A Stochastic Model of the Yeast Cell Cycle Reveals Roles for Feedback Regulation in Limiting Cellular Variability

Stochastic model of vesicular stomatitis virus replication reveals mutational effects on virion production

We present the first complete stochastic model of vesicular stomatitis virus (VSV) intracellular replication. Previous models developed to capture VSV's intracellular replication have either been ODE-based or have not represented the complete replicative cycle, limiting our ability to understand the impact of the stochastic nature of early cellular infections on virion production between cells and how these dynamics change in response to mutations. Our model accurately predicts changes in mean virion production in gene-shuffled VSV variants and can capture the distribution of the number of viruses produced. This model has allowed us to enhance our understanding of intercellular variability in virion production, which appears to be influenced by the duration of the early phase of infection, and variation between variants, arising from balancing the time the genome spends in the active state, the speed of incorporating new genomes into virions, and the production of viral components. Being a stochastic model, we can also assess other effects of mutations beyond just the mean number of virions produced, including the probability of aborted infections and the standard deviation of the number of virions produced. Our model provides a biologically interpretable framework for studying the stochastic nature of VSV replication, shedding light on the mechanisms underlying variation in virion production. In the future, this model could enable the design of more complex viral phenotypes when attenuating VSV, moving beyond solely considering the mean number of virions produced.

Multidimensional characterization of inducible promoters and a highly light-sensitive LOV-transcription factor

The ability to independently control the expression of different genes is important for quantitative biology. Using budding yeast, we characterize GAL1pr, GALL, MET3pr, CUP1pr, PHO5pr, tetOpr, terminator-tetOpr, Z₃EV, blue-light inducible optogenetic systems El222-LIP, El222-GLIP, and red-light inducible PhyB-PIF3. We report kinetic parameters, noise scaling, impact on growth, and the fundamental leakiness of each system using an intuitive unit, maxGAL1. We uncover disadvantages of widely used tools, e.g., nonmonotonic activity of MET3pr and GALL, slow off kinetics of the doxycycline- and estradiol-inducible systems tetOpr and Z₃EV, and high variability of PHO5pr and red-light activated PhyB-PIF3 system. We introduce two previously uncharacterized systems: strongLOV, a more light-sensitive El222 mutant, and ARG3pr, which is induced in the absence of arginine or presence of methionine. To demonstrate fine control over gene circuits, we experimentally tune the time between cell cycle Start and mitosis, artificially simulating near-wild-type timing. All strains, constructs, code, and data ( https://promoter-benchmark.epfl.ch/ ) are made available.

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.

Transcriptional Fluctuations Govern the Serum-Dependent Cell Cycle Duration Heterogeneities in Mammalian Cells

Mammalian cells exhibit a high degree of intercellular variability in cell cycle period and phase durations. However, the factors orchestrating the cell cycle duration heterogeneities remain unclear. Herein, by combining cell cycle network-based mathematical models with live single-cell imaging studies under varied serum conditions, we demonstrate that fluctuating transcription rates of cell cycle regulatory genes across cell lineages and during cell cycle progression in mammalian cells majorly govern the robust correlation patterns of cell cycle period and phase durations among sister, cousin, and mother-daughter lineage pairs. However, for the overall cellular population, alteration in the serum level modulates the fluctuation and correlation patterns of cell cycle period and phase durations in a correlated manner. These heterogeneities at the population level can be fine-tuned under limited serum conditions by perturbing the cell cycle network using a p38-signaling inhibitor without affecting the robust lineage-level correlations. Overall, our approach identifies transcriptional fluctuations as the key controlling factor for the cell cycle duration heterogeneities and predicts ways to reduce cell-to-cell variabilities by perturbing the cell cycle network regulations.

Emergent Bistable Switches from the Incoherent Feed-Forward Signaling of a Positive Feedback Loop

Bistability is intrinsically connected to various decision making processes in living systems. The operating principles of a bistable switch, generated from a positive feedback loop, are well understood both in natural and synthetic settings. However, the fate of dynamic modularity of a positive feedback loop is unknown when it is connected to another dynamically modular signaling motif. In order to address this, here we investigate feed-forward signaling of a positive feedback loop to determine the fate of a bistable switch under such signaling. Using the potential energy based high-throughput bifurcation analysis method, we uncover that in addition to the conventional bistability the hybrid motifs generate various emergent bistable switches, namely mushroom and isola switches, which are not produced by the individual motifs. Using random parameter sampling, network perturbation, and phase plane analysis, we establish the design principles of such emergent behaviors. Incoherent feed-forward signaling of a positive feedback loop with distinct regulatory thresholds of the two arms of the feed-forward loop are the key requirements for such emergent behaviors. Our calculations show that the specific types of atypical bistable responses depend on the logic gate configuration of the signals. However, the emergent bistable behaviors of the hybrid networks do not depend on the nature of the positive feedback loop.

Scaling of intrinsic noise in an autocratic reaction network

Biochemical reactions in living cells often produce stochastic trajectories due to the fluctuations of the finite number of the macromolecular species present inside the cell. A significant number of computational and theoretical studies have previously investigated stochasticity in small regulatory networks to understand its origin and regulation. At the systems level regulatory networks have been determined to be hierarchical resembling social networks. In order to determine the stochasticity in networks with hierarchical architecture, here we computationally investigated intrinsic noise in an autocratic reaction network in which only the upstream regulators regulate the downstream regulators. We studied the effects of the qualitative and quantitative nature of regulatory interactions on the stochasticity in the network. We established an unconventional scaling of noise with average abundance in which the noise passes through a minimum indicating that the network can be noisy both in the low and high abundance regimes. We determined that the bursty kinetics of the trajectories are responsible for such scaling. The scaling of noise remains intact for a mixed network that includes democratic subnetworks within the autocratic network.

Random Parametric Perturbations of Gene Regulatory Circuit Uncover State Transitions in Cell Cycle

Many biological processes involve precise cellular state transitions controlled by complex gene regulation. Here, we use budding yeast cell cycle as a model system and explore how a gene regulatory circuit encodes essential information of state transitions. We present a generalized random circuit perturbation method for circuits containing heterogeneous regulation types and its usage to analyze both steady and oscillatory states from an ensemble of circuit models with random kinetic parameters. The stable steady states form robust clusters with a circular structure that are associated with cell cycle phases. This circular structure in the clusters is consistent with single-cell RNA sequencing data. The oscillatory states specify the irreversible state transitions along cell cycle progression. Furthermore, we identify possible mechanisms to understand the irreversible state transitions from the steady states. We expect this approach to be robust and generally applicable to unbiasedly predict dynamical transitions of a gene regulatory circuit.

Genetic interactions derived from high-throughput phenotyping of 6589 yeast cell cycle mutants

Over the last 30 years, computational biologists have developed increasingly realistic mathematical models of the regulatory networks controlling the division of eukaryotic cells. These models capture data resulting from two complementary experimental approaches: low-throughput experiments aimed at extensively characterizing the functions of small numbers of genes, and large-scale genetic interaction screens that provide a systems-level perspective on the cell division process. The former is insufficient to capture the interconnectivity of the genetic control network, while the latter is fraught with irreproducibility issues. Here, we describe a hybrid approach in which the 630 genetic interactions between 36 cell-cycle genes are quantitatively estimated by high-throughput phenotyping with an unprecedented number of biological replicates. Using this approach, we identify a subset of high-confidence genetic interactions, which we use to refine a previously published mathematical model of the cell cycle. We also present a quantitative dataset of the growth rate of these mutants under six different media conditions in order to inform future cell cycle models.

Clb3-centered regulations are recurrent across distinct parameter regions in minimal autonomous cell cycle oscillator designs

Some biological networks exhibit oscillations in their components to convert stimuli to time-dependent responses. The eukaryotic cell cycle is such a case, being governed by waves of cyclin-dependent kinase (cyclin/Cdk) activities that rise and fall with specific timing and guarantee its timely occurrence. Disruption of cyclin/Cdk oscillations could result in dysfunction through reduced cell division. Therefore, it is of interest to capture properties of network designs that exhibit robust oscillations. Here we show that a minimal yeast cell cycle network is able to oscillate autonomously, and that cyclin/Cdk-mediated positive feedback loops (PFLs) and Clb3-centered regulations sustain cyclin/Cdk oscillations, in known and hypothetical network designs. We propose that Clb3-mediated coordination of cyclin/Cdk waves reconciles checkpoint and oscillatory cell cycle models. Considering the evolutionary conservation of the cyclin/Cdk network across eukaryotes, we hypothesize that functional ("healthy") phenotypes require the capacity to oscillate autonomously whereas dysfunctional (potentially "diseased") phenotypes may lack this capacity.

Qualitative and quantitative nature of mutual interactions dictate chemical noise in a democratic reaction network

The functions of a living cell rely on a complex network of biochemical reactions that allow it to respond against various internal and external cues. The outcomes of these chemical reactions are often stochastic due to intrinsic and extrinsic noise leading to population heterogeneity. The majority of calculations of stochasticity in reaction networks have focused on small regulatory networks addressing the role of timescales, feedback regulations, and network topology in propagation of noise. Here we computationally investigated chemical noise in a network with democratic architecture where each node is regulated by all other nodes in the network. We studied the effects of the qualitative and quantitative nature of mutual interactions on the propagation of both intrinsic and extrinsic noise in the network. We show that an increased number of inhibitory signals lead to ultrasensitive switching of average and that leads to sharp transition of intrinsic noise. The intrinsic noise exhibits a biphasic power-law scaling with the average, and the scaling coefficients strongly correlate with the strength of inhibitory signal. The noise strength critically depends on the strength of the interactions, where negative interactions attenuate both intrinsic and extrinsic noise.

A hybrid stochastic model of the budding yeast cell cycle

The growth and division of eukaryotic cells are regulated by complex, multi-scale networks. In this process, the mechanism of controlling cell-cycle progression has to be robust against inherent noise in the system. In this paper, a hybrid stochastic model is developed to study the effects of noise on the control mechanism of the budding yeast cell cycle. The modeling approach leverages, in a single multi-scale model, the advantages of two regimes: (1) the computational efficiency of a deterministic approach, and (2) the accuracy of stochastic simulations. Our results show that this hybrid stochastic model achieves high computational efficiency while generating simulation results that match very well with published experimental measurements.

Investigation of chemical noise in multisite phosphorylation chain using linear noise approximation

Quantitative and qualitative nature of chemical noise propagation in biochemical reaction networks depend crucially on the topology of the networks. Multisite reversible phosphorylation-dephosphorylation of target proteins is one such recurrently found topology that regulates host of key functions in living cells. Here we analytically calculated the stochasticity in multistep reversible chemical reactions by determining variance of phosphorylated species at the steady state using linear noise approximation to investigate the effect of mass action and Michaelis-Menten kinetics on the noise of phosphorylated species. We probed the dependence of noise on the number of phosphorylation sites and the equilibrium constants of the reaction equilibria to investigate the chemical noise propagation in the multisite phosphorylation chain.

Dichotomous Nature of Bistability Generated by Negative Cooperativity in Receptor-Ligand Binding

Positive cooperativity in receptor-ligand binding plays an important role in cell signaling as it generates an ultrasensitive response, a requirement for nonlinear phenomena such as bistability and oscillations in feedback regulated reaction networks. On the other hand, negative cooperativity typically produces a hyperbolic response and is thus less explored. However, recently negative cooperativity was shown to generate an ultrasensitive response under the condition of strong ligand affinity. In this work, we have used mathematical modeling to investigate the effect of negative cooperativity in receptor-ligand interaction on the bistability in a positive feedback regulatory motif. We systematically investigated the effect of negative cooperativity, modifying the two equilibrium constants of the receptor-ligand binding, on the robustness and tunability of bistability. We show that in the regime where negative cooperativity exhibits robust bistability, positive cooperativity results in poor bistability and vice versa. Further we find that the robustness and tunability of bistability depend crucially on the stability of singly and doubly engaged receptors. Our modeling highlights the ability of negative cooperativity to produce complex phenomena with potential applications in designing synthetic devices or in explaining experimental observations in cell biology.

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.

Systems biology: perspectives on multiscale modeling in research on endocrine-related cancers

Drawing on concepts from experimental biology, computer science, informatics, mathematics and statistics, systems biologists integrate data across diverse platforms and scales of time and space to create computational and mathematical models of the integrative, holistic functions of living systems. Endocrine-related cancers are well suited to study from a systems perspective because of the signaling complexities arising from the roles of growth factors, hormones and their receptors as critical regulators of cancer cell biology and from the interactions among cancer cells, normal cells and signaling molecules in the tumor microenvironment. Moreover, growth factors, hormones and their receptors are often effective targets for therapeutic intervention, such as estrogen biosynthesis, estrogen receptors or HER2 in breast cancer and androgen receptors in prostate cancer. Given the complexity underlying the molecular control networks in these cancers, a simple, intuitive understanding of how endocrine-related cancers respond to therapeutic protocols has proved incomplete and unsatisfactory. Systems biology offers an alternative paradigm for understanding these cancers and their treatment. To correctly interpret the results of systems-based studies requires some knowledge of how in silico models are built, and how they are used to describe a system and to predict the effects of perturbations on system function. In this review, we provide a general perspective on the field of cancer systems biology, and we explore some of the advantages, limitations and pitfalls associated with using predictive multiscale modeling to study endocrine-related cancers.

A transcriptome-wide analysis deciphers distinct roles of G1 cyclins in temporal organization of the yeast cell cycle

Oscillating gene expression is crucial for correct timing and progression through cell cycle. In Saccharomyces cerevisiae, G1 cyclins Cln1-3 are essential drivers of the cell cycle and have an important role for temporal fine-tuning. We measured time-resolved transcriptome-wide gene expression for wild type and cyclin single and double knockouts over cell cycle with and without osmotic stress. Clustering of expression profiles, peak time detection of oscillating genes, integration with transcription factor network dynamics, and assignment to cell cycle phases allowed us to quantify the effect of genetic or stress perturbations on the duration of cell cycle phases. Cln1 and Cln2 showed functional differences, especially affecting later phases. Deletion of Cln3 led to a delay of START followed by normal progression through later phases. Our data and network analysis suggest mutual effects of cyclins with the transcriptional regulators SBF and MBF.

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

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.

Mathematical modeling identifies Lck as a potential mediator for PD-1 induced inhibition of early TCR signaling

Programmed cell death-1 (PD-1) is an inhibitory immune checkpoint receptor that negatively regulates the functioning of T cell. Although the direct targets of PD-1 were not identified, its inhibitory action on the TCR signaling pathway was known much earlier. Recent experiments suggest that the PD-1 inhibits the TCR and CD28 signaling pathways at a very early stage ─ at the level of phosphorylation of the cytoplasmic domain of TCR and CD28 receptors. Here, we develop a mathematical model to investigate the influence of inhibitory effect of PD-1 on the activation of early TCR and CD28 signaling molecules. Proposed model recaptures several quantitative experimental observations of PD-1 mediated inhibition. Model simulations show that PD-1 imposes a net inhibitory effect on the Lck kinase. Further, the inhibitory effect of PD-1 on the activation of TCR signaling molecules such as Zap70 and SLP76 is significantly enhanced by the PD-1 mediated inhibition of Lck. These results suggest a critical role for Lck as a mediator for PD-1 induced inhibition of TCR signaling network. Multi parametric sensitivity analysis explores the effect of parameter uncertainty on model simulations.

Accuracy Analysis of Hybrid Stochastic Simulation Algorithm on Linear Chain Reaction Systems

Noise in cellular systems is often modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), but the low efficiency of the SSA limits its application to large biochemical networks. To improve the efficiency of stochastic simulations, Haseltine and Rawlings (HR) proposed a hybrid algorithm, which combines ordinary differential equations for traditional deterministic models and the SSA for stochastic models. In this paper, accuracy of the HR hybrid method is studied based on a linear chain reaction system. Mathematical analysis and numerical results both show that the HR hybrid method is accurate if either the quantity of reactant molecules in fast reactions is above a certain threshold, or the reaction rates of fast reactions are much larger than those of slow reactions. This analysis also shows that the HR hybrid method approximates the chemical master equation well for a much greater region in system parameter space than the slow-scale SSA and the stochastic quasi-steady-state assumption methods.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

Parallel arrangements of positive feedback loops limit cell-to-cell variability in differentiation

Cellular differentiations are often regulated by bistable switches resulting from specific arrangements of multiple positive feedback loops (PFL) fused to one another. Although bistability generates digital responses at the cellular level, stochasticity in chemical reactions causes population heterogeneity in terms of its differentiated states. We hypothesized that the specific arrangements of PFLs may have evolved to minimize the cellular heterogeneity in differentiation. In order to test this we investigated variability in cellular differentiation controlled either by parallel or serial arrangements of multiple PFLs having similar average properties under extrinsic and intrinsic noises. We find that motifs with PFLs fused in parallel to one another around a central regulator are less susceptible to noise as compared to the motifs with PFLs arranged serially. Our calculations suggest that the increased resistance to noise in parallel motifs originate from the less sensitivity of bifurcation points to the extrinsic noise. Whereas estimation of mean residence times indicate that stable branches of bifurcations are robust to intrinsic noise in parallel motifs as compared to serial motifs. Model conclusions are consistent both in AND- and OR-gate input signal configurations and also with two different modeling strategies. Our investigations provide some insight into recent findings that differentiation of preadipocyte to mature adipocyte is controlled by network of parallel PFLs.