Quantifying the impact of ecological memory on the dynamics of interacting communities.

Ecological memory refers to the influence of past events on the response of an ecosystem to exogenous or endogenous changes. Memory has been widely recognized as a key contributor to the dynamics of ecosystems and other complex systems, yet quantitative community models often ignore memory and its implications. Recent modeling studies have shown how interactions between community members can lead to the emergence of resilience and multistability under environmental perturbations. We demonstrate how memory can be introduced in such models using the framework of fractional calculus. We study how the dynamics of a well-characterized interaction model is affected by gradual increases in ecological memory under varying initial conditions, perturbations, and stochasticity. Our results highlight the implications of memory on several key aspects of community dynamics. In general, memory introduces inertia into the dynamics. This favors species coexistence under perturbation, enhances system resistance to state shifts, mitigates hysteresis, and can affect system resilience both ways depending on the time scale considered. Memory also promotes long transient dynamics, such as long-standing oscillations and delayed regime shifts, and contributes to the emergence and persistence of alternative stable states. Our study highlights the fundamental role of memory in communities, and provides quantitative tools to introduce it in ecological models and analyse its impact under varying conditions.

Deciphering Interactions Within a 4-Strain Riverine Bacterial Community.

The dynamics of a community of four planktonic bacterial strains isolated from river water was followed in R2 broth for 72 h in batch experiments. These strains were identified as Janthinobacterium sp., Brevundimonas sp., Flavobacterium sp. and Variovorax sp. 16S rRNA gene sequencing and flow cytometry analyses were combined to monitor the change in abundance of each individual strain in bi-cultures and quadri-culture. Two interaction networks were constructed that summarize the impact of the strains on each other's growth rate in exponential phase and carrying capacity in stationary phase. The networks agree on the absence of positive interactions but also show differences, implying that ecological interactions can be specific to particular growth phases. Janthinobacterium sp. was the fastest growing strain and dominated the co-cultures. However, its growth rate was negatively affected by the presence of other strains 10 to 100 times less abundant than Janthinobacterium sp. In general, we saw a positive correlation between growth rate and carrying capacity in this system. In addition, growth rate in monoculture was predictive of carrying capacity in co-culture. Taken together, our results highlight the necessity to take growth phases into account when measuring interactions within a microbial community. In addition, evidence that a minor strain can greatly influence the dynamics of a dominant one underlines the necessity to choose population models that do not assume a linear dependency of interaction strength to abundance of other species for accurate parameterization from such empirical data.

Magnesium maintains the length of the circadian period in Arabidopsis.

The circadian clock coordinates the physiological responses of a biological system to day and night rhythms through complex loops of transcriptional/translational regulation. It can respond to external stimuli and adjust generated circadian oscillations accordingly to maintain an endogenous period close to 24 h. However, the interaction between nutritional status and circadian rhythms in plants is poorly understood. Magnesium (Mg) is essential for numerous biological processes in plants, and its homeostasis is crucial to maintain optimal development and growth. Magnesium deficiency in young Arabidopsis thaliana seedlings increased the period of circadian oscillations of the CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1) promoter (pCCA1:LUC) activity and dampened their amplitude under constant light in a dose-dependent manner. Although the circadian period increase caused by Mg deficiency was light dependent, it did not depend on active photosynthesis. Mathematical modeling of the Mg input into the circadian clock reproduced the experimental increase of the circadian period and suggested that Mg is likely to affect global transcription/translation levels rather than a single component of the circadian oscillator. Upon addition of a low dose of cycloheximide to perturb translation, the circadian period increased further under Mg deficiency, which was rescued when sufficient Mg was supplied, supporting the model's prediction. These findings suggest that sufficient Mg supply is required to support proper timekeeping in plants.

Stochastic simulation of a model for circadian rhythms in plants.

Circadian clocks allow living organisms to anticipate and adapt to the daily variations of the environment. The interlocked feedback loops of the transcription factors network in the plant clock generate oscillations with expression peaks at specific times of the day. In this work, we explore the effect of molecular noise on the behavior of the plant circadian clock through numerical simulations. The influence of system size, photoperiod, and mutations of clock genes on the robustness of the oscillations are discussed. Our simulations show that the oscillations remain robust when the mRNA and protein levels are in the range of a few hundreds molecules. Entrainment by light-dark cycles enhances the robustness compared to constant conditions. Multiple light inputs and inter-cellular coupling also contribute to the robustness of the oscillations. The comparison between deterministic and stochastic simulations of single and double mutants shows that stochasticity does not qualitatively affect the behaviour of mutants but that they do not have the same robustness to noise. Finally, the model shows that noise can induce transitions between two limit cycles in a birhythmic clock mutant.

The Goodwin Oscillator and its Legacy.

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

Genome-scale metabolic modeling of Acetobacter pasteurianus 386B reveals its metabolic adaptation to cocoa fermentation conditions.

Acetobacter pasteurianus 386B has been selected as a candidate functional starter culture to better control the cocoa fermentation process. Previously, its genome has been sequenced and a genome-scale metabolic model (GEM) has been reconstructed. To understand its metabolic adaptation to cocoa fermentation conditions, different flux balance analysis (FBA) simulations were performed and compared with experimental data. In particular, metabolic flux distributions were simulated for two phases that characterize the growth of A. pasteurianus 386B under cocoa fermentation conditions, predicting a switch in respiratory chain usage in between these phases. The possible influence on the resulting energy production was shown using a reduced version of the GEM. FBA simulations revealed the importance of the compartmentalization of the ethanol oxidation reactions, namely in the periplasm or in the cytoplasm, and highlighted the potential role of ethanol as a source of carbon, energy, and NADPH. Regarding the latter, the physiological function of a proton-translocating NAD(P)+ transhydrogenase was further investigated in silico. This study revealed the potential of using a GEM to simulate the metabolism of A. pasteurianus 386B, and may provide a general framework toward a better physiological understanding of functional starter cultures in food fermentation processes.

Revisiting a skeleton model for the mammalian cell cycle: From bistability to Cdk oscillations and cellular heterogeneity.

A network of cyclin-dependent kinases (Cdks) regulated by multiple negative and positive feedback loops controls progression in the mammalian cell cycle. We previously proposed a detailed computational model for this network, which consists of four coupled Cdk modules. Both this detailed model and a reduced, skeleton version show that the Cdk network is capable of temporal self-organization in the form of sustained Cdk oscillations, which correspond to the orderly progression along the different cell cycle phases G1, S (DNA replication), G2 and M (mitosis). We use the skeleton model to revisit the role of positive feedback (PF) loops on the dynamics of the mammalian cell cycle by showing that the multiplicity of PF loops extends the range of bistability in the isolated Cdk modules controlling the G1/S and G2/M transitions. Resorting to stochastic simulations we show that, through their effect on the range of bistability, multiple PF loops enhance the robustness of Cdk oscillations with respect to molecular noise. The model predicts that a rise in the total level of Cdk1 also enlarges the domain of bistability in the isolated Cdk modules as well as the range of oscillations in the full Cdk network. Surprisingly, stochastic simulations indicate that Cdk1 overexpression reduces the robustness of Cdk oscillations towards molecular noise; this result is due to the increased distance between the two branches of the bistable switch at higher levels of Cdk1. At intermediate levels of growth factor stochastic simulations show that cells may randomly switch between cell cycle arrest and cell proliferation, as a consequence of fluctuations. In the presence of Cdk1 overexpression, these transitions occur even at low levels of growth factor. Extending stochastic simulations from single cells to cell populations suggests that stochastic switches between cell cycle arrest and proliferation may provide a source of heterogeneity in a cell population, as observed in cancer cells characterized by Cdk1 overexpression.

Signalling-modulated gene regulatory networks in early mammalian development.

Early mammalian embryo is a paradigm of dynamic, self-organised process. It involves gene expression, cell division and intercellular signalling. How these processes interact to ensure reproducible development is being often investigated by modelling, which allows to dissect the mechanisms controlling cell fate decisions. In this work, we present two models based on ordinary differential equations describing the first and second specification processes in the mouse embryo. Together, they describe the cell fate decisions leading to the first three cell lineages which form the blastocyst 4.5 days after fertilisation: the trophectoderm, the epiblast and the primitive endoderm. Both specifications rely on multistability, and signalling allows the selection of the appropriate steady-state. In addition to the gene regulatory network, the first specification process is indeed controlled by the Hippo pathway, which is itself controlled by cell polarity and cell-to-cell contacts. This leads to a spatially organised arrangement of cells. The second specification process is controlled by Fgf signalling and leads to a salt and pepper distribution of the two cell types. We discuss the respective mechanisms and their physiological implications.

Gata6, Nanog and Erk signaling control cell fate in the inner cell mass through a tristable regulatory network.

During blastocyst formation, inner cell mass (ICM) cells differentiate into either epiblast (Epi) or primitive endoderm (PrE) cells, labeled by Nanog and Gata6, respectively, and organized in a salt-and-pepper pattern. Previous work in the mouse has shown that, in absence of Nanog, all ICM cells adopt a PrE identity. Moreover, the activation or the blockade of the Fgf/RTK pathway biases cell fate specification towards either PrE or Epi, respectively. We show that, in absence of Gata6, all ICM cells adopt an Epi identity. Furthermore, the analysis of Gata6(+/-) embryos reveals a dose-sensitive phenotype, with fewer PrE-specified cells. These results and previous findings have enabled the development of a mathematical model for the dynamics of the regulatory network that controls ICM differentiation into Epi or PrE cells. The model describes the temporal dynamics of Erk signaling and of the concentrations of Nanog, Gata6, secreted Fgf4 and Fgf receptor 2. The model is able to recapitulate most of the cell behaviors observed in different experimental conditions and provides a unifying mechanism for the dynamics of these developmental transitions. The mechanism relies on the co-existence between three stable steady states (tristability), which correspond to ICM, Epi and PrE cells, respectively. Altogether, modeling and experimental results uncover novel features of ICM cell fate specification such as the role of the initial induction of a subset of cells into Epi in the initiation of the salt-and-pepper pattern, or the precocious Epi specification in Gata6(+/-) embryos.

Modeling the photoperiodic entrainment of the plant circadian clock.

The circadian clock is an endogenous 24 hour rhythm that helps organisms anticipate and adapt to daily and seasonal variations in environment, such as the day/night cycle or changing temperatures. The plant clock is a complex network of transcription factors that regulate each other, forming interlocked feedback loops. Most of its components are light-regulated in some way, making the system highly sensitive to changes in light conditions. Here, we explore the mechanisms by which the plant clock adapts to changing day length. We first present some experimental data illustrating the variety of behaviors found in seedlings exposed to external day/night cycles different from 24h. We then use a mathematical model to characterize the response of the clock to a wide range of external cycle lengths and photoperiods. We show the existence of several domains of periodic entrainment with different ratios between the external cycle length and the period of the clock, and the presence of quasiperiodic and chaotic behaviors outside of the entrainment range. We simulate knockout mutants with impaired clock function and theoretical variants with diminished light sensitivity to highlight the role of a complex network and multiple light inputs in keeping the clock entrained over a wide range of conditions.

Cell Fate Specification Based on Tristability in the Inner Cell Mass of Mouse Blastocysts.

During development, interactions between transcription factors control the specification of different cell fates. The regulatory networks of genetic interactions often exhibit multiple stable steady states; such multistability provides a common dynamical basis for differentiation. During early murine embryogenesis, cells from the inner cell mass (ICM) can be specified in epiblast (Epi) or primitive endoderm (PrE). Besides the intracellular gene regulatory network, specification is also controlled by intercellular interactions involving Erk signaling through extracellular Fgf4. We previously proposed a model that describes the gene regulatory network and its interaction with Erk signaling in ICM cells. The model displays tristability in a range of Fgf4 concentrations and accounts for the self-organized specification process observed in vivo. Here, we further investigate the origin of tristability in the model and analyze in more detail the specification process by resorting to a simplified two-cell model. We also carry out simulations of a population of 25 cells under various experimental conditions to compare their outcome with that of mutant embryos or of embryos submitted to exogenous treatments that interfere with Fgf signaling. The results are analyzed by means of bifurcation diagrams. Finally, the model predicts that heterogeneities in extracellular Fgf4 concentration play a primary role in the spatial arrangement of the Epi/PrE cells in a salt-and-pepper pattern. If, instead of heterogeneities in extracellular Fgf4 concentration, internal fluctuations in the levels of expression of the transcription factors are considered as a source of randomness, simulations predict the occurrence of unrealistic switches between the Epi and the PrE cell fates, as well as the evolution of some cells toward one of these states without passing through the previous ICM state, in contrast to what is observed in vivo.

A model for the epigenetic switch linking inflammation to cell transformation: deterministic and stochastic approaches.

Recently, a molecular pathway linking inflammation to cell transformation has been discovered. This molecular pathway rests on a positive inflammatory feedback loop between NF-κB, Lin28, Let-7 microRNA and IL6, which leads to an epigenetic switch allowing cell transformation. A transient activation of an inflammatory signal, mediated by the oncoprotein Src, activates NF-κB, which elicits the expression of Lin28. Lin28 decreases the expression of Let-7 microRNA, which results in higher level of IL6 than achieved directly by NF-κB. In turn, IL6 can promote NF-κB activation. Finally, IL6 also elicits the synthesis of STAT3, which is a crucial activator for cell transformation. Here, we propose a computational model to account for the dynamical behavior of this positive inflammatory feedback loop. By means of a deterministic model, we show that an irreversible bistable switch between a transformed and a non-transformed state of the cell is at the core of the dynamical behavior of the positive feedback loop linking inflammation to cell transformation. The model indicates that inhibitors (tumor suppressors) or activators (oncogenes) of this positive feedback loop regulate the occurrence of the epigenetic switch by modulating the threshold of inflammatory signal (Src) needed to promote cell transformation. Both stochastic simulations and deterministic simulations of a heterogeneous cell population suggest that random fluctuations (due to molecular noise or cell-to-cell variability) are able to trigger cell transformation. Moreover, the model predicts that oncogenes/tumor suppressors respectively decrease/increase the robustness of the non-transformed state of the cell towards random fluctuations. Finally, the model accounts for the potential effect of competing endogenous RNAs, ceRNAs, on the dynamics of the epigenetic switch. Depending on their microRNA targets, the model predicts that ceRNAs could act as oncogenes or tumor suppressors by regulating the occurrence of cell transformation.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment.

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

Modeling the effect of cell division on genetic oscillators.

Many genetic oscillators (circadian clocks, synthetic oscillators) continue to oscillate across the cell division cycle. Since cell divisions create discontinuities in the dynamics of genetic oscillators the question about the resilience of oscillations and the factors that contribute to the robustness of the oscillations may be raised. We study here, through stochastic simulations, the effect of the cell division cycle on genetic oscillations using the Repressilator-a genetic oscillator developed in the context of synthetic biology. We consider intrinsic noise (molecular noise due to the limited number of molecules) and extrinsic noise (variability in the cell division time and in the partition of the molecules into daughter cells, cell-cell variability in kinetic parameters, etc). Our numerical simulations show that, although noisy, oscillations are quite resilient to cell division and that cell-cell heterogeneity may be the main source of variability observed experimentally. Finally, similar simulations performed with another model, the Goodwin model, show that oscillations may be entrained and synchronized by cell division. This highlights the influence of the clock architecture on the robustness of genetic oscillations. Our approach provides a general framework to study the effect of cell division on dynamical systems and several possible extensions are described.

Effect of network architecture on synchronization and entrainment properties of the circadian oscillations in the suprachiasmatic nucleus.

In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus constitutes the central circadian pacemaker. The SCN receives light signals from the retina and controls peripheral circadian clocks (located in the cortex, the pineal gland, the liver, the kidney, the heart, etc.). This hierarchical organization of the circadian system ensures the proper timing of physiological processes. In each SCN neuron, interconnected transcriptional and translational feedback loops enable the circadian expression of the clock genes. Although all the neurons have the same genotype, the oscillations of individual cells are highly heterogeneous in dispersed cell culture: many cells present damped oscillations and the period of the oscillations varies from cell to cell. In addition, the neurotransmitters that ensure the intercellular coupling, and thereby the synchronization of the cellular rhythms, differ between the two main regions of the SCN. In this work, a mathematical model that accounts for this heterogeneous organization of the SCN is presented and used to study the implication of the SCN network topology on synchronization and entrainment properties. The results show that oscillations with larger amplitude can be obtained with scale-free networks, in contrast to random and local connections. Networks with the small-world property such as the scale-free networks used in this work can adapt faster to a delay or advance in the light/dark cycle (jet lag). Interestingly a certain level of cellular heterogeneity is not detrimental to synchronization performances, but on the contrary helps resynchronization after jet lag. When coupling two networks with different topologies that mimic the two regions of the SCN, efficient filtering of pulse-like perturbations in the entrainment pattern is observed. These results suggest that the complex and heterogeneous architecture of the SCN decreases the sensitivity of the network to short entrainment perturbations while, at the same time, improving its adaptation abilities to long term changes.

Modeling the Circadian Control of the Cell Cycle and Its Consequences for Cancer Chronotherapy.

The mammalian cell cycle is governed by a network of cyclin/Cdk complexes which signal the progression into the successive phases of the cell division cycle. Once coupled to the circadian clock, this network produces oscillations with a 24 h period such that the progression into each phase of the cell cycle is synchronized to the day-night cycle. Here, we use a computational model for the circadian clock control of the cell cycle to investigate the entrainment in a population of cells characterized by some variability in the kinetic parameters. Our numerical simulations showed that successful entrainment and synchronization are only possible with a sufficient circadian amplitude and an autonomous period close to 24 h. Cellular heterogeneity, however, introduces some variability in the entrainment phase of the cells. Many cancer cells have a disrupted clock or compromised clock control. In these conditions, the cell cycle runs independently of the circadian clock, leading to a lack of synchronization of cancer cells. When the coupling is weak, entrainment is largely impacted, but cells maintain a tendency to divide at specific times of day. These differential entrainment features between healthy and cancer cells can be exploited to optimize the timing of anti-cancer drug administration in order to minimize their toxicity and to maximize their efficacy. We then used our model to simulate such chronotherapeutic treatments and to predict the optimal timing for anti-cancer drugs targeting specific phases of the cell cycle. Although qualitative, the model highlights the need to better characterize cellular heterogeneity and synchronization in cell populations as well as their consequences for circadian entrainment in order to design successful chronopharmacological protocols.

Metabolic models of human gut microbiota: Advances and challenges.

The human gut is a complex ecosystem consisting of hundreds of microbial species interacting with each other and with the human host. Mathematical models of the gut microbiome integrate our knowledge of this system and help to formulate hypotheses to explain observations. The generalized Lotka-Volterra model has been widely used for this purpose, but it does not describe interaction mechanisms and thus does not account for metabolic flexibility. Recently, models that explicitly describe gut microbial metabolite production and consumption have become popular. These models have been used to investigate the factors that shape gut microbial composition and to link specific gut microorganisms to changes in metabolite concentrations found in diseases. Here, we review how such models are built and what we have learned so far from their application to human gut microbiome data. In addition, we discuss current challenges of these models and how these can be addressed in the future.

Starvation responses impact interaction dynamics of human gut bacteria Bacteroides thetaiotaomicron and Roseburia intestinalis.

Bacterial growth often alters the environment, which in turn can impact interspecies interactions among bacteria. Here, we used an in vitro batch system containing mucin beads to emulate the dynamic host environment and to study its impact on the interactions between two abundant and prevalent human gut bacteria, the primary fermenter Bacteroides thetaiotaomicron and the butyrate producer Roseburia intestinalis. By combining machine learning and flow cytometry, we found that the number of viable B. thetaiotaomicron cells decreases with glucose consumption due to acid production, while R. intestinalis survives post-glucose depletion by entering a slow growth mode. Both species attach to mucin beads, but only viable cell counts of B. thetaiotaomicron increase significantly. The number of viable co-culture cells varies significantly over time compared to those of monocultures. A combination of targeted metabolomics and RNA-seq showed that the slow growth mode of R. intestinalis represents a diauxic shift towards acetate and lactate consumption, whereas B. thetaiotaomicron survives glucose depletion and low pH by foraging on mucin sugars. In addition, most of the mucin monosaccharides we tested inhibited the growth of R. intestinalis but not B. thetaiotaomicron. We encoded these causal relationships in a kinetic model, which reproduced the observed dynamics. In summary, we explored how R. intestinalis and B. thetaiotaomicron respond to nutrient scarcity and how this affects their dynamics. We highlight the importance of understanding bacterial metabolic strategies to effectively modulate microbial dynamics in changing conditions.

Application of comparative genomics of Acetobacter species facilitates genome-scale metabolic reconstruction of the Acetobacter ghanensis LMG 23848T and Acetobacter senegalensis 108B cocoa strains.

Acetobacter species play an import role during cocoa fermentation. However, Acetobacter ghanensis and Acetobacter senegalensis are outcompeted during fermentation of the cocoa pulp-bean mass, whereas Acetobacter pasteurianus prevails. In this paper, an in silico approach aimed at delivering some insights into the possible metabolic adaptations of A. ghanensis LMG 23848T and A. senegalensis 108B, two candidate starter culture strains for cocoa fermentation processes, by reconstructing genome-scale metabolic models (GEMs). Therefore, genome sequence data of a selection of strains of Acetobacter species were used to perform a comparative genomic analysis. Combining the predicted orthologous groups of protein-encoding genes from the Acetobacter genomes with gene-reaction rules of GEMs from two reference bacteria, namely a previously manually curated model of A. pasteurianus 386B (iAp386B454) and two manually curated models of Escherichia coli (EcoCyc and iJO1366), allowed to predict the set of reactions present in A. ghanensis LMG 23848T and A. senegalensis 108B. The predicted metabolic network was manually curated using genome re-annotation data, followed by the reconstruction of species-specific GEMs. This approach additionally revealed possible differences concerning the carbon core metabolism and redox metabolism among Acetobacter species, pointing to a hitherto unexplored metabolic diversity. More specifically, the presence or absence of reactions related to citrate catabolism and the glyoxylate cycle for assimilation of C2 compounds provided not only new insights into cocoa fermentation but also interesting guidelines for future research. In general, the A. ghanensis LMG 23848T and A. senegalensis 108B GEMs, reconstructed in a semi-automated way, provided a proof-of-concept toward accelerated formation of GEMs of candidate functional starter cultures for food fermentation processes.

Initial source of heterogeneity in a model for cell fate decision in the early mammalian embryo.

During development, cells from a population of common progenitors evolve towards different fates characterized by distinct levels of specific transcription factors, a process known as cell differentiation. This evolution is governed by gene regulatory networks modulated by intercellular signalling. In order to evolve towards distinct fates, cells forming the population of common progenitors must display some heterogeneity. We applied a modelling approach to obtain insights into the possible sources of cell-to-cell variability initiating the specification of cells of the inner cell mass into epiblast or primitive endoderm cells in early mammalian embryo. At the single-cell level, these cell fates correspond to three possible steady states of the model. A combination of numerical simulations and bifurcation analyses predicts that the behaviour of the model is preserved with respect to the source of variability and that cell-cell coupling induces the emergence of multiple steady states associated with various cell fate configurations, and to a distribution of the levels of expression of key transcription factors. Statistical analysis of these time-dependent distributions reveals differences in the evolutions of the variance-to-mean ratios of key variables of the system, depending on the simulated source of variability, and, by comparison with experimental data, points to the rate of synthesis of the key transcription factor NANOG as a likely initial source of heterogeneity.