Oscillations, Вistability and Waves in Вiochemical and Cellular Systems

Symposium organized by Albert Goldbeter, June 14 – 20,1997



Edward C. Сох (University of Princeton), Jacques Demongeot (Université J.-Fournier, Grenoble), Rémi Dumollard (Station zoologique de Villefranche-sur-Mer),Geneviève Dupont (Université Libre de Bruxelles (ULB)), Albert Goldbeter (ULB), Guidi M. Gianluca (Université d’Urbine et ULB), José Halloу (ULB), Jean-François Hervagault  (Université de Compiègne), Benno Hess (Max-Planck Institute, Heidelberg), Joel Keizer (University of California, Davis), Michel Laurent (Université de Paris-Sud), Jean-Pierre Mazat (Université de Bordeaux-II), Stefan Müller (Otto-van-Guericke-Universitt, Magdeburg), Vidyanand Nanjundiah (Indian Institute of Science, Bangalore), Bela (Тechnical University of Budapest ), Obeyesekere Mandri N. (University of Texas, Houston), Olsen Lars Folke Novak (Odense University, Danemark), Jean-Pierre Ozil (Institut national de la recherche agronomique (INRA), Jouy-en Josas), Christian Sardet (Station zoologique de Villefranche-sur-Mer), Lee A. Segel (The Weizmann Institute of Science, Rehovot, Israel), James Sneyd (University of Michigan, Ani Arbor), John J. Tyson (Virginia Рolytechnic Institute and State University, Blacksburg), Cornelis Weijer (University of Dundee, Great Britain). 



The study of the molecular mechanism of biological rhythms has much to gain from a combined approach based on experimental observations nid on the analysis of theoretical models built according to the results of such experiments. Among the different rhythms known in biology, those which originate at the cellular level from various modes of regulation are particularly amenable to such an approach сombining experimental and theoretical aspects, because the mole­cular bases of rhythmic behaviour are then known in sufficient detail to allow the construction of biologically relevant models. Such models allow us to explore the precise conditions for the occurrence of instabilities which lead to oscillations and to the related phenomena of bistability or wave propagation.

The first goal of this workshop, which gathered 22 partici­pants, was to address the mechanism of the best-known examples of biochemical oscillations at the cellular level :

  1. Calcium oscillations : The study of this phenomenon has considerably developed over the last ten years. These oscillatiоns represent one of the most widespread periodic pheno­mena at the cellular level. The mechanism of calcium oscilla­tions was examined as well as their functions, рarticularlу the role they play in the development of the egg after fertilization.
  2. Glycolytic oscillatioпs : These oscillations, whose experi­mental study has recently made significant progress, represent for long the prototype of oscillatory behaviour resulting from enzymatic regulation.
  3. Oscillations of cyclic AMP : Periodic signals of cycliс AMР govern aggregation of the slime mould Dictyostelium discoideuт after starvation. In addition to their role in cell to сеll communication, these oscillations represent a prototype for pulsatile hormone signalling.

Other biochemical oscillators were considered, including the peroxydase reaction and the enzymatic cascade that drive the eukaryotic cell cycle. Particular emphasis was given to the mechanism of periodic activation of the cyclin-dependent kinases in somatic cells and yeast.

A second goal of the workshop was to highlight, for each of these systems, the close link between oscillations and wave propagation at the intracellular or extracellular level. The ехреrimental and theoretical study of calcium waves has rapidly developed in the last five years. Such waves were recently demonstrated experimentally in glycolyzing yeast extracts. Waves of cyclic AMP in the course of aggregation of Diclyostelium amoebae remain a prototype for spatiotemporal organization in cell biology ; further insight into the origin of these waves has recently been gained from theoretical studies.

Finally, a third goal of the workshop was to examine the phe­nomenon of bistability in biochemical systems. Bistability, i.e. the coexistence between two stable steady states in the same experimental conditions or for a given set of parameter values, is also closely related to the occurrence of oscillations. Different biochemical examples of bistability, known experimentally or predicted by theoretical models, were examined. Included was the recent analysis of conformational transitions of prion proteins in terms of bistability, and a discussion of dynamics invol­ving multiple steady states in immunology.


The abstracts of the presentations are given below, in the order followed during the workshop.

Vidyanand Nanjundiab : Cyclic AMP oscillations in Dictyostelium : theory and observations

The social amoeba Dictyostelium discoideum makes the tran­sition from unicellular to multicellular lifе via oscillatory aggre­gative movements. These movements are in response to the periodic synthesis, release and relay by individual cells of the chemoattractant cyclic AMP. Thanks to a combination of elegant modelling and experimentation, cAMP osсillаtiоns in Dictyostelium have become a paradigm for the role of temporal periodicity and spatial wave propagation in developmental biolоgy. But when observations made on synchronized suspen­sions are exarnined in dеtail, certain features that may or may not fall outside the scope of existing models call out for atten­tion. Among them are: (a) the occurence of apparently cAMP­-independent oscillations; (b) the question of whether an extra-cellular feedback step forms an essential component of the oscillatory loop ; (c) the string temperature-dependence of the period of oscillation ; (d) the secular increase in oscillation fre­quency as a function of developmental age ; and (e) the enor­mous range of cell densities over which oscillations can be observed.

Points (c), (d) and (e) might be taken care of by tinkering with existing models but (a) and (b) are troublesome. A purely intracellular calcium oscillator that can be coupled facultatively to the cAMP oscillator offers a possible solution.

Cornelis, J. Weljer : Control of co-ordinated cell move­ment during morphogenesis in Dictyostelium

Cell movement plays an important role in morphogenetic processes such as gastrulation, neurulation and formation of the brain. Its co-ordination most likely requires long range signaling, however little is known about the signals and mechanisms involved. We study the control of differential сеll movement in an evolutionary simple organism, the cellular slime mould Diclyostelium discoideum which stands on the threshold between unicellular and multicellular lifе. At the vegetative stage the cells live as solitary amoebae in the soil, which feed on bacteria and multiply by binary fission. However under starvation conditions a muldcellular developmental cycle is induced in which up to 100,000 cells aggregate chemotactically towards an aggregation centre to form a slug which then can move to a new habitat where it forms a fruiting body, consis­ting of a stalk supporting a spore head. The spores germinate and the cycle starts again.

During aggregation the cells communicate over large dis­tances via propagating waves of the chemotatractant cyclic AMP. These waves are initiated in the aggregation centre and propagate outwards as concentric or spiral waves, instructing the cells to move towards the aggregation centre. We have developed digital image processing methods to analyse the dynamics of the wave propagation (cell-cell communication) process in vivo. Novel image processing methods allow us to track individual transformant cells expressing the green fluo­rescent protein GFP as a cell type specific reporter gene. We analyse changes in speed and direction of cell movement as well as changes in cell shape at the single сеll level in a multi-сellular organism. We have now shown that the same principles that govern aggregation, i.e. spiral wave propagation and chemotactic cell movement, also control the morphogenesis of the later developmental stages. The rounds are characterised by multi-armed spirals while in slugs the chemotactic signal propagates as a three dimensional spiral wave in the front and transforms into planar waves in the back.

This transformation in wave shape results from differences in the excitablity of the prestalk and prespore cell types located in the front and the back of the slug respectively. These cells types arisе first in random positions in the mound and then sort out chemotactically to form an axial pattern in the slug. This cell sorting then feeds back onto the wave propagation system. We perform detailed mode calculations to understand how such complex wave forms can arise in three dimensional excitable media and how the cell movement feeds back on the pattern of wave propagation. This has given us some insight in the basic cellular principles of multi-cellular cell sorting, i. e. сеll type specific differences in excitability as well as differences in chernotactic responsiveness. We are now analysing the consequences of perturbations of the signalling mechanism on mor­phogenesis via the study wave propagation and cell movement in cAMP receptor, G protein, and chemotaxis and motility mutants.

José Halloy : Desynchronization of cells on the developmental path triggers the formation of spiral waves of CAMP in Dictyostelium

The aggregation of Dictyosteliuт discoideum amoebae after starvation provides one of the best examples of intercellular periodic signalling governing spatiotemporal pattern formation at the supracellular level. This transition from a unicellular to a multicellular stage of the life cycle occurs by a chemotactic res­ponse to cyclic AMP (cAMP) signals emitted by aggregation centers. In the first part, we present a model including the effect of both Gs and Gi proteins, desensitization of the receptor by a reversible phosphorylation and autocatalysis of the cAMP syn­thesis. The incorporation of G proteins allows to account for the response and adaptation to increasing concentration of suc­cessive cAMP stimuli when the autocatalysis of cAMP is supres­sed as in the exprimental observations. The model suggests that a mecanism slowing down the GTPase activity of G proteins may be involved. One striking effect of G proteins is to extend the sensitivity of the system to low concentration of cAMP. G proteins can enhance by two orders of magnitude the ligand concentration domain which gives a cellular response. Inorder to obtain oscillations, it is necessary to inсludе some non-linea­rity either at the level of the receptor, or at the level of the соupling between G proteins and adenylate cyсlasе. The model presents oscillations in the concentration of cAMP, the fraction of phosphorylated receptor and the fraction of active Gi and Gs proteins.

In the second part, we investigate a physiologically plausible mechanism for the spontaneous formation of concentric (tar­get) and spiral waves of cAMP in D. discoideuт. Whilе it is rela­tively easy to account for the formation of target waves of cAMP in the course of aggregation after starvation, the origin of spiral waves remains obscure. The scenario relies on the develop­mental path associated with the continuous changes in the acti­vity of enzymes such as adenylate cyclase and phosphodieste­rase observed during the hours that follow starvation. These changes bring the cells successively from a nonexcitable state to an excitable state in which they relay suprathreshold cAMP pulses, and then to autonomous oscillations of cAMP, before the system returns to an excitable state. By analyzing a model for cAMP signaling based on receptor desensitization, we show that the desynchronization of cells on this developmental path triggers the formation of fully developed spirals of cAMP. Developmental paths which do not correspond to the sequence of dynamic transitions no relay-relay-oscillations-relay are less able or fail to give risе to the formation of spirals.

Edward C. Cox : The evolution of spiral waves in the social amoeba Dictyostelium

Endogenous pattern formation in biological systems is wides­pread. During embryogenesis and development in most plants and animais, сеll division, growth, and morphogenesis are all intertwined. In the cellular slime molds, however, morphogenesis occurs after сеll division has been completed, and one can focus on a particular aspect of development, for example cell-cell signaling, without the complications usually associated with cell division. Starved amoebae of Didyostelium discoideum can be harvested and allowed to settle in a uniform layer on an agar surface. Under these conditions, the initial distribu­tion of amoebae is random. From this initial state, order arises as a few cells begin to send out pulses of the morphogen cyclic adenosine monophosphate (cAMP). cAMP acts to attract аddi- tional amoebae, and thus the pulsaltile cells act to found territo­ries of from a few dozen to many tens of thousands of cipe­rating, social, individuals. The aggregate then undergoes a series of morphogenetic movements that culminate in the for-ration of a motile slug which can crawl many hundreds of slug lengths before fit finally differentiates further into a fruiting body of dead stalk and ive spore cels. The spores germinate in the presence of a new food source and the life cycle begins anew. Starving amoebae in a thin layer on an agar surface have ail of the properties of an excitable medium : when a few indi­vidual cells cross a threshold in cAMP concentration, they begin a self sustaining oscillation which produces a wave of cAMP that spreads outward from the excited cell. Cells in the surround respond by moving against the direction of cAMP, producing characteristic circular and spiral waves as territories begin to form.

Very little is known about how spirals and circles emerge from the initial conditions. Here we show that the level of an inhibitor of a key enzyme in the cAMP synthesis pathway, the inhibitor of cAMP phosphodiesterase, has a major influence on the kinds of patterns observed. We find that by regulating the inhibitor level in a stochastic way in either space or time, the evolution of target and spiral patterns can be plausibly mode­led, and that deletions of the inhibitor gene change the system in such a way that spirals rarely form and propagate. (Work done with Eirikur Pallson)

Stefan C. Мüller : Traveling waves in a уеast extract and in cultures of Dictyostelium discoideum

Biological self-organization was investigated in a yeast extra­ct and in cultures of the slime mold Dictyostelium discoideum. In both systems traveling reaction-diffusion waves occur in res­ponse to oscillatory reactions. Glycolytic degradation of sugar in a yeast extract leads to the spontaneous formation of NADH and proton waves. Manipulation of the adenine nucleotide pool by addition of purified plasma membrane ATPase favors the formation of both reaction-diffusion waves aid phase waves, indicating that the energy charge has an important impact for the dynamics of glycolytic excitation waves. When affecting the lower part of glycolyses by pyruvate addition the frequency of wave generation was increased with concomitant formation of rotating NADH and proton spirals.

During morphogenesis of Dictyostelium discoideum circular and spiral shaped aggregation patterns of motile amoeboid cells form inresponse to traveling cAMP waves. Velocity analysis of the cell movements reveals that the cAMP waves guide the cells towards the site of wave initiation along optimized trajectories. The minimization of aggregation paths is based on a mecha­nism exploiting general properties of excitation waves. Resulting aggregation territories are reminescent of Voronoi dia­gramms. Thus, the cAMP waves mediate the spatio-temporal cell-cell communication that enables the formation of aggrega­tion centers and generation of fruiting bodies. In both systems the dynamics of excitation waves share great similarity with those patterns, known from the chemical Beloousov- Zhabotinsky reaction, pointing toward a general mechanism. (Work done with Thomas Mair and Oliver Steinbock)

Lars Folke Olfen : Oscillations in pеroxidasе-catalyzеd reactions and their poteпtial fuпction in vivo

The peroxidase-oxidase (PO) reaction is an oscillating bio­chemical reaction involving a single enzyme. It entails the oxi­dation of an organic substrate (e.g. Nicotinamide Adenine Dinucleotide Hydride (NADH) by molecular oxygen, catalyzed by peroxidase. In the presence of certain phenols or aromatic amines and methylene blue the reaction exhibits sustained per­iodic oscillations, quasiperiodic oscillations and chaos when NADH and oxygen are continuously supplied to the reaction mixture. As for chaotic dynamics, two different paths to chaos have been identified so far : a period-doubling route, and a per­iod-adding route. In the former path, which dominates at pH around 5, a change in one parameter, e.g. the flow rate of NADH, leads to successive multiplications by two of the period length at certain transition points. After the reaction has under­gone a number of such period-doublings chaotic behaviour sets in. In the period-adding path, which dominates around pH 6, one encounters transitions from simple periodic oscillations of large amplitude through oscillations consisting of one large and one small amplitude to oscillations with one large and two small amplitudes, etc. Hence, at each transition one more small amplitude oscillation is added to the periodic pattern. Beyond a region consisting of one large and four small amplitude oscil­lations chaotic behaviour sets in. The experimental observations of periodic oscillаtiоns and chaos can be simulated by a mathematical model of the reaction. The model is based on approximately 15 known second order reaction steps with experimentally determined rate constants.

The PO reaction occurs in a number of contexts in intact organisms, and many different peroxidases from different sources are known to catalyze the reaction. Furthermore, the aromatic compounds which trigger oscillatory behaviour are abundant in many cells. Therefore, it is tempting to conclude that the oscillatory dynamics does occur naturally, although this has never been verified experimentally. However, in vitro studies of crude horseradish extracts have demonstrated oscillatory dynamics thus lending support to the idea that the oscillatiоns have a physiological function.

Benno Hess : Molecular and cellular foundations of space and time patterns in biology

This special evening lecture dealt with the molecular and ce!­lular foundations of dynamic time and space pattern in biology, the functions of which were discussed elsewhere in the workshop. The identification of these dynamic regimes helps to understand complex behavior in biological systems, such as reaction overshoot or undershoot phenomena, excitability, limit cycle oscillations, quasiperiodic and chaotic domains, and control of oscillations by periodic forcing. It is interesting to note that with increasing complexity of the biochemical networks a reduction of the large number of rate equations for the detailed proceses can be achieved ; this leads to a description of the decisive steps in a few equations that dlsplay the overall dynamic features. The period of the oscillatory regimes thus described covers the range between a few 10 ms up to a 24 hours scale.

The properties of spatial waves of biochemical messenger results from the autocatalytic elements in the underlying bio­chemical reaction network, which trigger a spatial propagation from a local source throughout a spatial territory of a given size. Such an excitability is best analysed in case of the chemical model of the Zhabotinsky reaction and in the case of the aggre­gation pattern of the slime mold Dictyostelium dfscofdeum dis­cussed by others in this workshop. Proper reaction-diffusion equations describing the linear or curved spatial propagation process yield for both of these systems a linear relationship, the eikonal equation, linking the propagation velocity aid the cur­vature of a travelling messenger wave. This equation was experimentally tested and applied to understand spiral pattern for­mation also in other systems like the myocardial excitation or calcium waves. An important result of these studies is the pre­diction of the minimal radius below which the propagation of circular or spiral waves will flot take place. This minimal radius presents a first global size parameter in biology for understan­ding the physical sources of intercellular and intracellular orga­nization.

A discussion of new theoretical results pertaining to the microscopic level pointed to the occurrence of coherent spi-king in self-organizing molecular networks. String diffusional mixing and short delivery times typical for micrometer and sub-micrometer reaction volumes in intracellular vesicles lead to a situation where the turnover times of individual enzyme molecules become the largest characteristic time of chemical and biochemical kinetics. Under these conditions, small popu­lations of cross-regulating identical allosteric enzymes form molecular networks that exhibit varions forms of self organized, coherent collective dynamics, which microscopically display spiking time patterns in the order of their turnover time. This suggests that in the future it will be necessary to consider time patterns not only within the macroscopic, but also within the microscopic time and space scales.

Jean François Hervagault : Bistability and irreversible transitions in biochemical systems : theoretical and expe­rimental aspects

Substrate cycles are topological enzyme organizations ubi­quitously distributed throughout metabolic pathways. The more familiar feature related to bistability is the emergence of reversible hysteretic transitions between two functionally different branches of steady states in response to a change in a control parameter value : these abrupt transitions occur for two different threshold values of the parameter considered.

Under realistic conditions, these (non self releasing) triggers may loose their ability to switch reversiby from one steady state to the other. If one of the limit points either disappears or is no more physically accessible, then, a transition between the two branches of steady states is only possible one time and from that moment on, the system is unable to trace back its pre­vious history even when the contraints are released. The occurrence of these so-called irreversible transitions (IT) is demonstrated theoretically through the graphical analysis of a generic model binary substrate cycle. Two experimental systems are described and in which IT have been observed. One of the latter models is also able to predict non-connected branches of alternative steady state (BLS) where transitions between the two stable branches of a bistable are physically impossible. The case of the Са2+/ calmodulin-dependent protein kina­se II (CaMKII) is also briefly described. The рlаusiblе roles of IT and BLS are discussed in the frame of metabolic and signalling regulations.

Gianluca M. Guidi : Bistability in the isocitrate dehydrogenase reaction : a theoretical and experimental study

Experiments show that the enzyme NADP-isocitrate dehy­drogenase exhibits, under certain conditions, activation by its product NADPH. Moreover, the substrate NADP+ competes with NADPH for the activator site, but does not activate the enzyme, and NADPH competes with NADP+ for the catalytic site when it is present in concentrations much higher than those which produсе the activation phenomenon. The existence of a minimum and a maximum turnover number for the enzyme raises the possibility of non-linear dynamic behaviour. Experimental observations suggest (Carlier М.-F., 1976, Thèse de doctorat en sciences, Paris) that if it is coupled to another enzyme which transforms NADPH in NADP+, the production of NADPH could attain two stable states, one at low and the other at high NADPH concentration, depending on the initial concentration of NADPH.

We have conducted a theoretical and numerical analysis of a simplified model of this system. The results show the theoreti­cal possibility for the occurrence of multistability for parameter values that were derived from experiments (Carlier М.-F., Pantalon D., Вiochemistry Vol. 15, n° 8, 1961-6, 197() . We have further studied the occurrence of multistability as a function of the total concentration of NADP+ plus NADPH and of the concentration of isocitrate dehydrogenase. It appears that, depending on the values of these parameters, the system can either admit a single steady state or display bistability, in which case two stable steady states are separated by an unstable state. Beside the above-described behaviour, owing to its autoactivation property, NADP-isocitrate dehydrogenase reaction could display sustained oscillations when taking place in an open system. We show that with a constant input of isocitrate in the bi-еnzymаtiс system described above, different types of sustained oscillations can be observed. (Work done with Albert Goldbeter).

Jean Pierre Ozil : Role of Са2+ oscillations in the mammalian egg activatioп : experimeпtal approach

Most of the work on calcium waves and oscillations in cel­lular systems focus on their mechanisms while few studies are dedicated to reveal their functions. In the case of mammalian egg, the sperm causes egg activation by an increase in calcium excitability that provokes sustained calcium oscillations. While the transduction mechanism by which the sperm causes cal­cium oscillations is becoming clearer, it is interesting to have a better understanding about their effect at later developmental stages.

Cytosolic calcium oscillations in somatic cells are known to affect gene activation either through amplitude or frequency modulation. In case of mammalian egg development, the fun­damental stakes are to appreciate how oocytes use the fre­quency or amplitude mode of signalling and how the regime of calcium signal triggered by fertilization will co-ordinate the spatial and temporal organization of early events that lead to gene expression at later stages. In order to take part in this scientifiс challenge, we have developed a specific technology that allows us to modulate artifiсiallу the calcium signal both in amplitude and frequency during the early phases of the deve­lopmental processes.

By submitting mouse and rabbit eggs to various regimes of calcium stimulation with our technology, we found that the number of calcium pulses given during parthenogenetic activa­tion or alter fertilization have a clear effect on the ability of trea­ted eggs to develop up to preimplantation or to post imрlantation stage. Parthenogenetic rabbit eggs treated by 5 calcium pulses and transferred to pseudopregnant individuals does give an implantation rate at day 10 of pregnancy of 17 %, while those treated by 15 pulses have an implantation rate of 31%. Moreover, when fertilized egg are submitted to a series of calcium stimulation during the first cell cycle, the implantation rate at day 10 of pregnancy increased from 22 % to 47%.

Based on these results we get the picture of an enlarged func­tional role of pulsatile calcium signaling. The developmental process at later stages is very sensitive to limited calcium signal during the early phases post fertilization. Thus the dynamical control of the egg calcium oscillator permitted by our techno­logy will allow us to gain more knowledge on what is actually happening during egg activation leading up to gene expression at later stages and also how temporal events optimize this cel­lular process.

Christian Sardet : Calcium oscillations in eggs

Calcium oscillations originally discovered in mouse eggs are now known to be an essentiel activation signal in somatic cells and a wide variery of eggs of protostomes and deuterostomes (other mammaIs and ascidians) . We discuss what is known of the periodic calcium signals characterizing oocyte maturation and fertilization that seem to be restricted to eggs completing meiosis. More specifically, we have analysed the periodic cal­cium signals we have discovered in the eggs of ascidians (uro­chordates, tunicates) at fertilization.

In ascidians, the initial calcium wave triggered by the sperm is followed by two series of stereotyped periodic calcium waves. The localization of pacemakers and the pattern of oscil­lations strictly correspond to phases of reorganizations of the egg mediated by the cytoskeleton and controled by the level of factors that control the cell cycle. In addition we show that some of these calcium-driven events (cortical contraction) are important in the establishment of the polarity of the embryo. (Work done with Fabrice Roegiers, Rémi Dumollard, and Alex Mc Dougall).

Geпeviève Dupont: Oscillations of cytosolic Са2 modeling their role in egg activation and the effect of IP3 metabolism

Intracellular Са2+ oscillations have been observed in a many сеll types in response to external stimulation by a hormone or a neurotransmitter. Such repetitive Са2+ spikes have also been observed after fertilization of mammalian eggs. Several studies have shown that such repetitive Са2+ spikes орtimizе activation and further development of the egg. In the first part of the talk, we theoretically investigate the link between sperm-induced Са2+ oscilations and the resumption of the cell cycle. The model is based on the experimentaIly observed key role of Са2+ calmodulin kinase II and on the existence of two regulatory processes, differing by their time scales, between Са2+ increases and resumption of the cell cycle.

In the second part of the talk, we focus on the mechanism of Ca2+ oscillatiоns. It is well-known that inositol 1,4,5-trisphosphate (InsP3) triggers the release of Са2+ from internal stores. Many models, based on an autocatalytic regulation by which cytosolic Са2+ activates its own release from internal stores, have been proposed to account for Са2+ oscillations. Here, we quantitativelу incorporate InsР3 metabolism in such a model for Са2+ oscilla­tions. We were thus able to recover the observation made in Chinese Hamster ovary cells that surexpression of 3-kinase or 5­phosphatase, the two enzymes responsible for InsPз degradation, has very different effects on ATP-induсed Caz+ oscillations.

Jеаn-Pierre Mazat : A mathematical model of the Ca2+ oscillations and spikes generated by mitochondria

Са2+ is an ubiquitous intracellular messenger that controls an exceptional wide range of cellular processes. In order to асcount for the fact that a single messenger is able to mediate so many diverse cellular processes, it has been proposed that the message is encoded in the temporal and/or spatial delivery of the Са2+ signal. This hypothesis has found an increasing experi­mental support in the recent years. As a matter of fact, the intra­cellular Са2+ message takes the form of ‘blips’, ‘quarks’, ‘putts’, ‘sparks’, oscillаtiоns and waves, which, at least regarding the lat­ter, are based on the operation of Са2+-induсеd Са2+ release mechanisms mainlу attributed to the endoplasmic reticulum (ER) and its Са2+ release channels. However, recent evidences have underscored a key role of mitochondria in cell Са2+ signaling, and shown that these organelles modulate the Сa2+ signals primarily emitted from the ER. We have experimentally shown that mitochondria are able to generate Са2+ release spikes based on a purely mitochondrial Са2+-induced Са2+ release mechanism (mCICR). Since the operation of mCICR appears to be tightly dependent both on the amplitude and the frequency of the incoming triggering Са2+ signals, mitochondria appear as organelles capable to actively contribute to the AM- and FM- encoding of the intracellular Са2+ messages.

We present here a kinetic model of mCICR simulating the Са2+ osсillatiоns and spikes generated by mitochondria. The core of the model is based on (i) the gating of the mitochondrial permeability transition pore (РТР) by matrix pH, and (ii) the partial inhibition of the respiratory rate consecutive to РТР opening. Some predictions of the model are presented, particularlу regarding the amplitude-and frequency-dependence of mCICR triggering by extramitochondrial Са2+. We discuss the relevance of mCICR to the physiology of the spath-temporal encoding of Са2+ signals. (Work done with François Ichas aid Vitalii Selivanov)

Joel Keizer : Four modes for transmissioп of calcium waves in living cells

Using a modular approach, in which kinetic models of various mechanisms of calcium handling in cells are fine tuned to in vivo and in vitro measurements before combining them into whole-cell models, we explore modes of transmission of calcium waves in mature and immature frog eggs and in car­diac myocytes. Two modes of transmission are found in imma­ture eggs, where the inositol 1,4,5-trisphosphate receptor (IP3R) controls release of calcium from the endoplasmic reti­culum (ER) . The first mode corresponds to an excitable phy­siological state of the cytорlast and results in solitary « trigger » waves that can appear as circular or spiral waves in 2 dimen­sions. In this mode the wave speed is proportional to the squa­re root of the diffusion constant of calcium. The second mode occurs when the state of the cytoplasm is oscillatory and becau­se of the small size of the buffered diffusion constant for cal­cium, the wave speed is only weakly dependent on diffusion. A different situation is found in the mature frog egg, where the sperm-induced fertilization wave is a propagating front, suggesting that the cytosol is bistable and for which the speed is again proportional to the square root of the diffusion constant. Finally for cardiac myocytes, in which calcium release from the sacroplasmic reticulum (SR) is dominated by clusters of ryano­dine receptors spaced at regular intervals, the cytosol is exci­table, which leads to the random production of calcium « sparks » that can merge into waves with a speed proportional to the diffusion constant.

James Sneyd : Intercellular calcium waves : mechanisms, models and tests

Intercellular calcium waves are seen in very many cell types, and are a ubiquitous method of intercellular communication and coordination. In response to a mechanical stimulus, glial cell cultures exhibit an intercellular calcium wave, followed by asynchronous intracellular calcium oscillations. We have previously presented a model for these mechanically-stimulated waves. The major assumption of the model was that the inter-cellular wave is caused by the passive diffusion of inositol (1,4,5)-trisphosphate (IP3) from the stimulated cell. Recently, two of us (Andreja Strahonja aid Michael Sanderson) have per­formed a series of experiments designed to test this model; here, we compare these results to model predictions.

According to the model, the cells that display intracellular oscil­lations after the wave occur in a band a certain distance from the stimulated cell. The distance of this band from the stimula­ted cell can be modified by imposing a background concentration of IP3. Experimental results соnfirm these model pre­dictions. Further, the model predicts that different oscillatory patterns occur at specified distances from the stimulated сеll, predictions which are also confirmed by experiment. We conclude that mechanically-stimulated intercellular calcium waves are the result of passive diffusion of IP3 through gap junctions. (Work done in collaboration with Matt Wilkins, Andreja Strahonja, and Michael Sanderson).

John J. Tysoп and Bela Novak : Evolution of the eukaryotic cell cyсle

Growth and division of all modern organisms (except bac­teria) are coordinated by a complex molecular machine based on « cyclin-dependent kinases » (CDKs) : a family of protein kinases whose members function in combination with a regu­latory subunit from a different protein family, called cyclins. CDK activity depends on (1) cyclin availability (i.e., cyclin syn­thesis and degradation), (2) phosphorylation state of the kina­se subunit (carried out by auxiliary protein kinases and phos­phatases), and (3) binding to stoichiometric inhibitors (whose abundance is, in turn, controlled by synthesis, degradation, and phosphorylation) . In a mammalian cell there are, at last count, about 7 CDKs, 12 cyclins, 10 inhibitors, and an array of auxi­liary kinases and phosphatases. The full network of biochemi­cal interactions, even for the simplest cells (unicellular yeasts), is daunting in its complexity.

Undoubtedly the cell-cycle control mechanism for the ear­liest eukaryotic cells, when they first evolved from bacterial ancestors, must have been considerably simpler. We suppo­se that it consisted of a single CDK, a single type of cyclin, and only one mode of regulation. This primitive system must have been able to control the basic events of chromosome replication and partitioning that characterize the eukaryotic cell cycle in contrast to the much different steps of bacterial division. These basic events are (1) initiation of DNA replica­tion at multiple origins on multiple chromosomes, once and only once per cycle, and (2) accurate segregation of sister chromatids at anaphase, so that sister cells get one and only one copy of each chromosome after division. These two major transitions, which we call « Start » and « Finish », are controlled by « checkpoints » : halting points in the cell cycle where the сеll makes sure (1) that it has grown large enough to warrant another round of DNA replication and division, and (2) that all chromosomes are aligned on the metaphase plate so that sister chromatid separation can proceed properly.

Our model of the primitive chromosome-cycle control mechanism is based on a suggestion by Kir Nasrnyth that eukaryotic cells can persist in two « self maintaining » states : an interphase state with low CDK activiry, and a replicating state with high CDK activity. These states are further characterized by the activity of the « Anaphase Promoting Complex » (АPC), which is a multi-protein complex that determines the stabili­ty of the cyclin subunit of CDK/cyclin dimers. When the APC is « on », cyclin subunits are rapidly degraded and CDK subu­nits are rendered inactive. When the АPC is « off », cyсlin subu­nits accumulate, combine with CDK monomers, and create active dimers. The two alternative states are self maintaining, by Nasmyth’s hypothesis, because, not only does APC inacti­vate CDK by destroying its cyclin partner, but also CDК inactivates APC (presumably) by phosphorylating some crucial component.

We presented a mathematical model of this simple mechanism, studied its properties by analytical and numerical tech­niques, and proved that it can indeed provide all the necessary controls for a primitive eukaryotic mode of replication. We showed that the cell cycle control system is not an autono­mous oscillator, as many have supposed, but a machine that flips between two stable steady states (the « checkpoints ») by a process of saddle-node bifurcations. The Start transition is driven by cell growth, and the Finish transition by chromo­some replication and alignment. Then we speculated on the evolutionary forces that might have expanded this primitive mechanism into the much more complex network in modem cells. We backed up our ideas with model calcula­tions, showing how each level of additional controls provi­ded new options for the proliferation of eukaryotic cells, without jeopardizing the integrity of the underlying logic of Start and Finish transitions. Our models were justified by соmparison to modern cells which exhibit different variations on the underlying theme, especially wild-type yeasts and lab mutants, and early division cycles of fertilized eggs.

Mandri Obeyesekere : G1 phase kinetics of the cell cyсle : тultiрle attractors and hysteresis

A large body of work has focused on understanding the regu­lation of the G1/S transition in eukaryotes. Study of сеll cycle progression has lead to the elucidation of a variety of proteins which are required for appropriate entry into S-phase. These proteins include the G1 cyclins, D-type cyclins, cyclin E, and cyclin A, and their associated kinases, сdk4 and cdk2, which together form active complexes required for entry into S-phase. The retinoblastima tumor suppressor protein, RB, is the princi­pal known G1 substrate for active сdk/cyclin complexes. RB interacts with the E2F-famIly of transcription factors and sup­presses E2F-mediated transcription. E2F can promote the expression of genes involved in DNA synthesis and antry into S-phase.

A mathematical model integrating the roles of cyclin D, cdk4, cyclin E, cdk2, E2F, and RB in control of the G1 phase of the сеll cycle is presented. Experimental results described with murine embryo flbroblasts (MEFs), either Rb+/+ or Rb-/- , and with the RB-deficient osteosarcoma cell line, Saos-2, served as the basis for the formulation of this mathematical model. A model employing the known interactions of these six proteins does NOT reproduce the experimental observations described in the MEFs. The appropriate modeling of G1 requires the inclusion of a sensing mechanism which adjusts the activity of cyclin E/сdk2 in response to both RB concentration and grow­th factors. Incorporation of this sensing mechanism into the model allows it to reproduce most of the experimental results observed in Saos-2 cells, Rb-/- MEFs, and Rb+/+MEFs. The model also makes specific predictions which have not been tested experimentally. This model suggests that there exist two confi­gurations of сеll behavior for cells with non-zero RB levels, a proliferative (cyclic) as well as a non-proliferative stage, with no changes in the parameters. i.e., two basins of attraction. It is shown that these multiple attractors arise due to the double pathways of phosphorylation of RB. Furthermore, hysteresis is observable with respect to growth factors, which is a parameter in this model. The sensitivity of the growth factormechanism as well as the suggested sensing mechanism on observing hysterisis is discussed.

Lee A. Segel : Multiple attractors in immunology : theory and experimeпt

Multiple attractors are a common feature in immune network models, which treat of the interaction of different cell-types, and of their secretents. « Large networks » attempt to deal with the overall behavior of the immune system. They are necessarily highly simplified, athough recent work has begun to approach realism. The relevance of the theoretical results is controversial, although certain laboratories conduct experiments that are influenced by the theory bу contrast, « small networks » treat specific situations where only a few cell types are thought to be involved. The influence on experiment is conse­quently greater. One « small network » example where the pre­sence of multiple attractors is particularly evident is a « reverse engineering » approach to « T cell vaccination » as a treatment for autoimmune disease. Here « normalcy », « vaccination » and « disease » are modeled as steady states of the governing dyna­mical system. The equations are selected to provide observed transitions that are obtained by perturbations to the steady state. In spite of the biological simplicity of the model, surprising phe­nomena are predicted, and have been found experimentally.

Michel Laurent : Вistabilitу and the species barrier in prion diseases : Stepping across the threshold or not

The infections agent of transmissible spongiform encephalo­pathies is thought to be a cellular protein, the prion protein, which undergoes, under some circumstances, a dramatic con­formational change leading to pathogenesis. The conversion between the normal and pathogenic isoforms corresponds to an autocatalytic mechanism and the metabolism of the prion protein exhibits switches between a normal, stable steady state and a pathogenic one. When the disease can be transmitted bet­ween two species, a primary infection from a heterologous donor has to be followed by two passages in the same host species so that the incubation period is stabilized. Sometimes, no pathogenic isoform of the prion protein is detected after the first passage, although corresponding brain extracts remain infectious.

The observation that three and only three passages are nee­ded in order to stabilize the strain strongly suggests that, during the course of the primary infection by the heterologous don­nor, an intermediary conformational species is formed. Within this assumption, a common mechanism involving only confor­mational changes of the prion protein can give an unifying interpretation of the problem of species barrier, lag charaсteris­tics and apparent lack of detection of the pathogenic isoform after the first passage in experiments dealing with interspecies transmission of prion diseases.



Vidya Nanjundiah – Cyclic AMP osсillаtiоns in Dictyostelium : theory and observations

Cornelis Weijer – The signals that control cell movement during Dictyostelium development

Edward C. Сох – Selection for spiral waves in the social amoebae Dictyostetium

José Halloy – Desynchronization of cells on the developmental path triggers the formation of spiral waves of cAMP in Dictyostetium

Stefan Müller – Traveling waves in yeast extracts and in cultures of Dictyostelium discoideum

Lars Folke Olsen – Oscillations in peroхidase-сatalyzed reactions and their potential function in vivo

Jean-François Hervagault – Bistability and irreversible transitions in biochemical systems : theoretical and experimental aspects

Gianluca L. Guidi – Bistability in the isocitrate dehydrogenase reaction : a theoreti­cal and experimental study

Benno Hess – Molecular and cellular foundations of time and space patterns in biology

Jean-Pierre Ozil – Role of Ca2+ oscillations in the activation of mammalian egg: experimental approach

Christian Sardet – Са2+ waves and oscillations in eggs

Geneviève Dupont – Oscillations of cytosolic Са2+ : modeling their role in egg activation and the effect of IP3 metabolism.

Jean-Pierre Mazat – A mathematical model of Са2+ oscillations and Са2+ spike generated by mitochondria

Joel Keizer – Four modes for transmission of Са2+ waves in living cells

James Sneyd – Intercellular Са2+ waves : mechanisms, models and tests

John J. Tyson and Bela Novak – Evolution of the eukaryotic cell cycle

Mandri Obeyesekere – G1 phase kinetics of the сеll cycle : multiple attractors and hysteresis

Lee A. Segel – Multiple attractors in immunology : theory and experiment

Michel Laurent – Bistability in prion diseases : interpretation of the problem of the species barrier

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