A theory for controlling cell cycle dynamics using a reversibly binding inhibitor

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.

Application of Lie Symmetry to a Mathematical Model that Describes a Cancer Sub-Network

In this paper, a mathematical model of a cancer sub-network is analysed from the view point of Lie symmetry methods. This model discusses a human cancer cell which is developed due to the dysfunction of some genes at the R-checkpoint during the cell cycle. The primary purpose of this paper is to apply the techniques of Lie symmetry to the model and present some approximated solutions for the three-dimensional system of first-order ordinary differential equations describing a cancer sub-network. The result shows that the phosphatase gene (Cdc25A) regulates the cyclin-dependent kinases inhibitor (P27Kip1). Furthermore, this research discovered that the activity that reverses the inhibitory effects on cell cycle progression at the R-checkpoint initiates a pathway.

"Life-bearing molecules" versus "life-embodying systems": Two contrasting views on the what-is-life (WIL) problem persisting from the early days of molecular biology to the post-genomic cell- and organism-level biology

"What is life?" is an ultimate biological quest for the principle that makes organisms alive. This 'WIL problem' is not, however, a simple one that we have a straightforward strategy to attack. From the beginning, molecular biology tried to identify molecules that bear the essence of life: the double helical DNA represented replication, and enzymes were micro-actuators of biological activities. A dominating idea behind these mainstream biological studies relies on the identification of life-bearing molecules, which themselves are models of life. Another, prevalent idea emphasizes that life resides in the whole system of an organism, but not in some particular molecules. The behavior of a complex system may be considered to embody the essence of life. The thermodynamic view of life system in the early 20th century was remodeled as physics of complex systems and systems biology. The two views contrast with each other, but they are no longer heritage of the historical dualism in biology, such as mechanism/materialism versus vitalism, or reductionism versus holism. These two views are both materialistic and mechanistic, and act as driving forces of modern biology. In reality, molecules function in a context of systems, whereas systems presuppose functional molecules. A key notion to reconcile this conflict is that subjects of biological studies are given before we start to study them. Cell- or organism-level biology is destined to the dialectic of molecules and systems, but this antagonism can be resolved by dynamic thinking involving biological evolution.

Modeling and simulation of biological systems using SPICE language

The article deals with BB-SPICE (SPICE for Biochemical and Biological Systems), an extension of the famous Simulation Program with Integrated Circuit Emphasis (SPICE). BB-SPICE environment is composed of three modules: a new textual and compact description formalism for biological systems, a converter that handles this description and generates the SPICE netlist of the equivalent electronic circuit and NGSPICE which is an open-source SPICE simulator. In addition, the environment provides back and forth interfaces with SBML (System Biology Markup Language), a very common description language used in systems biology. BB-SPICE has been developed in order to bridge the gap between the simulation of biological systems on the one hand and electronics circuits on the other hand. Thus, it is suitable for applications at the interface between both domains, such as development of design tools for synthetic biology and for the virtual prototyping of biosensors and lab-on-chip. Simulation results obtained with BB-SPICE and COPASI (an open-source software used for the simulation of biochemical systems) have been compared on a benchmark of models commonly used in systems biology. Results are in accordance from a quantitative viewpoint but BB-SPICE outclasses COPASI by 1 to 3 orders of magnitude regarding the computation time. Moreover, as our software is based on NGSPICE, it could take profit of incoming updates such as the GPU implementation, of the coupling with powerful analysis and verification tools or of the integration in design automation tools (synthetic biology).

Discerning mechanistically rewired biological pathways by cumulative interaction heterogeneity statistics

Changes in response of a biological pathway could be a consequence of either pathway rewiring, changed input, or a combination of both. Most pathway analysis methods are not designed for mechanistic rewiring such as regulatory element variations. This limits our understanding of biological pathway evolution. Here we present a Q-method to discern whether changed pathway response is caused by mechanistic rewiring of pathways due to evolution. The main innovation is a cumulative pathway interaction heterogeneity statistic accounting for rewiring-specific effects on the rate of change of each molecular variable across conditions. The Q-method remarkably outperformed differential-correlation based approaches on data from diverse biological processes. Strikingly, it also worked well in differentiating rewired chaotic systems, whose dynamics are notoriously difficult to predict. Applying the Q-method on transcriptome data of four yeasts, we show that pathway interaction heterogeneity for known metabolic and signaling pathways is indeed a predictor of interspecies genetic rewiring due to unbalanced TATA box-containing genes among the yeasts. The demonstrated effectiveness of the Q-method paves the way to understanding network evolution at the resolution of functional biological pathways.

A data-driven, mathematical model of mammalian cell cycle regulation

Few of >150 published cell cycle modeling efforts use significant levels of data for tuning and validation. This reflects the difficultly to generate correlated quantitative data, and it points out a critical uncertainty in modeling efforts. To develop a data-driven model of cell cycle regulation, we used contiguous, dynamic measurements over two time scales (minutes and hours) calculated from static multiparametric cytometry data. The approach provided expression profiles of cyclin A2, cyclin B1, and phospho-S10-histone H3. The model was built by integrating and modifying two previously published models such that the model outputs for cyclins A and B fit cyclin expression measurements and the activation of B cyclin/Cdk1 coincided with phosphorylation of histone H3. The model depends on Cdh1-regulated cyclin degradation during G1, regulation of B cyclin/Cdk1 activity by cyclin A/Cdk via Wee1, and transcriptional control of the mitotic cyclins that reflects some of the current literature. We introduced autocatalytic transcription of E2F, E2F regulated transcription of cyclin B, Cdc20/Cdh1 mediated E2F degradation, enhanced transcription of mitotic cyclins during late S/early G2 phase, and the sustained synthesis of cyclin B during mitosis. These features produced a model with good correlation between state variable output and real measurements. Since the method of data generation is extensible, this model can be continually modified based on new correlated, quantitative data.

Creating biological nanomaterials using synthetic biology

Synthetic biology is a new discipline that combines science and engineering approaches to precisely control biological networks. These signaling networks are especially important in fields such as biomedicine and biochemical engineering. Additionally, biological networks can also be critical to the production of naturally occurring biological nanomaterials, and as a result, synthetic biology holds tremendous potential in creating new materials. This review introduces the field of synthetic biology, discusses how biological systems naturally produce materials, and then presents examples and strategies for incorporating synthetic biology approaches in the development of new materials. In particular, strategies for using synthetic biology to produce both organic and inorganic nanomaterials are discussed. Ultimately, synthetic biology holds the potential to dramatically impact biological materials science with significant potential applications in medical systems.

Mathematical modeling of fission yeast Schizosaccharomyces pombe cell cycle: exploring the role of multiple phosphatases

Cell cycle is the central process that regulates growth and division in all eukaryotes. Based on the environmental condition sensed, the cell lies in a resting phase G0 or proceeds through the cyclic cell division process (G1→S→G2→M). These series of events and phase transitions are governed mainly by the highly conserved Cyclin dependent kinases (Cdks) and its positive and negative regulators. The cell cycle regulation of fission yeast Schizosaccharomyces pombe is modeled in this study. The study exploits a detailed molecular interaction map compiled based on the published model and experimental data. There are accumulating evidences about the prominent regulatory role of specific phosphatases in cell cycle regulations. The current study emphasizes the possible role of multiple phosphatases that governs the cell cycle regulation in fission yeast S. pombe. The ability of the model to reproduce the reported regulatory profile for the wild-type and various mutants was verified though simulations.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s11693-011-9090-7) contains supplementary material, which is available to authorized users.

Modelling the size and composition of fruit, grain and seed by process-based simulation models

Understanding what determines the size and composition of fruit, grain and seed in response to the environment and genotype is challenging, as these traits result from several linked processes controlled at different levels of organization, from the subcellular to the crop level, with subtle interactions occurring at or between the levels of organization. Process-based simulation models (PBSMs) implement algorithms to simulate metabolic and biophysical aspects of cell, tissue and organ behaviour. In this review, fruit, grain and seed PBSMs describing the main phases of growth, development and storage metabolism are discussed. From this concurrent work, it is possible to identify generic storage organ processes which can be modelled similarly for fruit, grain and seed. Spatial heterogeneity at the tissue and whole-plant level is found to be a key consideration in modelling the effects of the environment and genotype on fruit, grain and seed end-use value. In the future, PBSMs may well become the main link between studies at the molecular and whole-plant levels. To bridge this phenotype-to-genotype gap, future models need to remain plastic without becoming overparameterized.

The ERECTA gene controls spatial and temporal patterns of epidermal cell number and size in successive developing leaves of Arabidopsis thaliana

BACKGROUND AND AIMS

ERECTA has been identified as a pleiotropic regulator of developmental and physiological processes in Arabidopsis thaliana. Previous work demonstrated a role for ERECTA in the control of compensation between epidermal cell expansion and division in leaves.

METHODS

In this work, spatial and temporal analyses of epidermal cell division and expansion were performed on successive developing vegetative leaves of Arabidopsis thaliana in both ERECTA and erecta lines, LER and Ler, respectively, to understand how the ERECTA gene regulates compensation between these two processes.

KEY RESULTS

The loss of ERECTA function leads to a low cell expansion rate in all zones of a leaf and in all successive leaves of a plant. This low cell expansion rate is counterbalanced by an increase in the duration of cell division. As a consequence, the ERECTA mutation eliminates the tip to base cellular gradient generally observed in the leaf epidermis and also flattens the heteroblastic changes in epidermal cell area and number within a rosette. Ablation of floral buds eliminates the heteroblastic changes in cellular patterns in an ERECTA-dependent manner.

CONCLUSIONS

The results provide a detailed description of changes in leaf growth dynamics and cellular variables in both LER and Ler. Altogether they suggest that ERECTA influences leaf cellular development in relation to whole plant ontogeny.

Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle

The cell cycle is regulated by a large number of enzymes and transcription factors. We have developed a modular description of the cell cycle, based on a set of interleaved modular feedback loops, each leading to a cyclic behavior. The slowest loop is the E2F transcription and ubiquitination, which determines the cycling frequency of the entire cell cycle. Faster feedback loops describe the dynamics of each Cyclin by itself. Our model shows that the cell cycle progression as well as the checkpoints of the cell cycle can be understood through the interactions between the main E2F feedback loop and the driven Cyclin feedback loops. Multiple models were proposed for the cell cycle dynamics; each with differing basic mechanisms. We here propose a new generic formalism. In contrast with existing models, the proposed formalism allows a straightforward analysis and understanding of the dynamics, neglecting the details of each interaction. This model is not sensitive to small changes in the parameters used and it reproduces the observed behavior of the transcription factor E2F and different Cyclins in continuous or regulated cycling conditions. The modular description of the cell cycle resolves the gap between cyclic models, solely based on protein-protein reactions and transcription reactions based models. Beyond the explanation of existing observations, this model suggests the existence of unknown interactions, such as the need for a functional interaction between Cyclin B and retinoblastoma protein (Rb) de-phosphorylation.

Design of regulation and dynamics in simple biochemical pathways

Complex regulation of biochemical pathways in a cell is brought about by the interaction of simpler regulatory structures. Among the basic regulatory designs, feedback inhibition of gene expression is the most common motif in gene regulation and a ubiquitous control structure found in nature. In this work, we have studied a common structural feature (delayed feedback) in gene organisation and shown, both theoretically and experimentally, its subtle but important functional role in gene expression kinetics in a negatively auto-regulated system. Using simple deterministic and stochastic models with varying levels of realism, we present detailed theoretical representations of negatively auto-regulated transcriptional circuits with increasing delays in the establishment of feedback of repression. The models of the circuits with and without delay are studied analytically as well as numerically for variation of parameters and delay lengths. The positive invariance, boundedness of the solutions, local and global asymptotic stability of both the systems around the unique positive steady state are studied analytically. Existence of transient temporal dynamics is shown mathematically. Comparison of the two types of model circuits shows that even though the long-term dynamics is stable and not affected by delays in repression, there is interesting variation in the transient dynamical features with increasing delays. Theoretical predictions are validated through experimentally constructed gene circuits of similar designs. This combined theoretical and experimental study helps delineate the opposing effects of delay-induced instability, and the stability-enhancing property of negative feedback in the pathway behaviour, and gives rationale for the abundance of similar designs in real biochemical pathways.

Computational systems biology of the cell cycle

One of the early success stories of computational systems biology was the work done on cell-cycle regulation. The earliest mathematical descriptions of cell-cycle control evolved into very complex, detailed computational models that describe the regulation of cell division in many different cell types. On the way these models predicted several dynamical properties and unknown components of the system that were later experimentally verified/identified. Still, research on this field is far from over. We need to understand how the core cell-cycle machinery is controlled by internal and external signals, also in yeast cells and in the more complex regulatory networks of higher eukaryotes. Furthermore, there are many computational challenges what we face as new types of data appear thanks to continuing advances in experimental techniques. We have to deal with cell-to-cell variations, revealed by single cell measurements, as well as the tremendous amount of data flowing from high throughput machines. We need new computational concepts and tools to handle these data and develop more detailed, more precise models of cell-cycle regulation in various organisms. Here we review past and present of computational modeling of cell-cycle regulation, and discuss possible future directions of the field.

A simple time delay model for eukaryotic cell cycle

We propose a seven variable model with time delay in one of the variables for the cell cycle in higher eukaryotes. The model consists of four important phosphorylation-dephosphorylation (P-D) cycles that govern the cell cycle, namely Pre-MPF-MPF, Cdc25P-Cdc25, Wee1P-Wee1 and APCP-APC. Other variables are cyclin, free cyclin dependent kinase (Cdk) and mass. The mass acts as a G2/M checkpoint and the checkpoint is represented by a saddle node loop bifurcation. The key feature of the model is that a time lag has been introduced in the activation of anaphase promoting complex (APC) by maturation promoting factor (MPF). This is effected by treating MPF as a time-delayed variable in the activation step of APC. The time lag acts as a spindle checkpoint. Absence of time delay induces a bistability in our model. Time delay also brings about variability in G1 phase timings. The model also reproduces the mutant phenotype experiments on wee1 cells. Stochasticity has been introduced in the model to simulate the dependence of the cycle time on cell birth length. Mutant phenotypes in the stochastic model reproduce the experimental observations better than the deterministic model.

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.

Dynamics of the cell cycle: checkpoints, sizers, and timers

We have developed a generic mathematical model of a cell cycle signaling network in higher eukaryotes that can be used to simulate both the G1/S and G2/M transitions. In our model, the positive feedback facilitated by CDC25 and wee1 causes bistability in cyclin-dependent kinase activity, whereas the negative feedback facilitated by SKP2 or anaphase-promoting-complex turns this bistable behavior into limit cycle behavior. The cell cycle checkpoint is a Hopf bifurcation point. These behaviors are coordinated by growth and division to maintain normal cell cycle and size homeostasis. This model successfully reproduces sizer, timer, and the restriction point features of the eukaryotic cell cycle, in addition to other experimental findings.

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.

Biological timing and the clock metaphor: oscillatory and hourglass mechanisms

Living organisms have developed a multitude of timing mechanisms--"biological clocks." Their mechanisms are based on either oscillations (oscillatory clocks) or unidirectional processes (hourglass clocks). Oscillatory clocks comprise circatidal, circalunidian, circadian, circalunar, and circannual oscillations--which keep time with environmental periodicities--as well as ultradian oscillations, ovarian cycles, and oscillations in development and in the brain, which keep time with biological timescales. These clocks mainly determine time points at specific phases of their oscillations. Hourglass clocks are predominantly found in development and aging and also in the brain. They determine time intervals (duration). More complex timing systems combine oscillatory and hourglass mechanisms, such as the case for cell cycle, sleep initiation, or brain clocks, whereas others combine external and internal periodicities (photoperiodism, seasonal reproduction). A definition of a biological clock may be derived from its control of functions external to its own processes and its use in determining temporal order (sequences of events) or durations. Biological and chemical oscillators are characterized by positive and negative feedback (or feedforward) mechanisms. During evolution, living organisms made use of the many existing oscillations for signal transmission, movement, and pump mechanisms, as well as for clocks. Some clocks, such as the circadian clock, that time with environmental periodicities are usually compensated (stabilized) against temperature, whereas other clocks, such as the cell cycle, that keep time with an organismic timescale are not compensated. This difference may be related to the predominance of negative feedback in the first class of clocks and a predominance of positive feedback (autocatalytic amplification) in the second class. The present knowledge of a compensated clock (the circadian oscillator) and an uncompensated clock (the cell cycle), as well as relevant models, are briefly re viewed. Hourglass clocks are based on linear or exponential unidirectional processes that trigger events mainly in the course of development and aging. An important hourglass mechanism within the aging process is the limitation of cell division capacity by the length of telomeres. The mechanism of this clock is briefly reviewed. In all clock mechanisms, thresholds at which "dependent variables" are triggered play an important role.