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.

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.

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.

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.

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.

Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy.

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.

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.

Design principles of biochemical oscillators.

Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions. Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism.

A stochastic model of size control in the budding yeast cell cycle.

BACKGROUND: Cell size is a key characteristic that significantly affects many aspects of cellular physiology. There are specific control mechanisms during cell cycle that maintain the cell size within a range from generation to generation. Such control mechanisms introduce substantial variabilities to important properties of the cell cycle such as growth and division. To quantitatively study the effect of such variability in progression through cell cycle, detailed stochastic models are required. RESULTS: In this paper, a new hybrid stochastic model is proposed to study the effect of molecular noise and size control mechanism on the variabilities in cell cycle of the budding yeast Saccharomyces cerevisiae. The proposed model provides an accurate, yet computationally efficient approach for simulation of an intricate system by integrating the deterministic and stochastic simulation schemes. The developed hybrid stochastic model can successfully capture several key features of the cell cycle observed in experimental data. In particular, the proposed model: 1) confirms that the majority of noise in size control stems from low copy numbers of transcripts in the G1 phase, 2) identifies the size and time regulation modules in the size control mechanism, and 3) conforms with phenotypes of early G1 mutants in exquisite detail. CONCLUSIONS: Hybrid stochastic modeling approach can be used to provide quantitative descriptions for stochastic properties of the cell cycle within a computationally efficient framework.

CrossPlan: systematic planning of genetic crosses to validate mathematical models.

Motivation: Mathematical models of cellular processes can systematically predict the phenotypes of novel combinations of multi-gene mutations. Searching for informative predictions and prioritizing them for experimental validation is challenging since the number of possible combinations grows exponentially in the number of mutations. Moreover, keeping track of the crosses needed to make new mutants and planning sequences of experiments is unmanageable when the experimenter is deluged by hundreds of potentially informative predictions to test. Results: We present CrossPlan, a novel methodology for systematically planning genetic crosses to make a set of target mutants from a set of source mutants. We base our approach on a generic experimental workflow used in performing genetic crosses in budding yeast. We prove that the CrossPlan problem is NP-complete. We develop an integer-linear-program (ILP) to maximize the number of target mutants that we can make under certain experimental constraints. We apply our method to a comprehensive mathematical model of the protein regulatory network controlling cell division in budding yeast. We also extend our solution to incorporate other experimental conditions such as a delay factor that decides the availability of a mutant and genetic markers to confirm gene deletions. The experimental flow that underlies our work is quite generic and our ILP-based algorithm is easy to modify. Hence, our framework should be relevant in plant and animal systems as well. Availability and implementation: CrossPlan code is freely available under GNU General Public Licence v3.0 at https://github.com/Murali-group/crossplan. Supplementary information: Supplementary data are available at Bioinformatics online.

Modeling the dynamic behavior of biochemical regulatory networks.

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network.

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

Dilution and titration of cell-cycle regulators may control cell size in budding yeast.

The size of a cell sets the scale for all biochemical processes within it, thereby affecting cellular fitness and survival. Hence, cell size needs to be kept within certain limits and relatively constant over multiple generations. However, how cells measure their size and use this information to regulate growth and division remains controversial. Here, we present two mechanistic mathematical models of the budding yeast (S. cerevisiae) cell cycle to investigate competing hypotheses on size control: inhibitor dilution and titration of nuclear sites. Our results suggest that an inhibitor-dilution mechanism, in which cell growth dilutes the transcriptional inhibitor Whi5 against the constant activator Cln3, can facilitate size homeostasis. This is achieved by utilising a positive feedback loop to establish a fixed size threshold for the Start transition, which efficiently couples cell growth to cell cycle progression. Yet, we show that inhibitor dilution cannot reproduce the size of mutants that alter the cell's overall ploidy and WHI5 gene copy number. By contrast, size control through titration of Cln3 against a constant number of genomic binding sites for the transcription factor SBF recapitulates both size homeostasis and the size of these mutant strains. Moreover, this model produces an imperfect 'sizer' behaviour in G1 and a 'timer' in S/G2/M, which combine to yield an 'adder' over the whole cell cycle; an observation recently made in experiments. Hence, our model connects these phenomenological data with the molecular details of the cell cycle, providing a systems-level perspective of budding yeast size control.

Model-driven experimental approach reveals the complex regulatory distribution of p53 by the circadian factor Period 2.

The circadian clock and cell cycle networks are interlocked on the molecular level, with the core clock loop exerting a multilevel regulatory role over cell cycle components. This is particularly relevant to the circadian factor Period 2 (Per2), which modulates the stability of the tumor suppressor p53 in unstressed cells and transcriptional activity in response to genotoxic stress. Per2 binding prevents Mdm2-mediated ubiquitination of p53 and, therefore, its degradation, and oscillations in the peaks of Per2 and p53 were expected to correspond. However, our findings showed that Per2 and p53 rhythms were significantly out-of-phase relative to each other in cell lysates and in purified cytoplasmic fractions. These seemingly conflicting experimental data motivated the use of a combined theoretical and experimental approach focusing on the role played by Per2 in dictating the phase of p53 oscillations. Systematic modeling of all possible regulatory scenarios predicted that the observed phase relationship between Per2 and p53 could be simulated if (i) p53 was more stable in the nucleus than in the cytoplasm, (ii) Per2 associates to various ubiquitinated forms of p53, and (iii) Per2 mediated p53 nuclear import. These predictions were supported by a sevenfold increase in p53's half-life in the nucleus and by in vitro binding of Per2 to the various ubiquitinated forms of p53. Last, p53's nuclear shuttling was significantly favored by ectopic expression of Per2 and reduced because of Per2 down-regulation. Our combined theoretical/mathematical approach reveals how clock regulatory nodes can be inferred from oscillating time course data.

Deciphering the Dynamics of Interlocked Feedback Loops in a Model of the Mammalian Circadian Clock.

Mathematical models of fundamental biological processes play an important role in consolidating theory and experiments, especially if they are systematically developed, thoroughly characterized, and well tested by experimental data. In this work, we report a detailed bifurcation analysis of a mathematical model of the mammalian circadian clock network developed by Relógio et al., noteworthy for its consistency with available data. Using one- and two-parameter bifurcation diagrams, we explore how oscillations in the model depend on the expression levels of its constituent genes and the activities of their encoded proteins. These bifurcation diagrams allow us to decipher the dynamics of interlocked feedback loops by parametric variation of genes and proteins in the model. Among other results, we find that REV-ERB, a member of a subfamily of orphan nuclear receptors, plays a critical role in the intertwined dynamics of Relógio's model. The bifurcation diagrams reported here can be used for predicting how the core clock network responds-in terms of period, amplitude and phases of oscillations-to different perturbations.

GraphSpace: stimulating interdisciplinary collaborations in network biology.

SUMMARY: Networks have become ubiquitous in systems biology. Visualization is a crucial component in their analysis. However, collaborations within research teams in network biology are hampered by software systems that are either specific to a computational algorithm, create visualizations that are not biologically meaningful, or have limited features for sharing networks and visualizations. We present GraphSpace, a web-based platform that fosters team science by allowing collaborating research groups to easily store, interact with, layout and share networks. AVAILABILITY AND IMPLEMENTATION: Anyone can upload and share networks at http://graphspace.org. In addition, the GraphSpace code is available at http://github.com/Murali-group/graphspace if a user wants to run his or her own server. CONTACT: murali@cs.vt.edu. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

JigCell Model Connector: building large molecular network models from components.

The growing size and complexity of molecular network models makes them increasingly difficult to construct and understand. Modifying a model that consists of tens of reactions is no easy task. Attempting the same on a model containing hundreds of reactions can seem nearly impossible. We present the JigCell Model Connector, a software tool that supports large-scale molecular network modeling. Our approach to developing large models is to combine smaller models, making the result easier to comprehend. At the base, the smaller models (called modules) are defined by small collections of reactions. Modules connect together to form larger modules through clearly defined interfaces, called ports. In this work, we enhance the port concept by defining three types of ports. An output port is linked to an internal component that will send a value. An input port is linked to an internal component that will receive a value. An equivalence port is linked to an internal component that will both receive and send values. Not all modules connect together in the same way; therefore, multiple connection options need to exist.