The bistable mitotic switch in fission yeast.

In favorable conditions, eukaryotic cells proceed irreversibly through the cell division cycle (G1-S-G2-M) in order to produce two daughter cells with the same number and identity of chromosomes of their progenitor. The integrity of this process is maintained by "checkpoints" that hold a cell at particular transition points of the cycle until all requisite events are completed. The crucial functions of these checkpoints seem to depend on irreversible bistability of the underlying checkpoint control systems. Bistability of cell cycle transitions has been confirmed experimentally in frog egg extracts, budding yeast cells and mammalian cells. For fission yeast cells, a recent paper by Patterson et al. (2021) provides experimental evidence for an abrupt transition from G2 phase into mitosis, and we show that these data are consistent with a stochastic model of a bistable switch governing the G2/M checkpoint. Interestingly, our model suggests that their experimental data could also be explained by a reversible/sigmoidal switch, and stochastic simulations confirm this supposition. We propose a simple modification of their experimental protocol that could provide convincing evidence for (or against) bistability of the G2/M transition in fission yeast.

The oscillation of mitotic kinase governs cell cycle latches in mammalian cells.

The mammalian cell cycle alternates between two phases - S-G2-M with high levels of A- and B-type cyclins (CycA and CycB, respectively) bound to cyclin-dependent kinases (CDKs), and G1 with persistent degradation of CycA and CycB by an activated anaphase promoting complex/cyclosome (APC/C) bound to Cdh1 (also known as FZR1 in mammals; denoted APC/C:Cdh1). Because CDKs phosphorylate and inactivate Cdh1, these two phases are mutually exclusive. This 'toggle switch' is flipped from G1 to S by cyclin-E bound to a CDK (CycE:CDK), which is not degraded by APC/C:Cdh1, and from M to G1 by Cdc20-bound APC/C (APC/C:Cdc20), which is not inactivated by CycA:CDK or CycB:CDK. After flipping the switch, cyclin E is degraded and APC/C:Cdc20 is inactivated. Combining mathematical modelling with single-cell timelapse imaging, we show that dysregulation of CycB:CDK disrupts strict alternation of the G1-S and M-G1 switches. Inhibition of CycB:CDK results in Cdc20-independent Cdh1 'endocycles', and sustained activity of CycB:CDK drives Cdh1-independent Cdc20 endocycles. Our model provides a mechanistic explanation for how whole-genome doubling can arise, a common event in tumorigenesis that can drive tumour evolution.

A dynamical model of growth and maturation in Drosophila.

The decision to stop growing and mature into an adult is a critical point in development that determines adult body size, impacting multiple aspects of an adult's biology. In many animals, growth cessation is a consequence of hormone release that appears to be tied to the attainment of a particular body size or condition. Nevertheless, the size-sensing mechanism animals use to initiate hormone synthesis is poorly understood. Here, we develop a simple mathematical model of growth cessation in Drosophila melanogaster, which is ostensibly triggered by the attainment of a critical weight (CW) early in the last instar. Attainment of CW is correlated with the synthesis of the steroid hormone ecdysone, which causes a larva to stop growing, pupate, and metamorphose into the adult form. Our model suggests that, contrary to expectation, the size-sensing mechanism that initiates metamorphosis occurs before the larva reaches CW; that is, the critical-weight phenomenon is a downstream consequence of an earlier size-dependent developmental decision, not a decision point itself. Further, this size-sensing mechanism does not require a direct assessment of body size but emerges from the interactions between body size, ecdysone, and nutritional signaling. Because many aspects of our model are evolutionarily conserved among all animals, the model may provide a general framework for understanding how animals commit to maturing from their juvenile to adult form.

Evolutionary Stability of Small Molecular Regulatory Networks That Exhibit Near-Perfect Adaptation.

Large-scale protein regulatory networks, such as signal transduction systems, contain small-scale modules ('motifs') that carry out specific dynamical functions. Systematic characterization of the properties of small network motifs is therefore of great interest to molecular systems biologists. We simulate a generic model of three-node motifs in search of near-perfect adaptation, the property that a system responds transiently to a change in an environmental signal and then returns near-perfectly to its pre-signal state (even in the continued presence of the signal). Using an evolutionary algorithm, we search the parameter space of these generic motifs for network topologies that score well on a pre-defined measure of near-perfect adaptation. We find many high-scoring parameter sets across a variety of three-node topologies. Of all possibilities, the highest scoring topologies contain incoherent feed-forward loops (IFFLs), and these topologies are evolutionarily stable in the sense that, under 'macro-mutations' that alter the topology of a network, the IFFL motif is consistently maintained. Topologies that rely on negative feedback loops with buffering (NFLBs) are also high-scoring; however, they are not evolutionarily stable in the sense that, under macro-mutations, they tend to evolve an IFFL motif and may-or may not-lose the NFLB motif.

Turing-pattern model of scaffolding proteins that establish spatial asymmetry during the cell cycle of Caulobacter crescentus.

The crescent-shaped bacterium Caulobacter crescentus divides asymmetrically into a sessile (stalked) cell and a motile (flagellated) cell. This dimorphic cell division cycle is driven by the asymmetric appearance of scaffolding proteins at the cell's stalk and flagellum poles. The scaffolding proteins recruit enzyme complexes that phosphorylate and degrade a master transcription factor, CtrA, and the abundance and phosphorylation state of CtrA control the onset of DNA synthesis and the differentiation of stalked and flagellated cell types. In this study, we use a Turing-pattern mechanism to simulate the spatiotemporal dynamics of scaffolding proteins in Caulobacter and how they influence the abundance and intracellular distribution of CtrA∼P. Our mathematical model captures crucial features of wild-type and mutant strains and predicts the distributions of CtrA∼P and signaling proteins in mutant strains. Our model accounts for Caulobacter polar morphogenesis and shows how spatial localization and phosphosignaling cooperate to establish asymmetry during the cell cycle.

Feedback in the ß-catenin destruction complex imparts bistability and cellular memory.

Wnt ligands are considered classical morphogens, for which the strength of the cellular response is proportional to the concentration of the ligand. Herein, we show an emergent property of bistability arising from feedback among the Wnt destruction complex proteins that target the key transcriptional co-activator ß-catenin for degradation. Using biochemical reconstitution, we identified positive feedback between the scaffold protein Axin and the kinase glycogen synthase kinase 3 (GSK3). Theoretical modeling of this feedback between Axin and GSK3 suggested that the activity of the destruction complex exhibits bistable behavior. We experimentally confirmed these predictions by demonstrating that cellular cytoplasmic ß-catenin concentrations exhibit an "all-or-none" response with sustained memory (hysteresis) of the signaling input. This bistable behavior was transformed into a graded response and memory was lost through inhibition of GSK3. These findings provide a mechanism for establishing decisive, switch-like cellular response and memory upon Wnt pathway stimulation.

A continuous-time stochastic Boolean model provides a quantitative description of the budding yeast cell cycle.

The cell division cycle is regulated by a complex network of interacting genes and proteins. The control system has been modeled in many ways, from qualitative Boolean switching-networks to quantitative differential equations and highly detailed stochastic simulations. Here we develop a continuous-time stochastic model using seven Boolean variables to represent the activities of major regulators of the budding yeast cell cycle plus one continuous variable representing cell growth. The Boolean variables are updated asynchronously by logical rules based on known biochemistry of the cell-cycle control system using Gillespie's stochastic simulation algorithm. Time and cell size are updated continuously. By simulating a population of yeast cells, we calculate statistical properties of cell cycle progression that can be compared directly to experimental measurements. Perturbations of the normal sequence of events indicate that the cell cycle is 91% robust to random 'flips' of the Boolean variables, but 9% of the perturbations induce lethal mistakes in cell cycle progression. This simple, hybrid Boolean model gives a good account of the growth and division of budding yeast cells, suggesting that this modeling approach may be as accurate as detailed reaction-kinetic modeling with considerably less demands on estimating rate constants.

Nucleation of stem cell domains in a bistable activator-inhibitor model of the shoot apical meristem.

Shoot apical meristems (SAMs) give rise to all above-ground tissues of a plant. Expansion of meristematic tissue is derived from the growth and division of stem cells that reside in a central zone of the SAM. This reservoir of stem cells is maintained by expression of a transcription factor WUSCHEL that is responsible for the development of stem cells in the central zone. WUSCHEL expression is self-activating and downregulated by a signaling pathway initiated by CLAVATA proteins, which are upregulated by WUSCHEL. This classic activator-inhibitor network can generate localized patterns of WUSCHEL activity by a Turing instability provided certain constraints on reaction rates and diffusion constants of WUSCHEL and CLAVATA are satisfied, and most existing mathematical models of nucleation and confinement of stem cells in the SAM rely on Turing's mechanism. However, Turing patterns have certain properties that are inconsistent with observed patterns of stem cell differentiation in the SAM. As an alternative mechanism, we propose a model for stem cell confinement based on a bistable-switch in WUSCHEL-CLAVATA interactions. We study the bistable-switch mechanism for pattern formation in a spatially continuous domain and in a discrete cellularized tissue in the presence of a non-uniform field of a rapidly diffusing hormone. By comparing domain formation by Turing and bistable-switch mechanisms in these contexts, we show that bistable switching provides a superior account of nucleation and confinement of the stem cell domain under reasonable assumptions on reaction rates and diffusion constants.

Time-keeping and decision-making in the cell cycle.

Cell growth, DNA replication, mitosis and division are the fundamental processes by which life is passed on from one generation of eukaryotic cells to the next. The eukaryotic cell cycle is intrinsically a periodic process but not so much a 'clock' as a 'copy machine', making new daughter cells as warranted. Cells growing under ideal conditions divide with clock-like regularity; however, if they are challenged with DNA-damaging agents or mitotic spindle disrupters, they will not progress to the next stage of the cycle until the damage is repaired. These 'decisions' (to exit and re-enter the cell cycle) are essential to maintain the integrity of the genome from generation to generation. A crucial challenge for molecular cell biologists in the 1990s was to unravel the genetic and biochemical mechanisms of cell cycle control in eukaryotes. Central to this effort were biochemical studies of the clock-like regulation of 'mitosis promoting factor' during synchronous mitotic cycles of fertilized frog eggs and genetic studies of the switch-like regulation of 'cyclin-dependent kinases' in yeast cells. In this review, we uncover some secrets of cell cycle regulation by mathematical modelling of increasingly more complex molecular regulatory networks of cell cycle 'clocks' and 'switches'.

BubR1 recruitment to the kinetochore via Bub1 enhances spindle assembly checkpoint signaling.

During mitosis, unattached kinetochores in a dividing cell activate the spindle assembly checkpoint (SAC) and delay anaphase onset by generating the anaphase-inhibitory mitotic checkpoint complex (MCC). These kinetochores generate the MCC by recruiting its constituent proteins, including BubR1. In principle, BubR1 recruitment to signaling kinetochores should increase its local concentration and promote MCC formation. However, in human cells BubR1 is mainly thought to sensitize the SAC to silencing. Whether BubR1 localization to signaling kinetochores by itself enhances SAC signaling remains unknown. Therefore, we used ectopic SAC activation (eSAC) systems to isolate two molecules that recruit BubR1 to the kinetochore, the checkpoint protein Bub1 and the KI and MELT motifs in the kinetochore protein KNL1, and observed their contribution to eSAC signaling. Our quantitative analyses and mathematical modeling show that Bub1-mediated BubR1 recruitment to the human kinetochore promotes SAC signaling and highlight BubR1's dual role of strengthening the SAC directly and silencing it indirectly.

Mitotic kinase oscillation governs the latching of cell cycle switches.

In 1996, Kim Nasmyth1 proposed that the eukaryotic cell cycle is an alternating sequence of transitions from G1 to S-G2-M and back again. These two phases correlate to high activity of cyclin-dependent kinases (CDKs) that trigger S-G2-M events and CDK antagonists that stabilize G1 phase. We associated these "alternative phases" with the coexistence of two stable steady states of the biochemical reactions among CDKs and their antagonists.2,3 Transitions between these steady states (G1-to-S and M-to-G1) are driven by "helper" proteins. The fact that the transitions are irreversible is guaranteed by a "latching" property of the molecular switches, as we have argued in previous publications.4,5 Here, we show that if the latch is broken, then the biochemical reactions can swing back-and-forth across the transitions; either G1-S-G1-S … (periodic DNA replication without mitosis or cell division) or M-(G1)-M-(G1) … (periodic Cdc14 release, without fully exiting mitosis). Using mathematical modeling of the molecular control circuit in budding yeast, we provide a fresh account of aberrant cell cycles in mutant strains: endoreplication in the clb1-5Δ strain6 and periodic release and resequestration of Cdc14 (an "exit" phosphatase) in the CLB2kdΔ strain.7,8 In our opinion, these "endocycles" are not autonomous oscillatory modules that must be entrained by the CDK oscillator6,7 but rather inadvertent and deleterious oscillations that are normally suppressed by the CDK latching-gate mechanism.8.

Mathematical analysis of robustness of oscillations in models of the mammalian circadian clock.

Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger's interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, [Formula: see text], must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1-10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger's 'single negative feedback' (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable [Formula: see text] to ~2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger's observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (~24 h period) with [Formula: see text] ≥ 1 nM. Our results provide testable predictions for future experimental studies.

From the Belousov-Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation.

In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.

Understanding virtual patients efficiently and rigorously by combining machine learning with dynamical modelling.

Individual biological organisms are characterized by daunting heterogeneity, which precludes describing or understanding populations of 'patients' with a single mathematical model. Recently, the field of quantitative systems pharmacology (QSP) has adopted the notion of virtual patients (VPs) to cope with this challenge. A typical population of VPs represents the behavior of a heterogeneous patient population with a distribution of parameter values over a mathematical model of fixed structure. Though this notion of VPs is a powerful tool to describe patients' heterogeneity, the analysis and understanding of these VPs present new challenges to systems pharmacologists. Here, using a model of the hypothalamic-pituitary-adrenal axis, we show that an integrated pipeline that combines machine learning (ML) and bifurcation analysis can be used to effectively and efficiently analyse the behaviors observed in populations of VPs. Compared with local sensitivity analyses, ML allows us to capture and analyse the contributions of simultaneous changes of multiple model parameters. Following up with bifurcation analysis, we are able to provide rigorous mechanistic insight regarding the influences of ML-identified parameters on the dynamical system's behaviors. In this work, we illustrate the utility of this pipeline and suggest that its wider adoption will facilitate the use of VPs in the practice of systems pharmacology.

Computational modeling of unphosphorylated CtrA:Cori binding in the Caulobacter cell cycle.

In the alphaproteobacterium, Caulobacter crescentus, phosphorylated CtrA (CtrA∼P), a master regulatory protein, binds directly to the chromosome origin (Cori) to inhibit DNA replication. Using a mathematical model of CtrA binding at Cori site [d], we provide computational evidence that CtrAU can displace CtrA∼P from Cori at the G1-S transition. Investigation of this interaction within a detailed model of the C. crescentus cell cycle suggests that CckA phosphatase may clear Cori of CtrA∼P by altering the [CtrAU]/[CtrA∼P] ratio rather than by completely depleting CtrA∼P. Model analysis reveals that the mechanism allows for a speedier transition into S phase, stabilizes the timing of chromosome replication under fluctuating rates of CtrA proteolysis, and may contribute to the viability of numerous mutant strains. Overall, these results suggest that CtrAU enhances the robustness of chromosome replication. More generally, our proposed regulation of CtrA:Cori dynamics may represent a novel motif for molecular signaling in cell physiology.

Modeling and analysis of the macronutrient signaling network in budding yeast.

Adaptive modulation of the global cellular growth state of unicellular organisms is crucial for their survival in fluctuating nutrient environments. Because these organisms must be able to respond reliably to ever varying and unpredictable nutritional conditions, their nutrient signaling networks must have a certain inbuilt robustness. In eukaryotes, such as the budding yeast Saccharomyces cerevisiae, distinct nutrient signals are relayed by specific plasma membrane receptors to signal transduction pathways that are interconnected in complex information-processing networks, which have been well characterized. However, the complexity of the signaling network confounds the interpretation of the overall regulatory "logic" of the control system. Here, we propose a literature-curated molecular mechanism of the integrated nutrient signaling network in budding yeast, focusing on early temporal responses to carbon and nitrogen signaling. We build a computational model of this network to reconcile literature-curated quantitative experimental data with our proposed molecular mechanism. We evaluate the robustness of our estimates of the model's kinetic parameter values. We test the model by comparing predictions made in mutant strains with qualitative experimental observations made in the same strains. Finally, we use the model to predict nutrient-responsive transcription factor activities in a number of mutant strains undergoing complex nutrient shifts.

Natural antisense transcript of Period2, Per2AS, regulates the amplitude of the mouse circadian clock.

In mammals, a set of core clock genes form transcription-translation feedback loops to generate circadian oscillations. We and others recently identified a novel transcript at the Period2 (Per2) locus that is transcribed from the antisense strand of Per2 This transcript, Per2AS, is expressed rhythmically and antiphasic to Per2 mRNA, leading to our hypothesis that Per2AS and Per2 mutually inhibit each other's expression and form a double negative feedback loop. By perturbing the expression of Per2AS, we found that Per2AS transcription, but not transcript, represses Per2 However, Per2 does not repress Per2AS, as Per2 knockdown led to a decrease in the Per2AS level, indicating that Per2AS forms a single negative feedback loop with Per2 and maintains the level of Per2 within the oscillatory range. Per2AS also regulates the amplitude of the circadian clock, and this function cannot be solely explained through its interaction with Per2, as Per2 knockdown does not recapitulate the phenotypes of Per2AS perturbation. Overall, our data indicate that Per2AS is an important regulatory molecule in the mammalian circadian clock machinery. Our work also supports the idea that antisense transcripts of core clock genes constitute a common feature of circadian clocks, as they are found in other organisms.

Computational modeling of chromosome re-replication in mutant strains of fission yeast.

Typically cells replicate their genome only once per division cycle, but under some circumstances, both natural and unnatural, cells synthesize an overabundance of DNA, either in a disorganized manner ("overreplication") or by a systematic doubling of chromosome number ("endoreplication"). These variations on the theme of DNA replication and division have been studied in strains of fission yeast, Schizosaccharomyces pombe, carrying mutations that interfere with the function of mitotic cyclin-dependent kinase (Cdk1:Cdc13) without impeding the roles of DNA-replication loading factor (Cdc18) and S-phase cyclin-dependent kinase (Cdk1:Cig2). Some of these mutations support endoreplication, and some overreplication. In this paper, we propose a dynamical model of the interactions among the proteins governing DNA replication and cell division in fission yeast. By computational simulations of the mathematical model, we account for the observed phenotypes of these re-replicating mutants, and by theoretical analysis of the dynamical system, we provide insight into the molecular distinctions between overreplicating and endoreplicating cells. In the case of induced overproduction of regulatory proteins, our model predicts that cells first switch from normal mitotic cell cycles to growth-controlled endoreplication, and ultimately to disorganized overreplication, parallel to the slow increase of protein to very high levels.

Mechanisms of signalling-memory governing progression through the eukaryotic cell cycle.

As cells pass through each replication-division cycle, they must be able to postpone further progression if they detect any threats to genome integrity, such as DNA damage or misaligned chromosomes. Once a 'decision' is made to proceed, the cell unequivocally enters into a qualitatively different biochemical state, which makes the transitions from one cell cycle phase to the next switch-like and irreversible. Each transition is governed by a unique signalling network; nonetheless, they share a common characteristic of bistable behaviour, a hallmark of molecular memory devices. Comparing the cell cycle signalling mechanisms acting at the restriction point, G1/S, G2/M and meta-to-anaphase transitions, we deduce a generic network motif of coupled positive and negative feedback loops underlying each transition.

Cell cycle control and environmental response by second messengers in Caulobacter crescentus.

BACKGROUND: Second messengers, c-di-GMP and (p)ppGpp, are vital regulatory molecules in bacteria, influencing cellular processes such as biofilm formation, transcription, virulence, quorum sensing, and proliferation. While c-di-GMP and (p)ppGpp are both synthesized from GTP molecules, they play antagonistic roles in regulating the cell cycle. In C. crescentus, c-di-GMP works as a major regulator of pole morphogenesis and cell development. It inhibits cell motility and promotes S-phase entry by inhibiting the activity of the master regulator, CtrA. Intracellular (p)ppGpp accumulates under starvation, which helps bacteria to survive under stressful conditions through regulating nucleotide levels and halting proliferation. (p)ppGpp responds to nitrogen levels through RelA-SpoT homolog enzymes, detecting glutamine concentration using a nitrogen phosphotransferase system (PTS Ntr). This work relates the guanine nucleotide-based second messenger regulatory network with the bacterial PTS Ntr system and investigates how bacteria respond to nutrient availability. RESULTS: We propose a mathematical model for the dynamics of c-di-GMP and (p)ppGpp in C. crescentus and analyze how the guanine nucleotide-based second messenger system responds to certain environmental changes communicated through the PTS Ntr system. Our mathematical model consists of seven ODEs describing the dynamics of nucleotides and PTS Ntr enzymes. Our simulations are consistent with experimental observations and suggest, among other predictions, that SpoT can effectively decrease c-di-GMP levels in response to nitrogen starvation just as well as it increases (p)ppGpp levels. Thus, the activity of SpoT (or its homologues in other bacterial species) can likely influence the cell cycle by influencing both c-di-GMP and (p)ppGpp. CONCLUSIONS: In this work, we integrate current knowledge and experimental observations from the literature to formulate a novel mathematical model. We analyze the model and demonstrate how the PTS Ntr system influences (p)ppGpp, c-di-GMP, GMP and GTP concentrations. While this model does not consider all aspects of PTS Ntr signaling, such as cross-talk with the carbon PTS system, here we present our first effort to develop a model of nutrient signaling in C. crescentus.