Periodic patterns in biochemical reactions

Quarterly Reviews of Biophysics , (), pp. –. Printed in the United Kingdom

# Cambridge University Press

Periodic patterns in biochemical reactions

BENNO HESS

Max-Planck-Institut fuX r Medizinische Forschung Heidelberg, Germany

Whenever fundamental features of living systems and their molecular basis are

reviewed, the problem of timing, of time setting or free open-end running

times is only marginally on the desk of research agendas, although the finite

ageing as one of the features resulting from time markers is known since long.

With the discovery of cellular and most important of cellfree oscillatory

processes new concepts and experimental techniques were designed to

approach these questions more directly leading not only to a better

understanding of timing but strongly contributed to concepts for spatial

pattern generation. As given in the list of contents major items in the field of

intracellular and intercellular periodic reactions are reviewed in Sections – in

terms of specific properties of various systems and in Section in summing

important features common to all oscillatory stems in chemistry and biology.

Section draws attention to the problem of patterning in the mesoscopic

domains of living systems, which is so basic in terms of the volume dimensions

specific for the cellular and subcellular reaction compartments in biology. The

last chapter sets some marks on urgent problems currently approached by the

combined methods of molecular genetics, biochemistry and computer

technologies.

.

. :

± Cell-free glycolysis

± Intact cells

.. Yeast

.. Myocytes

.. β-Cells of the islets of pancreas

.

. Temporal oscillations

. Calcium waves

. Physiology

.

.

Benno Hess

. DICTYOSTELIUM DISCOIDEUM

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.

.

.

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About years ago the discovery of oscillation of the peroxidase reaction, of

glycolysis and, simultaneously and independently, of the malonic acid oxidation

reaction (Belousov–Zhabotinsky reaction [BZ reaction]) prompted a long-range

investigation of the mechanisms and fundamental significance of this new

dynamic state. Today it should be remembered that the general concept of

chemical and biological processes at that time relied on classical reaction kinetics

within the linear domain of equilibrium thermodynamics. It was soon realised

that oscillations of chemical and biochemical reaction systems result from the

nonlinear properties of their complex underlying mechanisms, a topic which was

then outside the scope of the current views on chemical–biochemical processes.

This research coincided with the extension of thermodynamics to open and far-

from-equilibrium conditions, which yielded the basis of macroscopic mechanisms

of self-organisation. In a first step the work of Onsager, Meixner and Prigogine

had built up a consistent phenomenological theory of irreversible processes. Later

it was found that outside the linear domain of interactions, a thermodynamic

threshold exists beyond which steady states of the thermodynamic branch may

become unstable and may be replaced by new classes of behaviour having

completely different properties. To describe these the concept of dissipative

structures was introduced and here the observation of chemical and biological

oscillations served as experimental proof. Indeed, spatial and temporal

organization is a fundamental property of living systems and of great interest for

chemical reaction systems in general. The temporal organisation in the living

world in terms of oscillatory and rhythmic phenomena covers a time range of over

orders of magnitude, which sets the range of spatial organisation of biological

matter (Glansdorff & Prigogine, ; Nicolis & Prigogine, ).

This development was accompanied by a rediscovery of the importance of

autocatalysis – originally established by Lotka years ago – in the general

function of biological systems. Simple autocatalysis in many variations of

mechanisms such as feed back and feed forward interactions were found to be an

essential part of the regulation in biology. At the same time, the theory of allosteric

enzymes as cooperative ligand–protein interaction was discovered and strongly

enhanced our understanding of cellular regulation (Hess et al. ).

In addition it is important to know that new physical techniques in combination


with more powerful computational and mathematical methodologies were

developed. These allowed the analysis of complex nonlinear dissipative processes

under non-invasive conditions. The refinement of classical spectrophotometric

techniques and the development of D spectroscopy and visualisation procedures

yielded new handles to search for mechanisms of time-dependent spatial dynamics

and evolution (Mu$ ller et al. ).

The earlier development in the field has been treated in review articles and

books (Hess & Boiteux, ; Hess et al. ; Goldbeter & Nicolis, ;

Gerisch, ; Goldbeter & Caplan, ; Nicolis & Prigogine ; Hess et al.

; Berridge & Rapp, ; Field & Burger, ; Goldbeter, , ) as

well as in extensive conference reports (Chance et al. ; Berridge et al. ).

Based on this documentation, this review will deal with newer developments in the

field and focus first on intracellular reaction networks and later on multicellular

interactions, especially on the dynamic properties of glycolysis, on intracellular

calcium oscillations, on slime-mould colonies and on a few other systems as

examples of temporal and spatiotemporal order in biological systems. Since many

fundamental discoveries of general importance were made in quite different

excitable systems, the review will cover a multiplicity of biochemical reaction

systems of various cellular classes, but which nevertheless show analogous

properties.

The topic of chaotic macroscopic behaviour will only be treated marginally (see

Degn et al. ). In addition, spatial ordering systems will be discussed in

relation to pattern formation in the BZ reaction. Important results of the latter

work have been reviewed elsewhere (Markus et al. ; Goldbeter, ; Mu$ ller& Plesser, ).

. :

. Cell-free glycolysis

After the discovery of glycolytic oscillations in yeast, in yeast extracts and later in

extracts of heart, mechanistic studies led to the identification of phosphofructo-

kinase (PFK) (EC \\ ; ATP; -fructose--phosphat--phosphotransferase)

as the enzymic source, which periodically generates its products ADP and

fructose,-bisphosphate. Furthermore, the mechanism of propagation of the

periodic change of activity of PFK through the adenylate kinase–pyruvate kinase

system and a proper source along the enzyme reaction sequence of glycolysis was

elucidated. The allosteric properties of PFK were found to be responsible for its

periodic operation (for ref. see Section ). Since fructose ,-bisphosphate is one

of the most potent activators of PFK, the response of glycolytic oscillations in cell-

free cytoplasmic extracts of the yeast Saccharomyces cerevisiae was analysed and

found to react to micromolar concentrations of the activator by a pronounced

decrease of both the amplitude and the period. Oscillations of the endogenuous

concentrations of the activator were also observed. However, the minute amounts

of the endogenuous activator levels in the range of ± µ and its phase

relationship relative to other metabolites exclude an essential role in the oscillatory

Benno Hess

0·01E400˝

320˝

–90

0mM-h–1

Fig. . NADH absorbance (upper trace) in a yeast extract entrained by the }-harmonic of

a periodic glucose injection rate (lower trace). T!¯ s, T«¯ s. Resulting period: T¯

, T«¯ s (from Boiteux et al. ).

mechanism, a conclusion which is also supported by studies of a mutant strain of

yeast lacking detectable amounts of the compound although showing well

sustained oscillations in extract as well as in intact cell (Yuan et al. ).

More recently the quantitative analysis of oscillating glycolytic metabolites,

described earlier, was complemented by calorimetric measurements in oscillating

glycolytic yeasts extracts. It was found that under oscillating conditions the

NADH maximum is perfectly in phase with the maximum heat production rate.

Furthermore, it was found that high metabolic fluxes coincided with low

amplitudes and with high frequencies yielding an activation energy of Ea¯ ± kJ

mol−" and a mean Q"!

value of ±³±. The calorimetrically determined reaction

enthalpy ∆H of the glucose breakdown showed two distinct groups of extract

preparations with ®±³± and ®±³± kJ mol−" respectively, compared

to ®± kJ mol−" as predicted from theoretical calculations. This result indicates

secondary reactions interfering with the energy flux balance in the latter case, a

matter to be explored in the future. These experiments also did not support the

predicted rate limiting effect of GAPDH in oscillating extracts (Plesser et al. ;

Mu$ ller & Plesser, ; Kreuzberg & Betz, ; Plesser & Lamprecht, ;

Drong et al. ; Grospietsch et al. ).

Earlier results prompted a general study of the principal dynamic states of

glycolysis, and it could be shown that oscillatory domains in an extract of yeast as

well as in intact yeast cells are induced with a variety of steady rates of substrate

input. Later the question was asked, whether glycolysis could also be entrained by

a periodic source of substrate and whether it reacts to random perturbation. In

accord with the kinetic properties of a simple allosteric model for the periodic

PFK reaction, experiments with glycolysing yeast extracts showed that stochastic

variation of the rate of substrate input after a short steady rate of injection leads

to sustained periodic behaviour with irregular wave forms and a stable period. The

oscillations settle around the autonomous period of the system in a narrow range.

Indeed, under stochastic input conditions, model and experiment showed that

glycolysis has the intrinsic property of a narrow band-path filter centred at the


Table . Interaction of the glycolytic oscillator with a periodic source of substrate

(From Boiteux et al. )

Relation between T«and T

!Interaction

T«}T!E}n (n¯,…) Entrainment by the }n sub-harmonic of the input

frequency

±%T«}T!%± Entrainment by the fundamental frequency of the input

±!T«}T!!± No entrainment

T«}T!" Double periodicity: separation of autonomous and input

frequencies

mean autonomous frequency, keeping the period stable in spite of short time

variations of the source rate.

By periodic addition of substrate, the glycolytic system is readily entrained,

whereby the oscillation period of glycolysis synchronises with the period of

substrate input. Furthermore, synchronisation to a subharmonic of the driving

frequency of the periodic input is found. This is observed, if the driving frequency

is near to an integral multiple of the frequency being recorded, when a continuous

input rate is used. This phenomenon is also known as subharmonic resonance or

frequency division. Fig. illustrates the entrainment by the }-harmonic in a

record of NADH absorbance of a yeast extract driven by the periodic glucose

injection rate. The ranges of dynamic interactions of the glycolytic oscillator with

a periodic source are summarised in Table , demonstrating the domains of

coupling in comparison with the model studies. The observation of subharmonic

synchronisation in glycolytic oscillations proves the nonlinear nature of the

glycolytic oscillator, the mechanism of which has been analysed and relies on the

allosteric properties of PFK coupled to the adenylate kinase system. These

experiments were the first demonstration of periodic forcing of chemical reactions

in general (Boiteux et al. ).

Subsequent theoretical studies revealed the occurrence of more complex

dynamic phenomena such as chaos and coexistence of two oscillatory regimes in

the case of an autonomous smaller system consisting of two enzymes coupled in

series and activated by their respective products (Goldbeter & Decroly, ). On

the basis of a glycolytic model system a rich variety of time patterns corresponding

to different periodic, quasi-periodic and chaotic attractors were found. These

patterns undergo complex hysteresis loops when bifurcation parameters are

slowly changed by modulating the input amplitude. With this technique up to

four attractors coexisting in phase space were identified. The time patterns

corresponding to coexisting attractors can be switched into one another by

triggering the system with short substrate pulses (Hess & Markus, ; Markus

& Hess, ).

The occurrence of chaotic regimes in the complex process of glycolysis was

analysed theoretically in detail. Most of the theoretical results were confirmed by

a large number of biochemical experiments with glycolysing yeast extracts.

Benno Hess

Entrainment : Quasiperiodicity (a) : Quasiperiodicity (b) :

Chaos (a) :

Chaos (c) :

Chaos (b) :

Chaos (d) :

30 min

F

Vin

F

Vin

F

Vin

Fig. . Experimental time pattern of entrained ( :), quasiperiodic and chaotic oscillations,

obtained by measuring the NADH fluorescence (F) of glycolysing yeast extract (upper

course) under sinusoidal glucose input flux (lower course) (Vin

¯ input flux, amplitude and

frequency are varied) (from Hess et al. ).

Measurements of the time course of NADH fluorescence yielded autonomous

oscillations as well as quasiperiodicity and chaos. Fig. demonstrates examples of

different types of oscillations resulting from varying driving modes in the form of

periodic input fluxes with different amplitudes and frequencies. Indeed, period-

doubling cascades up to period seven with chaotic windows were recorded, and a

thorough study of the basins of attraction was undertaken. The evidence of these

experimental regimes of deterministic chaos was obtained by stroboscopic transfer

plots admitting a period of three transfer processes, by reconstruction of attractors

in D phase space, by Poincare! sections at varying input flux phases of

reconstructed strange attractors following a stretch-fold-press process (similar to

the baker’s transformation), and by determination of the maximum Lyapunov

exponent from reconstructed attractors. The information dimension of the

attractors reconstructed from the experiment was found to be always smaller than

three, indicating that three phase variables are sufficient to describe the complex

dynamics of glycolysis, in spite of the much larger number of metabolites involved

(Markus et al. , a, b ; Hess & Markus, ).

In order to display the complex property of such a system, novel graphical

techniques have been developed. In particular, a technique based on two


[PE

P]

(mM

)

(mM

)

[F6P

]

54·

80·3 0·2 0·1 0

[ADP]

(mM)

1·2

1·0

0·8

0·6

0·4

0·2

[ATP](mM)

3·2

3·4

2·3 2·4 2·5

Fig. . Strange attractor displayed in a rotating trapezium (Hess & Markus, ). For

details see text.

conservation laws allowing the display of four component concentrations of the

glycolytic model as well as the input phase on one single picture, by use of a

rotating trapezium demonstrates the order of complexity in terms of the number

of independent and dependent variables. Fig. presents a chaotic attractor

obtained with this technique. For each point of this trajectory the concentration

of adenosine-«-diphosphate (ADP), of adenosine-«-triphosphate (ATP), of

phosphoenolypyruvate (PEP), and of fructose--phosphate (FP) are given at the

upper edge, lower edge, inner edge (rotation axis), and outer edge of the

trapezium, respectively. The concentrations are determined by drawing a straight

line, passing through the point of the trajectory and parallel to the rotating axis for

ADP and ATP to the upper trapezium edge for PEP and the lower trapezium edge

for FP. The input phase is given by the angle of rotation of the trapezium (Hess

& Markus, ).

The observation of classical steady states, of periodic and quasiperiodic states

and more recently of macroscopic chaos (¯deterministic chaos) illustrates the

variety of dynamic options of living systems. From the latter state the Lyapunov

dimension can be obtained yielding the number of variables for its quantitative

description, which is of great heuristic value for analytical procedures. Physiology

will provide an answer to the questions, how the three regimes are controlled and

regulated in intact cells and cellular networks, and how regimes are selected in the

process of long-range evolution. After all, the classical steady states of laboratory

experimentation do not seem to occur in the open natural environments of

Benno Hess

biological systems (Markus & Hess, ; Hess, ). In addition, it was found

that the free energy dissipation under oscillatory conditions is lower than in the

steady state and that at entrainment the dissipation is even lower than under

condition of autonomous oscillation (Markus & Hess, ; Ross & Schell, ).

In an earlier study the question was asked whether oscillating cell-free

glycolysis, as an excitable medium, might also display spatial structures under

proper conditions. A theoretical study of an allosteric enzyme model indicated

general boundary conditions for the occurrence of a spatial distribution pattern

(Goldbeter, ). Indeed, the development of D optical techniques allowing

one to continuously record the fluorescence and absorption of intermediates

(NADH) in thin layers of a reactive medium resulted in the demonstration of

oscillating and propagating NADH patterns, which reacted to the addition of

small pulses of ATP and AMP (Boiteux & Hess, ). Only recently circular

travelling NADH- and proton-waves in an organelle-free glycolysing yeast extract

have been detected and found to exhibit collisional mutual annihilation (Fig. a,

b, see facing p. ). Furthermore, the formation of rotating spirals on increase of

the adenosine-«-monophosphate (AMP) concentration was found. Controlled

waves were initiated by local injection of fructose ,-bisphosphate, the strong

activator of PFK, indicating the crucial role of this allosterically regulated enzyme

in the control of the dynamics of this pattern, which is analogous to the pattern of

calcium waves in heart cells and frog eggs (see below) (Mair & Mu$ ller, ; see

also Yuan et al. ).

. Intact cells

.. Yeast

The microfluorometric observation of glucose-induced NADH-oscillation with a

fairly constant amplitude and small damping factor in a single yeast cell (S.

carlsbergensis) in presence of cyanide, clearly showed that the oscillatory regime of

glycolysis of many yeast cells is not dependent on a high density population of

interacting yeast cells, but rather results from the singular glycolytic property of

one single yeast cell. Also, it was found that in a large population of cells no gross

heterogeneity with respect to the oscillatory state of each individual cell is

detectable. These results indicated than an intercellular synchronisation must

occur in oscillating yeast cell populations (Chance et al. ).

Only recently the function of acetaldehyde as the intercellular coupling and

synchronisation variable in a yeast cell population under conditions of glycolytic

oscillation has been identified. The extracellular acetaldehyde concentration

oscillates at the frequency of the intracellular glycolytic oscillation. The

dependence of the phase shift on the acetaldehyde concentration and on the phase

of its addition proves this intermediate as being the exclusive synchronising agent

(Richard et al., ).

The intracellular coupling of oscillating glycolysis to other cellular functions in

yeast cells has not been analysed in great detail. Nevertheless, the coupling of

periodic glycolysis to the plasma membrane potential could well be demonstrated


NADH-fluorescence

I.

II.

III.

PotentialRHODAMINE - fluorescence

RHODAMINE-NADH

fluorescence

60“

NADH

Potential

Pote

ntia

l

Fig. . A cutout of steady oscillations of glycolysis as well as the plasma membrane potential

induced by addition of glucose. The insert represents a phase plane plot of the two variables

recorded in the figure. The plot indicates the presence of three different dynamic states of

glycolysis and the plasma membrane potential.

in yeast cells with glycolysis as the only ATP-generating system. In this case

glycolysis generates ATP which drives the proton translocating H+-ATPase of the

plasma membrane. During the process a proton gradient is built up setting up a

plasma membrane potential with or without companion movement of other ions.

The simultaneous time analysis of the change of the activity of glycolysis and the

plasma membrane potential is based on a record of NADH fluorescence as an

indicator of glycolysis and rhodamine G fluorescence as an indicator of the

plasma membrane potential.

Evidence that the plasma membrane H+-translocating ATPase is directly

involved in the generation of the plasma membrane potential comes from a study

in which the two processes were uncoupled by appropriate inhibitors. Uncoupling

agents such as sodium azide and pentachlorophenol and the specific inhibitors of

the ATPase diethylstilbestrol and vanadate ions inhibit the oscillation of the

plasma membrane potential. On the other hand, the influence of the proton

translocating system on glycolysis can be documented by affecting the plasma

membrane potential from outside the cell using appropriate cations or electrical

field perturbation.

A cutout steady oscillations of glycolysis as well as the plasma membrane

potential induced by addition of glucose is shown in Fig. using a double

Benno Hess

fluorimeter that selects fluorescence emission of rhodamine G as well as NADH.

The record in Fig. illustrates the membrane potential oscillation between and

% of the total rhodamine G fluorescence and the NADH fluorescence

oscillating between and % of total with a period of about s. Comparing

the maxima of the oscillations, it is seen that both components are running with

the same frequency. The relationship between the rhodamine G and NADH-

fluorescence over one period can be studied by a phase plane plot (see insert of Fig.

) in which three different phases during one limit cycle can be detected. In phase

I the two functions run simultaneously towards NAD and a membrane potential

maximum. Phase II is controlled by a relatively slow rate of reduction of NAD

during the initial part of the transition from NAD to NADH maximum.

However, this is not reflected in the plasma membrane potential trace which

evolves during this phase along an autonomous path. On the other hand, in phase

III glycolysis seems to be running on its own ahead of the membrane potential

change. It should be added that under non-oscillating conditions potassium,

calcium and lantanum ions induce oscillations of both the plasma membrane

potential as well as glycolysis. These results indicate that ‘chemical resonance’ is

clearly observed between the two highly nonlinear processes (Hess et al. ).

.. Myocytes

Recently, the linkage of periodic changes in membrane ionic current to intrinsic

oscillation of energy metabolism was discovered in guinea pig cardiomyocytes

(O’Rourke et al. ). The result of these experiments strongly revise an earlier

suggestion for cardiac dynamics: namely the operation of two oscillator circuits,

an internal, subcellular oscillator – the calcium oscillator – and a membrane

oscillator, the first one driving the surface membrane oscillator as well as other

functions in cardiac pacemaker oscillations (see Tsien et al. ; Tsien & Tsien,

). The new experiments show that the subcellular NADH oscillations are due

to the glycolytic oscillator which controls the membrane potential oscillations, and

that the membrane oscillator is not caused by pacemaker currents or an internal

calcium oscillator.

Studies of the sensitivity of membrane ionic currents and the oscillation of

NADH with appropriate inhibitors and, furthermore, studies of the correlation of

oscillating NADH, the ATP-sensitive potassium current and the depolarization-

evoked intracellular calcium transients causing repetitive contraction showed that

glycolysis acts as the primary source of the membrane current oscillation. In

accord with earlier experiments on yeast (Boiteux et al. )," the oscillatory

regime of glycolysis in myocytes covered a critical range of the glycolytic rate,

above and below of which no oscillations are observed. The oscillations neither

require voltage changes nor do they rely on feedback control by intracellular

calcium. The authors suggest that the ATP-sensitive potassium channel is the

target of the periodic glycolytic ATP production. Furthermore, it is concluded

that oscillations of energy metabolism might have a function in modulating

" Oscillating glycolysis in cell-free extracts of heart has been extensively studied earlier (for review see

Hess & Boiteux, ).


cardiac excitability and intracellular calcium homeostasis. In addition, the

contribution of the oscillatory regime to the contractile and electrical dysfunctions

associated with myocardial ischaemia is suggested (for cardiac spiral excitation

waves see Section , p. ). These studies are of interest because of the

observation, that in skeletal muscle extracts the oscillating state of glycolysis –

compared to the steady state – is advantageous for the regulation of the

carbohydrate utilisation and the maintenance of a high [ATP]}[ADP] ratio

(Tornheim et al. ), a condition which also might hold in cardiac muscle.

.. β-Cells of the islets of pancreas

Glucose-induced glycolytic oscillation has been shown to generate oscillatory

regimes of the membrane potential in β-cells of pancreatic islets, the frequency of

which is a function of the glucose concentration (Matthews & O’Connor, ).

Consequently the bursting pattern of membrane depolarisation controls the

pattern of insulin secretion. Indeed, in a reconstituted system a direct link

between the glucose-induced metabolite changes and the free calcium levels could

be demonstrated, suggesting that glycolytic oscillations and the ATP}ADP ratio

are driving the oscillation of the cytosolic free Ca#+ β-cell membrane potential and

the insulin release. In order to understand this complex relationship a model was

suggested linking glycolytic ATP oscillation to the periodic insulin secretion via

the ATP-sensitive K+ channel and the Ca#+ channel as well as the Ca#+ ATPase

in a double pathway controlling the cytosolic free calcium. This model invokes a

simultaneous action of the membrane oscillator as well as the cytosolic calcium

oscillator (Corkey et al. ) (see below).

In order to understand these complex interactions, yielding finally the controlled

pulsatile insulin release observed experimentally, a deterministic, rather simplified

model was analysed on the basis of an intrinsic calcium feedback (Chay & Keizer,

, ). However, it was soon realised that any mechanism of the pulsatile

insulin secretion has to meet an essential physiological feature of islet cells, namely

the collective properties of β-cells. These cells function as a tightly coupled

cellular assembly which was described in a first approach in a stochastic model

implying a critical number of reactive β-cells (Sherman & Rinzel, ).

All cases of cellular oscillations reported so far demonstrate the direct linkage

of metabolic oscillations that yield oscillations of the ATP}ADP}AMP system

which couples to membrane potential functions implying calcium as an ionic,

timespecific signal transducer (see Pralong et al. ). Here, it is suggested that

the collective function of cellular or intracellular entities might well be an

obligatory requirement of temporal functions in physiology, some aspects of

which will be touched on further below.

.

. Temporal oscillations

More than a decade ago, the availability of sensitive techniques to analyse the

dynamic behaviour of the level of free calcium in intact and single cells led to the

Benno Hess

Table . Calcium oscillations in a variety of cells

(adapted from Berridge )

Cell Stimulus Period (s)

Rat myocyte Caffeine ±–Astrocytes TPA ±–Parotid gland Carbachol Lacrimal gland Acetylcholine –Gonadotropes GnRH β-cells Carbamylcholine –Mouse oocytes TPA –Rat hepatocytes Vasopressin –Macrophages Cell spreading –Xenopus oocytes Acetycholine HeLa cells Histamine –L cells — Smooth muscle Phenylephrine or histamine –Fibroblasts (REF ) Gramicidinvasopressin –Endothelial cells Histamine –B lymphocytes Antigen –Hamster eggs Fertilisation Sympathetic neurones K+ depolarisation and caffeine –Sympathetic ganglion Caffeine ca. Mouse oocytes Fertilisation –

discovery of calcium oscillations in a large variety of cells, an early summary of

which is given in Table from . Since that time an ever increasing number

of observations of sustained calcium oscillations have been reported occurring, as

given in the table, in response to appropriate stimuli. The table emphasises the

large range of frequencies observed, although in most cases the frequencies are

comparable to those obtained in oscillating glycolytic extracts or glycolysis in

intact cells. The wide occurrence of calcium oscillations stresses its general

significance for intracellular signal transmission and for functional coupling and

synchronisation of multiple targets. In contrast to electrically excitable neural

media, calcium oscillation in non-electrically excitable cells persists, if voltage-

clamped by electrodes (Gray, ).

The most remarkable feature of intracellular calcium oscillations is their

frequency response to the level of extracellular calcium and agonists concentration.

A typical record is given in Fig. , illustrating the dependency of the calcium

frequency in hepatocytes on the vasopressin concentration. Indeed, these

observations show for the first time directly not only calcium spikes in single cells,

but also a true biochemical frequency encoding, analogous to the frequency

modulation of trains of action potentials (Cuthbertson & Cobbold, ; Woods

et al. ).

This review does not allow a detailed presentation of all experimental reports as


800

400

200

10 20 30 40 50 60 70

0·4 nM Vasopressin 0·6 nM 0·9 nM

Fig. . Ca-transients in a hepatocyte (see text, from Woods et al. ).

well as model studies, which can be retrieved in a number of reviews (Jacob, ;

Petersen & Wakui, ; Tsien & Tsien, ; Cuthbertson & Cobbold, ;

Berridge & Dupont, ; Petersen et al. ; Goldbeter, ), but rather

focuses on general properties, which are common to all oscillating systems.

Berridge & Dupont classified the biochemical mechanisms, underlying calcium

oscillations on the basis of their cellular location as membrane oscillator and

cytosolic oscillator, although they point to the fact that in a number of cases an

interaction of both oscillator sources occurs (Berridge & Dupont, ). Calcium

oscillations in Dictyostelium discoideum and its interaction with cAMP oscillations

should be mentioned as an example where both a membrane-bound and a

cytosolic oscillator are involved, although the exact mechanism is still obscure

(Wurster et al. ). Fig. illustrates the complex feedback circuits of four

different loop structures involving intracellular calcium stores, in three cases of

which the essential participation of inositol triphosphate as second messenger has

been described (Berridge, ).

A minimal circuitry for a signal-induced calcium oscillator, based on the self-

amplified release of calcium from intracellular stores, implying components given

for the cytosolic oscillator of Fig. , was presented by Dupont and Goldbeter and

later extended to cover the variation of the inositol-,,-triphosphate (IP$)

receptor level and sensitivity. The minimal two-variable model was subjected to

a phase plane analysis and the criteria of the Bendixon theorem to identify

oscillatory regimes (Minorsky, ; Dupont & Goldbeter, ; Dupont et al.

). A detailed comparison of the properties of the model with experimental

observations shows that the simplified model presents a unified explanation for

experimental observations in a variety of cell types, such as the control of the

frequency of calcium oscillations by the external stimulus or extracellular calcium,

Benno Hess

Membrane oscillators

VOCs

(a) Extracellular (b)

Agonist

Capacitative

Empty store

Agonist

DG/PKC

Sinusoidal

Cytosolic oscillators

Agonist

(c) (d)

Baseline

Intracellular

Distribution of oscillatory mechanisms in different cell types

Lacrimal glandLymphocyleParotid gland

Gonadotrophs (spont)â-cell (Glucose)Smooth muscle cellsS-A node

Gonadotrophs (GnRH)â-cell (ACh)AstrocytesPancreatic cellsEndothelial cellsHepatocytesEggsXenopusNeuronsMacrophagesAdrenal glomerulosaMast cellsFibroblastsMesangial cellsSmooth muscleAvian salt glandBlood plateletsMegakaryocyte

Receptors for generating InsP3

Ion channels

Ca2+ pumps Positive and negative signalling pathways

The flux of Ca2+ responsible for generating a Ca2+ spike

Formation of InsP3 and flow of ions

K+ Ca2+

InsP3Ca2+

Ca2+

Ca2+ InsP3

Ca2+

Ca2+

InsP3

Ca2+

Ca2+

Fig. . Major mechanisms responsible for generating calcium oscillations (from Berridge &

Dupont, ). (a) Periodic opening of voltage-operated channels (VOCs) controlled by the

activity of potassium channels regulating membrane potential. (b) A capacitative mechanism

implies the periodic opening of a calcium release activated channel resulting from a positive

stimulus by a messenger derived from empty stores and a negative calcium feedback loop.

(c) Negative feedback loop operated through the protein kinase C (PKC}DG¯diacylglycerol) controlling the oscillatory regime of calcium release from internal stores. (d )

Baseline calcium spikes resulting from periodic opening of the inositol triphosphate receptor

(InsP$R; InsP

$¯ IP

$) through the operation of the positive feedback of calcium-induced

calcium release (CICR).

the correlation of latency with periods of calcium oscillations obtained at different

levels of stimulation, as well as the effect of a transient increase in IP$, phase shift

and transient suppression of calcium oscillation by calcium pulses and the

propagation of calcium waves.

The very nature of the oscillation-generating mechanisms as a highly nonlinear

phenomenon results in a complex relationship between the frequency and the

essential variable parameters of the system. An early study of this relationship was

presented in a glycolytic model relating the oscillation frequency to the substrate

input rate as well as the enzyme concentration (Goldbeter & Lefever, ; see

also Goldbeter & Nicolis, ). However, at that time, because of the limited

knowledge of the biochemical network structures involved, the problems of


S

R

Y

ZA

IP3

Ca2+

Fig. . Model for signal-induced calcium oscillations, based on the self-amplified release of

calcium from intracellular stores. The external signal (S) binds to membrane receptor (R)

and thereby triggers the synthesis of Ins(, , )P$

(¯ IP$) ; the latter messenger elicits the

release of calcium from Ins(, , )P$-sensitive store (hatched domain) whose calcium

content (A) is shown to produce then a constant, net influx of cytosolic calcium (Z). The

latter is pumped into an insensitive store (Y); calcium in this store is transported into the

cytosol, in a process activated by cytosolic calcium (Z). Other arrows refer to calcium influx

into and extrusion from the cell (from Dupont & Goldbeter, ).

frequency control and stabilisation of biological oscillations in general could not

be treated. Only recently, a mechanism for frequency control based on protein

phosphorylation driven by intracellular calcium oscillation was presented.

A detailed kinetic study of a protein phosphorylation model was based on an

extension of the model given in Fig. by incorporating the function of a catalytic

protein, which occurs in two interconvertible forms as commonly known in

enzymology, namely an activated, phosphorylated and a deactivated,

dephosphorylated form. The interconversion is catalysed by a phosphorylating

protein kinase and by a dephosphorylating phosphatase, both being enzymes of

the Michaelis–Menten type. If the protein kinase reaction is activated by calcium

in a cooperative manner – decisive for the width of the control function – and if

the level of calcium is oscillating, the fraction of the activated protein kinase varies

periodically (Goldbeter et al. ; Dupont & Goldbeter, a).

This type of circuitry simply transmits an extracellular steady agonist signal

into intracellular calcium oscillations, which in consequence leads to a generation

of periodic activity changes of a ‘master’ protein kinase which executes a

frequency-based coordination of many cellular target functions. Thus, a steady

external signal activates frequency-encoded cellular functions with increased

extracellular agonist concentrations leading to a frequency increase of changes in

intracellular protein kinase activity. The detailed kinetic analysis of this model

shows that its properties are largely independent of the model on which the

generation of calcium oscillations in living cells is based (Dupont & Goldbeter,

b).

Among the known calcium activable protein kinases, the multifunctional

calcium-calmodulin dependent protein kinase (CaM kinase) is of special interest

Benno Hess

because of the manifold of its substrates (for details see Braun & Shulman, ).

Given the complexity of calcium interactions observed in many cellular species, it

is difficult to visualise a unique mechanism for the generation of calcium

oscillations and its physiological transduction towards acceptor functions, which

utilise a calcium frequency signal or serve as ‘solitary spike detector’ (Meyer &

Stryer, ).

The development of a general theory of frequency encoding in excitable systems

by hormonal stimuli allowed the study of intracellular calcium dynamics. It was

found that three distinct modes exist, by which frequency encoding can be realised

by changing a single parameter for each case. Calcium oscillations were found to

be a modulation of the time of the recovery phase, whereas the amplitude and

width of the spike is unchanged. This model holds for cells, which operate with

only one IP$-sensitive calcium store (Tang & Othmer, a).

A critical comparison of six signal flow schemes including an application of

analytical and numerical mathematical tools on relevant rate equations was

recently presented summarising the limits for reduction to obtain limit cycle type

oscillations in given reaction networks. Also, it was pointed out that in biological

systems – in contrast to chemical systems – the significance of informational

coupling has to be considered, because important variables controlling feedback

functions are not interconvertible and do not show up in the actual fluxes, but

most be considered in terms of signal flow schemes (Stucki & Somogyi, ).

. Calcium waves

In a multitude of experimental observations of calcium waves in various cells

was summarised by Jaffe (). This work showed velocity ranges in the order

of µm s−" in activated eggs and in the order of µm s−" in other cells at room

temperature with a width of cells in the range between and µm. The data

were tested as planar waves by Luther’s equation resulting in estimates for

Medaka eggs, hepatocytes and myocytes well comparable to experimental

observation. Independent analytical and numerical studies of empirical models of

the calcium propagation mechanisms, e.g. in amphibian eggs (Cheer et al. )

and cardiac cells (Backx et al. ) should also be mentioned.

When later the observation of concentric and spiral calcium waves in Xenopus

laevis oocytes was reported, the determination of essential wave parameters

became feasible (Lechleiter et al. ). Designing a discrete dynamical model in

the form of a cellular automaton, originally developed for the study of spiral waves

observed in the chemical BZ reaction (Markus & Hess, ), the authors

converted their observations into a simple model that defines for each automaton

cell three different states, a receptive, an excited as well as a refractory state, a

common feature for the simulation of excitable media. This approach allows to

compare experiments and model calculations and to identify the parameters of the

Eikonal equation. It showed that the propagating species of the wave is calcium

and not IP$, fitting with the CICR model (Berridge & Irvine, ) and yielded

the minimal critical radius (R) for propagation of focal calcium waves of ± µm,


the effective diffusion coefficient for the propagating signal of ±¬−' cm# s−"

and the absolute refractory period for calcium stores of ± s. As expected, waves

propagating with undiminished amplitude and annihilation of colliding wave

fronts were found (Lechleiter et al. ).

Here we would like to point to the critical core size (R) of the calcium spiral in

the order of µm and the velocity wavelength of µm, both of which indicate

the lower limit of a cellular size allowing observable waves to occur and not being

washed out within the cellular space, because the wavelength is below the critical

radius of the excitable system’s core (see below). Since the kinematic theory for

excitable media (Mikhailov, ) predicts for maximal chemical dispersion and

frequency a minimal spatial period of Lmin

¯ π}kmax

and Rmin

¯Lmin

}π, it is

expected that true calcium waves occur only in cells large enough to build up

proper waves.

A number of theoretical reaction–diffusion models have been studied to

describe intracellular calcium waves. They served to discriminate between various

biochemical feedback circuits describing possible regulatory mechanisms for the

initiation and propagation of calcium spikes. In each model analysed so far, a

positive feedback was invoked with the propagation process implying an

intracellular relay sequence consisting of proper calcium stores, each of them

waiting in an excitable state for a calcium signal targeting in and yielding a new

pulse. The calcium-induced calcium release model (CICR) with two distinguished

calcium pools, both sensitive to IP$

and calcium with two variables, namely the

concentration of calcium in the cytosol and a calcium-sensitive calcium store, was

studied by incorporation of a proper diffusion term for Ca#+ into the rate equation

for the CICR model (Dupont & Goldbeter, ).

The solution of the reaction–diffusion equations for D propagation yields the

correct magnitude of the wave propagation rate near µm s−" with a period of

oscillation of s, which is observed in cardiomyocytes (Takamatsu & Wier, ).

It is important to note that the diffusion coefficient for calcium was assumed as

¬−' cm# s−" (Backx et al. ). This value is near the diffusion coefficient

(±¬−' cm# s−") which was obtained by analysing the propagation of calcium

waves in oocytes using a cellular automaton.

The result of a numerical integration of a D spatial propagation model of

calcium waves in cardiac cells is shown in Fig. (see facing p. ) (Dupont &

Goldbeter, a, b, ). It illustrates the sequence of spatial transitions with

equidistantly distributed calcium pools from top to bottom being initiated at the

left side of the top trace (red¯ ± µ calcium concentration, dark blue¯ µ

calcium concentration) over µm distance during a time of ± s. Inspection of

the coloured transition scaling the concentration change shows that the front of

the calcium wave is steep and straight (in the order of ± µ µm−" for the third

panel from top), whereas the tail levels off in a long stretch. In case of a

homogeneous distribution of the calcium stores the waves are more smeared out.

Depending on the boundary conditions also echo waves were observed (Dupont

& Goldbeter, ). Furthermore, in accord with experiments, spiral waves,

created in single cardiac cells by the cellular nucleus, have been modelled (Dupont

Benno Hess

et al. ). It is interesting to note that the steep wave front and the long tail are

also seen in case of propagating waves of the BZ reaction.

Reviewing the number of other models presented in the literature the robustness

of the CICR model is remarkable for its capacity to fit a broad range of parameter

values composing the extended dynamic parameter space. Also, the spatial scale

of this model fits well with the size of cells in which calcium propagation waves

are observed such as oocytes and cardiac cells. In those cases, the relationship

between the spatial diameter of the cellular size (either D or D) agrees with the

limits set by the spatial propagation wavelength of the model and fulfils the

requirements of the kinematic theory mentioned above. In case of so-called

calcium tide as seen in hepatocytes and endothelial cells and also in smaller oocytes

– where the theoretical limit is reached – it might well be that an extension of the

propagation space is obtained by its reduction to a D propagation along the

surface of the living cell, although in the case of oocytes D- and D-effects might

well be mixed.

Extensions of the simple CICR model to more complex regulatory networks of

the system are useful to fit the various experimentally observed calcium waveforms

and their transition times. Also, variation in the model properties with respect to

the number of pools could well offer an understanding for the variability of

calcium oscillations observed in a large variety of living cellular systems (Dupont

& Goldbeter, ). Furthermore, the interaction of calcium waves with

mechanical waves on the surface of eggs should be considered, although a

biochemical coupling mechanism is currently not at hand (Cheer et al. ).

. Physiology

In recent years, numerous studies revealed the spatial inhomogeneity of periodic

calcium dynamics in a variety of cellular processes under control of calcium

signals, such as cell development and growth, muscle contraction, hormonal

secretion and neuronal function (Tsien & Tsien, ; Meyer & Stryer, ;

Berridge, ). This was indicated by the observation of an occurrence of

intracellular hotspots or microdomains on a submicron to micron scale as well as

intra- and intercellular waves over distances up to mm (Silver et al. ; Llina! set al. ; Allbritton & Meyer, ). These observations corroborated detailed

studies of intracellular coupling mechanisms, involving calcium release out of

vesicles, calcium pumps, calcium entry as well as intra- and intercellular

propagation mechanisms (for secretory cells see Tepikin & Peterson, ; Kasai

et al. ; Thorn et al. ). Furthermore, mechanisms of intracellular

messenger cross-talk modulating calcium oscillations were studied in hepatocytes

(Somogyi et al. ) and oocytes (Yao & Parker, ). Of special interest is the

observation of an intracellular coupling of calcium oscillation and metabolic

indicators localised in mitochondria with calcium oscillations in the cytosolic

compartment, which demonstrates that calcium oscillation by controlling the

activity of key enzymes in metabolism might well coordinate in time and phase

metabolic and specific cellular target functions (Pralong et al. ).


The transition from a resting near-equilibrium state to an oscillatory state,

triggered by an external receptorbound signal – whether a single spike, a steady or

periodic, or stochastic forcing function (see Boiteux et al. ) – could be

essential for the activation of multiple cellular functions. Since cytosolic calcium

stores are bound to vesicular compartments and calcium-binding proteins, a

coordinating function of periodic calcium dynamics might link compartment and

storage structures of living cells. Here, the recent observation of a synchronisation

phenomenon of synaptic boutons is of importance, demonstrating the coordinated

calcium fluctuations and oscillations at the neuromuscular junction that coordinate

the regulation of transmitter release (Melamed et al. ).

General concepts for an understanding of the purpose and specificity of

frequency patterns in cellular and intercellular functions are based on various

arguments pointing to an advantage of a structured time pattern over a single rise

of time-independent concentrations of hormones and intracellular messengers.#

Here, it is obvious that calcium spikes are more resistant to noise than a single

graded rise in its concentration. Furthermore, it has been pointed out that

different frequency patterns of calcium spikes could selectively activate calcium-

binding proteins responding by their differences in calcium affinities, and also,

that local short-lasting calcium spikes could well exert only a localised function

with pulsed waves dying out within a critical length in distance from their point

of generation. Here, indeed, repetitive pulsing would be essential (Petersen,

personal communication, , see also Section ). In addition, it should be

noted, that a variation of the phase angle of oscillating time patterns might serve

as additional parameter for useful coordination (Hess & Boiteux, ; Friel &

Tsien, ).

Considering the significance of chemical waves with respect to the timescale to

cover a macroscopic spatial territory in comparison to simple diffusion, it is

important to note that the pulsed propagation mechanism yields a fast signal

transmission in form of a sharp concentration gradient over a given distance. The

analysis of a soluble allosteric enzyme model showed that, at the amplitude plateau

region of a wave, the time required to travel at a constant rate over a distance of

¬−# cm is about min. On the other hand, the time required by a wavefront

to cover a similar distance by diffusion alone is about h (Goldbeter, ). (The

relevance of the number of the various reacting messenger molecules per volume

and their diffusion coefficients for the intracellular traffic time and the critical

length of coupled reaction–diffusion propagation is discussed in Section .)

The extension of such models towards intercellular messenger propagation

through a linear set of neighbouring cells linked, for instance, by gap junctions led

to the early suggestion that a spatially coherent behaviour of an assembly of cells

could well be induced by the synchronisation of an enzymic reaction operating in

the limit cycle domain (Goldbeter, ).

# An obligatory pulse-activation has been discovered earlier, when it was found that the aggregation

dynamics of Dictyostelium discoideum could only be triggered or phaseshifted by a pulsed addition of cAMP

(Gerisch & Hess, ).

Benno Hess

.

The cytoskeleton of eukaryotes consists of microtubules, which are composed of

tubulins as building blocks and a number of associated proteins. Investigations of

the complex mechanism of its assembly and breakdown during cellular growth,

differentiation and mitosis led to the recognition of a dynamic microtubule

instability based on the coexistence of growth and shrinkage of the polymer thread

(Mitchison & Kirschner, ). Soon it was found by time-resolved X-ray or light

scattering techniques that a synchronised population of microtubules in vitro

readily settles into an oscillatory state, which is typically represented in Fig. a

showing the dynamics of a solution of oscillating microtubules as X-ray intensity

(z axis) as a function of scattering angle vector (x axis) and time (y axis) with a

periodicity of C min. The overall assembly (central scatter left) is initiated by a

temperature jump. The subsidiary maxima and minima indicate the microtubules

and the oligomers respectively. The overall reaction cycle of the assembly–

disassembly process is GTP-dependent; after binding to tubulin, GTP is

hydrolysed after assembly of the microtubules. (Mandelkow et al. ;

Obermann et al. ; Mandelkow & Mandelkow, ).

A minimal model of the microtubule oscillations in the form of a set of

differential equations, based on the reaction cycle with negative feedback

properties, yields the time course of the seven independent variables and describes

qualitatively the experimental observations obtained by the X-ray scattering

technique. In order to obtain a perfect fit, a mechanism accounting also for the

desynchronisation of the microtubules and an experimental selection out of

several possible options was needed (Marx & Mandelkow, ).

Solutions of tubulin and GTP in the oscillating regime readily generate

dissipative structures such as travelling waves of microtubule assembly and

disassembly as well as polygonal networks. A typical time series of travelling

waves, shown in Fig. b, was observed in a thin layer of tubulin solution by two-

dimensional u.v. spectrophotometry at nm, the pseudocolour green to light

blue corresponding to maxima in the microtubule assembly. These waves broadly

resemble the trigger waves of the BZ reaction, for which three conditions

analogous to the ones applicable here have been invoked: () the solution must be

in an excitable state, () the reaction is started at a nucleation site proceeding

autocatalytically by diffusion coupling and () waves occur because the initial

reaction is followed by a transient refractory state. In addition, the occurrence of

phase waves in oscillatory microtubule media must also be considered and should

be analysed. The experiments so far reported demonstrate quite clearly that

cytoskeletal proteins can form dynamical spatial structures by themselves, even in

the absence of cellular organising centres (Mandelkow et al. ).

The functional relationship between the in vitro observation of microtubule

oscillation and its intracellular function as an essential part of the cytoskeletal

architecture has not been clarified. Although the dynamic instability of single

microtubules in intact cells has been established as a functional part of cellular life,

an oscillating state of a synchronised population of microtubules has not been seen


in vivo. Also, the time scale of the in vitro oscillation, although concentration

dependent, is not in the range expected for global cell cycle events. On the other

hand, since it has been reported that chromosomes attached to microtubules

oscillate in the minute range during metaphase congression (Bajer, ) and that

chromosome movements are promoted by disassembly of microtubules in vitro

(Coue et al. ; Mandelkow & Mandelkov, ), the dynamic domain of

microtubule oscillations might well have a direct parallel in living cells. This

function could be controlled by the overall tubulin concentration or its periodic

spatial distribution.

The search for understanding functions of oscillatory regimes must also

consider the fact that the existence of temporal oscillations is a prerequisite for the

generation of chemical waves which precede the formation of stationary periodic

concentration patterns (see Nicolis & Prigogine, ). In model experiments the

generation of striped patterns of microtubule concentrations has been described

with distance scales which include the range of dimensions of living cells. The

pattern morphology was found to be dependent on the microtubule concentration

with the stripe periodicity decreasing with increasing tubulin concentration. This

observation points to a spatial limit, in terms of the critical length, for such a self-

ordering mechanism: namely, the concentration of the microtubules as well as

the spatial dimension of a given cell relative to the spatial wavelength given by its

diffusion and reactivity (Tabony, ; Hess & Mikhailov, a ; see also

Section ). This approach might well yield a uniform mechanism for a quasi-static

ordering of cellular cytostructures and needs to be further explored.

.

The process of cell division is one of the most decisive and intricate events in the

life cycle of eukaryotes. The precise runabout, its spatial distribution and

organisation of matter within two dividing volume entities occurs with utmost

biological timing and coordination. If unrestrained, pathological developments

ensue. A model of the cell division process relying on an intrinsic chemical

oscillator was put forward quite early (Rashevsky, , see ), and the

experimental observation and theoretical analysis of a phase sensitivity in the

division cycle of Physarum suggested an oscillator-driven mitosis control

(Kauffman & Wille, ). More recently, however, experimental advances in our

understanding of the decisive enzymic network of a mitotic oscillator have become

available (for details see Goldbeter, ).

The simplest form of a mitotic oscillator driving the alternation between the

interphase and mitosis of cell life has been found in amphibian eggs. It consists of

a periodic synthesis and breakdown of a maturation-promoting factor (MPF),

which is a heterodimer composed of cyclin B and a protein kinase, called

cdckinase. The dissociation of MPF into its two components results in its

inactivation, which correlates with the end of mitosis and sets the cellular

interphase. Within this phase new active cyclin builds up for rebinding and

reactivation of the cdckinase to give MPF, which finally triggers the new mitotic

Benno Hess

process as well as cyclin breakdown. The rates of cyclin synthesis and its

proteolytic breakdown control the dynamic regime for cyclin function. On the

other hand cdckinase is controlled by reversible covalent modification: by tyrosin

dephosphorylation of cdckinase the MPF complex is activated and stimulates the

mitotic process and at its end the kinase is inactivated by rephosphorylation. In

addition, it should be noted that the activation of cdckinase might be of

autocatalytic nature. The detailed experimental observations in many laboratories

also on a variety of other biological species reveal a much more complex regulatory

network, which cannot be discussed here at length and should be extracted

elsewhere (see Goldbeter, , ; see Cold Spring Harbor Symposium on

Quantitative Biology, vol. LVI, ). However, model studies show that with

relatively simple assumptions general properties of a mitotic oscillator can well be

simulated.

Over the years, a variety of controlling circuitries for the mitotic transition

involving a multitude of cellular functions have been studied each yielding some

features of a mitotic oscillator. One group of studies (Hyver & Le Guyader, ;

Norel & Agur, ; Tyson, ) relies on the role of autocatalysis as the source

of sustained oscillations of MPF. These models display properties as exemplified

by the classical Brusselator (Lefever & Nicolis, ), which can settle on three

different domains: a steady state, an oscillatory state and an excitable switch, the

latter state would allow for travelling mitotic waves (Tyson & Keener, ;

Tyson, ), which have been observed experimentally.

The other approach applies earlier results obtained in a study of the dynamic

properties of reversible covalent modification systems based on classical

Michaelis–Menten kinetics ruling a cascade of reversible converter enzymes such

as kinases and phosphatases. Such systems display cooperativity analogous to

allosteric systems – but in complete absence of classical allostericity – with

cooperativity and Hill coefficient larger than unity. The non-linear amplification

property arises from a zero-order ultrasensitivity which originates from the

kinetics of the covalent modification cycles. Functionally, they show integrating

properties by amplifying low molecular input signals into outputs over orders of

magnitude and might control a multitude of secondary functions$ (Goldbeter &

Koshland, , ).

A minimal, bicyclic cascade model for the mitotic oscillator involving cyclin and

cdckinase in a feed-back loop of posttranslational modifications (see Fig. )

(Goldbeter, ) with proper threshold and time lag showed limit cycle

oscillation over a cyclin concentration range of ±–± m and a fraction of

active cdckinase between ± and ± with the waveform and period – for a given

set of parameters – matching experimental observations in various species. In

phase space the unique, closed trajectory runs around a nonequilibrium, unstable

steady state. The system of three kinetic equations describing the network of Fig.

has a modular structure which can be analysed by separation of the two reaction

cycles. Their study shows that the time lag and the threshold resulting from

covalent modification are essential for the onset of mitotic oscillation. The

$ See also above the calcium oscillation model p. .


Cyclin

cdc25

Vi Vd

V1

V2

wee 1

MM+

X+ X

V3

V4M = active cdc2 kinaseX = active cyclin protease

Fig. . Minimal cascade model for mitotic oscillations (Goldbeter, ).

threshold is given by the steepness% of the activation curves for the generation of

cdckinase (M) and the active cyclin protease (X), below of which the system stops

to oscillate. Also, the threshold sets the sensitivity toward small cyclin

concentration changes. The simple model based on covalent modification does not

require a positive feedback loop and has all properties of a biochemical switch, it

is robust and would increase its robustness with the addition of more covalent

modification cycles (Goldbeter, , ).

This model was extended to analyse the implication of autocatalysis showing

that the latter one is not required to yield sustained oscillations of the mitotic

cycle, although confirming that an autocatalysic circuit yields oscillations in the

absence of zero-order ultrasensitivity. Thus, experimental studies are necessary to

support one of the two alternatives as sources of non-linearity, although the latter

one seems to be the more robust circuit structure. The cascade model also allows

the study of interactions controlling the mitotic oscillations in relation to the

control of cellular proliferation mainly in terms of G"}S and G

#}S transitions

(Goldbeter & Guilmot, ), and the regulatory coupling of the covalent cycle

– especially via cdckinase – via calcium. However, it should be mentioned that a

direct relationship between cellular calcium oscillations (see Section ) and the

mitotic oscillator cannot be expected because the frequency scales of the two

processes differ by orders of magnitude, in the ms period and the min period

range, which do not yield a synchronisation. Rather, a frequency-dependent

build-up triggering mechanisms could be imagined. Furthermore, the model

suggests mechanisms by which gene products could interfere with mitosis by

interactions with the cascade (Goldbeter, ; for the biochemistry of

interactions of the cycles with calcium see also the review by Lu & Means, ).

In general, the cascade model has been found of great heuristic value. In order

to understand the most complex timing mechanism in somatic cells in comparison

% Related to the zero-order ultrasensitivity (Goldbeter & Koshland, ).

Benno Hess

with the simpler embryonic cells an extended system, namely a double, cdc-cdk

oscillator has been suggested, which could control the proper timing of the M and

S phases. Also, the cascade principle was found to be applicable to simulate a

mechanism for the circadian oscillations in the period protein in Drosophila

(Goldbeter, , ).

. DICTYOSTELIUM DISCOIDEUM

At the end of growth and sporulation, cells of the slime mould Dictyostelium

discoideum aggregate in response to chemotactic stimuli. This process is an

example of cellular self-organisation in spatial patterns by biochemical cell

communication. In a layer of about %–& randomly distributed identical cells,

a few cells become an aggregation centre and start the aggregation process by an

autonomous and pulsed release of cyclic AMP (cAMP) as chemotactic signal with

a pulse frequency of ±–± min−". The cells around a centre respond by oriented

cell movement, and also by producing new pulses to which the outer neighbouring

cells respond after a signal input}output delay of s. This cellular property

establishes a kind of relay network, analogous to neural systems. So, waves of

chemotactic pulses can be propagated over a distance of much larger than the

chemotactic action radius of an aggregation centre. In addition to concentric target

patterns rotating spirals are also observed.

In a variety of experiments, the oscillation of the concentrations of intracellular

and extracellular cAMP, intracellular cGMP, calcium and other molecules and

coupled periodic states of the receptors, channels and motility have been

demonstrated. A binding of cAMP to cAMP receptors is transduced via two G-

protein pathways, one leading to an activation of adenylate cyclase and the other

one to formation of IP$and guanylate cyclase activation (for review see Goldbeter

). The extracellular cAMP is rapidly destroyed by extracellular as well as cell-

bound phosphodiesterase. This function allows a repetitive clearance of all

unbound and bound cAMP, so that cAMP receptor sites become ready in time for

the binding of a new incoming cAMP pulse, which triggers the intracellular

response functions.

Periodic cellular activities and the action of cAMP can be recorded optically in

stirred cell suspensions. Titration with cAMP allows identification of the phase

sensitivity of the oscillation and thus the competent phase of the life cycle of the

slime mould cells. It is important to note that the cells, and their cAMP receptors

do not respond to concentrations of cAMP, but only to concentration gradients in

the nano- to micromolar range (Gerisch & Hess, ; Dinauer et al. ;

Devreotes, ).

The cAMP signalling system is described in a model developed by Martiel &

Goldbeter (), which accounts for the following processes: secretion of cAMP

by the cells, hydrolysis of cAMP by phosphodiesterase, binding of the secreted

cAMP to the membrane receptor with a resulting stimulation of cAMP

production, and desensitisation of the membrane receptor. The main components

of this model are cAMP and the receptor, which can exist in a sensitised and


desensitised state (Devreotes & Sherring, ; Vaughan & Devreotes, ).

This allows the description of active and passive phases of the cAMP synthesis.

The model leads to a system of three coupled nonlinear ordinary differential

equations describing the dynamic interaction of the intracellular cAMP (β), the

extracellular cAMP (γ) and the fraction of active membrane receptors (ρ). With

appropriately chosen parameter sets for the rate constants, Martiel & Goldbeter

() were able to model both oscillations and relay of cAMP signals, in

agreement with experimental investigations. A treatment of models and

experiments of more complex dynamic states such as birhythmicity and bursting

as well as aperiodic oscillations and the route to deterministic chaos is beyond the

scope of this review (See Goldbeter, ).

In order to model cAMP waves in monolayered cultures of Dictyostelium cells

on agar surfaces, where the secreted cAMP diffuses through the aqueous

extracellular medium, the Martiel–Goldbeter model has to be supplemented by

terms describing the diffusion of the extracellular cAMP (γ). Tyson et al. ()

proposed a reduced system of two reaction–diffusion equations which can be

written in the following form (Tyson et al. ; Tyson & Murray, ) :

¥γ}¥t¯ ε∆γ}ε [sφ (ρ,γ)®γ]

¥γ}¥t¯®f"(γ) ρf

#(γ) (®ρ),

5

6

7

8

()

where ε is a scaling parameter, s, φ, f"and f

#are specified in Tyson et al. (),

∆ is the Laplacian operator describing the diffusion of the variable γ.

From this type of equation, a linear relationship, the Eikonal equation, between

the propagation velocity and the curvature of travelling cAMP waves has been

derived (Zykov, , see ; Tyson et al. ) and was experimentally tested

(Foerster et al. ) :

N¯ c®DK. ()

N is the normal velocity, c the velocity of plane waves, D the diffusion coefficient

of the autocatalytic species cAMP, and K is the curvature of the waves.

The main statements of eqn () are: () for negative curvature the normal

velocity increases with increasing curvature; () for positive curvature, it decreases

with increasing curvature; () there exists a minimal radius below which

propagation of circular waves will not take place.

Furthermore, eqn () has, under quite general conditions, periodic wave

solutions that satisfy a dispersion relation (Dockery et al. ; Foerster et al.

; Tyson & Keener, ).

c¯ s(T ). ()

This relationship expresses the dependence of the propagation velocity (c) on the

temporal period of the wave train (T ). It shows an increase in the velocity of wave

propagation with increasing period of wave initiation T, reaching an asymptotic

value cmax

as T!¢. Below a minimal value of T, no wave trains can exist because

the membrane receptor cannot become sensitive again between the successive

Benno Hess

wave trains. It should be added that σ is a complex function of the reaction kinetics

and the scaling parameter ε.

The curvature and spiral geometry in the aggregation pattern of Dictyostelium

discoideum was experimentally recorded using a dark-field equipment combined

with video techniques. A computerised image processing allows the analysis of

wave collision structures, expending concentric circles and rotating spirals in

terms of wave velocity and front geometry. In these studies the linear relationship

between the normal velocity and the curvature of the wave fronts predicted by the

reaction–diffusion model was verified and the proportionality factor, which in this

case is a diffusion coefficient of the chemical signal transmitter cAMP, was

determined to be ±¬−& cm# s−". The critical radius of wave propagation was

roughly estimated from measurements of the positively curved circular waves. It

was found to be approximately mm which means that up to cells could fill

the space to form the centre of an aggregation structure. The geometrical

parameters of spiral wave patterns were also obtained and led to an estimation of

the core radius to be approximately mm (Foerster et al. b).

Comparison of the model and experimental results showed that, in spiral waves,

curvature is not negligible in the core region and therefore they must satisfy both

the curvature–velocity relationship and the dispersion relation. Consequently,

spiral waves have characteristic c, T values which satisfy both conditions. These

values are functions of the size of the spiral core. From a numerically computed

spiral wave, Tyson and co-worker (Tyson & Keener, ; Tyson et al. ;

Tyson & Murray, ) calculated a core radius r!E µm; the value measured

by overlaying successive contour maps is r!E µm. This extended core region

containing a high amount of core cells supports the suggestion of Durston (),

that the core, the region around which the spiral is rotating, is built by a loop of

cells around which a continuous circulation of an excitation wave takes place, a

concept which was later followed extensively (see below).

The studies, furthermore, showed that the spiral pattern of Dictyostelium

discoideum resembles more an involute of a circle than an Archimedian spiral. It

is interesting to note that these studies provide quantitative evidence that the

mechanism of the cAMP signal-relaying systems underlies general conditions of

excitable media. They strongly support the concept that the theoretical

investigation of excitable media, as mainly done for simpler systems, e.g. the

chemical BZ reaction (see below), allow predictions for systems of higher

complexity like the Dictyostelium discoideum cell population.

Detailed comparison between the simplified model of the waves in the

aggregation territory of spores of Dictyostelium discoideum given above and a

multitude of experimental data show, that, although global features of the

biological phenomenon are well represented, some properties are not and require

extension implying, for instance, receptor modification, adaptation and relay

phenomena (Monk & Othmer, , ). These authors present and analyse in

great detail a continuum model based on reciprocal interaction loops between the

localised cAMP as well as calcium signals. They treat for the first time the

dynamics of single cells as well as of the whole cell population, and show the


initiation of spiral waves and that travelling waves of cAMP do not result from

Turing (diffusive) instabilities and that a detailed comparison of their results

(including the dispersion relation) with those of other authors (Tyson et al. ;

Tyson & Murray, ) gives a much better fit of the experimental data. Further

refinement of simulation studies was achieved on the basis of a detailed G-protein

mechanism, including adaptation features, for signal transduction, which allowed

the prediction of excitation to cAMP stimuli, sustained oscillation, spiral waves

and target patterns depending on the developmental stage of the cells (Tang &

Othmer, b).

The properties of a reaction–diffusion model of the FitzHugh–Nagumo-type

for the simulation cAMP waves and a continuity equation for the aggregation

motion have been studied in order to understand the cell streaming process in

more detail (Vasiev et al. ). Their numerical analysis led to the conclusion that

the aggregation pattern is formed as a result of front instabilities due to the

dependence of wave velocity on the density of the amoebae, which by their spike

gradients set the velocity of the cAMP wave. Indeed, this instability may also lead

to wave defects and, depending on its size, to the generation of spiral waves.

The mechanisms of the complex stream pattern formation is also described in

another model extended from the original cAMP model given above, not implying

the complex calcium interaction, but including a time-dependent adaptation

response (Ho$ fer et al. ). This model is based on reaction–diffusion-chemotaxis

– including random cell migration – and provides not only a good description, but

it also accords well with the numbers obtained from the eikonal equation with an

initial core size of mm (see above) and predicts a locking of the core dynamics

into a cell loop as well as a continuous shrinkage of the core radius to a lower limit,

both options depending on the excitability of the medium. This important study

links the earlier model investigations on the dynamic properties of large single cell

territories moving toward a centre to the next step of models aiming at a

description of later phases of the life cycle of Dictyostelium discoideum, such as the

slug formation and migration.

In a detailed experimental study of the cell movements during the slug

formation phase it was again found that, in this phase of the life cycle of

Dictyostelium discoideum, periodic signals and chemotaxis control the

morphogenesis of the slug. This process consists of two distinct territories within

the slug tube, each of them moving along the slug migration axis in a coherent

periodic timing typical of excitable media: in the prestalk zone the chemotactic

signal propagates as a D scroll wave, while in the prespore zone a planar wave

propagation is recorded. Indeed, the slug formation and its intrinsic cell migration

are based on a twisted scroll wave propagation mechanism. The cell migration

direction is opposite to the signal propagation – just as in the early cell aggregation

phase of the life cycle (see above) (Siegert & Weijer, ).

The experimental observations of the complex motion pattern within the slug

organised as a highly excitable prestalk zone of a scroll wave signal propagation

and the low excitable prespore zone of a planar wave front were analysed

numerically in a first approach on the basis of a simplified, but generic

Benno Hess

two-variable model, developed for the study of excitable media, in general

(Barkley, ). This study yielded a first explanation for the mechanism of a

dynamic ‘two-territory’ self-organisation within one multicellular body in terms

of a well coordinated collective cellular motion resulting from propagating cAMP

waves, which order the whole system. The model calculation showed that the

transition from a D wave motion to a D planar wave can be due to a change in

excitability – involving different minimal wavelength and wave speed – along the

long slug body axis. Thus, because of too large a wavelength due to lower

excitability, a scroll wave does not fit the prespore zone (Steinbock et al. ).

Further development of these results led to an extended model study of the

realistic cAMP relay mechanism, which describes the periodic spatial propagation

pattern of cAMP during the aggregation phase of the life cycle of Dictyostelium

discoideum (Martiel & Goldbeter, ). The analysis of this three-variable model,

based on biochemical data, showed that the dynamics of the controlled self-

organisation of the growing and moving slug result from a simple, dynamic cAMP

pattern and that the spatial transition from one territory to the other is due to a

controlled reduction of the number of signal relaying cells to about % in the

prespore zone relative to the prestalk zone. This result indicates that the

competent cell density controls the influence of the dispersion term [see above and

eqn (), p. ] on the curved to planar wave transition as a bifurcating parameter

and points to the critical core size of the spiral in the slug. Indeed, it is suggested,

that the mechanism of the dynamic control of cell density in the transition region

might well be realised by a direct expression of specific genes sensitive to the low

and rather steady cAMP concentration in the core of the spiral (Bretschneider et

al. ; Siegert & Weijer, ).

The spatial dynamics and organisation of a multicellular body such as the

mound of a slime mould is one of the most intricate problems in cell biology, and

the mould can be considered as a general model of spatial development and

differentiation. The wave propagation in mounds was recently recorded with

sensitive optical dark-field and image-processing techniques in great detail. These

studies allowed one to follow in mounds composed of $–& individual cells the

ordered motions of waves and detect the occurrence of concentric rings, which

might break into counter-clockwise rotating spirals of one or more (" )& arms

(see Fig. a) of varying stability, frequency (periods of – min) and propagation

velocity of – mm min−" depending on their progressing local state: thus, the

generation, fusion and decay of waves are seen in the process of development. As

expected in the case of several pacemakers generated simultaneously in a near

neighbourhood, annihilation of colliding wave occurs whenever proper spatial

phases meet. Again, as mentioned above, the cell movement was found to be

opposite to the propagation direction of the cAMP signal. This search and trial

process, depending on the excitability state of the prestalk cell always results in the

formation of a stable centre core in the tip of a mound with a size of – mm

in diameter& (Siegert & Weijer, ; Rietdorf et al. ).

& F. Siegert, personal communication.


A recent detailed model for the mound formation simulates cell movements on

the basis of the earlier fluid flow concept (Odell & Bonner, ) with the

chemotactic parameters of the cell population as well as the frictional and adhesive

forces and the internal pressure activated and controlled by cAMP-waves. Again,

the FitzHugh–Nagumo equations are used to describe the cAMP-kinetics, and the

cell movements and its density changes are based on the Navier–Stokes equations

and its conservation equation, respectively (Vasiev et al. ).

The model describes the development from the D randomly distributed cell

aggregation territory up to the D mound stage It considers especially the

formation of the cell-free core during early aggregation in terms of the density-

dependent excitation parameter – the dispersion. Furthermore it takes into

account the meandering mounds and ring transformation. A typical comparison

between the experimental observations and the model is given in Fig.

demonstrating two snapshots of the spiral motion at the tip for both cases [(A) and

(B), respectively]. This type of studies also yielded an order of magnitude of the

core size in the range of – mm, well comparable to the size observed for the

core size of D field of the slime mould as well as for the D fields of the BZ

reaction (Foerster et al. , ). It also allows a comparison of the dynamic

cell distribution patterns in dependence of their relative excitability and dispersion

parameter as a function of density. The meandering feature is well analysed in case

of the BZ reaction, demonstrating its dependency of the excitability state of the

system and the dispersion parameter.

All these studies suggest that the simple principle of a periodic D signal

generation, propagation and reception – of most likely cAMP – followed by

chemotactic motion, governs the morphogenetic phases of the life cycle of

Dictyostelium discoideum. In the future, a linkage between the model describing

the dynamic properties of large single cell territories moving toward a centre and

models aiming at a description of the following steps of the life cycle of

Dictyostelium discoideum, such as the tip formation and prestalk–prespore

differentiation, the slug formation and migration, the culmination and finally the

terminal differentiation, might be achieved and ultimately yield a stepwise and

perhaps unique model description of a full life cycle of a biological species. Such

a goal implies a detailed understanding of the mechanisms of coupling the various

developmental stages to the cell cycle phases ruling the cell sorting and

differentiation processes and finally the overall sizes of the mound and the slug,

until the next life cycle is triggered (see Maeda, , and also Section ). An

important step in this direction is the recent model, which demonstrates a critical,

positive genetic feedback between the cAMP signalling and the expression of

genes encoding the signal transduction and response system. Using a hybrid

automata-continuum scheme for modelling and appropriate computer simulation,

the direct coupling of genetic expression to the spiral pattern formation, as a

source of pattern instability, was suggested to play an important role in the

developmental mechanism (Levine et al. ).

Benno Hess

.

The neuromuscular network of cardiac tissue exhibits all features of excitable

media analogous to those observed in the amoebae of Dictyostelium discoideum (see

Section ), the chicken retina (see Section ), or the chemical BZ reaction (see

Section ). Although the macroscopic dynamics of all these media are well

comparable, the underlying intracellular or intercellular biochemical reaction

mechanisms differ depending on the nature of the various biological species

involved. Because of their common intrinsic reaction circuits, namely their

autocatalysis and}or feedback connectivities, they can be described by similar

reaction–diffusion equations with nonlinear and singular kinetics.

The multicellular excitation network of the heart, the pulses for which originate

from the sinus node and propagate throughout the quasi-D neuromuscular mass,

relies on its netted glycolytic as well as ion channel and pumping system located

in each individual myocyte (see above Section , .). In addition, it should be

noted that the cellular structure of the sinus node, as a beating signal producing

device, differs qualitatively from the neuromuscular signal propagation network,

which receives the trigger signal from the node and is composed of millions of

rather uniform cells interwoven by a capillary net and connective tissue throughout

the heart. The latter structures do not hinder the propagation of the excitation

waves.

Since the observation of rotating excitation in myocardial tissue (Allessie et al.

) analogous to rotating waves in other systems mentioned above, such

spatially propagating patterns have been theoretically studied by a number of

groups. Following a numerical study based on the FitzHugh–Nagumo equation to

simulate rotating spiral waves in a D excitable heart muscle medium (Pertsov et

al. ), a general theory of rotating spiral waves in excitable media was

developed analytically yielding results well comparable with the numerical data

and also allowing one to understand the dispersion of a nonlinear plane wave and

the curvature effect on the propagation of waves in D media (Tyson & Keener,

, ), and, by use of the eikonal equation (see Section ), yielding a

comparison with experimental observations.

Self-sustaining vortex-like excitation waves could be initiated experimentally in

normal isolated ventricular muscle. The waves proceed in a clockwise rotating

manner with a revolution time of ms and a wavefront propagation speed of

±–± mm ms−" depending on the location at the nonuniform waveform. The

centre of the vortex has an average core size of ±– mm composed of a relatively

small number of cells with no activity, and the axis runs parallel to the fibre

orientation. The size and the position of the core was found to be stable during

true pivoting, and it was anchored near a small artery. Furthermore, spontaneous

instabilities of the rotating waves were observed resulting in a drifting of the core

resulting in a Doppler shift of the period of rotation. Collision of rotating waves

leads to annihilation of waves analogous to other excitable systems. These

experimental observations were accompanied by computer simulations of wave

propagation in generic excitable media based on the FitzHugh–Nagumo


equations. They suggest the occurrence of a D vortex-like re-entrant excitation

phenomenon in the heart muscle system analogous to the spiral waves in other

systems. Furthermore, it was indicated that future experimentation might lead to

a common mechanism for monomorphic and polymorphic tachycardias

(Davidenko et al. , ; Pertsov et al. ).

Experimental instability of the electrochemical propagation pattern in normal

heart leading to the formation of vortices and turbulent electrical activity were

observed in the presence of a partial blockade of membrane sodium channels or

high frequency of excitation in thin epicardial muscle slices, if an unexcitable

obstacle is induced. The formation of vortices was found to be similar to vortex

shedding in turbulent hydrodynamic flows (Cabo et al. ).

These experiments point to the spatial limit of an unhindered pulse propagation

in normal heart. As reported earlier in theoretical studies (Winfree, ;

Mikhailov, ; Panfilov & Keener, ) and in experimental observations of

the BZ reaction, the relationship between the width of a wave front (L) and the

critical diameter for excitation (dc) sets the boundary for stable propagation.

Whenever the width of a wavefront is higher than the critical diameter stable

propagation prevails. In the opposite case a wave becomes unstable, with the

boundary given by the equality of the two parameters, which have been derived

from the reaction mechanism of the cerium catalysed BZ reaction (Nagy-

Ungvarai et al. a, b). This relationship was also supported by computer

calculations (Pertsov et al. ).

These studies give an estimate of the critical size and geometry of the many

possible anatomical obstacles in normal and pathological heart tissue. Thus, the

minimum number of muscular cells constituting a core of a vortex is of interest.

In addition to the activation of the wave front curvature and the vortex geometry,

however, the electrical thresholds of the myocardium for stimulation and

fibrillation have to be considered. In this respect it has been found theoretically

that a single aberrant cell cannot serve as an ectopic focus in normal myocardium

– which would also be too small with respect to the critical radius given above, but

a dense focal set of roughly cells has been computed as a local source to

overcome the threshold (Winfree, , a ; see also Zykov, , ).

In this context, recently, the problem of geometry and the critical thickness of

the muscle and its significance for the mechanism of cardiac fibrillation became a

matter of debate. The notion that several D rotors of electrical activity become

unstable if the heart thickness exceeds a critical value (Winfree a, a) was

challenged because of its contradiction to experimental observations, which show

that a single moving rotor displays an ECG pattern as observed in fibrillation.

Furthermore, experimental evidence demonstrates that a large number of rotors or

a critical thickness are not necessary for fibrillation to occur and also that the

observation of a Doppler effect explains the typical frequency spectra of fibrillation

(Gray et al. ). Also, a computer study of the chirality in D and turbulence

in D based on the FitzHugh–Nagumo equation showed a strong influence of the

geometry of the excitable medium (Panfilov & Hogeweg, , ).

Furthermore we note that the vulnerability of the cardiac muscle network needs

Benno Hess

careful consideration, especially, since model studies have revealed special

propagation properties of the excitable media, which are not visible in isolated

cells, and also a linkage of macroscopic features of waves to sodium conduction

functions (Winfree, a ; Starmer et al. ). All these studies show that

currently useful models for the understanding of propagation mechanisms in

complex biological media, either under natural or pathological conditions, are

slowly emerging.

.

In general, spatial pattern formation occurs in two different regimes of

reaction–diffusion systems. First, there is the classical Turing pattern, which –

following an earlier study (Rashevsky, , see ) – was found in a detailed

mathematical analysis of the onset of instability in a simple reaction–diffusion

system and suggested as the chemical basis of morphogenesis (Turing, ).

Turing structures are characterised by critical wavelengths dependent only on

intrinsic parameters and not on the geometry of a reacting system in contrast to

the second case of dissipative structures, we deal with in this review primarily,

evolving from complex nonlinear reaction networks, which readily settle with

proper input on periodic, quasiperiodic and chaotic states giving rise to spatial

patterns (Glansdorff & Prigogine, ; Nicolis & Prigogine, ).

It was only recently that stationary and nonstationary Turing structures have

been generated in chemical systems giving regular striped and hexagonal patterns

in D and D and studied in great detail (Castets et al. ; DeKepper et al.

; Ouyang & Swinney, ; Boissonade et al. ). However, such systems

are still difficult to handle, theoretically and experimentally, especially because of

the spatial dependency of the control parameters. Up to now experimental

evidence for macroscopic as well as mesoscopic Turing instabilities in biological

systems, as discussed in the literature (Murray, ; Meinhardt, ) has not

been found. A theoretical study of the generation of Turing structures in a

biochemical system demonstrated its occurrence on a mesoscopic scale and was

found to be stable to molecular fluctuations. Using a reaction lattice gas automaton

for simulation, a study of Selkov’s model for glycolytic oscillation showed that

spatial symmetry breaking in the ATP concentration within a cell cytoplasma

develops for a critical Turing length in the range of typical cell dimensions. It was

suggested that such a mechanism might result in a global breaking of energy

distribution in a cell (Hasslacher et al. ). Also, it was pointed out, that such

Turing instabilities might be involved in the mechanism of pattern formation in

cell membranes (Lengyel & Epstein, ).

Today, the experimental and theoretical prototype of dissipative spatial patterns

are the spirals of many reactive media. The geometric and kinematic parameters

of a dynamic spiral have first been explored and quantified experimentally in great

detail in case of the spiral-shaped waves of the BZ reaction, making use of a

combination of D optical methods and computerized video techniques with high

spatial and temporal resolution. A comprehensive software package for the


representation of the D spectrophotometric data array allowed the extraction of

profiles of transmitted light intensities, its conversion into concentration and

pseudocolour presentations and the fitting for specific isoconcentration lines and

gradients, as well as the D to D transformation of chemical gradients for spatial

and perspective quantification (Mu$ ller et al. a, , a, b ; Kramarczyk,

). Recently, a quantitative optical tomography for the analysis of D chemical

waves has been developed, allowing the calculation of the D distribution of

reaction components such as torroidal waves from optical projection in a set-up

realized as outlined by A. T. Winfree (Stock & Mu$ ller, ; Winfree et al. ).

The dynamical and structural investigation of spiral propagating waves of the

BZ reaction with the redox couple ferroin}ferriin as catalyst and optical indicator

revealed the motion of a perfect spiral form of isoconcentration lines, which could

be described by an Archimedian spiral or by the involute of a circle. A remarkable

constant velocity of mm s−" and a spatial stability of the spiral tip of ! µm

was found. The core of the spiral, namely the so-called black hole, has a size of

E ± mm in diameter, and it is a singular site with only very small intensity

modulation (Mu$ ller et al. b, c).

In addition to the spectrophotometric detection of the spatial distribution of

redox-catalysts in the wave pattern, the redox potential of catalysts and the Br-

activity has been followed by microelectrodes with a diameter of ±–± µm –

smaller than the critical radius of the system and thus not disturbing its wave

propagation – indicating the spatial relationship of both components along a wave

train and their dispersion relationship (Nagy-Ungvarai et al. ; Nagy-

Ungvarai & Hess, ). The analysis of the temperature dependence of the

curvature–velocity relationship of the waves showed, that with increasing

temperature the critical radius becomes small to a limit (see below) with an

activation energy of wave propagation of E¯ ±³± kJ −" and that of the

diffusion of the autocatalytic species of E¯ ±³± kJ −" (Foerster et al. ).

In this context, in the BZ reaction and in general, the competition between the

transport mechanisms of diffusion and convection as well as the linkage between

hydrodynamic and chemical pattern formation is of interest, and convective

patterns travelling with waves in a shallow layer of liquid BZ solution were

observed. If the chemical wavelength is large, each wave carries a pair of

convection rolls with a pronounced downward flow at the site of the front, with a

weak upward flow over a broad zone far away from the wave. At the surface this

generates a flow against the direction of wave propagation in front of the wave and

in the direction of propagation behind the wave. This convection is strongly

connected to the wave geometry and independent of evaporative surface cooling

(Miike et al. ). The local thermal inhomogeneities travelling with the

oxidation wave fronts have been measured by sensitive thermodetectors showing

that an exothermic reaction in the excitation front generates a local temperature

rise of mK. The fact that there is a localised rise of temperature excludes its

active participation in the mechanism of the convection processes induced by

chemical waves with the clear indication that the driving forces are produced by

local concentrations only (Bo$ ckmann et al. ).

Benno Hess

For a long time local physical perturbation techniques were used to trigger and

influence wave patterns and its conversion from target to spiral form in D. More

recently, by use of the light sensitive catalyst of the BZ reaction, the ruthenium

dipyridyl complex, local and global light pertubation techniques have been

applied, yielding a variety of light controlled macroscopic patterns (Kuhnert et al.

; Markus et al. ; Steinbock & Mu$ ller, a). The latter authors

achieved the generation of multiarmed (up to arms) spirals and their stable

anchoring in the case of meandering waves (Steinbock & Mu$ ller, b).

Furthermore, the creation of spiral waves by geometry (Agladze et al. ) and

their interaction with transversal gradient (Zhabotinsky et al. ) as well as the

observation of refraction and reflection of chemical waves (Zhabotinsky et al.

) and of helical waves resonance (Agladze et al. ) have been reported.

Chemical systems with the properties of excitable media offer a unique example

to explore the mechanistic requirements for stable and instable pattern formation,

serving also as a model for biological systems as treated in other sections of this

review. The dynamic reaction course of excitable media is characterised by three

sequential state phases, which in space appear as three spatial phases, namely an

excitable, an excited and a refractory phase, reflecting the underlying chemical

mechanism of a general autoregulated feedback circuit and indicating the

dispersive interphases. Such systems can be properly described by reaction–

diffusion coupling with the diffusion term of an autocatalytic species triggering the

spatial reaction progress of an excitation wave.

Among the many excitable reactions, the BZ reaction was found to be the most

useful example for suitable modifications. Early, the range of pH, of redox

potentials, as well as of other initial conditions were studied, mostly effecting the

essential autocatalytic reaction step, if not the course of the classical advance of the

BZ reaction (see Kapral & Showalter, ). Important for these investigations

was, under proper conditions, the observation of a spatial instability of the tip of

a spiral and its geometric trajectories, which started to meander spatially around

over more and more complex routes following a looping trail like an epicycle. A

typical set of tip motions at various proton concentrations is given in Fig.

demonstrating the large variety of the complex spatial dynamics composed of two

main frequencies (Plesser et al. ; Mu$ ller & Plesser, ). Detailed

experimental studies of the chemical waves using high redox potential metals and

low redox potential complex catalysts allowed one to test the rigidity of the

underlying chemical mechanism (Nagy-Ungvarai et al. ). The cerium system

was especially useful to analyse the lateral instabilities of a wave front, giving the

boundaries for the critical radius of the curvature and the pulse width for

transition to unstable wave fronts in the BZ reaction with the dispersive nearest

neighbour interactions (Nagy-Ungvarai et al. a, b, , a, b, ).

Theoretical studies led to a remarkable description of the essential parameters

fitting the experimental findings. There are two important relationships that have

to be considered for the understanding of the geometry and dynamic behaviour of

spatial waves: () the dependence of the normal velocity on the positive and

negative front curvatures expressing the correlation between the shape and the


1 mm

1 mm

H2SO4 = 0·15 M H2SO4 = 0·19 M

H2SO4 = 0·23 M H2SO4 = 0·26 M

1 mm

H2SO4 = 0·37 M

Fig. . Traces of the motion of the spiral tip in the BZ reaction differering in the initial

sulphuric acid concentration (from Plesser et al. ; see also Mu$ ller & Plesser, ).

propagation velocity of the wave fronts, and () the dispersion relation which

expresses the dependence of the propagation velocity on the frequency of the wave

initiation (Keener & Tyson, , ; Zykov, , see ; Mikhailov et al.

; Dockery et al. ; Keener, ). The description of the BZ reaction

kinetics was based on the so-called Oregonator model in the form of a normalised

reaction diffusion equation, obtained after reduction to two variables and leading

analytically to the expression of the eikonal equation, the dispersion relation and

to the minimal radius of the core of a spiral below which propagation of circular

waves will not take place. The experimental analysis of the positive and negative

curvature of wave formation and its critical core size in an excitable chemical

medium, namely the BZ reaction, verified these relationships with microscopic

resolution (Foerster et al. , a).

In order to understand the geometry of the transition to unstable wave fronts

– as reported for the cerium-catalysed BZ reaction – the theoretical relationship

between the critical radius (rcrit

) and the width of a wave front has to be considered

and compared to the experimental results. It was found that whenever the width

of the wave front is higher than the critical diameter, free ends of wave front can

grow together or form stable spirals, the opposite case leads to spreading,

disintegration and fragmentation of waves, the boundary being the equality

between both parameters. It should be pointed out that the mechanism of

fragmentation is closely related to a nucleation phenomenon. Furthermore, the

relationship between the width of the wave front and the critical diameter of the

medium is a figure of merit for the concept of low and high excitability of reaction

Benno Hess

media. This relationship is of importance for biological media, because it gives an

estimate of the stability boundaries for activating and inhibiting interactions

within excitable cellular nets (see above) (Nagy-Ungvarai et al. a, b).

An important investigation of the general dynamics of excitable wave systems

led to a simple kinematic theory describing the front kinematics of a wave, without

and with interactions between single neighbouring waves of a wave train, and also

considering sprouting of wave fronts, meandering, resonance and drift motions in

D as well as D (Brazhnik et al. ; Mikhailov, ; Mikhailov et al. ;

Mikhailov & Zykov, ).

The theoretical analysis of the complex motion of spiral tips in excitable media

yielded the discovery of several regimes of such behaviour (Mu$ ller & Plesser,

; see Kapral & Showalter, ), which could be classified by application of

a Fourier analysis technique. This study showed that in the Oregonator model up

to four frequency branches can be obtained, which determine as main structural

components the coarse structure and geometry of the Ma$ ander pattern, whereas

other frequencies only distort the epicycles to a certain extent. So far, no chaotic

regimes have been found. The spiral wave motion outside the core was found to

be independent of the complex tip trajectories and only determined by the

maximal structural Fourier component, namely the tip motion with the highest

circular speed (Plesser & Mu$ ller, ).

The vast literature on the D to D transition giving stable D patterns in

excitable media cannot be reviewed here. The early experiments already showed

the occurrence of remarkable, dynamic D structures in the form of helices and

scroll rings. Theoretical and experimental studies followed and contributed

intensively to our current understanding of D patterns in biochemical and

biological reaction systems (see Keener & Tyson, ; Tyson & Keener, ;

Jahnke et al. ; Fast et al. ; Markus & Hess, ; Winfree, b, ,

b, b, c).

In general, all studies of the large class of excitable media composed of chemical

or biological elements show, that, whether classical reaction diffusion waves of

chemical reactions in solution or on catalytic reaction surfaces (see Eiswirth &

Ertl, ; Imbihl & Ertl, ) or the various classes of nerve impulse or other

multicellular propagation waves involving cell–cell interfaces of biochemical

nature, they all share common properties as described by very similar underlying

mathematical equation systems, and illustrate the low dimensionality of general

mechanisms for the generation of complex structures.

.

The energetic conditions, under which living cells exist, create the possibility of

self-organisation as noted by Schro$ dinger (). Whereas the macroscopic

properties discussed so far are readily understood in terms of classical chemical

kinetics, the kind of self-organisation found within a living cell are not necessarily

only a reduced copy of what is characteristic for large macroscopic systems, and

the question can be asked whether significant changes in the expression of self-


organisation are brought about by going to much smaller length scales typical for

many cells. What are the boundaries for the developments of periodic patterns

within such scales?

An entire cell, or one of its closed compartments, in small volumes may

sometimes include only a few thousand molecules of a particular species, such as

enzymes, receptors, messengers or ions, such as Ca#+. Yet spatial and temporal

organisation and coordination are not necessarily rigidly controlled in such

processes – this is a natural consequence of interactions between the elements of

the system.

Recently, some criteria for self-organisation in chemical reactions involving

small numbers of molecules and occurring in relatively small spatial volumes have

been formulated beyond the classical scales dealing only with mean concentrations

of molecules (Hess & Mikhailov, a, ; Mikhailov & Hess, ). Since the

conditions for microscopic self-organisation are set by the relations between the

principal timescales of the physical and chemical processes involved, the analysis

defined first the estimates for their characteristic times.

Considering a biochemical system in a small volume formed by enzyme

molecules (B) whose activity is regulated by intermediate product molecules of a

smaller molecular weight, the mixing time tmix

– based on random diffusive

motion described by Fick’s law – is the time after which a molecule released at

some point inside the cell is found with about the same probability anywhere

within the cellular volume. This time is required to equilibrate chemical

concentrations throughout the cell. As is well known, this is roughly estimated as

tmix

¯L#}D, where D is the diffusion constant of molecules and L is the cell

diameter. For a small cell or a cellular compartment with diameter LE −% cm

and diffusion constant D¯ −' cm# s−" the characteristic mixing time, tmix

, is

about −# s.

The traffic time ttraffic

yields the time typically needed for two given molecules

(A and B), inside the cell, to meet if they are initially separated by a distance of

approximately the cell diameter. Using the equations of Smoluchowski’s theory of

diffusion-controlled coagulation (von Smoluchowski, ; Chandrasekar, ),

this characteristic time has been estimated (Hess & Mikhailov, b, , )

as ttraffic

¯L$}DR. In this estimate, which is valid when L(R, we have D¯D

AD

Band R¯R

AR

B, where R

Aand R

Bare the radii of molecules A and B

and DA

and DB

are their diffusion constants. If target molecules B are much larger

(and hence less mobile) than molecules A, as an approximation one can set R¯R

Band D¯D

A. In this case the traffic time is determined by the radii of the

targets and the mobility of the intermediate molecules. Note that then the

relationship ttraffic

C (L}R) tmix

holds.

To obtain a numerical estimate of the traffic time, an enzyme molecule B with

the size of the order of a few hundred AI ngstro$ ms (R¯ −' cm) is taken

considering an intermediate molecule A whose typical diffusion constant in a

water solution is about D¯ −' cm# s−". For small cells or compartments with

diameters L of about −% cm, substitution of these values yields a traffic time,

ttraffic

, of s. This result is remarkable: it tells us that any two molecules within

Benno Hess

a micrometer-size cell meet each other every second. Since the traffic time

depends strongly on the cell diameter (as a cube of L), under the same conditions

ttraffic

increases to tens of minutes for cells of diameter µm and to tens of hours

for large cells with diameters of µm. This strong sensitivity on the cell size

indicates that one should expect special kinetic regimes in small cells and cellular

compartments, and special machineries for large cells not being discussed here.

While the traffic time shows the time needed for a molecule to meet a given

target molecule, the transit time ttransit

characterises the typical time required

simply to meet one of the molecules of this kind present inside the cell. If the cell

contains N copies of target molecules B and they are independently distributed

over its volume, molecule A will touch the first of them in a time of about ttransit

¯ ttraffic

}N. In the context of enzymic reactions, the transit time has been

introduced by Dixon & Webb () : after this time a coenzyme (intermediate)

molecule finds one of the molecules of its target enzyme.

Of course, contact with another molecule does not yet necessarily trigger a

chemical reaction between the two. When the energetic barriers are high, most of

the collisions are not productive. However, if the barrier is low, a significant

fraction of collisions leads rapidly to a reaction event. Such fast chemical reactions

develop strong spatial correlations between the molecules inside a correlation

volume, and their rate is controlled by diffusion (Eigen, ). Enzymic reactions

inside a cell are optimised with respect to their yields. It is natural to expect that,

when the numbers of the enzyme and intermediate molecules involved in a

particular reaction in the cell are small, the majority of collisions should be

reactive. As a rough estimate, it is assumed below that any collision with the target

molecule entails a reaction event.

A molecule, released as a consequence of a reaction event at a certain spatial

location, moves randomly until it meets one of its target molecules and triggers the

next reaction event. These two events, which are causally correlated, will be

typically separated in space by the correlation length lc, which is the distance

passed by the intermediate molecule before it meets the first target. The

correlation length is thus estimated as lc¯ (Dt

transit)"#.

If the correlation length is much shorter than the cell diameter (lc'L), the

interior of the cell breaks into statistically uncorrelated spatial domains (Stanley,

). Since all intermediate molecules find their targets inside a domain of the

correlation length, the information about occurring reaction events remains

localised within it. It is worth noting that the condition lc!L can also be

expressed as ttransit

' tmix

in terms of the characteristic times of the reaction.

What happens if the opposite condition, ttransit

( tmixing

, is realised and hence

the correlation length lc

turns out to be larger than the cell size? Since the

intermediate molecule will then meet its target with equal probability anywhere in

the cell, statistical correlations between subsequent reaction events would extend

over the entire cell volume. In other words, the information about the occurrence

of a given reaction event, conveyed by the intermediate molecule, is not confined

in this case inside a certain spatial domain but can instead reach any region within

the cell.


Since the transit time can be expressed in terms of the mixing time as ttransit

¯(}N ) t

traffic¯ (L}NR) t

mix, where N is the total number of the considered target

molecules in the cell, the condition ttransit

( tmix

can be equivalently written as N

(Ncrit

, where the critical number of target molecules is Ncrit

¯L}R. Here L is

the cell diameter and R is the radius of the target enzyme. Thus, it can be

concluded that strong correlations extending over the entire cell (or a separate

cellular compartment) should be expected for biochemical subsystems with small

numbers of active molecules. For a small cell or a compartment with a diameter

of a few micrometers (L¯ −% cm) and molecules with diameters of a few

hundred AI ngstro$ ms (R¯ −' cm), the critical number of enzyme molecules

involved in a particular reaction (Ncrit

) is about .

Since the turnover time tturn

of the majority of enzymes operates in the interval

tturn

¯ −$–−# s, in small cells the characteristic turnover times may well have

the same order of magnitude as the mixing and the transient times that were

estimated above. It means that in these cases an enzymic reaction can no longer

be viewed as consisting of instantaneous reaction events. The time-resolved

physical processes in individual enzyme molecules must then be incorporated into

the complete kinetic description.

Whenever the turnover time is much larger than the transit and mixing times,

tturn

( ttransit

( tmix

(typical for cellular compartments) the overall rate of the

reaction of controlled not by diffusion, but by the speed of the catalytic turnover

of individual enzyme molecules. The intermediate regulatory molecules reach

their target enzymes within a time that is much shorter than the duration of a

catalytic event in a single enzyme molecule and the found targets may lie anywhere

within the reaction volume.

This kinetic regime is clearly very different from traditional chemical kinetics.

In effect, one has here a population of active elements (enzyme molecules) which

undergo cyclic internal evolution and, while performing it, receive and release

various intermediate molecules that can influence the speed and the nature of

processes inside these elements. The intermediate regulatory molecules, released

by one enzyme, go to the others and thus convey information about the current

states of enzyme molecules. This can be viewed as communication between the

enzyme molecules. The entire communicating population undergoes collective

evolution. It was suggested that biochemical subsystems demonstrating this kind

of behaviour should be called molecular networks (Hess & Mikhailov, b,

).

The closest analogue to molecular networks is provided by neural nets. Here

one has a population of active cells (neurons) that communicate by sending and

receiving electrical signals through a set of synaptic connections (McCulloch &

Pitts, ). Other examples are microbiological populations with chemical cell-

to-cell signalling (Gerisch et al. ) and insect societies (Pasteels & Deneudourg,

). Related models are used to describe aspects of collective behaviour in

human societies (see Haken & Mikhailov, ).

The relationship tturn

( ttransit

( tmix

between characteristic times of an enzymic

reaction in a small spatial volume represents a necessary condition for coherent

Benno Hess

regime of a molecular network. To demonstrate that biochemical reactions in

small spatial volumes, such as cellular compartments, can indeed proceed in a

regime characterised by strong correlations between individual reaction events, a

theoretical study of an irreversible product-activated reaction including only

allosteric enzyme molecules has been performed (Hess & Mikhailov, ). When

the strength of allosteric regulation is increased and the system’s parameters are

properly chosen, this system goes from the state of molecular chaos to a highly

ordered periodic spiking regime. The spikes are produced by synchronous firing

of a few groups of enzyme molecules. Under spiking conditions, each enzyme

group collectively behaves as an excitable element (i.e. similar to a neuron) and the

groups form a closed functional loop along which the excitation can indefinitely

circulate. The transition from molecular chaos to coherent spiking can be

understood as emergence of a functional structure in the originally uniform

population.

Since the process of an enzymic reaction in this kinetic regime cannot be

described by classical rate equations, a microscopic stochastic theory is needed.

Here the evolution of the reaction system is described by a stochastic algorithm

(Hess & Mikhailov, ), specifying the probabilities of binding of a substrate or

a regulatory molecule to the enzyme, as well as the probability of the product

decay.

When the allosteric regulation is weak, only a random enzymic activity has been

found where the phases of each catalytic cycle of the individual enzyme are

uniformly distributed. This regime can thus be described as molecular chaos (see

Fig. a). If the strength of the allosteric regulation is increased, coherent periodic

spiking sets in (Fig. b). It is interesting to note that the population of enzyme

molecules breaks then into two synchronous groups with the phase shift about a

half of the cycle duration. Indeed, the product generation rate displays spiking

with a period about twice shorter than the duration of a single cycle. The numbers

of enzymes in these two groups and the heights of the spikes are fluctuating to a

certain extent and enzymes occasionally change from one group to another.

It is important to note that rapid microscopic spikes are principally different

from slow macroscopic rate oscillations and do not transform into them as the

reaction volume is increased. The rate oscillations are macroscopic. Their period

is determined not by the duration of catalytic cycles in individual enzymes, but by

reaction rates of the system. Moreover, no kinetic oscillations are possible for the

reaction scheme, which was used in the above microscopic analysis.

It should be added that spiking disappears and the stationary state with random

fluctuation is established whenever the traffic time, controlling the cooperativity

parameter, is increased. The traffic time is proportional to the reaction volume.

Even while the conditions of the model discussed above are still fulfilled, spiking

is replaced by molecular chaos for larger volumes. When the delays due to

different spatial locations of the enzymes are taken into account, this would further

wash out the coherence. Thus, spikes do not continuously transform into classical

rate oscillations. They belong to a different branch, characterised by the presence

of microscopic coherence in the reacting system.


time, ms100 200 300

30n

20

10

a

n

30

20

10

b

time, ms100 200 300

Fig. . Transition from molecular chaos to a coherent spiking regime of the enzymic

reaction. The generation rate of product molecules, i.e. the number n of enzymes releasing a

product molecule per time step, is shown as a function of time for two different allosteric

regulation strengths (a) a¯ (molecular chaos) and (b) a¯ (spiking). The volume of

linear size ± mm contains n¯ enzymes. A histogram of this pattern shows two

components (Hess & Mikhailov, ).

The robustness of spiking is remarkable and points out to the general

significance of this phenomenon, which can easily be extended to more complex

reaction networks. Such coherent regimes of chemical reactions in very small

spatial volumes may be essential for molecular physiological processes in the

living cells. The mathematical analogues of such phenomena are seen in the

ensembles of strong interacting classical nonlinear oscillators where synchronously

oscillating clusters are known to occur spontaneously (Golomb et al. ;

Mikhailov & Hess, ).

New experimental techniques for investigations of the chemical reaction

dynamics at the level of single molecules in intact cells as well as in isolated states

have recently been proposed for the analysis of RNA (Eigen & Rigler, ) and

myosin (Funatsu et al. ) as well as of lactate dehydrogenase (Xue & Yeung,

). They would allow testing of the microscopic molecular-network behaviour

under in situ conditions. Furthermore, a more detailed analysis of the observed hot

spots of the calcium excitation wave (Petersen, ) might already show that here

the spiking phenomenon is involved.

Benno Hess

.

The fast development of research in the fields reviewed above spreads into many

sections of molecular and cellular biology, some special perspectives of which

should be discussed here.

In the field of molecular mechanics, recently, a model of the proton channel-

stator machinery of the flagellae motor of bacteria was presented. It shows the

execution of limit cycle oscillation of the stator states as part of the coupling device

which links the energy transduction from the proton channel with the flagellae

rotor in a mechanism well based on the current functional and structural

knowledge of the system (Elston et al. ) and ready for further experimental

testing.

There are many oscillating phenomena in cellular biology, for which detailed

biochemical mechanisms are currently not at hand. Yet, a few reaction systems

should be mentioned, that illustrate the fast advances in this area. Based on the

observation of clock mutants of Drosophila melanogaster (Konopka & Benzer,

) and the functions of the ‘period’ (per) and ‘timeless’ (tim) proteins (Hardin

et al. ; Sehgal et al. ) in the complex feedback interactions controlling the

gene transcription process, a delicate biochemical model has been studied fitting

well the many experimental data (Goldbeter, , Leloup & Goldbeter, ).

Here, for the first time, the process of control of transcription and translation has

been found to display circadian time windows. It involves complex formation,

phosphorylation and dephosphorylation functions and the rates of nuclear entry

and exit of the per–tim complex. The model with its ten variables demonstrates

limit cycle oscillations in the amount of the per and tim mRNAs and the two

proteins. The robustness of the model, their parameters and periodic domain, also

in terms of displaying the light-induced entrainment and resetting of the clock

observed experimentally (Lee et al. ; Myers et al. ; Zeng et al. ) is

remarkable and suggests, that this approach is well applicable to the clock problem

in higher organisms (see also Vitaterna et al. ).

In general, neuronal systems exemplify the properties of biological excitable

media, although relatively few investigations on the initiation, propagation and the

pathways of dynamic excitatory waves are currently at hand. The spreading

cortical waves have been analysed by reaction–diffusion equations (Tuckwell,

; Winfree, d). When spreading waves in the retina representing a quasi

two-dimensional receptor system were discovered, their properties could be

analysed in more detail (see Martins-Ferreira & Oliveira Castro, ; Oliveira

Castro & Martins-Ferreira, ; Gorelova & Bures, ; Bures et al. ).

Recently, evoked spreading depression waves in isolated retinal layers of chicken

have directly been recorded by optical techniques. In these studies, circular wave

fronts and rotating spiral waves with a meandering tip but no epicycles, and wave

interactions have been found, as expected for excitable media. The propagation

rate was in the range of µ s−" and the rotation time about ± min (Dahlem &

Mu$ ller, ). The underlying mechanism of the process, involving release and

propagation of K+ ions and its metabolic intracellular restoration is not


understood, although a simple Huxley equation system was found to give an

estimate of the sizes of the leading centre and the critical mass (Bures et al. ).

The general significance and origin of spreading depression waves has been

discussed in terms of possible pathogenic processes leading to a general, perhaps

irreversible distortion of the multicellular coordination of neuronal functions.

A review of the oscillatory discharge of individual neurons in the visual cortex

and the corpus geniculatum laterale (Ghose & Freeman, ; Neuenschwander

& Singer, ) or other nuclei of the brain and of the stimulus-dependent

transient oscillatory synchronisation of neuronal populations in the visual cortex

(Engel et al. ) is outside the scope of this discussion. However, these studies

shed some light on possible mechanistic views and the intricate problem of long-

and short-term oscillatory states as carrier signals for temporal coding in brain or

other biological systems. Indeed, whether intra- or intercellular, the general role

of pulsatile signalling in long-range transduction functions is bound to be on the

agenda for future research (Golomb et al. ; Singer, ). Here, the question

of the efficiency of periodic signals vs. random or chaotic signalling – not only in

receptor desensitisation – is of interest (Meyer & Stryer, ; Li & Goldbeter,

). Furthermore, the function of the well established temporal and spatial

coexistence of periodic, aperiodic and chaotic state for patterning and boundary

domains is to be explored.

Finally, the principal difference between chemical and biological systems

should clearly be kept in mind. Whereas chemical systems lack intrinsic properties

necessary to exhibit stable specific positional information of reacting entities such

as spatial position markers, living cells and their nets respond to multiple spatial

gradients and specific nearest neighbour interactions by activating in due time

within the circadian and cell cycle limits gene families for memorizing spatial

position markers, whenever competence in morphogenesis or in multicellular

evolution is required.

.

I gratefully acknowledge the critical reading of this manuscript by my long time

colleagues Albert Goldbeter, Alexander Mikhailov and Stefan Mu$ ller, and also I

would like to thank Florian Siegert for allowing me to use so far unpublished

material and to reproduce Figs a, b.

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