Michael C. Mackey

PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS

A large number of human diseases are characterized by changes in the qualitative dynamics of physiological control systems: Systems that normally oscillate. stop oscillating, or begin to oscillate in a new and unexpected fashion, and systems that normally d o not oscillate, begin oscillating. These changes in qualitative dynamics often have a sudden onset, and in many instances it has not been possible to identify the factors that lead to the disease. By dynarnical disease we mean a disease that occurs in an intact physiological control system operating in a range of control parameters that leads to abnormal dynamics and human p a t h ~ l o g y . ~ ’ In this paper, the changes in qualitative dynamics associated with the onset of the disease are identified with bifurcations in the dynamics of mathematical models of the physiological control systems. We shall consider in some detail dynamical diseases in the respiratory and haematopoietic systems. Our starting point is the ordinary differential equation

Feedback Regulation in the Lactose Operon: A Mathematical Modeling Study and Comparison with Experimental Data

A mathematical model for the regulation of induction in the lac operon in Escherichia coli is presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; beta-galactosidase, which is involved in the conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We have paid particular attention to the estimation of the parameters in the model. We have tested our model against two sets of beta-galactosidase activity versus time data, as well as a set of data on beta-galactosidase activity during periodic phosphate feeding. In all three cases we find excellent agreement between the data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation in the model dynamics.

Age-structured and two-delay models for erythropoiesis

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

Oscillations in cyclical neutropenia: new evidence based on mathematical modeling

We present a dynamical model of the production and regulation of circulating blood neutrophil number. This model is derived from physiologically relevant features of the hematopoietic system, and is analysed using both analytic and numerical methods. Supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles are shown to exist. We make the estimation of kinetic parameters for dogs and then apply the model to cyclical neutropenia (CN) in the grey collie, a rare disorder in which oscillations in all blood cell counts are found. We conclude that the major cause of the oscillations in CN is an increased rate of apoptosis of neutrophil precursors which leads to a destabilization of the hematopoietic stem cell compartment.

Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors

This paper develops a price adjustment model for a single commodity market with state dependent production and storage delays. Conditions for the equilibrium price to be stable are derived in terms of a variety of economic parameters. When stability of the equilibrium price is lost a Hopf bifurcation occurs, giving rise to an oscillatory commodity price with a period between two and four times the equilibrium production-storage delay.

A simple model for phase locking of biological oscillators.

SummaryA mathematical model is presented for phase locking of a biological oscillator to a sinusoidal stimulus. Analytical, numerical and topological considerations are used to discuss the patterns of phase locking as a function of the amplitude of the sinusoidal stimulus and the relative frequencies of the oscillator and the sinusoidal stimulus. The sorts of experimental data which are needed to make comparisons between theory and experiment are discussed.

Nonlinear Dynamics in Physiology and Medicine

This book deals with the application of mathematics in modeling and understanding physiological systems, especially those involving rhythms. It is divided roughly into two sections. In the first part of the book, the authors introduce ideas and techniques from nonlinear dynamics that are relevant to the analysis of biological rhythms. The second part consists of five in-depth case studies in which the authors use the theoretical tools developed earlier to investigate a number of physiological processes: the dynamics of excitable nerve and cardiac tissue, resetting and entrainment of biological oscillators, the effects of noise and time delay on the pupil light reflex, pathologies associated with blood cell replication, and Parkinsonian tremor. One novel feature of the book is the inclusion of classroom-tested computer exercises throughout, designed to form a bridge between the mathematical theory and physiological experiments. This book will be of interest to students and researchers in the natural and physical sciences wanting to learn about the complexities and subtleties of physiological systems from a mathematical perspective. The authors are members of the Centre for Nonlinear Dynamics in Physiology and Medicine. The material in this book was developed for use in courses and was presented in three Summer Schools run by the authors in Montreal.

Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications

Of 24 published clinical reports of periodic chronic myelogenous leukaemia (PCML), 21 had sufficient data to analyse for periodicity, and 12 showed significant periodicity ( p  0.05) using the Lomb periodogram. Leucocyte oscillations had periods T ranging from 37 to 83 d. When data were also reported for platelets and reticulocytes there was no significant difference between their periods and those of the leucocytes. These data and their analysis provide strong circumstantial evidence for a haemopoietic stem cell origin of PCML. Namely, the known chromosomal changes in CML patients may, on occasion, also be accompanied by a destabilization resulting in an oscillatory efflux into the leucocyte, platelet and erythrocytic pathways.

Cell kinetic status of haematopoietic stem cells

The haematopoietic stem cell (HSC) population supports a tremendous cellular production over the course of an animal’s lifetime, e.g. adult humans produce their body weight in red cells, white cells and platelets every 7 years, while the mouse produces about 60% of its body weight in the course of a 2 year lifespan. Understanding how the HSC population carries this out is of interest and importance, and a first step in that understanding involves the characterization of HSC kinetics. Using previously published continuous labelling data (of Bradford et al. 1997 and Cheshier et al. 1999 ) from mouse HSC and a standard G0 model for the cell cycle, the steady state parameters characterizing these HSC populations are derived. It is calculated that in the mouse the differentiation rate ranges between about 0.01 and 0.02, the rate of cell re‐entry from G0 back into the proliferative phase is between 0.02 and 0.05, the rate of apoptosis from the proliferative phase is between 0.07 and 0.23 (all units are days−1), and the duration of the proliferative phase is between 1.4 and 4.3 days. These values are compared with previously obtained values derived from the modelling by Abkowitz and colleagues of long‐term haematopoietic reconstitution in the cat ( Abkowitz et al. 1996 ) and the mouse ( Abkowitz et al. 2000 ). It is further calculated using the estimates derived in this paper and other data on mice that between the HSC and the circulating blood cells there are between 17 and 19.5 effective cell divisions giving a net amplification of between ~170 000 and ~720 000.

The dynamics of production and destruction: Analytic insight into complex behavior

This paper analytically explores the properties of simple differential-difference equations that represent dynamic processes with feedback dependent on prior states of the system. Systems with pure negative and positive feedback are examined, as well as those with mixed (positive/negative) feedback characteristics. Very complex time dependent behaviors may arise from these processes. Indeed, the same mechanism may, depending on system parameters and initial conditions, produce simple, regular, repetitive patterns and completely irregular random-like fluctuations.For the differential-delay equations considered here we prove the existence of: (i) stable and unstable limit cycles, where the stable cycles may have an arbitrary number of extrema per period; and (ii) chaos, meaning the presence of infinitely many periodic solutions of different period and of infinitely many irregular and mixing solutions.