The dynamics of cell cycle regulation

Achieving optimal growth through product feedback inhibition in metabolism

Recent evidence suggests that the metabolism of some organisms, such as Escherichia coli, is remarkably efficient, producing close to the maximum amount of biomass per unit of nutrient consumed. This observation raises the question of what regulatory mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. We analyze several representative metabolic modules—starting from a linear pathway and advancing to a bidirectional pathway and metabolic cycle, and finally to integration of two different nutrient inputs. In each case, our mathematical analysis shows that product-feedback inhibition is not only homeostatic but also, with appropriate feedback connections, can minimize futile cycling and optimize fluxes. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pool sizes can be restricted if feedback inhibition is ultrasensitive. Indeed, the multi-layer regulation of metabolism by control of enzyme expression, enzyme covalent modification, and allostery is expected to result in such ultrasensitive feedbacks. To experimentally test whether the qualitative predictions from our analysis of feedback inhibition apply to metabolic modules beyond linear pathways, we examine the case of nitrogen assimilation in E. coli, which involves both nutrient integration and a metabolic cycle. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the network and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes in nutrient-switching experiments.

Bacteria live in remarkably diverse environments and constantly adapt to changing nutrient conditions. Recent evidence suggests that some bacteria, such as E. coli, are extraordinarily efficient in producing biomass under a variety of different nutrient conditions. This observation raises the question of what physical mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. Product-feedback inhibition is a metabolic regulatory scheme in which an end product inhibits the first dedicated step of the chain of reactions leading to its own synthesis. Our mathematical analysis of several representative metabolic modules suggests that simple feedback inhibition can indeed allow for optimal and efficient biomass production. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pools can be restricted if feedback inhibition is ultrasensitive. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the nitrogen assimilation network in E. coli and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes seen in experiments.

Global entrainment of transcriptional systems to periodic inputs

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.

The activities of living organisms are governed by complex sets of biochemical reactions. Often, entrainment to certain external signals helps control the timing and sequencing of reactions. An important open problem is to understand the onset of entrainment and under what conditions it can be ensured in the presence of uncertainties, noise, and environmental variations. In this paper, we focus mainly on transcriptional systems, modeled by Ordinary Differential Equations. These are basic building blocks for more complex biochemical systems. However, the results that we obtain are of more generality. To illustrate this generality, and to emphasize the use of our techniques in synthetic biology, we discuss the entrainment of a Repressilator circuit and the synchronization of a network of Repressilators. We answer the following two questions: 1) What are the dynamical mechanisms that ensure the entrainment to periodic inputs in transcriptional modules? 2) Starting from natural systems, what properties can be used to design novel synthetic biological circuits that can be entrained? For some biological systems which are always “in contact” with a continuously changing environment, entrainment may be a “desired” property. Thus, answering the above two questions is of fundamental importance. While entrainment may appear obvious at first thought, it is not a generic property of nonlinear dynamical systems. The main result of our paper shows that, even if the transcriptional modules are modeled by nonlinear ODEs, they can be entrained by any (positive) periodic signal. Surprisingly, such a property is preserved if the system parameters are varied: entrainment is obtained independently of the particular biochemical conditions. We prove that combinations of the above transcriptional module also show the same property. Finally, we show how the developed tools can be applied to design synthetic biochemical systems guaranteed to exhibit entrainment.

A system-based comparison of gene expression reveals alterations in oxidative stress, disruption of ubiquitin-proteasome system and altered cell cycle regulation after exposure to cadmium and methylmercury in mouse embryonic fibroblast

Environmental and occupational exposures to heavy metals such as methylmercury (MeHg) and cadmium (Cd) pose significant health risks to humans, including neurotoxicity. The underlying mechanisms of their toxicity, however, remain to be fully characterized. Our previous studies with Cd and MeHg have demonstrated that the perturbation of the ubiquitin-proteasome system (UPS) was associated with metal-induced cytotoxicity and apoptosis. We conducted a microarray-based gene expression analysis to compare metal-altered gene expression patterns with a classical proteasome inhibitor, MG132 (0.5 microM), to determine whether the disruption of the UPS is a critical mechanism of metal-induced toxicity. We treated mouse embryonic fibroblast cells at doses of MeHg (2.5 microM) and Cd (5.0 microM) for 24 h. The doses selected were based on the neutral red-based cell viability assay where initial statistically significant decreases in variability were detected. Following normalization of the array data, we employed multilevel analysis tools to explore the data, including group comparisons, cluster analysis, gene annotations analysis (gene ontology analysis), and pathway analysis using GenMAPP and Ingenuity Pathway Analysis (IPA). Using these integrated approaches, we identified significant gene expression changes across treatments within the UPS (Uchl1 and Ube2c), antioxidant and phase II enzymes (Gsta2, Gsta4, and Noq1), and genes involved in cell cycle regulation pathways (ccnb1, cdc2a, and cdc25c). Furthermore, pathway analysis revealed significant alterations in genes implicated in Parkinson's disease pathogenesis following metal exposure. This study suggests that these pathways play a critical role in the development of adverse effects associated with metal exposures.

A mathematical model for cell size control in fission yeast

Experimental investigations of cell size control in fission yeast Schizosaccharomyces pombe have illustrated that the cell cycle features 'sizer' and 'timer' phases which are distinguished by a growth rate changing point. Based on current biological knowledge of fission yeast size control, we propose here a model of ordinary differential equations (ODEs) for a possible explanation of the facts and control mechanism which is coupled with the cell cycle. Simulation results of the ODE model are demonstrated to agree with experimental data for the wild type and the cdc2-33 mutant. We show that the coupling of cell growth to cell division by translational control may account for observed properties of size control in fission yeast. As the translational control in the expression of cycle proteins Cdc13 and Cdc25 constructs positive feedback loops, the dynamical activities of the key components undergoes a rapid rising after a preliminary stage of slow increase. The coupling of this dynamical behavior to the elongation of the cell naturally gives rise to a rate change point and to 'sizer' and 'timer' phases, which characterize the cell cycle of fission yeast.

Modeling Dose-response at Low Dose: A Systems Biology Approach for Ionization Radiation

For ionization radiation (IR) induced cancer, a linear non-threshold (LNT) model at very low doses is the default used by a number of national and international organizations and in regulatory law. This default denies any positive benefit from any level of exposure. However, experimental observations and theoretical biology have found that both linear and J-shaped IR dose-response curves can exist at those very low doses. We develop low dose J-shaped dose-response, based on systems biology, and thus justify its use regarding exposure to IR. This approach incorporates detailed, molecular and cellular descriptions of biological/toxicological mechanisms to develop a dose-response model through a set of nonlinear, differential equations describing the signaling pathways and biochemical mechanisms of cell cycle checkpoint, apoptosis, and tumor incidence due to IR. This approach yields a J-shaped dose response curve while showing where LNT behaviors are likely to occur. The results confirm the hypothesis of the J-shaped dose response curve: the main reason is that, at low-doses of IR, cells stimulate protective systems through a longer cell arrest time per unit of IR dose. We suggest that the policy implications of this approach are an increasingly correct way to deal with precautionary measures in public health.

A theory for the arrangement of sensory organs in Drosophila

We study the arrangements of recurved bristles on the anterior wing margin of wild-type and mutant Drosophila. The epidermal or neural fate of a proneural cell depends on the concentrations of proteins of the achaete-scute complex. At puparium formation, concentrations of proteins are nearly identical in all cells of the anterior wing and each cell has the potential for neural fate. In wild-type flies, the action of regulatory networks drives the initial state to one where a bristle grows out of every fifth cell. Recent experiments have shown that the frequency of recurved bristles can be made to change by adjusting the mean concentrations of the zinc-finger transcription factor Senseless and the micro-RNA miR-9a. Specifically, mutant flies with reduced levels of miR-9a exhibit ectopic bristles, and those with lower levels of both miR-9a and Senseless show regular organization of recurved bristles, but with a lower periodicity of 4. We argue that these characteristics can be explained assuming an underlying Turing-type bifurcation whereby a periodic pattern spontaneously emerges from a uniform background. However, bristle patterns occur in a discrete array of cells, and are not mediated by diffusion. We argue that intracellular actions of transmembrane proteins such as Delta and Notch can play a role of diffusion in destabilizing the homogeneous state. In contrast to diffusion, intercellular actions can be activating or inhibiting; further, there can be lateral cross-species interactions. We introduce a phenomenological model to study bristle arrangements and make several model-independent predictions that can be tested in experiments. In our theory, miRNA-9a is one of the components of the underlying network and has no special regulatory role. The loss of periodicity in its absence is due to the transfer of the system to a bistable state.

Computational systems biology and dose-response modeling in relation to new directions in toxicity testing

The new paradigm envisioned for toxicity testing in the 21st century advocates shifting from the current animal-based testing process to a combination of in vitro cell-based studies, high-throughput techniques, and in silico modeling. A strategic component of the vision is the adoption of the systems biology approach to acquire, analyze, and interpret toxicity pathway data. As key toxicity pathways are identified and their wiring details elucidated using traditional and high-throughput techniques, there is a pressing need to understand their qualitative and quantitative behaviors in response to perturbation by both physiological signals and exogenous stressors. The complexity of these molecular networks makes the task of understanding cellular responses merely by human intuition challenging, if not impossible. This process can be aided by mathematical modeling and computer simulation of the networks and their dynamic behaviors. A number of theoretical frameworks were developed in the last century for understanding dynamical systems in science and engineering disciplines. These frameworks, which include metabolic control analysis, biochemical systems theory, nonlinear dynamics, and control theory, can greatly facilitate the process of organizing, analyzing, and understanding toxicity pathways. Such analysis will require a comprehensive examination of the dynamic properties of "network motifs"--the basic building blocks of molecular circuits. Network motifs like feedback and feedforward loops appear repeatedly in various molecular circuits across cell types and enable vital cellular functions like homeostasis, all-or-none response, memory, and biological rhythm. These functional motifs and associated qualitative and quantitative properties are the predominant source of nonlinearities observed in cellular dose response data. Complex response behaviors can arise from toxicity pathways built upon combinations of network motifs. While the field of computational cell biology has advanced rapidly with increasing availability of new data and powerful simulation techniques, a quantitative orientation is still lacking in life sciences education to make efficient use of these new tools to implement the new toxicity testing paradigm. A revamped undergraduate curriculum in the biological sciences including compulsory courses in mathematics and analysis of dynamical systems is required to address this gap. In parallel, dissemination of computational systems biology techniques and other analytical tools among practicing toxicologists and risk assessment professionals will help accelerate implementation of the new toxicity testing vision.

A bistable switch underlying B-cell differentiation and its disruption by the environmental contaminant 2,3,7,8-tetrachlorodibenzo-p-dioxin

The differentiation of B cells into antibody-secreting plasma cells upon antigen stimulation, a crucial step in the humoral immune response, is disrupted by 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD). Several key regulatory proteins in the B-cell transcriptional network have been identified, with two coupled mutually repressive feedback loops among the three transcription factors B-cell lymphoma 6 (Bcl-6), B lymphocyte-induced maturation protein 1(Blimp-1), and paired box 5 (Pax5) forming the core of the network. However, the precise mechanisms underlying B-cell differentiation and its disruption by TCDD are not fully understood. Here we show with a computational systems biology model that coupling of the two feedback loops at the Blimp-1 node, through parallel inhibition of Blimp-1 gene activation by Bcl-6 and repression of Blimp-1 gene deactivation by Pax5, can generate a bistable switch capable of directing B cells to differentiate into plasma cells. We also use bifurcation analysis to propose that TCDD may suppress the B-cell to plasma cell differentiation process by raising the threshold dose of antigens such as lipopolysaccharide required to trigger the bistable switch. Our model further predicts that high doses of TCDD may render the switch reversible, thus causing plasma cells to lose immune function and dedifferentiate to a B cell-like state. The immunotoxic implications of these predictions are twofold. First, TCDD and related compounds would disrupt the initiation of the humoral immune response by reducing the proportion of B cells that respond to antigen and differentiate into antibody-secreting plasma cells. Second, TCDD may also disrupt the maintenance of the immune response by depleting the pool of available plasma cells through dedifferentiation.

Scalable steady state analysis of Boolean biological regulatory networks

BACKGROUND

Computing the long term behavior of regulatory and signaling networks is critical in understanding how biological functions take place in organisms. Steady states of these networks determine the activity levels of individual entities in the long run. Identifying all the steady states of these networks is difficult due to the state space explosion problem.

METHODOLOGY

In this paper, we propose a method for identifying all the steady states of Boolean regulatory and signaling networks accurately and efficiently. We build a mathematical model that allows pruning a large portion of the state space quickly without causing any false dismissals. For the remaining state space, which is typically very small compared to the whole state space, we develop a randomized traversal method that extracts the steady states. We estimate the number of steady states, and the expected behavior of individual genes and gene pairs in steady states in an online fashion. Also, we formulate a stopping criterion that terminates the traversal as soon as user supplied percentage of the results are returned with high confidence.

CONCLUSIONS

This method identifies the observed steady states of boolean biological networks computationally. Our algorithm successfully reported the G1 phases of both budding and fission yeast cell cycles. Besides, the experiments suggest that this method is useful in identifying co-expressed genes as well. By analyzing the steady state profile of Hedgehog network, we were able to find the highly co-expressed gene pair GL1-SMO together with other such pairs.

AVAILABILITY

Source code of this work is available at http://bioinformatics.cise.ufl.edu/palSteady.html twocolumnfalse].

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs.

Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.

A quantitative systems view of the spindle assembly checkpoint

The idle assembly checkpoint acts to delay chromosome segregation until all duplicated sister chromatids are captured by the mitotic spindle. This pathway ensures that each daughter cell receives a complete copy of the genome. The high fidelity and robustness of this process have made it a subject of intense study in both the experimental and computational realms. A significant number of checkpoint proteins have been identified but how they orchestrate the communication between local spindle attachment and global cytoplasmic signalling to delay segregation is not yet understood. Here, we propose a systems view of the spindle assembly checkpoint to focus attention on the key regulators of the dynamics of this pathway. These regulators in turn have been the subject of detailed cellular measurements and computational modelling to connect molecular function to the dynamics of spindle assembly checkpoint signalling. A review of these efforts reveals the insights provided by such approaches and underscores the need for further interdisciplinary studies to reveal in full the quantitative underpinnings of this cellular control pathway.

Investigating the two-moment characterisation of subcellular biochemical networks

While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.

The ups and downs of p53: understanding protein dynamics in single cells

Cells living in a complex environment must constantly detect, process and appropriately respond to changing signals. Therefore, all cellular information processing is dynamic in nature. As a consequence, understanding the process of signal transduction often requires detailed quantitative analysis of dynamic behaviours. Here, we focus on the oscillatory dynamics of the tumour suppressor protein p53 as a model for studying protein dynamics in single cells to better understand its regulation and function.

How to synchronize biological clocks

This paper is concerned with a novel algorithm to study networks of biological clocks. A new set of conditions is established that can be used to verify whether an existing network synchronizes or to give guidelines to construct a new synthetic network of biological oscillators that synchronize. The methodology uses the so-called contraction theory from dynamical system theory and Gershgorin disk theorem. The strategy is validated on two examples: a model of glycolisis in yeast cells and a synthetic network of Repressilators that synchronizes.

Simple molecular networks that respond optimally to time-periodic stimulation

Background

Bacteria or cells receive many signals from their environment and from other organisms. In order to process this large amount of information, Systems Biology shows that a central role is played by regulatory networks composed of genes and proteins. The objective of this paper is to present and to discuss simple regulatory network motifs having the property to maximize their responses under time-periodic stimulations. In elucidating the mechanisms underlying these responses through simple networks the goal is to pinpoint general principles which optimize the oscillatory responses of molecular networks.

Results

We took a look at basic network motifs studied in the literature such as the Incoherent Feedforward Loop (IFFL) or the interlerlocked negative feedback loop. The former is also generalized to a diamond pattern, with network components being either purely genetic or combining genetic and signaling pathways. Using standard mathematics and numerical simulations, we explain the types of responses exhibited by the IFFL with respect to a train of periodic pulses. We show that this system has a non-vanishing response only if the inter-pulse interval is above a threshold. A slight generalisation of the IFFL (the diamond) is shown to work as an ideal pass-band filter. We next show a mechanism by which average of oscillatory response can be maximized by bursting temporal patterns. Finally we study the interlerlocked negative feedback loop, i.e. a 2-gene motif forming a loop where the nodes respectively activate and repress each other, and show situations where this system possesses a resonance under periodic stimulation.

Conclusion

We present several simple motif designs of molecular networks producing optimal output in response to periodic stimulations of the system. The identified mechanisms are simple and based on known network motifs in the literature, so that that they could be embodied in existing organisms, or easily implementable by means of synthetic biology. Moreover we show that these designs can be studied in different contexts of molecular biology, as for example in genetic networks or in signaling pathways.

Temporal organization of the cell cycle

The coordination of growth, DNA replication and division in proliferating cells can be adequately explained by a 'clock + checkpoint' model. The clock is an underlying cyclical sequence of states; the checkpoints ensure that the cycle proceeds without mistakes. From the molecular complexities of the control system in modern eukaryotes, we isolate a simple network of positive and negative feedbacks that embodies a 'clock + checkpoints'. The model accounts for the fundamental physiological properties of mitotic cell divisions, evokes a new view of the meiotic program, and suggests how the control system may have evolved in the first place.

Reverse engineering models of cell cycle regulation

From general considerations of the basic physiological properties of the cell division cycle, we deduce what the dynamical properties of the underlying molecular control system must be. Then, taking a few hints from the biochemistry of cyclin-dependent kinases (the master regulators of the eukaryotic cell cycle), we guess what molecular mechanisms must be operating to produce the desired dynamical properties of the control system.

Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law

The dynamical systems arising from gene regulatory, signalling and metabolic networks are strongly nonlinear, have high-dimensional state spaces and depend on large numbers of parameters. Understanding the relation between the structure and the function for such systems is a considerable challenge. We need tools to identify key points of regulation, illuminate such issues as robustness and control and aid in the design of experiments. Here, I tackle this by developing new techniques for sensitivity analysis. In particular, I show how to globally analyse the sensitivity of a complex system by means of two new graphical objects: the sensitivity heat map and the parameter sensitivity spectrum. The approach to sensitivity analysis is global in the sense that it studies the variation in the whole of the model's solution rather than focusing on output variables one at a time, as in classical sensitivity analysis. This viewpoint relies on the discovery of local geometric rigidity for such systems, the mathematical insight that makes a practicable approach to such problems feasible for highly complex systems. In addition, we demonstrate a new summation theorem that substantially generalizes previous results for oscillatory and other dynamical phenomena. This theorem can be interpreted as a mathematical law stating the need for a balance between fragility and robustness in such systems.

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.

The role of dynamic stimulation pattern in the analysis of bistable intracellular networks

Bistable systems play an important role in the functioning of living cells. Depending on the strength of the necessary positive feedback one can distinguish between (irreversible) "one-way switch" or (reversible) "toggle-switch" type behavior. Besides the well- established steady-state properties, some important characteristics of bistable systems arise from an analysis of their dynamics. We demonstrate that a supercritical stimulus amplitude is not sufficient to move the system from the lower (off-state) to the higher branch (on-state) for either a step or a pulse input. A switching surface is identified for the system as a function of the initial condition, input pulse amplitude and duration (a supercritical signal). We introduce the concept of bounded autonomy for single level systems with a pulse input. Towards this end, we investigate and characterize the role of the duration of the stimulus. Furthermore we show, that a minimal signal power is also necessary to change the steady state of the bistable system. This limiting signal power is independent of the applied stimulus and is determined only by systems parameters. These results are relevant for the design of experiments, where it is often difficult to create a defined pattern for the stimulus. Furthermore, intracellular processes, like receptor internalization, do manipulate the level of stimulus such that level and duration of the stimulus is conducive to characteristic behavior.

Boolean network model predicts cell cycle sequence of fission yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.

Thermodynamics of natural selection I: Energy flow and the limits on organization

This is the first of three papers analyzing the representation of information in the biosphere, and the energetic constraints limiting the imposition or maintenance of that information. Biological information is inherently a chemical property, but is equally an aspect of control flow and a result of processes equivalent to computation. The current paper develops the constraints on a theory of biological information capable of incorporating these three characterizations and their quantitative consequences. The paper illustrates the need for a theory linking energy and information by considering the problem of existence and reslience of the biosphere, and presents empirical evidence from growth and development at the organismal level suggesting that the theory developed will capture relevant constraints on real systems. The main result of the paper is that the limits on the minimal energetic cost of information flow will be tractable and universal whereas the assembly of more literal process models into a system-level description often is not. The second paper in the series then goes on to construct reversible models of energy and information flow in chemistry which achieve the idealized limits, and the third paper relates these to fundamental operations of computation.

Probabilistic modeling of DNA mismatch repair effects on cell cycle dynamics and iododeoxyuridine-DNA incorporation

Previous studies in our laboratory have described increased and preferential radiosensitization of mismatch repair-deficient (MMR(-)) HCT116 colon cancer cells with 5-iododeoxyuridine (IUdR). Indeed, our studies showed that MMR is involved in the repair (removal) of IUdR-DNA, principally the G:IU mispair. Consequently, we have shown that MMR(-) cells incorporate 25% to 42% more IUdR than MMR(+) cells, and that IUdR and ionizing radiation (IR) interact to produce up to 3-fold greater cytotoxicity in MMR(-) cells. The present study uses the integration of probabilistic mathematical models and experimental data on MMR(-) versus MMR(+) cells to describe the effects of IUdR incorporation upon the cell cycle for the purpose of increasing IUdR-mediated radiosensitivity in MMR(-) cells. Two computational models have been developed. The first is a stochastic model of the progression of cell cycle states, which is applied to experimental data for two synchronized isogenic MMR(+) and MMR(-) colon cancer cell lines treated with and without IUdR. The second model defines the relation between the percentage of cells in the different cell cycle states and the corresponding IUdR-DNA incorporation at a particular time point. These models can be combined to predict IUdR-DNA incorporation at any time in the cell cycle. These mathematical models will be modified and used to maximize therapeutic gain in MMR(-) tumors versus MMR(+) normal tissues by predicting the optimal dose of IUdR and optimal timing for IR treatment to increase the synergistic action using xenograft models and, later, in clinical trials.

SRC family kinases and receptors: analysis of three activation mechanisms by dynamic systems modeling

Src family kinases (SFKs) interact with a number of cellular receptors. They participate in diverse signaling pathways and cellular functions. Most of the receptors involved in SFK signaling are characterized by similar modes of regulation. This computational study discusses a general kinetic model of SFK-receptor interaction. The analysis of the model reveals three major ways of SFK activation: release of inhibition by C-terminal Src kinase, weakening of the inhibitory intramolecular phosphotyrosine-SH2 interaction, and amplification of a stimulating kinase activity. The SFK model was then extended to simulate interaction with growth factor and T-cell receptors. The modular SFK signaling system was shown to adapt to the requirements of specific signaling contexts and yield qualitatively different responses in the different simulated environments. The model also provides a systematic overview of the major interactions between SFKs and various cellular signaling systems and identifies their common properties.

HAUSP-regulated switch from auto- to p53 ubiquitination by Mdm2 (in silico discovery)

Stability of the 'guardian of the genome' tumor suppressor protein p53 is regulated predominantly through its ubiquitination. The ubiquitin-specific protease HAUSP plays an important role in this process. Recent experiments showed that p53 demonstrates a differential response to changes in HAUSP which nature and significance are not understood yet. Here a data-driven mathematical model of the Mdm2-mediated p53 ubiquitination network is presented which offers an explanation for the cause of such a response. The model predicts existence of the HAUSP-regulated switch from auto- to p53 ubiquitination by Mdm2. This switch suggests a potential role of HAUSP as a downstream target of stress signals in cells. The model accounts for a significant amount of experimental data, makes predictions for some rate constants, and can serve as a building block for the larger model describing a complex dynamic response of p53 to cellular stresses.

Current approaches to gene regulatory network modelling

Many different approaches have been developed to model and simulate gene regulatory networks. We proposed the following categories for gene regulatory network models: network parts lists, network topology models, network control logic models, and dynamic models. Here we will describe some examples for each of these categories. We will study the topology of gene regulatory networks in yeast in more detail, comparing a direct network derived from transcription factor binding data and an indirect network derived from genome-wide expression data in mutants. Regarding the network dynamics we briefly describe discrete and continuous approaches to network modelling, then describe a hybrid model called Finite State Linear Model and demonstrate that some simple network dynamics can be simulated in this model.

Conceptual modeling in systems biology fosters empirical findings: the mRNA lifecycle

One of the main obstacles to understanding complex biological systems is the extent and rapid evolution of information, way beyond the capacity individuals to manage and comprehend. Current modeling approaches and tools lack adequate capacity to model concurrently structure and behavior of biological systems. Here we propose Object-Process Methodology (OPM), a holistic conceptual modeling paradigm, as a means to model both diagrammatically and textually biological systems formally and intuitively at any desired number of levels of detail. OPM combines objects, e.g., proteins, and processes, e.g., transcription, in a way that is simple and easily comprehensible to researchers and scholars. As a case in point, we modeled the yeast mRNA lifecycle. The mRNA lifecycle involves mRNA synthesis in the nucleus, mRNA transport to the cytoplasm, and its subsequent translation and degradation therein. Recent studies have identified specific cytoplasmic foci, termed processing bodies that contain large complexes of mRNAs and decay factors. Our OPM model of this cellular subsystem, presented here, led to the discovery of a new constituent of these complexes, the translation termination factor eRF3. Association of eRF3 with processing bodies is observed after a long-term starvation period. We suggest that OPM can eventually serve as a comprehensive evolvable model of the entire living cell system. The model would serve as a research and communication platform, highlighting unknown and uncertain aspects that can be addressed empirically and updated consequently while maintaining consistency.

Dynamical modeling of syncytial mitotic cycles in Drosophila embryos

Immediately following fertilization, the fruit fly embryo undergoes 13 rapid, synchronous, syncytial nuclear division cycles driven by maternal genes and proteins. During these mitotic cycles, there are barely detectable oscillations in the total level of B-type cyclins. In this paper, we propose a dynamical model for the molecular events underlying these early nuclear division cycles in Drosophila. The model distinguishes nuclear and cytoplasmic compartments of the embryo and permits exploration of a variety of rules for protein transport between the compartments. Numerical simulations reproduce the main features of wild-type mitotic cycles: patterns of protein accumulation and degradation, lengthening of later cycles, and arrest in interphase 14. The model is consistent with mutations that introduce subtle changes in the number of mitotic cycles before interphase arrest. Bifurcation analysis of the differential equations reveals the dependence of mitotic oscillations on cycle number, and how this dependence is altered by mutations. The model can be used to predict the phenotypes of novel mutations and effective ranges of the unmeasured rate constants and transport coefficients in the proposed mechanism.

Network motifs: theory and experimental approaches

Transcription regulation networks control the expression of genes. The transcription networks of well-studied microorganisms appear to be made up of a small set of recurring regulation patterns, called network motifs. The same network motifs have recently been found in diverse organisms from bacteria to humans, suggesting that they serve as basic building blocks of transcription networks. Here I review network motifs and their functions, with an emphasis on experimental studies. Network motifs in other biological networks are also mentioned, including signalling and neuronal networks.

Ab initio phenomenological simulation of the growth of large tumor cell populations

In a previous paper we have introduced a phenomenological model of cell metabolism and of the cell cycle to simulate the behavior of large tumor cell populations (Chignola and Milotti 2005 Phys. Biol. 2 8). Here we describe a refined and extended version of the model that includes some of the complex interactions between cells and their surrounding environment. The present version takes into consideration several additional energy-consuming biochemical pathways such as protein and DNA synthesis, the tuning of extracellular pH and of the cell membrane potential. The control of the cell cycle, which was previously modeled by means of ad hoc thresholds, has been directly addressed here by considering checkpoints from proteins that act as targets for phosphorylation on multiple sites. As simulated cells grow, they can now modify the chemical composition of the surrounding environment which in turn acts as a feedback mechanism to tune cell metabolism and hence cell proliferation: in this way we obtain growth curves that match quite well those observed in vitro with human leukemia cell lines. The model is strongly constrained and returns results that can be directly compared with actual experiments, because it uses parameter values in narrow ranges estimated from experimental data, and in perspective we hope to utilize it to develop in silico studies of the growth of very large tumor cell populations (10(6) cells or more) and to support experimental research. In particular, the program is used here to make predictions on the behavior of cells grown in a glucose-poor medium: these predictions are confirmed by experimental observation.

The stochastic behavior of a molecular switching circuit with feedback

Background

Using a statistical physics approach, we study the stochastic switching behavior of a model circuit of multisite phosphorylation and dephosphorylation with feedback. The circuit consists of a kinase and phosphatase acting on multiple sites of a substrate that, contingent on its modification state, catalyzes its own phosphorylation and, in a symmetric scenario, dephosphorylation. The symmetric case is viewed as a cartoon of conflicting feedback that could result from antagonistic pathways impinging on the state of a shared component.

Results

Multisite phosphorylation is sufficient for bistable behavior under feedback even when catalysis is linear in substrate concentration, which is the case we consider. We compute the phase diagram, fluctuation spectrum and large-deviation properties related to switch memory within a statistical mechanics framework. Bistability occurs as either a first-order or second-order non-equilibrium phase transition, depending on the network symmetries and the ratio of phosphatase to kinase numbers. In the second-order case, the circuit never leaves the bistable regime upon increasing the number of substrate molecules at constant kinase to phosphatase ratio.

Conclusion

The number of substrate molecules is a key parameter controlling both the onset of the bistable regime, fluctuation intensity, and the residence time in a switched state. The relevance of the concept of memory depends on the degree of switch symmetry, as memory presupposes information to be remembered, which is highest for equal residence times in the switched states.

Reviewers

This article was reviewed by Artem Novozhilov (nominated by Eugene Koonin), Sergei Maslov, and Ned Wingreen.

Unravelling the world of cis-regulatory elements

Genome-wide comparisons indicate that only studying the coding regions will not be enough for explaining the biological complexity of an organism, while the genetic variants and the epigenetic differences of cis-regulatory elements are crucial to elucidate many complicated biological phenomena. Their various regulatory functions also play indispensable roles in forming organismal polymorphism. Recent studies showed that the cis-regulatory elements can regulate gene expression as nuclear organizers, and involve in functional noncoding transcription and produce regulatory noncoding RNA molecules. Novel high-throughput strategies and in silico analysis make a great amount data of cis-regulatory elements available. Particularly, the computational methods could help to combine reductionist studies with network biomedical investigations, and begin the era to understand organismal regulatory events at systems biology level.

A phenomenological approach to the simulation of metabolism and proliferation dynamics of large tumour cell populations

A major goal of modern computational biology is to simulate the collective behaviour of large cell populations starting from the intricate web of molecular interactions occurring at the microscopic level. In this paper we describe a simplified model of cell metabolism, growth and proliferation, suitable for inclusion in a multicell simulator, now under development (Chignola R and Milotti E 2004 Physica A 338 261-6). Nutrients regulate the proliferation dynamics of tumour cells which adapt their behaviour to respond to changes in the biochemical composition of the environment. This modelling of nutrient metabolism and cell cycle at a mesoscopic scale level leads to a continuous flow of information between the two disparate spatiotemporal scales of molecular and cellular dynamics that can be simulated with modern computers and tested experimentally.

Sensitivity and control analysis of periodically forced reaction networks using the Green's function method

A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green's function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network's parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically.

An atypical active cell death process underlies the fungicidal activity of ciclopirox olamine against the yeast Saccharomyces cerevisiae

Ciclopirox olamine (CPO), a fungicidal agent widely used in clinical practice, induced in Saccharomyces cerevisiae an active cell death (ACD) process characterized by changes in nuclear morphology and chromatin condensation associated with the appearance of a population in the sub-G(0)/G(1) cell cycle phase and an arrest delay in the G(2)/M phases. This ACD was associated neither with intracellular reactive oxygen species (ROS) signaling, as revealed by the use of different classes of ROS scavengers, nor with a terminal deoxynucleotidyl transferase-mediated dUTP nick end labeling (TUNEL)-positive phenotype. Furthermore, CPO-induced cell death seems to be dependent on unknown protease activity but independent of the apoptotic regulators Aif1p and Yca1p and of autophagic pathways involving Apg5p, Apg8p and Uth1p. Our results show that CPO triggers in S. cerevisiae an atypical nonapoptotic, nonautophagic ACD with as yet unknown regulators.

A proposal for robust temperature compensation of circadian rhythms

The internal circadian rhythms of cells and organisms coordinate their physiological properties to the prevailing 24-h cycle of light and dark on earth. The mechanisms generating circadian rhythms have four defining characteristics: they oscillate endogenously with period close to 24 h, entrain to external signals, suffer phase shifts by aberrant pulses of light or temperature, and compensate for changes in temperature over a range of 10 degrees C or more. Most theoretical descriptions of circadian rhythms propose that the underlying mechanism generates a stable limit cycle oscillation (in constant darkness or dim light), because limit cycles quite naturally possess the first three defining properties of circadian rhythms. On the other hand, the period of a limit cycle oscillator is typically very sensitive to kinetic rate constants, which increase markedly with temperature. Temperature compensation is therefore not a general property of limit cycle oscillations but must be imposed by some delicate balance of temperature dependent effects. However, "delicate balances" are unlikely to be robust to mutations. On the other hand, if circadian rhythms arise from a mechanism that concentrates sensitivity into a few rate constants, then the "balancing act" is likely to be more robust and evolvable. We propose a switch-like mechanism for circadian rhythms that concentrates period sensitivity in just two parameters, by forcing the system to alternate between a stable steady state and a stable limit cycle.

Time scale and dimension analysis of a budding yeast cell cycle model

Background

The progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle.

Results

By calculating the eigenvalues of the Jacobian of kinetic differential equations we have determined the stability of the cell cycle trajectories of the Chen model. Based on the sign of the real part of the eigenvalues, the cell cycle can be divided into excitation and relaxation periods. During an excitation period, the cell cycle control system leaves a formerly stable steady state and, accordingly, excitation periods can be associated with irreversible cell cycle transitions like START, entry into mitosis and exit from mitosis. During relaxation periods, the control system asymptotically approaches the new steady state. We also show that the dynamical dimension of the Chen's model fluctuates by increasing during excitation periods followed by decrease during relaxation periods. In each relaxation period the dynamical dimension of the model drops to one, indicating a period where kinetic processes are in steady state and all concentration changes are driven by the increase of cytoplasmic growth.

Conclusion

We apply two numerical methods, which have not been used to analyze biological control systems. These methods are more sensitive than the bifurcation analysis used before because they identify those transitions between steady states that are not controlled by a bifurcation parameter (e.g. cell mass). Therefore by applying these tools for a cell cycle control model, we provide a deeper understanding of the dynamical transitions in the underlying molecular network.

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.

A tuneable attractor underlies yeast respiratory dynamics

Our understanding of the molecular structure and function in the budding yeast, Saccharomyces cerevisiae, surpasses that of all other eukaryotic cells. However, the fundamental properties of the complex processes and their control systems have been difficult to reconstruct from detailed dissection of their molecular components. Spontaneous oscillatory dynamics observed in self-synchronized continuous cultures is pervasive, involves much of the cellular network, and provides unique insights into integrative cell physiology. Here, in non-invasive experiments in vivo, we exploit these oscillatory dynamics to analyse the global timing of the cellular network to show the presence of a low-order chaotic component. Although robust to a wide range of environmental perturbations, the system responds and reacts to the imposition of harsh environmental conditions, in this case low pH, by dynamic re-organization of respiration, and this feeds upwards to affect cell division. These complex dynamics can be represented by a tuneable attractor that orchestrates cellular complexity and coherence to the environment.

Inverse bifurcation analysis: application to simple gene systems

Background

Bifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks. In addition to the more traditional forward problem of determining the mapping from parameter space to the space of model behavior, the inverse problem of determining model parameters to result in certain desired properties of the bifurcation diagram provides an attractive methodology for addressing important biological problems. These include understanding how the robustness of qualitative behavior arises from system design as well as providing a way to engineer biological networks with qualitative properties.

Results

We demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. This formulation allows for an iterative solution procedure based on performing a sequence of eigen-system computations and one-parameter continuations of solutions, the latter being a standard capability in existing numerical bifurcation software. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved.

Modelling in molecular biology: describing transcription regulatory networks at different scales

Approaches to describe gene regulation networks can be categorized by increasing detail, as network parts lists, network topology models, network control logic models or dynamic models. We discuss the current state of the art for each of these approaches. We study the relationship between different topology models, and give examples how they can be used to infer functional annotations for genes of unknown function. We introduce a new simple way of describing dynamic models called finite state linear model (FSLM). We discuss the gap between the parts list and topology models on one hand, and network logic and dynamic models, on the other hand. The first two classes of models have reached a genome-wide scale, while for the other model classes high-throughput technologies are yet to make a major impact.

Analysis of a generic model of eukaryotic cell-cycle regulation

We propose a protein interaction network for the regulation of DNA synthesis and mitosis that emphasizes the universality of the regulatory system among eukaryotic cells. The idiosyncrasies of cell cycle regulation in particular organisms can be attributed, we claim, to specific settings of rate constants in the dynamic network of chemical reactions. The values of these rate constants are determined ultimately by the genetic makeup of an organism. To support these claims, we convert the reaction mechanism into a set of governing kinetic equations and provide parameter values (specific to budding yeast, fission yeast, frog eggs, and mammalian cells) that account for many curious features of cell cycle regulation in these organisms. Using one-parameter bifurcation diagrams, we show how overall cell growth drives progression through the cell cycle, how cell-size homeostasis can be achieved by two different strategies, and how mutations remodel bifurcation diagrams and create unusual cell-division phenotypes. The relation between gene dosage and phenotype can be summarized compactly in two-parameter bifurcation diagrams. Our approach provides a theoretical framework in which to understand both the universality and particularity of cell cycle regulation, and to construct, in modular fashion, increasingly complex models of the networks controlling cell growth and division.

Periodic forcing of a mathematical model of the eukaryotic cell cycle

In a differential equation model of the molecular network governing cell growth and division, cell cycle phases and transitions through checkpoints are associated with certain bifurcations of the underlying vector field. If the cell cycle is driven by another rhythmic process, interactions between forcing and bifurcations lead to emergent orbits and oscillations. In this paper, by varying the amplitude and frequency of forcing of the synthesis rates of regulatory proteins and the mass growth rate in a minimal model of the eukaryotic cell cycle, we study changes of the probability distributions of interdivision time and mass at division. By computing numerically the Lyapunov exponent of the model, we show that the splitting of probability distributions is associated with mode-locked solutions. We also introduce a simple, integrate-and-fire model to analyze mode locking in the cell cycle.

Origins and evolution of cell phenotypes in breast tumors

This study presents a stochastic model that correlates genomic instability with tumor formation. The model describes the time- and space-variant volumetric concentrations of cancer cells of various phenotypes in a breast tumor. The cells of epithelial origin in the cancerous breast tissue are classified into four different phenotypes, normal epithelial cells and the grade 1, grade 2 and grade 3 cancer cell types with increasing potential for growth and invasion. Equations governing the time course of volumetric concentrations of cell phenotypes are derived by using the principle of conservation of mass. Cell migration into and from the stroma is taken into account. The transformations between cell phenotypes are due to genetic inheritance and chromosome aberrations. These transformations are assumed to be stochastic functions of the local cell concentration. The simulations of the model for planar geometry replicate the shapes of human breast tumors and capture the time history of tumor growth in animal models. Simulations point to transformation of tumor cell population from heterogeneous compositions to a single phenotype at advanced stages of invasive tumors. Systematic variations of model parameters in the computations indicate the important roles the migration capacity, proliferation rate, and phenotype transition probability play in tumor growth. The model developed provides realistic simulations for standard breast cancer therapies and can be used in the optimization studies of chemotherapy, radiotherapy, hormone therapy and emerging individualized therapies for cancer.

Models of cell signaling pathways

Cellular signaling circuits handle an enormous range of computations. Beyond the housekeeping, replicating and other functions of individual cells, signaling circuits must implement the immensely complex logic of development and function of multicellular organisms. Computer models are useful tools to understand this complexity. Recent studies have extended such models to include electrical, mechanical and spatial details of signaling, and to address the stochastic effects that arise when small numbers of molecules interact. Increasing numbers of models have been developed in close conjunction with experiments, and this interplay gives a deeper and more reliable insight into signaling function.