A novel clustering approach to bipartite investor-startup networks

We propose a novel similarity-based clustering approach to venture capital investors that takes as input the bipartite graph of funding interactions between investors and startups and returns clusterings of investors built upon 5 characteristic dimensions. We first validate that investors are clustered in a meaningful manner and present methods of visualizing cluster characteristics. We further analyze the temporal dynamics at the cluster level and observe a meaningful second-order evolution of the sectoral investment trends. Finally, and surprisingly, we report that clusters appear stable even when running the clustering algorithm with all but one of the 5 characteristic dimensions, for instance observing geography-focused clusters without taking into account the geographical dimension or sector-focused clusters without taking into account the sectoral dimension, suggesting the presence of significant underlying complex investment patterns.

Using natural language processing to find research topics in Living Machines conferences and their intersections with Bioinspiration & Biomimetics publications

The number of published scientific articles is increasing dramatically and makes it difficult to keep track of research topics. This is particularly difficult in interdisciplinary research areas where different communities from different disciplines are working together. It would be useful to develop methods to automate the detection of research topics in a research domain. Here we propose a natural language processing (NLP) based method to automatically detect topics in defined corpora. We start by automatically generating a global state of the art of Living Machines conferences. Our NLP-based method classifies all published papers into different clusters corresponding to the research topic published in these conferences. We perform the same study on all papers published in the journals Bioinspiration & Biomimetics and Soft Robotics. In total this analysis concerns 2099 articles. Next, we analyze the intersection between the research themes published in the conferences and the corpora of these two journals. We also examine the evolution of the number of papers per research theme which determines the research trends. Together, these analyses provide a snapshot of the current state of the field, help to highlight open questions, and provide insights into the future.

Automated optimization of multilevel models of collective behaviour: application to mixed society of animals and robots

Animal societies exhibit complex dynamics that require multi-level descriptions. They are difficult to model, as they encompass information at different levels of description, such as individual physiology, individual behaviour, group behaviour and features of the environment. The collective behaviour of a group of animals can be modelled as a dynamical system. Typically, models of behaviour are either macroscopic (differential equations of population dynamics) or microscopic (such as Markov chains, explicitly specifying the spatio-temporal state of each individual). These two kind of models offer distinct and complementary descriptions of the observed behaviour. Macroscopic models offer mean field description of the collective dynamics, where collective choices are considered as the stable steady states of a nonlinear system governed by control parameters leading to bifurcation diagrams. Microscopic models can be used to perform computer simulations or as building blocks for robot controllers, at the individual level, of the observed spatial behaviour of animals. Here, we present a methodology to translate a macroscopic model into different microscopic models. We automatically calibrate the microscopic models so that the resulting simulated collective dynamics fit the solutions of the reference macroscopic model for a set of parameter values corresponding to a bifurcation diagram leading to multiple steady states. We apply evolutionary algorithms to simultaneously optimize the parameters of the models at different levels of description. This methodology is applied, in simulation, to an experimentally validated shelter-selection problem solved by gregarious insects and robots. Our framework can be used for multi-level modelling of collective behaviour in animals and robots.

Tracing the origins of SARS-COV-2 in coronavirus phylogenies: a review

SARS-CoV-2 is a new human coronavirus (CoV), which emerged in China in late 2019 and is responsible for the global COVID-19 pandemic that caused more than 97 million infections and 2 million deaths in 12 months. Understanding the origin of this virus is an important issue, and it is necessary to determine the mechanisms of viral dissemination in order to contain future epidemics. Based on phylogenetic inferences, sequence analysis and structure-function relationships of coronavirus proteins, informed by the knowledge currently available on the virus, we discuss the different scenarios on the origin-natural or synthetic-of the virus. The data currently available are not sufficient to firmly assert whether SARS-CoV2 results from a zoonotic emergence or from an accidental escape of a laboratory strain. This question needs to be solved because it has important consequences on the risk/benefit balance of our interactions with ecosystems, on intensive breeding of wild and domestic animals, on some laboratory practices and on scientific policy and biosafety regulations. Regardless of COVID-19 origin, studying the evolution of the molecular mechanisms involved in the emergence of pandemic viruses is essential to develop therapeutic and vaccine strategies and to prevent future zoonoses. This article is a translation and update of a French article published in Médecine/Sciences, August/September 2020 (10.1051/medsci/2020123).

Supplementary Information

The online version of this article (10.1007/s10311-020-01151-1) contains supplementary material, which is available to authorized users.

[Tracing the origins of SARS-COV-2 in coronavirus phylogenies]

SARS-CoV-2 is a new human coronavirus (CoV), which emerged in People's Republic of China at the end of 2019 and is responsible for the global Covid-19 pandemic that caused more than 540 000 deaths in six months. Understanding the origin of this virus is an important issue and it is necessary to determine the mechanisms of its dissemination in order to be able to contain new epidemics. Based on phylogenetic inferences, sequence analysis and structure-function relationships of coronavirus proteins, informed by the knowledge currently available, we discuss the different scenarios evoked to account for the origin - natural or synthetic - of the virus. On the basis of currently available data, it is impossible to determine whether SARS-CoV-2 is the result of a natural zoonotic emergence or an accidental escape from experimental strains. Regardless of its origin, the study of the evolution of the molecular mechanisms involved in the emergence of this pandemic virus is essential to develop therapeutic and vaccine strategies.

Strategies to modulate zebrafish collective dynamics with a closed-loop biomimetic robotic system

The objective of this study is to integrate biomimetic robots into small groups of zebrafish and to modulate their collective behaviours. A possible approach is to have the robots behave like sheepdogs. In this case, the robots would behave like a different species than the fish and would present different relevant behaviours. In this study, we explore different strategies that use biomimetic zebrafish behaviours. In past work, we have shown that robots biomimicking zebrafish can be socially integrated into zebrafish groups. We have also shown that a fish-like robot can modulate the rotation choice of zebrafish groups in a circular set-up. Here, we further study the modulation capabilities of such robots in a more complex set-up. To do this, we exploit zebrafish social behaviours we identified in previous studies. We first modulate collective departure by replicating the leadership mechanisms with the robot in a set-up composed of two rooms connected by a corridor. Then, we test different behavioural strategies to drive the fish groups towards a predefined target room. To drive the biohybrid groups towards a predefined choice, they have to adopt some specific fish-like behaviours. The first strategy is based on a single robot using the initiation behaviour. In this case, the robot keeps trying to initiate a group transition towards the target room. The second strategy is based on two robots, one initiating and one staying in the target room as a social attractant. The third strategy is based on a single robot behaving like a zebrafish but staying in the target room as a social attractant. The fourth strategy uses two robots behaving like zebrafish but staying in the target room. We conclude that robots can modulate zebrafish group behaviour by adopting strategies based on existing fish behaviours. Under these conditions, robots enable the testing of hypotheses about the behaviours of fish.

Collective departures and leadership in zebrafish

In social animals, morphological and behavioural traits may give to some individuals a stronger influence on the collective decisions, even in groups assumed to be leaderless such as fish shoals. Here, we studied and characterized the leadership in collective movements of shoals of zebrafish Danio rerio by observing groups of 2, 3, 5, 7 and 10 zebrafish swimming in a two resting sites arena during one hour. We quantified the number of collective departures initiated by each fish and the number of attempts that they made. To do so, we developed an automated pipeline that analysed the individual trajectories generated by the tracking software. For all shoal sizes, the leadership was distributed among several individuals. However, it was equally shared among all the fish in some shoals while other groups showed a more asymmetrical sharing of the initiation of collective departures. To quantify this distribution, we computed the entropy associated with the time series of the identity of all initiators for each experiment and confirmed the presence of a continuum between a homogeneous and a heterogeneous distribution of the leadership. While some fish led more departures than others, an individual analysis showed that all fish had actually the same success rate to lead the shoal out of a resting site after an attempt. Thus, some individuals monopolized the leadership by attempting more often than others to exit a resting site. Finally, we highlight that the intra-group ranking of a fish for the initiative is correlated to its intra-group ranking for the average speed with mobile individuals more prone to lead the shoal. These results demonstrate that the collective behaviour of a shoal can be mainly driven by a subset of individuals even in the absence of higher influence of a fish on its congeneers.

Time dynamics and invariant subnetwork structures in the world cereals trade network

The development of industrial agriculture has enabled a sharp increase in food trade at the global scale. Worldwide trade underpins food security by distributing food surpluses to food deficient countries. The study of agricultural product flows can provide insights on the complex interactions between exporting and importing countries and the resulting network structures. Commercial partnerships between countries can be modelled using a complex network approach. Based on the detailed trade matrices from FAO covering the period from 1986 to 2013, we present an analysis of the world cereal trade in terms of weighted and directed networks. The network nodes are the countries and the links are the trades of agricultural products in mass. We reveal the changing topology and degree distribution of the world network during the studied period. We distinguish three entangled subnetwork structures when considering the temporal stability of the trades. The three subnetworks display distinct properties and a differential contribution in total trade. Trades of uninterrupted activity over the 28-year study period compose the backbone network which accounts for two thirds of all traded mass and is scale-free. Inversely, two thirds of the trades only have one or two consecutive years of activity and define the transient subnetwork which displays random growth and accounts for very little traded mass. The trades of intermediate duration display an exponential growth both in numbers and in traded mass and define the intermediate subnetwork. The topology of each subnetwork is a time invariant. The identification of invariant structures is a useful basis for developing prospective agri-food network modelling to assess their resilience to perturbations and shocks.

Robots mediating interactions between animals for interspecies collective behaviors

Self-organized collective behavior has been analyzed in diverse types of gregarious animals. Such collective intelligence emerges from the synergy between individuals, which behave at their own time and spatial scales and without global rules. Recently, robots have been developed to collaborate with animal groups in the pursuit of better understanding their decision-making processes. These biohybrid systems make cooperative relationships between artificial systems and animals possible, which can yield new capabilities in the resulting mixed group. However, robots are currently tailor-made to successfully engage with one animal species at a time. This limits the possibilities of introducing distinct species-dependent perceptual capabilities and types of behaviors in the same system. Here, we show that robots socially integrated into animal groups of honeybees and zebrafish, each one located in a different city, allowing these two species to interact. This interspecific information transfer is demonstrated by collective decisions that emerge between the two autonomous robotic systems and the two animal groups. The robots enable this biohybrid system to function at any distance and operates in water and air with multiple sensorimotor properties across species barriers and ecosystems. These results demonstrate the feasibility of generating and controlling behavioral patterns in biohybrid groups of multiple species. Such interspecies connections between diverse robotic systems and animal species may open the door for new forms of artificial collective intelligence, where the unrivaled perceptual capabilities of the animals and their brains can be used to enhance autonomous decision-making, which could find applications in selective "rewiring" of ecosystems.

Loose social organisation of AB strain zebrafish groups in a two-patch environment

We study the collective behaviour of zebrafish shoals of different numbers of individuals (1, 2, 3, 5, 7, 10 and 20 AB zebrafish Danio rerio) in a constraint environment composed of two identical square rooms connected by a corridor. This simple set-up is similar to a natural patchy environment. We track the positions and the identities of the fish and compute the metrics at the group and at the individual levels. First, we show that the number of fish affects the behaviour of each individual in a group, the cohesion of the groups, the preferential interactions and the transition dynamics between the two rooms. Second, during collective departures, we show that the rankings of exit correspond to the topological organisations of the fish prior to their collective departure. This spatial organisation appears in the group a few seconds before a collective departure. These results provide new evidences on the spatial organisation of the groups and the effect of the number of fish on individual and collective behaviours in a patchy environment.

How mimetic should a robotic fish be to socially integrate into zebrafish groups?

Biomimetic robots are promising tools in animal behavioural studies. If they are socially integrated in a group of animals, they can produce calibrated social stimuli to test the animal responses. However, the design of such social robots is challenging as it involves both a luring capability including appropriate robot behaviours, and the acceptation of the robots by the animals as social companions. Here, we investigate the integration of a biomimetic robot driven by biomimetic behavioural models into a group of zebrafish (Danio rerio). The robot behaviours are based on a stochastic model linking zebrafish visual perception to individual behaviour and calibrated experimentally to correspond to the behaviour of zebrafish. We show that our robot can be integrated into a group of zebrafish, mimic their behaviour and exhibit similar collective dynamics compared to fish-only groups. This study shows that an autonomous biomimetic robot was enhanced by a biomimetic behavioural model so that it can socially integrate into groups of fish.

Synchronization and collective swimming patterns in fish (Hemigrammus bleheri)

In this work, we address the case of red nose tetra fish Hemigrammus bleheri swimming in groups in a uniform flow, giving special attention to the basic interactions and cooperative swimming of a single pair of fish. We first bring evidence of synchronization of the two fish, where the swimming modes are dominated by 'out-phase' and 'in-phase' configurations. We show that the transition to this synchronization state is correlated with the swimming speed (i.e. the flow rate), and thus with the magnitude of the hydrodynamic pressure generated by the fish body during each swimming cycle. From a careful spatio-temporal analysis corresponding to those synchronized modes, we characterize the distances between the two individuals in a pair in the basic schooling pattern. We test the conclusions of the analysis of fish pairs with a second set of experiments using groups of three fish. By identifying the typical spatial configurations, we explain how the nearest neighbour interactions constitute the building blocks of collective fish swimming.

Simple phalanx pattern leads to energy saving in cohesive fish schooling

The question of how individuals in a population organize when living in groups arises for systems as different as a swarm of microorganisms or a flock of seagulls. The different patterns for moving collectively involve a wide spectrum of reasons, such as evading predators or optimizing food prospection. Also, the schooling pattern has often been associated with an advantage in terms of energy consumption. In this study, we use a popular aquarium fish, the red nose tetra fish, Hemigrammus bleheri, which is known to swim in highly cohesive groups, to analyze the schooling dynamics. In our experiments, fish swim in a shallow-water tunnel with controlled velocity, and stereoscopic video recordings are used to track the 3D positions of each individual in a school, as well as their tail-beating kinematics. Challenging the widespread idea of fish favoring a diamond pattern to swim more efficiently [Weihs D (1973) Nature 241:290-291], we observe that when fish are forced to swim fast-well above their free-swimming typical velocity, and hence in a situation where efficient swimming would be favored-the most frequent configuration is the "phalanx" or "soldier" formation, with all individuals swimming side by side. We explain this observation by considering the advantages of tail-beating synchronization between neighbors, which we have also characterized. Most importantly, we show that schooling is advantageous as compared with swimming alone from an energy-efficiency perspective.

Collective behaviour in vertebrates: a sensory perspective

Collective behaviour models can predict behaviours of schools, flocks, and herds. However, in many cases, these models make biologically unrealistic assumptions in terms of the sensory capabilities of the organism, which are applied across different species. We explored how sensitive collective behaviour models are to these sensory assumptions. Specifically, we used parameters reflecting the visual coverage and visual acuity that determine the spatial range over which an individual can detect and interact with conspecifics. Using metric and topological collective behaviour models, we compared the classic sensory parameters, typically used to model birds and fish, with a set of realistic sensory parameters obtained through physiological measurements. Compared with the classic sensory assumptions, the realistic assumptions increased perceptual ranges, which led to fewer groups and larger group sizes in all species, and higher polarity values and slightly shorter neighbour distances in the fish species. Overall, classic visual sensory assumptions are not representative of many species showing collective behaviour and constrain unrealistically their perceptual ranges. More importantly, caution must be exercised when empirically testing the predictions of these models in terms of choosing the model species, making realistic predictions, and interpreting the results.

Strain differences in the collective behaviour of zebrafish (Danio rerio) in heterogeneous environment

Recent studies show differences in individual motion and shoaling tendency between strains of the same species. Here, we analyse collective motion and response to visual stimuli in two morphologically different strains (TL and AB) of zebrafish. For both strains, we observed 10 groups of 5 and 10 zebrafish swimming freely in a large experimental tank with two identical landmarks (cylinders or discs) for 1 h. We tracked the positions of the fish by an automated tracking method and compute several metrics at the group level. First, the probability of the presence shows that both strains avoid free space and are more likely to swim in the vicinity of the walls of the tank and the landmarks. Second, the analysis of landmarks occupancy shows that AB zebrafish are more present in their vicinity than TL ones and that both strains regularly transit from one to the other one with no preference on the long duration. Finally, TL zebrafish show a higher cohesion than AB zebrafish. Thus, environmental heterogeneity and duration of the trials allow to reveal individual and collective behavioural variabilities among different strains of zebrafish. These results provide a new insight into the need to take into account individual variability of zebrafish strains for studying collective behaviour.

Strain differences in the collective behaviour of zebrafish (Danio rerio) in heterogeneous environment

Recent studies show differences in individual motion and shoaling tendency between strains of the same species. Here, we analyse collective motion and response to visual stimuli in two morphologically different strains (TL and AB) of zebrafish. For both strains, we observed 10 groups of 5 and 10 zebrafish swimming freely in a large experimental tank with two identical landmarks (cylinders or discs) for 1 h. We tracked the positions of the fish by an automated tracking method and compute several metrics at the group level. First, the probability of the presence shows that both strains avoid free space and are more likely to swim in the vicinity of the walls of the tank and the landmarks. Second, the analysis of landmarks occupancy shows that AB zebrafish are more present in their vicinity than TL ones and that both strains regularly transit from one to the other one with no preference on the long duration. Finally, TL zebrafish show a higher cohesion than AB zebrafish. Thus, environmental heterogeneity and duration of the trials allow to reveal individual and collective behavioural variabilities among different strains of zebrafish. These results provide a new insight into the need to take into account individual variability of zebrafish strains for studying collective behaviour.

Transition between segregation and aggregation: the role of environmental constraints

Interactions between sub-groups (species, strains) have been reported in many species among many taxae. We propose a generic model based on earlier experiments accounting for both conspecific (or between individuals of the same strains) and heterospecific (or between strains) interactions. The model predicts different collective behaviours without any change of individuals’ algorithm as some key generic parameters such as the carrying capacity, the number of individuals involved and the strength of inter-attraction between sub-groups are varied. A key result is the possibility for sub-groups to segregate between patches and for transition between different patterns, even in absence of active agonistic behaviour. The model can be viewed as a network of feedbacks that is independent of the signals or cues involved in mixed groups interactions. Its predictions are therefore applicable to a wide spectrum of situations including social insects (inter castes interaction) and provides insights on possible mechanisms that can be at work.

A stochastic vision-based model inspired by zebrafish collective behaviour in heterogeneous environments

Collective motion is one of the most ubiquitous behaviours displayed by social organisms and has led to the development of numerous models. Recent advances in the understanding of sensory system and information processing by animals impels one to revise classical assumptions made in decisional algorithms. In this context, we present a model describing the three-dimensional visual sensory system of fish that adjust their trajectory according to their perception field. Furthermore, we introduce a stochastic process based on a probability distribution function to move in targeted directions rather than on a summation of influential vectors as is classically assumed by most models. In parallel, we present experimental results of zebrafish (alone or in group of 10) swimming in both homogeneous and heterogeneous environments. We use these experimental data to set the parameter values of our model and show that this perception-based approach can simulate the collective motion of species showing cohesive behaviour in heterogeneous environments. Finally, we discuss the advances of this multilayer model and its possible outcomes in biological, physical and robotic sciences.

Deterministic versus stochastic models for circadian rhythms

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression.Deterministic models based on genetic regulatory processes account for theoccurrence of circadian rhythms in constant environmental conditions (e.g.constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core modelwas previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens orhundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.

How molecular should your molecular model be? On the level of molecular detail required to simulate biological networks in systems and synthetic biology

The recent advance of genetic studies and the rapid accumulation of molecular data, together with the increasing performance of computers, led researchers to design more and more detailed mathematical models of biological systems. Many modeling approaches rely on ordinary differential equations (ODE) which are based on standard enzyme kinetics. Michaelis-Menten and Hill functions are indeed commonly used in dynamical models in systems and synthetic biology because they provide the necessary nonlinearity to make the dynamics nontrivial (i.e., limit-cycle oscillations or multistability). For most of the systems modeled, the actual molecular mechanism is unknown, and the enzyme equations should be regarded as phenomenological. In this chapter, we discuss the validity and accuracy of these approximations. In particular, we focus on the validity of the Michaelis-Menten function for open systems and on the use of Hill kinetics to describe transcription rates of regulated genes. Our discussion is illustrated by numerical simulations of prototype systems, including the Repressilator (a genetic oscillator) and the Toggle Switch model (a bistable system). We systematically compare the results obtained with the compact version (based on Michaelis-Menten and Hill functions) with its corresponding developed versions (based on "elementary" reaction steps and mass action laws). We also discuss the use of compact approaches to perform stochastic simulations (Gillespie algorithm). On the basis of these results, we argue that using compact models is suitable to model qualitatively biological systems.

Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations

The existence and stability of stable standing-wave patterns in an assembly of spatially distributed generic oscillators governed by a couple of complex Ginzburg-Landau equations, subjected to parametric forcing, are reported. The mechanism of a dispersion-induced pattern in dissipative oscillators parametrically forced near the degenerate Turing-Hopf bifurcation is also illustrated. We show that, when excitation occurs just after the Turing bifurcation and before the Hopf instability, the system exhibits a new type of stable standing-wave structures, namely the mixed-mode solutions. The Brussellator-model, parametrically forced below the threshold of oscillations, is analyzed as an example of calculation.

Self-organized service orchestration through collective differentiation

This paper shows how self-(*) mechanisms give rise to complex but predictable and therefore steerable global system behavior in a cooperative computing environment. The operation of and the interactions between a set of networked autonomic devices are simulated. These are used as access points to a number of services, have the ability to accept or delegate execution of the associated tasks, and can adjust their internal state in response to the demand. At initialization, all devices are assigned a random internal state, i.e., there is no correlation between their configuration and the tasks that they are expected to perform. The authors study the emergence of cooperation and find that it spontaneously occurs when specific conditions are met, which allow individual devices to focus on performing a single task, sacrificing their ability to efficiently perform others. A relatively simple model that can be completely and thoroughly analyzed was chosen so as to demonstrate how the methodology developed to study complex adaptive systems in biology can be a powerful tool when planning the deployment of large ensembles of interacting autonomic devices.

Individual discrimination capability and collective decision-making

Amplification is the main component of many collective phenomena in social and gregarious insects. In a society, individuals face a mixed palette of odours coming from different groups (lines, strains) and individuals present discrimination capabilities. However, often at the collective level, different groups may cooperate and act together. To understand this apparent contradiction, we use a model of food recruitment where each group of foragers have its own blend of pheromone trail that is partly recognized by the others groups. The model shows that a low level of recognition between signals is sufficient to produce a collaborative pattern between groups and that beyond a critical value of recognition, only the aggregation of all the groups around the same food source is observed. The comparison between this model and one describing the site selection by gregarious insects (e.g. cockroach) suggests that such collective response is a generic property of social phenomena governed by amplification processes.

Collegial decision making based on social amplification leads to optimal group formation

Group-living animals are often faced with choosing between one or more alternative resource sites. A central question in such collective decision making includes determining which individuals induce the decision and when. This experimental and theoretical study of shelter selection by cockroach groups demonstrates that choices can emerge through nonlinear interaction dynamics between equal individuals without perfect knowledge or leadership. We identify a simple mechanism whereby a decision is taken on the move with limited information and signaling and without comparison of available opportunities. This mechanism leads to optimal mean benefit for group individuals. Our model points to a generic self-organized collective decision-making process independent of animal species.

[Deterministic and stochastic models for circadian rhythms]

Circadian rhythms, characterized by a period of about 24h, are generated in nearly all living organisms by the negative autoregulation of clock gene expression. Deterministic models based on this genetic regulation account for circadian oscillations in constant environmental conditions (e.g., in constant darkness) and for entrainment of these rhythms by light-dark cycles. When the number of clock mRNA and protein molecules is low, it is necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. Indeed, it is possible that the autoregulatory mechanism of gene expression might not produce stable rhythms due to fluctuations if the number of molecules involved in the clock mechanism remains too low. We have compared the deterministic and stochastic approaches for a model based on the negative autoregulation of a clock gene. We show by means of stochastic simulations that robust circadian oscillations can already occur when the maximum number of mRNA and protein molecules is of the order of a few tens or hundreds, respectively. Furthermore, the results indicate that the cooperativity characterizing the repression of the transcription process strenghtens the robustness of circadian rhythms and that entrainment by light-dark cycles stabilizes the phase of the oscillations.

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.

The follicular automaton model: effect of stochasticity and of synchronization of hair cycles

Human scalp hair consists of a set of about 10(5)follicles which progress independently through developmental cycles. Each hair follicle successively goes through the anagen (A), catagen (C), telogen (T) and latency (L) phases that correspond, respectively, to growth, arrest and hair shedding before a new anagen phase is initiated. Long-term experimental observations in a group of ten male, alopecic and non-alopecic volunteers allowed determination of the characteristics of hair follicle cycles. On the basis of these observations, we previously proposed a follicular automaton model to simulate the dynamics of human hair cycles and the development of different patterns of alopecia [Halloy et al. (2000) Proc. Natl Acad. Sci. U.S.A.97, 8328-8333]. The automaton model is defined by a set of rules that govern the stochastic transitions of each follicle between the successive states A, T, L and the subsequent return to A. These transitions occur independently for each follicle, after time intervals given stochastically by a distribution characterized by a mean and a standard deviation. The follicular automaton model was shown to account both for the dynamical transitions observed in a single follicle, and for the behaviour of an ensemble of independently cycling follicles. Here, we extend these results and investigate additional properties of the model. We present a deterministic version of the follicular automaton. We show that numerical simulations of the stochastic version of the automaton yield steady-state level of follicles in the different phases which approach the levels predicted by the deterministic equations as the number of follicles progressively increases. Only the stochastic version can successfully reproduce the fluctuations of the fractions of follicles in each of the three phases, observed in small follicle populations. When the standard deviation is reduced or when the follicles become otherwise synchronized, e.g. by a periodic external signal inducing the transition of anagen follicles into telogen phase, large-amplitude oscillations occur in the fractions of follicles in the three phases. These oscillations are not observed in humans but are reminiscent of the phenomenon of moulting observed in a number of mammalian species.

Robustness of circadian rhythms with respect to molecular noise

We use a core molecular model capable of generating circadian rhythms to assess the robustness of circadian oscillations with respect to molecular noise. The model is based on the negative feedback exerted by a regulatory protein on the expression of its gene. Such a negative regulatory mechanism underlies circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. The model incorporates gene transcription into mRNA, translation of mRNA into protein, reversible phosphorylation leading to degradation of the regulatory protein, transport of the latter into the nucleus, and repression of gene expression by the nuclear form of the protein. To assess the effect of molecular noise, we perform stochastic simulations after decomposing the deterministic model into elementary reaction steps. The oscillations predicted by the stochastic simulations agree with those obtained with the deterministic version of the model. We show that robust circadian oscillations can occur already with a limited number of mRNA and protein molecules, in the range of tens and hundreds, respectively. Entrainment by light/dark cycles and cooperativity in repression enhance the robustness of circadian oscillations with respect to molecular noise.

Modeling the dynamics of human hair cycles by a follicular automaton

The hair follicle cycle successively goes through the anagen, catagen, telogen, and latency phases, which correspond, respectively, to hair growth, arrest, shedding, and absence before a new anagen phase is initiated. Experimental observations collected over a period of 14 years in a group of 10 male volunteers, alopecic and nonalopecic, allowed us to determine the characteristics of scalp hair follicle cycles. On the basis of these observations, we propose a follicular automaton model to simulate the dynamics of human hair cycles. The automaton model is defined by a set of rules that govern the stochastic transitions of each follicle between the successive states anagen, telogen, and latency, and the subsequent return to anagen. The transitions occur independently for each follicle, after time intervals given stochastically by a distribution characterized by a mean and a variance. The follicular automaton model accounts both for the dynamical transitions observed in a single follicle and for the behavior of an ensemble of independently cycling follicles. Thus, the model successfully reproduces the evolution of the fractions of follicle populations in each of the three phases, which fluctuate around steady-state or slowly drifting values. We apply the follicular automaton model to the study of spatial patterns of follicular growth that result from a spatially heterogeneous distribution of parameters such as the mean duration of anagen phase. When considering that follicles die or miniaturize after going through a critical number of successive cycles, the model can reproduce the evolution to hair patterns similar to well known types of diffuse or androgenetic alopecia.

The frequency encoding of pulsatility

Examples of pulsatile signalling abound in intercellular communication, suggesting that this phenomenon represents a major function of biological rhythms. Pulsatile signals can be encoded in terms of their frequency and prove more efficient than monotonous ones whenever constant stimulation induces desensitization of target cells. We address the main examples of frequency encoding of pulsatility, besides those of neuronal nature. Considered in turn are cAMP oscillations in the slime mould Dictyostelium discoideum, the pulsatile secretion of hormones such as gonadotropin-releasing hormone or growth hormone, intracellular Ca2+ oscillations, and circadian rhythms. Models based on receptor desensitization show the possibility of optimizing cellular responses to cAMP signals in Dictyostelium or to pulsatile hormonal stimulation. The models indicate how the optimal duration of the pulsatile signal and the optimal interval between successive pulses vary as a function of the rates or receptor desensitization and resensitization and of the maximum ligand level during stimulation. The frequency encoding of intracellular Ca2+ oscillations appears to rely on another molecular mechanism. Models based on protein phosphorylation by a Ca(2+)-calmodulin activated kinase show that the mean level of phosphorylated protein increases with the frequency of calcium spikes--which itself rises with the degree of stimulation--and that distinct levels of different phosphorylated proteins can be reached for a Ca2+ signal of given frequency.

Coexistence of multiple propagating wave-fronts in a regulated enzyme reaction model: link with birhythmicity and multi-threshold excitability

We analyze the spatial propagation of wave-fronts in a biochemical model for a product-activated enzyme reaction with non-linear recycling of product into substrate. This model was previously studied as a prototype for the coexistence of two distinct types of periodic oscillations (birhythmicity). The system is initially in a stable steady state characterized by the property of multi-threshold excitability, by which it is capable of amplifying in a pulsatory manner perturbations exceeding two distinct thresholds. In such conditions, when the effect of diffusion is taken into account, two distinct wave-fronts are shown to propagate in space, with distinct amplitudes and velocities, for the same set of parameter values, depending on the magnitude of the initial perturbation. Such a multiplicity of propagating wave-fronts represents a new type of coexistence of multiple modes of dynamic behavior, besides the coexistence involving, under spatially homogeneous conditions, multiple steady states, multiple periodic regimes, or a combination of steady and periodic regimes.

Modeling oscillations and waves of cAMP in Dictyostelium discoideum cells

We examine the theoretical aspects of temporal and spatiotemporal organization in the cAMP signaling system of Dictyostelium discoideum amoebae which aggregate in a wavelike manner after starvation, in response to pulses of cAMP emitted with a periodicity of several minutes by cells behaving as aggregation centers. We first extend the model based on receptor desensitization, previously proposed by Martiel and Goldbeter, by incorporating the role of G proteins in signal transduction. The extended model accounts for observations on the response of the signaling system to successive step increases in extracellular cAMP. In the presence of the positive feedback loop in cAMP synthesis, this model generates sustained oscillations in cAMP and in the fraction of active cAMP receptor, similar to those obtained in the simpler model where the role of the G proteins is not taken into account explicitly. We use the latter model to address the formation of concentric and spiral waves of cAMP in the course of D. discoideum aggregation. Previous analyses of the model showed that a progressive increase in the activity of adenylate cyclase and phosphodiesterase can account for the transitions no relay-relay-oscillations-relay observed in the experiments. We show that the degree of cellular synchronization on such a developmental path in parameter space markedly affects the nature of the spatial patterns generated by the model. These patterns range from concentric waves to a small number of large spirals, and finally to a large number of smaller spirals, as the degree of developmental desynchronization between cells increases.

Desynchronization of cells on the developmental path triggers the formation of spiral waves of cAMP during Dictyostelium aggregation

Whereas it is relatively easy to account for the formation of concentric (target) waves of cAMP in the course of Dictyostelium discoideum aggregation after starvation, the origin of spiral waves remains obscure. We investigate a physiologically plausible mechanism for the spontaneous formation of spiral waves of cAMP in D. discoideum. 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 that do not correspond to the sequence of dynamic transitions no relay-relay-oscillations-relay are less able or fail to give rise to the formation of spirals.

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.

Suppression of chaos by periodic oscillations in a model for cyclic AMP signalling in Dictyostelium cells

We investigate how the introduction of cells oscillating periodically affects the behaviour of a suspension of Dictyostelium discoideum amoebae undergoing chaotic oscillations of cyclic AMP. The analysis of a model indicates that a tiny proportion of periodic cells suffices to transform chaos into periodic oscillations in such suspensions. A similar result is obtained by forcing the aperiodic oscillations by a small-amplitude, periodic input of cyclic AMP. The results provide an explanation for the observation of regular oscillations in suspensions of a putatively chaotic mutant of Dictyostelium discoideum. More generally, the results show how chaos in biological systems may disappear through the coupling with periodic oscillations.