Biological and Biochemical Oscillators

Biological and Biochemical Oscillators compiles papers on biochemical and biological oscillators from a theoretical and experimental standpoint. This book discusses the oscillatory behavior, excitability, and propagation phenomena on membranes and membrane-like interfaces; two-dimensional analysis of chemical oscillators; and chemiluminescence in oscillatory oxidation reactions catalyzed. The problems associated with the computer simulation of oscillating systems; mechanism of single-frequency glycolytic oscillations; excitation wave propagation during heart fibrillation; and biochemical cycle of excitation are also elaborated. This compilation likewise covers the physiological rhythms in Saccharomyces cerevisiae populations; integral and indissociable property of eukaryotic gene-action systems; and role of actidione in the temperature jump response of the circadian rhythm in Euglena gracilis. This publication is valuable to biochemists interested in biochemical and biological oscillations.

• Language: English
• Publisher: Academic Press
• Release date: Jun 28, 2014
• ISBN: 9781483271194

Book preview

Hess

INTRODUCTION

Benno Hess

The great interest in biochemical oscillations derives from the fact that only recently has it become recognized that biochemical systems can generate self-sustained oscillations. Studies over the past decade have clearly shown that relatively simple enzyme reaction systems with appropriate coupling mechanisms for activation and inhibition, as well as suitable input and output rates, can generate oscillations with a wide variety of periods and waveforms. This discovery has revolutionized the study of the biological systems and has opened the way for their understanding at the molecular level. In addition, the availability of technical methods to analyze these instabilities has tremendously stimulated both theoretical and experimental work.

The experimental field of biochemical oscillations began with the measurement of intracellular components in studies of photosynthesis. The observation of cyclic changes in the concentrations of phosphoglycerate and ribulose diphosphate in 1955 led, in 1958, to a detailed kinetic theory of the self–oscillating mode of the dark reactions of photosynthesis. In non–photosynthetic cells the direct readout of intracellular components, such as NADH, showed overshoots (1952, 1954) and oscillations (1957) in glycolysis. In 1964 nearly continuous oscillations of NADH-fluorescence were reported in yeast cells. This observation was followed by the demonstration of continuous glycolytic oscillations in a cell–free system of yeast (1966). The development coincided with the observation of oscillatory ion movements in mitochondria (1965) where different metabolic functions, as well as components of the respiratory chain, displayed sustained periodic behavior.

The frequencies of biological rhythms cover a bandwidth of over ten orders of magnitude which reflects the temporal organization of the living world. It is interesting to see that the biochemical oscillations now fill the gap between the circadian oscillations and the neural frequencies observed in biological systems. Therefore, it was only proper that communication should be opened between those interested in biochemical oscillations and others who focused their attention on the numerous periodic phenomena such as diurnal rhythms, the circadian clock and other periodic activities of biological systems.

This volume of contributed papers is one result of the communication between researchers of various nationalities in the two fields. The papers summarise the major view points from a number of laboratories and clearly illustrate the present situation of the field from both a theoretical and experimental standpoint.

I

OSCILLATOR THEORY

Outline

Chapter 2: OSCILLATORY BEHAVIOR, EXCITABILITY AND PROPAGATION PHENOMENA ON MEMBRANES AND MEMBRANE-LIKE INTERFACES

Chapter 3: TWO-DIMENSIONAL ANALYSIS OF CHEMICAL OSCILLATORS

Chapter 4: STABILITY PROPERTIES OF METABOLIC PATHWAYS WITH FEEDBACK INTERACTIONS

OSCILLATORY BEHAVIOR, EXCITABILITY AND PROPAGATION PHENOMENA ON MEMBRANES AND MEMBRANE-LIKE INTERFACES

U.F. Franck,     Institut für Physikalische Chemie der Rhein.-Westf. Techn. Hochschule, Aachen, Germany

Publisher Summary

The quantitative treatment, on the basis of primary principles, of oscillating physico-chemical and biological systems is cumbersome. This is on account of their essentially non-linear nature and their complex kinetics involving a set of independent variables. The general principles that lead to oscillatory behavior are hard to recognize because each oscillating system requires a particular treatment. For these reasons, a phenomenological approach is proposed, which considers oscillating systems as black boxes whose kinetic properties are studied by directly measurable forces and fluxes, which exist between them and their environment. Sustained periodic transportation processes and/or chemical reactions occur only in energetically open systems, which are in contact with an appropriate environment. Physico-chemical and biological oscillations are always a common result of the kinetic properties of the system and the given environment. They are energetically brought about by two or more independent driving forces existing in the environment, or in the system, or in both of them.

The quantitative treatment, on the basis of primary principles, of oscillating physico-chemical and biological systems is extremely cumbersome. This is on account of their essentially non-linear nature and their complex kinetics involving a set of independent variables. Moreover the general principles which lead to oscillatory behavior are, in this way, hard to recognize because each oscillating system requires, as a rule, a particular treatment.

For these reasons a phenomenological approach is proposed in this paper which considers oscillating systems as black boxes whose kinetic properties are studied by directly measurable forces and fluxes which exist between them and their environment.

Sustained periodic transportation processes and/or chemical reactions occur, as we are aware, only in energetically open systems which are in contact with an appropriate environment. Physico-chemical and biological oscillations are therefore always a common result of the kinetic properties of the system and the given environment. They are energetically brought about by two or more independent driving forces existing in the environment, or in the system, or in both of them.

I One process systems

Energetically open systems containing only one kind of transportation process are not able to oscillate, but, as shown later, they can exhibit, under certain circumstances, excitation and propagation phenomena.

As already mentioned every energetically open system can be regarded as a black box which represents a kind of dipole (1–3). At the poles (these are the entry and the exit of the flux) the intensity of the flux, I, and the force, X, can be measured without knowing the special inner structure of the dipole.

The system under consideration, as well as its environment, are dipoles. In a given working situation they are connected together forming a flux circuit containing a source of driving forces, XO, inside of one or both of the dipoles (See Figure 1).

Figure 1 Black-box treatment of physico-chemical systems and their environments.

For all stationary states there exists, for the system as well as for its environment, a defined relationship between the force, X, and the conjugated flux, I. Both relationships can be measured independently of each other at the separated dipoles and can be plotted in the form of force-flux characteristics. In either case, the characteristic is the geometric locus of all stationary states in which the dipole in question can exist (Figure 1). Accordingly, their intersection represents the common stationary state of the system and the environment. All the other states are, as a function of the force, X, non–stationary and therefore they change with time.

Between the connected dipoles a transfer of energy takes place. Denoting the dipole having the higher strength force source, XO, as donator and the other dipole as acceptor, then the area of the rectangle formed by the coordinates of state X and I in Figure 1, can be directly interpreted as the power transferred from the donator to the acceptor under stationary conditions.

The distinction between acceptor and donator is important with respect to the definition of the sign of the flux and to the criteria of stability of the stationary states.

For all one-process systems there are valid corresponding graphs. Figure 2 shows examples of electric, hydraulic, thermal, diffusional and chemical systems. In the first case, the characteristic of the electric environment is the well-known load-line of electric circuits. This useful term may well be extended to all characteristics of energy transferring environments.

Figure 2 Force-flux characteristics for electric, hydraulic, thermal, diffusional and chemical systems.

Although the ensemble of characteristics of the system and its environment describes stationary states, direct information concerning the time-dependent behavior of the non-stationary states (as a function of X) can be derived from it. In non-stationary states there exists a defined difference between the stationary state values of the flux of the system and the flux of the environment (see Figure 3). The deviation of the flux supply of the environment, with respect to the stationary state value of the system, necessarily has to be balanced by the storage elements inside the system giving rise to a change of the force, X, with time, according to the well-known capacity equation (1):

Figure 3 Time-dependence of non-stationary states.

Here C denotes the capacity, in a generalized sense, e.g.:

electric systems - an influx of electric charges into an electric capacitance leads to an increase in voltage;

chemical systems - an influx of matter into a volume of solvent leads to an increase in concentration;

thermal systems - an influx of heat into, a heat capacitance leads to an increase in temperature.

In simple cases these generalized capacities are constant. They may, however, depend on the forces, but they are always positive as a consequence of the conservation laws.

, as a function of X (see Figure 3), then we obtain a relationship between the time derivative of X and X itself. Such a representation is called a dynamic diagram because it describes graphically the time behavior of the ensemble of the system and its environment, with respect to the variable of state, X.

The stationary states correspond to the intersections of the dynamic characteristic with the X-axis, according to the condition of stationarity:

Stationary states behave in a different manner with respect to disturbances by stimuli, depending on the dynamic nature of their non-stationary neighborhood in the dynamic diagram. If the dynamic characteristic has a negative slope at the X-axis intersection, then the stationary state is stable, i.e., after disturbances by stimuli the original stationary state is spontaneously restored by the system (see Figure 3). If, however, the slope is positive, then the stationary state is unstable. Here, already, the weakest disturbance leads to an autocatalytically-increasing deviation away from the original stationary state. With respect to the dynamic diagram the stability conditions (4) are:

Dynamic instability of one process in one of the dipoles is an essential, but not yet sufficient condition for oscillatory behavior. Therefore, in order to understand oscillation kinetics we have primarily to look for instability constellations in the systems in question. Instability arises, in general, when the dynamic characteristic is non-monotonic, giving the possibility of more than one intersection with the X-axis (4). Then we get an odd number of stationary states with alternating stability and instability as shown in Figure 4.

Figure 4 Stable and unstable states given by a non-monotonic dynamic characteristic.

In the case of forward inhibition instability (6), the acceptor dipole has a non-monotonic characteristic. In the case of backward activation instability, the donator dipole is non-monotonic. In either case we get the same type of dynamic characteristic as shown in Figure 5.

Figure 5 Forward inhibition and backward activation represented by force-flux characteristics.

Figure 6 shows how stable and unstable states are produced by intersections of force-flux characteristics. Stability arises when in the neighborhood to the right of a given stationary state the value of the stationary current of the acceptor is greater than that of the donator, or smaller in the neighborhood of the left, respectively. Instability arises under inverse conditions. With respect to the slopes of the force-flux characteristics these conditions are:

Figure 6 Stability and instability situations given by force-flux characteristics.

Stability: 

Instability: 

In most of the known systems with unstable states the non–monotonic force-flux characteristic is caused by resistances, R, (in a generalized sense) which depend on the driving force in an appropriate manner. Figure 7 shows the condition for the occurrence of non-monotonicity. In the R/X graph the R curve is steeper in a certain range of X than the geometric locus for I = constant. Hence the condition for non-monotonicity with respect for the force dependence of R becomes:

Figure 7 The condition of non-monotonicity.

forward inhibition: 

backward activation: 

Force-dependent resistances occur in systems (dipoles) in which high field strengths of forces arise, such as in membranes of all kind, electrolytic layers on electrodes, boundaries of crystal grains, junction zones in semiconductors and all other systems containing thin conducting interfaces. Figure 8 gives examples of such membrane-like interfaces exhibiting strong voltage-dependent electrical resistances. Similar examples are known for pressure-dependent volume fluxes and concentration-dependent fluxes of solutes. In the case of chemical systems, autocatalytic (backward activation) and autoinhibition (forward inhibition) reactions produce the analogous effects.

Figure 8 Membrane-like interfaces inhibiting non-monotonic force-flux characteristics.

All of the systems mentioned show non-monotonic force-flux behavior. As it will be explained later, they are able to produce oscillations under suitable additional conditions. In contact with appropriate environments they have one unstable and two stable stationary states. They are called bistable because they can exist in two distinct stable states. The transition from one stable state to the other is a trigger process obeying the all-or-nothing law of excitation. This behavior can easily be demonstrated by means of the dynamic diagram shown in Figure 9 (4).

Figure 9 The trigger process of a transition from one stable state into another stable state.

In the case of bistability the dynamic characteristic intersects the X-axis three times giving one unstable state in the middle and two stable states. During the period of the stimulus the situation of the dynamic characteristic is different from the situation under unirritated conditions. As shown in ). This kind of all-or-nothing mechanism of triggering is valid for transitions in both directions (X1→X2 and X2→X1) (see Figure 9b).

Another important property of bistable systems is their ability to propagate localized triggered state transitions over the entire surface of the triggerable interface, or, in the case of chemical reactions, through the entire volume of triggerable space (7–11). At the boundary between areas of different stable states, local fluxes arise as a consequence of the existing difference of force, which act as stimuli on each area (see Figure 10). The direction in which the propagation actually takes place depends on the quantities of the threshold strengths of both transitions. (12).

Figure 10 The propagation of state transition in a bistable system.

According to the nature of the triggerable systems the local fluxes may be electric currents, local diffusion fluxes, local heat fluxes, etc.

In summarizing the essential properties of bistable one–process systems we can state the following facts:

1. They can exist in two distinct stable stationary states.

2. They contain one unstable stationary state situated between the stable states. With respect to stimuli the unstable state plays a role as a critical state dividing the kinetic ranges of the two stable states.

3. The transition from one stable state to the other is a trigger process which obeys the all-or-nothing law.

4. They exhibit propagation phenomena.

II Multi-process Systems

As already mentioned oscillatory behavior occurs only in multi-process systems which contain more than one independent driving force. Because the moving particles which are driven through the interfaces by the driving forces have volume, mass, energy, and, in case of ions, electric charge, the different transportation processes are not independent of each other, (e.g. a driving pressure difference causes not only a volume flux but also a flux of mass and electric charge; a concentration difference causes not only a mass transportation but also a flux of volume and charge, etc.). In this way certain coupling effects exist between the simultaneous transportation processes. As a consequence the dynamic characteristic of the driving force in question also depends on the other driving forces, thus yeilding several varieties of dynamic characteristics according to the specific values of the other forces (4,2).

Figure 11 exemplifies the kinetic situation of a coupled two-process system with respect to the force-flux relationship and the dynamic characteristics of the two variables X and Y. The quantitative description now requires, accordingly, a set of two simultaneous differential equations (3):

Figure 11 The effects of the trigger and the recovery processes.

Oscillatory behavior, in particular, arises in two-process systems when two essential conditions, among others, are satisfied:

1. There must exist one process which exhibits bistability, as explained above.

2. The other process, which may well have a monotonic dynamic characteristic, must depend on the bistable process in such a way that all occurring all-or-nothing state transitions recover spontaneously in a delayed counteraction.

It is reasonable to denote the first process as the trigger process and the other as the recovery process and, accordingly, the responsible variables of force as the trigger variable, X, and the recovery variable, Y. Some important new properties arise in bistable systems due to the action of the recovery process. Figure 12 illustrates the recovery process of a state transition triggered by a superthreshold stimulus. State 1 corresponds to the general resting state, where X as well as Y are stationary stable. At the beginning of the stimulus the system jumps to state 2 and changes with time, reaching state 3 when the stimulus ceases. The system now jumps back to the initial characteristic at state 4 and changes with time to state 5. Here, unlike in Figure 9, this state is not purely stationary since the recovery variable, Y, is simultaneously non-stationary and changes with time, shifting the characteristic of X downward, as in Figure 12a, or upward, as in Figure 12b, until the system again becomes unstable at state 6 and recovers by self-triggering into the original resting state (7→8).

Figure 12 The recovery process of a monostable system.

This behavior is called monostability because only one state (1 and 8) is actually stationary in X and Y at the same time. The graph of X versus t for a triggered state transition, with self-recovery, is shown in Figure 12c and 12d. It corresponds to the well-known recordings of action potential spikes in nerves and muscles. The undershoot or overshoot, points 6 to 7 to 8 in Figures 12c and 12d respectively, is the result of the delayed readjustment of Y to its final resting state after the self-triggering at 6.

It also can happen, as shown in Figure 13, that the recovery variable shifts the characteristic of X so far that state 8 also becomes unstable and self-triggering to state 5 recommences. Then the system exhibits oscillatory behavior and no stable state exists where both the variables X and Y are simultaneously stationary.

Figure 13 The dynamic diagram of an oscillating system.

In this way, oscillations arise in two-process systems by successive self-triggering and recovery as a result of an appropriate kinetic coupling of both processes. If there are more than two coupled processes effective in the system, then the general feature of the kinetics is not much different from that of the two-process systems. Obviously, in these cases also, instability and recovery may play a similar role, but either of the properties may be a common result of several coupled