Additionally, a cascade of period-doubling bifurcations leading to chaos can coexist with a multiplicity of stable steady states. These results draw attention to the role of unbuffering as a feedback in biochemical systems.

Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark

Using traditional spectral analysis and recently developed non-linear methods, we analyze the incidence of six childhood diseases in Copenhagen, Denmark. In three cases, measles, mumps, rubella, the dynamics suggest low dimensional chaos. Outbreaks of chicken pox, on the other hand, conform to an annual cycle with noise superimposed. The remaining diseases, pertussis and scarlet fever, remain problematic. The real epidemics are compared with the output of a Monte Carlo analog of the SEIR model for childhood infections. For measles, mumps, rubella, and chicken pox, we find substantial agreement between the model simulations and the data.

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.