A model for a bistable biochemical trigger of mitosis

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.

Comprehensive Parameter Space Mapping of Cell Cycle Dynamics under Network Perturbations

Studies of quantitative systems and synthetic biology have extensively utilized models to interpret data, make predictions, and guide experimental designs. However, models often simplify complex biological systems and lack experimentally validated parameters, making their reliability in perturbed systems unclear. Here, we developed a droplet-based synthetic cell system to continuously tune parameters at the single-cell level in multiple dimensions with full dynamic ranges, providing an experimental framework for global parameter space scans. We systematically perturbed a cell-cycle oscillator centered on cyclin-dependent kinase (Cdk1), enabling comprehensive mapping of period landscapes in response to network perturbations. The data allowed us to challenge existing models and refine a new model that matches the observed response. Our analysis demonstrated that Cdk1 positive feedback inhibition restricts the cell cycle frequency range, confirming model predictions; furthermore, it revealed new cellular responses to the inhibition of the Cdk1-counteracting phosphatase PP2A: monomodal or bimodal distributions across varying inhibition levels, underscoring the complex nature of cell cycle regulation that can be explained by our model. This comprehensive perturbation platform may be generalizable to exploring other complex dynamic systems.

A Model for the Proliferation–Quiescence Transition in Human Cells

The process of revitalising quiescent cells in order for them to proliferate plays a pivotal role in the repair of worn-out tissues as well as for tissue homeostasis. This process is also crucial in the growth, development and well-being of higher multi-cellular organisms such as mammals. Deregulation of proliferation-quiescence transition is related to many diseases, such as cancer. Recent studies have revealed that this proliferation–quiescence process is regulated tightly by the Rb−E2F bistable switch mechanism. Based on experimental observations, in this study, we formulate a mathematical model to examine the effect of the growth factor concentration on the proliferation–quiescence transition in human cells. Working with a non-dimensionalised model, we prove the positivity, boundedness and uniqueness of solutions. To understand model solution behaviour close to bifurcation points, we carry out bifurcation analysis, which is further illustrated by the use of numerical bifurcation analysis, sensitivity analysis and numerical simulations. Indeed, bifurcation and numerical analysis of the model predicted a transition between bistable and stable states, which are dependent on the growth factor concentration parameter (GF). The derived predictions confirm experimental observations.

Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities

Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.

Bistability plays an important role in many biochemical processes and typically emerges from complex interaction patterns such as positive and double negative feedback loops. Here, we propose to theoretically study the effect of bistability in a larger interaction network. We explicitly incorporate a functional expression describing an S-shaped input-output curve in the model equations, without the need for considering the underlying biochemical events. This expression can be converted into a functional module for an ultrasensitive response, and a time delay is easily included as well. Exploiting the fact that several of these modules can easily be combined in larger networks, we construct a cell cycle model consisting of multiple bistable switches and show how this approach can account for a number of known properties of the cell cycle.

Leveraging autocatalytic reactions for chemical domain image classification

Autocatalysis is fundamental to many biological processes, and kinetic models of autocatalytic reactions have mathematical forms similar to activation functions used in artificial neural networks. Inspired by these similarities, we use an autocatalytic reaction, the copper-catalyzed azide-alkyne cycloaddition, to perform digital image recognition tasks. Images are encoded in the concentration of a catalyst across an array of liquid samples, and the classification is performed with a sequence of automated fluid transfers. The outputs of the operations are monitored using UV-vis spectroscopy. The growing interest in molecular information storage suggests that methods for computing in chemistry will become increasingly important for querying and manipulating molecular memory.

Ca2+-Induced Mitochondrial ROS Regulate the Early Embryonic Cell Cycle

While it is appreciated that reactive oxygen species (ROS) can act as second messengers in both homeostastic and stress response signaling pathways, potential roles for ROS during early vertebrate development have remained largely unexplored. Here, we show that fertilization in Xenopus embryos triggers a rapid increase in ROS levels, which oscillate with each cell division. Furthermore, we show that the fertilization-induced Ca²⁺ wave is necessary and sufficient to induce ROS production in activated or fertilized eggs. Using chemical inhibitors, we identified mitochondria as the major source of fertilization-induced ROS production. Inhibition of mitochondrial ROS production in early embryos results in cell-cycle arrest, in part, via ROS-dependent regulation of Cdc25C activity. This study reveals a role for oscillating ROS levels in early cell cycle regulation in Xenopus embryos.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

A robust and tunable mitotic oscillator in artificial cells

Single-cell analysis is pivotal to deciphering complex phenomena like heterogeneity, bistability, and asynchronous oscillations, where a population ensemble cannot represent individual behaviors. Bulk cell-free systems, despite having unique advantages of manipulation and characterization of biochemical networks, lack the essential single-cell information to understand a class of out-of-steady-state dynamics including cell cycles. Here, by encapsulating Xenopus egg extracts in water-in-oil microemulsions, we developed artificial cells that are adjustable in sizes and periods, sustain mitotic oscillations for over 30 cycles, and function in forms from the simplest cytoplasmic-only to the more complicated ones involving nuclear dynamics, mimicking real cells. Such innate flexibility and robustness make it key to studying clock properties like tunability and stochasticity. Our results also highlight energy as an important regulator of cell cycles. We demonstrate a simple, powerful, and likely generalizable strategy of integrating strengths of single-cell approaches into conventional in vitro systems to study complex clock functions.

Systems and synthetic biology approaches in understanding biological oscillators

Background

Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators.

Results

Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations.

Conclusion

With the recent development of biological and computational tools, both approaches have made significant achievements.

Cell-cycle transitions: a common role for stoichiometric inhibitors

The abrupt and irreversible transitions that drive cells through the DNA replication-division cycle are governed by molecular mechanisms that function as bistable “toggle” switches. A common theme of these switches is a network motif consisting of a “beleaguered” enzyme and its “domineering” substrate, locked in a feedback amplification loop.

The cell division cycle is the process by which eukaryotic cells replicate their chromosomes and partition them to two daughter cells. To maintain the integrity of the genome, proliferating cells must be able to block progression through the division cycle at key transition points (called “checkpoints”) if there have been problems in the replication of the chromosomes or their biorientation on the mitotic spindle. These checkpoints are governed by protein-interaction networks, composed of phase-specific cell-cycle activators and inhibitors. Examples include Cdk1:Clb5 and its inhibitor Sic1 at the G1/S checkpoint in budding yeast, APC:Cdc20 and its inhibitor MCC at the mitotic checkpoint, and PP2A:B55 and its inhibitor, alpha-endosulfine, at the mitotic-exit checkpoint. Each of these inhibitors is a substrate as well as a stoichiometric inhibitor of the cell-cycle activator. Because the production of each inhibitor is promoted by a regulatory protein that is itself inhibited by the cell-cycle activator, their interaction network presents a regulatory motif characteristic of a “feedback-amplified domineering substrate” (FADS). We describe how the FADS motif responds to signals in the manner of a bistable toggle switch, and then we discuss how this toggle switch accounts for the abrupt and irreversible nature of three specific cell-cycle checkpoints.

Positive Feedback Keeps Duration of Mitosis Temporally Insulated from Upstream Cell-Cycle Events

Cell division is characterized by a sequence of events by which a cell gives rise to two daughter cells. Quantitative measurements of cell-cycle dynamics in single cells showed that despite variability in G1-, S-, and G2 phases, duration of mitosis is short and remarkably constant. Surprisingly, there is no correlation between cell-cycle length and mitotic duration, suggesting that mitosis is temporally insulated from variability in earlier cell-cycle phases. By combining live cell imaging and computational modeling, we showed that positive feedback is the molecular mechanism underlying the temporal insulation of mitosis. Perturbing positive feedback gave rise to a sluggish, variable entry and progression through mitosis and uncoupled duration of mitosis from variability in cell cycle length. We show that positive feedback is important to keep mitosis short, constant, and temporally insulated and anticipate it might be a commonly used regulatory strategy to create modularity in other biological systems.

Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback

In this series of reviews, we are examining ultrasensitive responses, the switch-like input-output relationships that contribute to signal processing in a wide variety of signaling contexts. In the first part of this series, we explored one mechanism for generating ultrasensitivity, zero-order ultrasensitivity, where the saturation of two converting enzymes allows the output to switch from low to high over a tight range of input levels. In this second installment, we focus on three conceptually distinct mechanisms for ultrasensitivity: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. We also examine several related mechanisms and concepts, including cooperativity, reciprocal regulation, coherent feed-forward regulation, and substrate competition, and provide several examples of signaling processes where these mechanisms are known or are suspected to be applicable.

Ultrasensitivity part I: Michaelian responses and zero-order ultrasensitivity

Quantitative studies of signal transduction systems have shown that ultrasensitive responses - switch-like, sigmoidal input/output relationships - are commonplace in cell signaling. Ultrasensitivity is important for various complex signaling systems, including signaling cascades, bistable switches, and oscillators. In this first installment of a series on ultrasensitivity we survey the occurrence of ultrasensitive responses in signaling systems. We review why the simplest mass action systems exhibit Michaelian responses, and then move on to zero-order ultrasensitivity, a phenomenon that occurs when signaling proteins are operating near saturation. We also discuss the physiological relevance of zero-order ultrasensitivity to cellular regulation.

Synergistic dual positive feedback loops established by molecular sequestration generate robust bimodal response

Feedback loops are ubiquitous features of biological networks and can produce significant phenotypic heterogeneity, including a bimodal distribution of gene expression across an isogenic cell population. In this work, a combination of experiments and computational modeling was used to explore the roles of multiple feedback loops in the bimodal, switch-like response of the Saccharomyces cerevisiae galactose regulatory network. Here, we show that bistability underlies the observed bimodality, as opposed to stochastic effects, and that two unique positive feedback loops established by Gal1p and Gal3p, which both regulate network activity by molecular sequestration of Gal80p, induce this bimodality. Indeed, systematically scanning through different single and multiple feedback loop knockouts, we demonstrate that there is always a concentration regime that preserves the system's bimodality, except for the double deletion of GAL1 and the GAL3 feedback loop, which exhibits a graded response for all conditions tested. The constitutive production rates of Gal1p and Gal3p operate as bifurcation parameters because variations in these rates can also abolish the system's bimodal response. Our model indicates that this second loss of bistability ensues from the inactivation of the remaining feedback loop by the overexpressed regulatory component. More broadly, we show that the sequestration binding affinity is a critical parameter that can tune the range of conditions for bistability in a circuit with positive feedback established by molecular sequestration. In this system, two positive feedback loops can significantly enhance the region of bistability and the dynamic response time.

Hybrid modeling and simulation of stochastic effects on progression through the eukaryotic cell cycle

The eukaryotic cell cycle is regulated by a complicated chemical reaction network. Although many deterministic models have been proposed, stochastic models are desired to capture noise in the cell resulting from low numbers of critical species. However, converting a deterministic model into one that accurately captures stochastic effects can result in a complex model that is hard to build and expensive to simulate. In this paper, we first apply a hybrid (mixed deterministic and stochastic) simulation method to such a stochastic model. With proper partitioning of reactions between deterministic and stochastic simulation methods, the hybrid method generates the same primary characteristics and the same level of noise as Gillespie's stochastic simulation algorithm, but with better efficiency. By studying the results generated by various partitionings of reactions, we developed a new strategy for hybrid stochastic modeling of the cell cycle. The new approach is not limited to using mass-action rate laws. Numerical experiments demonstrate that our approach is consistent with characteristics of noisy cell cycle progression, and yields cell cycle statistics in accord with experimental observations.

Autocatalyses

Autocatalysis is a fundamental concept, used in a wide range of domains. From the most general definition of autocatalysis, that is, a process in which a chemical compound is able to catalyze its own formation, several different systems can be described. We detail the different categories of autocatalyses and compare them on the basis of their mechanistic, kinetic, and dynamic properties. It is shown how autocatalytic patterns can be generated by different systems of chemical reactions. With the notion of autocatalysis covering a large variety of mechanistic realizations with very similar behaviors, it is proposed that the key signature of autocatalysis is its kinetic pattern expressed in a mathematical form.

Ultrasensitivity in the Regulation of Cdc25C by Cdk1

Cdc25C is a critical component of the interlinked positive and double-negative feedback loops that constitute the bistable mitotic trigger. Computational studies have indicated that the trigger's bistability should be more robust if the individual legs of the loops exhibit ultrasensitive responses. Here, we show that in Xenopus extracts two measures of Cdc25C activation (hyperphosphorylation and Ser 287 dephosphorylation) are highly ultrasensitive functions of the Cdk1 activity; estimated Hill coefficients were 11 to 32. Some of Cdc25C's ultrasensitivity can be reconstituted in vitro with purified components, and the reconstituted ultrasensitivity depends upon multisite phosphorylation. The response functions determined here for Cdc25C and previously for Wee1A allow us to formulate a simple mathematical model of the transition between interphase and mitosis. The model shows how the continuously variable regulators of mitosis work collectively to generate a switch-like, hysteretic response.

Ultrasensitivity and positive feedback to promote sharp mitotic entry

In this issue, Trunnell et al. (2011) show that in mitotic entry the positive feedback that drives the activation of cyclin-dependent kinase (Cdk) involves a very ultrasensitive step of phosphorylation of Cdc25C by Cdk, thus strongly contributing to the switch-like behavior of this essential cell-cycle transition.

Regulated protein kinases and phosphatases in cell cycle decisions

Many aspects of cell physiology are controlled by protein kinases and phosphatases, which together determine the phosphorylation state of targeted substrates. Some of these target proteins are themselves kinases or phosphatases or other components of a regulatory network characterized by feedback and feed-forward loops. In this review we describe some common regulatory motifs involving kinases, phosphatases, and their substrates, focusing particularly on bistable switches involved in cellular decision processes. These general principles are applied to cell cycle transitions, with special emphasis on the roles of regulated phosphatases in orchestrating progression from one phase to the next of the DNA replication-division cycle.

Making cellular memories

The induction of a protracted response to a brief stimulus is a form of cellular memory. Here we describe the role of transcriptional regulation in both natural and synthetic memory networks and discuss the potential applications of engineered memory networks in medicine and industrial biotechnology.

Simple, realistic models of complex biological processes: positive feedback and bistability in a cell fate switch and a cell cycle oscillator

Here we review some of our work over the last decade on Xenopus oocyte maturation, a cell fate switch, and the Xenopus embryonic cell cycle, a highly dynamical process. Our approach has been to start with wiring diagrams for the regulatory networks that underpin the processes; carry out quantitative experiments to describe the response functions for individual legs of the networks; and then construct simple analytical models based on chemical kinetic theory and the graphical rate-balance formalism. These studies support the view that the all-or-none, irreversible nature of oocyte maturation arises from a saddle-node bifurcation in the regulatory system that drives the process, and that the clock-like oscillations of the embryo are built upon a hysteretic switch with two saddle-node bifurcations. We believe that this type of reductionistic systems biology holds great promise for understanding complicated biochemical processes in simpler terms.

Logical modelling of cell cycle control in eukaryotes: a comparative study

Dynamical modelling is at the core of the systems biology paradigm. However, the development of comprehensive quantitative models is complicated by the daunting complexity of regulatory networks controlling crucial biological processes such as cell division, the paucity of currently available quantitative data, as well as the limited reproducibility of large-scale experiments. In this context, qualitative modelling approaches offer a useful alternative or complementary framework to build and analyse simplified, but still rigorous dynamical models. This point is illustrated here by analysing recent logical models of the molecular network controlling mitosis in different organisms, from yeasts to mammals. After a short introduction covering cell cycle and logical modelling, we compare the assumptions and properties underlying these different models. Next, leaning on their transposition into a common logical framework, we compare their functional structure in terms of regulatory circuits. Finally, we discuss assets and prospects of qualitative approaches for the modelling of the cell cycle.

Robust, tunable biological oscillations from interlinked positive and negative feedback loops

A simple negative feedback loop of interacting genes or proteins has the potential to generate sustained oscillations. However, many biological oscillators also have a positive feedback loop, raising the question of what advantages the extra loop imparts. Through computational studies, we show that it is generally difficult to adjust a negative feedback oscillator's frequency without compromising its amplitude, whereas with positive-plus-negative feedback, one can achieve a widely tunable frequency and near-constant amplitude. This tunability makes the latter design suitable for biological rhythms like heartbeats and cell cycles that need to provide a constant output over a range of frequencies. Positive-plus-negative oscillators also appear to be more robust and easier to evolve, rationalizing why they are found in contexts where an adjustable frequency is unimportant.

Rapid cycling and precocious termination of G1 phase in cells expressing CDK1AF

In Xenopus embryos, the cell cycle is driven by an autonomous biochemical oscillator that controls the periodic activation and inactivation of cyclin B1-CDK1. The oscillator circuit includes a system of three interlinked positive and double-negative feedback loops (CDK1 -> Cdc25 -> CDK1; CDK1 -/ Wee1 -/ CDK1; and CDK1 -/ Myt1 -/ CDK1) that collectively function as a bistable trigger. Previous work established that this bistable trigger is essential for CDK1 oscillations in the early embryonic cell cycle. Here, we assess the importance of the trigger in the somatic cell cycle, where checkpoints and additional regulatory mechanisms could render it dispensable. Our approach was to express the phosphorylation site mutant CDK1AF, which short-circuits the feedback loops, in HeLa cells, and to monitor cell cycle progression by live cell fluorescence microscopy. We found that CDK1AF-expressing cells carry out a relatively normal first mitosis, but then undergo rapid cycles of cyclin B1 accumulation and destruction at intervals of 3-6 h. During these cycles, the cells enter and exit M phase-like states without carrying out cytokinesis or karyokinesis. Phenotypically similar rapid cycles were seen in Wee1 knockdown cells. These findings show that the interplay between CDK1, Wee1/Myt1, and Cdc25 is required for the establishment of G1 phase, for the normal approximately 20-h cell cycle period, and for the switch-like oscillations in cyclin B1 abundance characteristic of the somatic cell cycle. We propose that the HeLa cell cycle is built upon an unreliable negative feedback oscillator and that the normal high reliability, slow pace and switch-like character of the cycle is imposed by a bistable CDK1/Wee1/Myt1/Cdc25 system.

Ras nanoclusters: combining digital and analog signaling

Cellular signaling pathways respond to external inputs to drive pivotal cellular decisions. Far from being mere data relay systems, signaling cascades form complex interacting networks with multiple layers of feedback and feed-forward control loops regulated in both space and time. While it may be intuitively obvious that this complexity allows cells to assess and respond appropriately to a myriad of external cues, untangling the wires to understand precisely how complex networks function as control and computational systems presents a daunting challenge to theoretical and experimental biologists alike. In this review we have focused on activation of the canonical MAP kinase cascade by receptor tyrosine kinases (RTKs) in order to examine some of the fundamental design principles used to build biological circuits and control systems. In particular, we explore how cells can reconfigure signaling cascades to generate distinct biological outputs by utilizing the unique spatial constraints available in biological membranes.

Cyclin B1-Cdk1 activation continues after centrosome separation to control mitotic progression

Activation of cyclin B1–cyclin-dependent kinase 1 (Cdk1), triggered by a positive feedback loop at the end of G2, is the key event that initiates mitotic entry. In metaphase, anaphase-promoting complex/cyclosome–dependent destruction of cyclin B1 inactivates Cdk1 again, allowing mitotic exit and cell division. Several models describe Cdk1 activation kinetics in mitosis, but experimental data on how the activation proceeds in mitotic cells have largely been lacking. We use a novel approach to determine the temporal development of cyclin B1–Cdk1 activity in single cells. By quantifying both dephosphorylation of Cdk1 and phosphorylation of the Cdk1 target anaphase-promoting complex/cyclosome 3, we disclose how cyclin B1–Cdk1 continues to be activated after centrosome separation. Importantly, we discovered that cytoplasmic cyclin B1–Cdk1 activity can be maintained even when cyclin B1 translocates to the nucleus in prophase. These experimental data are fitted into a model describing cyclin B1–Cdk1 activation in human cells, revealing a striking resemblance to a bistable circuit. In line with the observed kinetics, cyclin B1–Cdk1 levels required to enter mitosis are lower than the amount of cyclin B1–Cdk1 needed for mitotic progression. We propose that gradually increasing cyclin B1–Cdk1 activity after centrosome separation is critical to coordinate mitotic progression.

When active, the enzyme cyclin B1–cyclin-dependent kinase 1 (Cdk1) commits a growing cell to the process of mitotic cell division and chromosome separation. Cyclin B1–Cdk1 activation is controlled in many ways, but once its activity rises above a certain level, further activation of cyclin B1–Cdk1 is catalyzed by a positive-feedback loop. This generates highly active cyclin B1–Cdk1 and triggers the start of mitosis, which can only be completed when cyclin B1–Cdk1 activity is properly shut off again. However, it is not clear how cyclin B1–Cdk1 activity develops in human cells or how the switch between its inactive and active states is controlled. Our work combines activation measurements with a kinetic model to study how cyclin B1–Cdk1 activity accumulates just before and during mitosis. We show that cyclin B1–Cdk1 activity develops gradually in early mitosis and that different activity levels are required for initiation of, and progression through, mitosis. We also demonstrate that once cyclin B1–Cdk1 activation is truly launched, it is bound to continue and will not lightly drop back again. We propose that the successive cyclin B1–Cdk1 activity levels by themselves may coordinate the progression through the distinct phases of mitosis.

The gradual increase of cyclin B1-Cdk1 activation in human cells is proposed to be critical for the progression of mitosis.

Mathematical modeling as a tool for investigating cell cycle control networks

Although not a traditional experimental "method," mathematical modeling can provide a powerful approach for investigating complex cell signaling networks, such as those that regulate the eukaryotic cell division cycle. We describe here one modeling approach based on expressing the rates of biochemical reactions in terms of nonlinear ordinary differential equations. We discuss the steps and challenges in assigning numerical values to model parameters and the importance of experimental testing of a mathematical model. We illustrate this approach throughout with the simple and well-characterized example of mitotic cell cycles in frog egg extracts. To facilitate new modeling efforts, we describe several publicly available modeling environments, each with a collection of integrated programs for mathematical modeling. This review is intended to justify the place of mathematical modeling as a standard method for studying molecular regulatory networks and to guide the non-expert to initiate modeling projects in order to gain a systems-level perspective for complex control systems.

Substrate competition as a source of ultrasensitivity in the inactivation of Wee1

The mitotic regulators Wee1 and Cdk1 can inactivate each other through inhibitory phosphorylations. This double-negative feedback loop is part of a bistable trigger that makes the transition into mitosis abrupt and decisive. To generate a bistable response, some component of a double-negative feedback loop must exhibit an ultrasensitive response to its upstream regulator. Here, we experimentally demonstrate that Wee1 exhibits a highly ultrasensitive response to Cdk1. Several mechanisms can, in principle, give rise to ultrasensitivity, including zero-order effects, multisite phosphorylation, and competition mechanisms. We found that the ultrasensitivity in the inactivation of Wee1 arises mainly through two competition mechanisms: competition between two sets of phosphorylation sites in Wee1 and between Wee1 and other high-affinity Cdk1 targets. Based on these findings, we were able to reconstitute a highly ultrasensitive Wee1 response with purified components. Competition provides a simple way of generating the equivalent of a highly cooperative allosteric response.

The role of modelling in identifying drug targets for diseases of the cell cycle

The cell cycle is implicated in diseases that are the leading cause of mortality and morbidity in the developed world. Until recently, the search for drug targets has focused on relatively small parts of the regulatory network under the assumption that key events can be controlled by targeting single pathways. This is valid provided the impact of couplings to the wider scale context of the network can be ignored. The resulting depth of study has revealed many new insights; however, these have been won at the expense of breadth and a proper understanding of the consequences of links between the different parts of the network. Since it is now becoming clear that these early assumptions may not hold and successful treatments are likely to employ drugs that simultaneously target a number of different sites in the regulatory network, it is timely to redress this imbalance. However, the substantial increase in complexity presents new challenges and necessitates parallel theoretical and experimental approaches. We review the current status of theoretical models for the cell cycle in light of these new challenges. Many of the existing approaches are not sufficiently comprehensive to simultaneously incorporate the required extent of couplings. Where more appropriate levels of complexity are incorporated, the models are difficult to link directly to currently available data. Further progress requires a better integration of experiment and theory. New kinds of data are required that are quantitative, have a higher temporal resolution and that allow simultaneous quantitative comparison of the concentration of larger numbers of different proteins. More comprehensive models are required and must accommodate not only substantial uncertainties in the structure and kinetic parameters of the networks, but also high levels of ignorance. The most recent results relating network complexity to robustness of the dynamics provide clues that suggest progress is possible.

Multisite M-phase phosphorylation of Xenopus Wee1A

The Cdk1 inhibitor Wee1 is inactivated during mitotic entry by proteolysis, translational regulation, and transcriptional regulation. Wee1 is also regulated by posttranslational modifications, and here we have identified five phosphorylation sites in the N-terminal domain of embryonic Xenopus Wee1A through a combination of mutagenesis studies and matrix-assisted laser desorption ionization-time of flight mass spectrometry. All five sites conform to the Ser-Pro/Thr-Pro consensus for proline-directed kinases like Cdks. Three of the sites (Ser 38, Thr 53, and Ser 62) are required for the mitotic gel shift, and at least two of these sites (Ser 38 and Thr 53) regulate the proteolysis of Wee1A during interphase. The other two sites (Thr 104 and Thr 150) are primarily responsible for the mitotic inactivation of Wee1A. Alanine mutants of Thr 150 or Thr 104 had an increased capacity to inhibit mitotic entry in cyclin B-treated interphase extracts, and Thr 150 was found to be transiently phosphorylated just prior to nuclear envelope breakdown in cycling egg extracts. These findings establish the phosphorylation-dependent direct inactivation of Wee1A as a critical mechanism for the promotion of M-phase entry. These results also show that multisite phosphorylation cooperatively inactivates Wee1A and cooperatively promotes Wee1A proteolysis.

Systems-Level Dissection of the Cell-Cycle Oscillator: Bypassing Positive Feedback Produces Damped Oscillations

The cell-cycle oscillator includes an essential negative-feedback loop: Cdc2 activates the anaphase-promoting complex (APC), which leads to cyclin destruction and Cdc2 inactivation. Under some circumstances, a negative-feedback loop is sufficient to generate sustained oscillations. However, the Cdc2/APC system also includes positive-feedback loops, whose functional importance we now assess. We show that short-circuiting positive feedback makes the oscillations in Cdc2 activity faster, less temporally abrupt, and damped. This compromises the activation of cyclin destruction and interferes with mitotic exit and DNA replication. This work demonstrates a systems-level role for positive-feedback loops in the embryonic cell cycle and provides an example of how oscillations can emerge out of combinations of subcircuits whose individual behaviors are not oscillatory. This work also underscores the fundamental similarity of cell-cycle oscillations in embryos to repetitive action potentials in pacemaker neurons, with both systems relying on a combination of negative and positive-feedback loops.

Subcellular Localization Determines MAP Kinase Signal Output

The Raf-MEK-ERK MAP kinase cascade transmits signals from activated receptors into the cell to regulate proliferation and differentiation. The cascade is controlled by the Ras GTPase, which recruits Raf from the cytosol to the plasma membrane for activation. In turn, MEK, ERK, and scaffold proteins translocate to the plasma membrane for activation. Here, we examine the input-output properties of the Raf-MEK-ERK MAP kinase module in mammalian cells activated in different cellular contexts. We show that the MAP kinase module operates as a molecular switch in vivo but that the input sensitivity of the module is determined by subcellular location. Signal output from the module is sensitive to low-level input only when it is activated at the plasma membrane. This is because the threshold for activation is low at the plasma membrane, whereas the threshold for activation is high in the cytosol. Thus, the circuit configuration of the module at the plasma membrane generates maximal outputs from low-level analog inputs, allowing cells to process and respond appropriately to physiological stimuli. These results reveal the engineering logic behind the recruitment of elements of the module from the cytosol to the membrane for activation.

Quantitative analysis of signaling networks

The response of biological cells to environmental change is coordinated by protein-based signaling networks. These networks are to be found in both prokaryotes and eukaryotes. In eukaryotes, the signaling networks can be highly complex, some networks comprising of 60 or more proteins. The fundamental motif that has been found in all signaling networks is the protein phosphorylation/dephosphorylation cycle--the cascade cycle. At this time, the computational function of many of the signaling networks is poorly understood. However, it is clear that it is possible to construct a huge variety of control and computational circuits, both analog and digital from combinations of the cascade cycle. In this review, we will summarize the great versatility of the simple cascade cycle as a computational unit and towards the end give two examples, one prokaryotic chemotaxis circuit and the other, the eukaryotic MAPK cascade.

Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems

It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable mitogen-activated protein kinase cascade.

Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2

In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, whose robust biochemical rhythm continues even when DNA replication or mitosis is blocked. At the core of the oscillator is a negative feedback loop; cyclins accumulate and produce active mitotic Cdc2-cyclin B; Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback, with active Cdc2-cyclin B stimulating its activator Cdc25 (refs 5-7) and inactivating its inhibitors Wee1 and Myt1 (refs 8-11). Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. Therefore, we examined whether Cdc2 activation is bistable. We confirm that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switch-like, as would be expected if Cdc2 activation were bistable. We also show that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.

Regulation of the mammalian cell cycle: a model of the G1-to-S transition

We have formulated a mathematical model for regulation of the G(1)-to-S transition of the mammalian cell cycle. This mathematical model incorporates the key molecules and interactions that have been identified experimentally. By subdividing these critical molecules into modules, we have been able to systematically analyze the contribution of each to dynamics of the G(1)-to-S transition. The primary module, which includes the interactions between cyclin E (CycE), cyclin-dependent kinase 2 (CDK2), and protein phosphatase CDC25A, exhibits dynamics such as limit cycle, bistability, and excitable transient. The positive feedback between CycE and transcription factor E2F causes bistability, provided that the total E2F is constant and the retinoblastoma protein (Rb) can be hyperphosphorylated. The positive feedback between active CDK2 and cyclin-dependent kinase inhibitor (CKI) generates a limit cycle. When combined with the primary module, the E2F/Rb and CKI modules potentiate or attenuate the dynamics generated by the primary module. In addition, we found that multisite phosphorylation of CDC25A, Rb, and CKI was critical for the generation of dynamics required for cell cycle progression.

Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability

Cell signaling systems that contain positive-feedback loops or double-negative feedback loops can, in principle, convert graded inputs into switch-like, irreversible responses. Systems of this sort are termed "bistable". Recently, several groups have engineered artificial bistable systems into Escherichia coli and Saccharomyces cerevisiae, and have shown that the systems exhibit interesting and potentially useful properties. In addition, two naturally occurring signaling systems, the p42 mitogen-activated protein kinase and c-Jun amino-terminal kinase pathways in Xenopus oocytes, have been shown to exhibit bistable responses. Here we review the basic properties of bistable circuits, the requirements for construction of a satisfactory bistable switch, and the recent progress towards constructing and analysing bistable signaling systems.

Bistability in the JNK cascade

BACKGROUND

Important signaling properties, like adaptation, oscillations, and bistability, can emerge at the level of relatively simple systems of signaling proteins. Here, we have examined the quantitative properties of one well-studied signaling system, the JNK cascade. We experimentally assessed the response of JNK to a physiological stimulus (progesterone) and a pathological stress (hyperosmolar sorbitol) in Xenopus laevis oocytes, a cell type that is well-suited to the quantitative analysis of cell signaling. Our aim was to determine whether JNK responses are graded (Michaelian) in character; ultrasensitive in character, resembling the responses of cooperative enzymes; or bistable and all-or-none in character.

RESULTS

The responses of JNK to both progesterone and sorbitol were found to be essentially all-or-none. Individual oocytes had either very high or very low JNK activities, with few oocytes possessing intermediate levels of JNK activity. Moreover, JNK activation was autocatalytic, indicating that the JNK cascade is either embedded in or downstream of a positive feedback loop. JNK also exhibited hysteresis, a form of biochemical memory, in its response to sorbitol. These findings indicate that the JNK cascade is part of a bistable signaling system in oocytes.

CONCLUSIONS

In Xenopus oocytes, JNK responds to physiological and pathological stimuli in an all-or-none manner. The JNK response shows all the hallmarks of a bistable response, including strong positive feedback and hysteresis. Bistability is a recurring theme in the biochemistry of oocyte maturation and early embryogenesis; the Mos/MEK/p42 MAPK cascade also exhibits bistable responses, and the Cdc2/cyclin B system is hypothesized to be bistable as well. However, the mechanisms underpinning the positive feedback and bistability in the three cases are different, suggesting that evolution has repeatedly converged upon bistability as a way of producing digital responses.

Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible

Xenopus oocyte maturation is an example of an all-or-none, irreversible cell fate induction process. In response to a submaximal concentration of the steroid hormone progesterone, a given oocyte may either mature or not mature, but it can exist in intermediate states only transiently. Moreover, once an oocyte has matured, it will remain arrested in the mature state even after the progesterone is removed. It has been hypothesized that the all-or-none character of oocyte maturation, and some aspects of the irreversibility of maturation, arise out of the bistability of the signal transduction system that triggers maturation. The bistability, in turn, is hypothesized to arise from the way the signal transducers are organized into a signaling circuit that includes positive feedback (which makes it so that the system cannot rest in intermediate states) and ultrasensitivity (which filters small stimuli out of the feedback loop, allowing the system to have a stable off-state). Here we review two simple graphical methods that are commonly used to analyze bistable systems, discuss the experimental evidence for bistability in oocyte maturation, and suggest that bistability may be a common means of producing all-or-none responses and a type of biochemical memory. (c) 2001 American Institute of Physics.

Synchronization by irregular inactivation

Many natural and technological systems have on/off switches. For instance, mitosis can be halted by biochemical switches which act through the phosphorylation state of a complex called mitosis promoting factor. If switching between the on and off states is periodic, chaos is observed over a substantial portion of the on/off time parameter plane. However, we have discovered that the chaotic state is fragile with respect to random fluctuations in the on time. In the presence of such fluctuations, two uncoupled copies of the system (e.g., two cells) controlled by the same switch rapidly synchronize.

How to make a biological switch

Some biological regulatory systems must "remember" a state for long periods of time. A simple type of system that can accomplish this task is one in which two regulatory elements negatively regulate one another. For example, two repressor proteins might control one another's synthesis. Qualitative reasoning suggests that such a system will have two stable states, one in which the first element is "on" and the second "off", and another in which these states are reversed. Quantitative analysis shows that the existence of two stable steady states depends on the details of the system. Among other things, the shapes of functions describing the effect of one regulatory element on the other must meet certain criteria in order for two steady states to exist. Many biologically reasonable functions do not meet these criteria. In particular, repression that is well described by a Michaelis-Menten-type equation cannot lead to a working switch. However, functions describing positive cooperativity of binding, non-additive effects of multiple operator sites, or depletion of free repressor can lead to working switches.

Alternating oscillations and chaos in a model of two coupled biochemical oscillators driving successive phases of the cell cycle

The animal cell cycle is controlled by the periodic variation of two cyclin-dependent protein kinases, cdk1 and cdk2, which govern the entry into the M (mitosis) and S (DNA replication) phases, respectively. The ordered progression between these phases is achieved thanks to the existence of checkpoint mechanisms based on mutual inhibition of these processes. Here we study a simple theoretical model for oscillations in cdk1 and cdk2 activity, involving mutual inhibition of the two oscillators. Each minimal oscillator is described by a three-variable cascade involving a cdk, together with the associated cyclin and cyclin-degrading enzyme. The dynamics of this skeleton model of coupled oscillators is determined as a function of the strength of their mutual inhibition. The most common mode of dynamic behavior, obtained under conditions of strong mutual inhibition, is that of alternating oscillations in cdk1 and cdk2, which correspond to the physiological situation of the ordered recurrence of the M and S phases. In addition, for weaker inhibition we obtain evidence for a variety of dynamic phenomena such as complex periodic oscillations, chaos, and the coexistence between multiple periodic or chaotic attractors. We discuss the conditions of occurrence of these various modes of oscillatory behavior, as well as their possible physiological significance.

Mathematical analysis of binary activation of a cell cycle kinase which down-regulates its own inhibitor

In mammalian cells, the heterodimeric kinase cyclin E/CDK2 (EK2) mediates cell cycle progress from G1 phase into S phase. The protein p27Kip1 (p27) binds to and inhibits EK2; but EK2 can phosphorylate p27, and that leads to the deactivation of p27, presumably liberating more EK2 and forming a positive-feedback loop. It has been proposed that this positive-feedback loop gives rise to binary (all-or-none) release of EK2 from its inactive complex with p27. Binary release suggests a bistable biochemical system in which a stable steady state with low EK2 activity is extinguished in a saddle-node bifurcation, causing the system to shift abruptly to a stable steady state with high EK2 activity. Two mathematical models are discussed, one in which free EK2 deactivates p27 in the EK2-p27 inhibitory complex as well as free p27, and one in which the rate of EK2-catalyzed deactivation of free p27 has saturable kinetics with respect to free p27. In general, if inhibitory binding is approximately in equilibrium, bistability requires that there be a potential unstable steady state where the reaction order of p27 deactivation is greater with respect to EK2 than with respect to p27.

Models of cell cycle control in eukaryotes

The molecular mechanisms of cell cycle control are now known in enough detail to warrant mathematical modeling by kinetic equations. Despite the repetitive nature of the cell division cycle, the most appropriate models emphasize steady-state solutions rather than limit cycle oscillations, because cells progress toward division by passing a series of checkpoints (steady states).

Mathematical model of the fission yeast cell cycle with checkpoint controls at the G1/S, G2/M and metaphase/anaphase transitions

All events of the fission yeast cell cycle can be orchestrated by fluctuations of a single cyclin-dependent protein kinase, the Cdc13/Cdc2 heterodimer. The G1/S transition is controlled by interactions of Cdc13/Cdc2 and its stoichiometric inhibitor, Rum1. The G2/M transition is regulated by a kinase-phosphatase pair, Wee1 and Cdc25, which determine the phosphorylation state of the Tyr-15 residue of Cdc2. The meta/anaphase transition is controlled by interactions between Cdc13/Cdc2 and the anaphase promoting complex, which labels Cdc13 subunits for proteolysis. We construct a mathematical model of fission yeast growth and division that encompasses all three crucial checkpoint controls. By numerical simulations we show that the model is consistent with a broad selection of cell cycle mutants, and we predict the phenotypes of several multiple-mutant strains that have not yet been constructed.