## Contents

Introduction |

Methods |

Specific pathways, specific models |

Specific models, specific questions |

Discussion |

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Review# Modelling dynamic processes in yeast

## Authors

### Edda Klipp

Corresponding author- E-mail address: klipp@molgen.mpg.de

- Max Planck Institute for Molecular Genetics, Computational Systems Biology, Ihnestrasse 63-73, 14195 Berlin, Germany

- Max Planck Institute for Molecular Genetics, Computational Systems Biology, Ihnestrasse 63-73, 14195 Berlin, Germany.

- First published: Full publication history
- DOI: 10.1002/yea.1544 View/save citation
- Cited by (CrossRef): 16 articles Check for updates
- Funding Information

Yeast molecular and cell biology has accumulated large amounts of qualitative and quantitative data of diverse cellular processes. The results are often summarized as verbal or graphical descriptions. Moreover, a series of mathematical models has been developed that should help to interpret such data, to integrate them into a coherent picture and to allow for an understanding of the underlying processes. Dynamic modelling of regulatory processes in yeast focuses on central carbon metabolism, on a number of selected signalling pathways and on cell cycle regulation. These models can explain questions of general relevance, such as whether the dynamics of a network can be understood from the combination of *in vitro* kinetics of its individual reactions. They help to elucidate complicated dynamic features, such as glycolytic oscillations, effects of feedback regulation or the optimal regulation of gene expression. The availability of comprehensive qualitative information, such as protein interactions or pathway composition, and sets of quantitative data make yeast a perfect model organism. Therefore, yeast-related data are often used to develop and examine computational approaches and modelling methods. Copyright © 2007 John Wiley & Sons, Ltd.

Introduction |

Methods |

Specific pathways, specific models |

Specific models, specific questions |

Discussion |

Life is change. To study and understand life, it is necessary to study genes, proteins or metabolites and networks thereof in static conditions, but this is not sufficient. Instead, we must learn to handle the dynamic action. Systems biology has been defined in many different ways, always claiming that its new quality compared to traditional biology is the analysis of systems and the interactions of their parts. One important direction in this approach is the integrated investigation of dynamic biological systems by experimental techniques and mathematical modelling.

Yeast is an ideal model organism for the integrated experimental and theoretical approach. It is harmless and easy to cultivate; it can be manipulated without ethical problems. Moreover, it is a eukaryote with extensive homology to higher organisms in many aspects. These advantages induce another advantage: since it is a highly employed laboratory organism, a vast amount of qualitative and quantitative data are available, ranging from detailed information about individual genes, proteins or pathways to complete DNA sequences (e.g. for *Saccharomyces cerevisiae*31 and *Schizosaccharomyces pombe*122) or gene expression datasets for all genes under various conditions25, 27, 101. These data, combined with a number of open questions and unresolved problems, are promising pre-conditions for modelling approaches. In addition, the open mind of the yeast community towards modelling induced the production of further data specifically produced for model quantification and testing46, 72.

Modelling of biochemical networks can help to integrate experimental knowledge into a coherent picture and to test, support or falsify hypotheses about the underlying biological mechanisms. The behaviour of complex systems is often hard to grasp by intuition, because our reasoning tends to follow simple causal chains: if feedback cycles come into play, or if the relative timing of processes makes a difference, then mathematical simulation may be more reliable than mere intuition. Modelling emphasizes the holistic aspects of signalling networks, which disappear if the components are studied separately in different ‘wet labs’ around the globe. Furthermore, once a model has been established, it can be used to test hypotheses or simulate experiments that would be hard or impossible to do in the laboratory.

Modelling itself is useful as a process, even if the resulting model is not satisfactory. It forces abstract thinking and the extraction of essential features of a process. It highlights aspects where our understanding of a matter is wrong or insufficient. It facilitates a unique description of our current knowledge–and of the gaps therein.

For the actual construction of a model, three major directions of discovery have been formulated: bottom-up, top-down and middle-out (term credited to Sydney Brenner, in Noble76). For dynamic models, the bottom-up approach is still prevalent, thanks to the complexity of the systems and to the fact that even the behaviour of individual pathways is rarely understood. The dynamics of small systems, say a set of a few metabolic reactions, are not trivial; even less trivial are large networks. Although the vision of a virtual cell is still around, this aim is not close, even for yeast. In general, successful dynamic modelling contains the following ingredients (Figure 1): first, formulation of a problem to solve (no problem, no useful model!); second, construction of a network or wiring scheme for the process and formulation of a set of mathematical equations, usually ordinary differential equations; third, model validation (can the model, in principle, give an answer to the posed questions?) and model verification (determine the parameters from experimental data and try to reproduce the input data); and last but not least, the prediction of new features, especially of experimentally testable effects, such as deletion or overexpression mutants, the outcome of changing experimental conditions or the effect of certain perturbations.

Cellular life combines various different biochemical processes, which have been considered separately in experimental research and in theoretical model building. These processes include metabolism, signalling, gene expression and the cell cycle. We will briefly describe these, highlight their differences and then discuss various models and their abilities to give new insights. First, however, the mathematical techniques and methods commonly used will be outlined.

Dynamic modelling of yeast biochemical networks employs discrete models such as Boolean networks and Petri nets, as well as continuous models, such as ordinary differential equation (ODE) systems. Simulation techniques for network dynamics can be deterministic or stochastic. ‘Deterministic’ means that if one state is known, then all following states are known in principle, while stochastic simulation calculates the probability of successive states. One may describe the dynamics of the same pathway with different approaches, which is illustrated in Figure 2 using the MAP kinase cascade as an example.

Boolean models in their basic form are deterministic models that assign discrete values to the state of a compound and discrete time points for updating this state. Update rules are based on a network of interactions between the nodes, i.e. the compounds. They are often used to describe gene expression events. A value of 0 denotes the off-state of a gene (no or low expression) and a value of 1 means the on-state (high expression). Boolean modelling, for example, has been applied to model yeast cell cycle progression4.

A Petri net is a mathematical representation of a (biochemical) network that consists of places (here indicating compounds), transitions (here indicating reactions), and directed arcs, which run between places and transitions. Places contain a number of tokens indicating the number of objects. A biochemical reaction is computed as the firing of a transition, i.e. the redistribution of tokens from one place to another via a transition according to specified rules.

ODE systems are the most frequently applied technique for modelling of dynamic processes56. They describe the temporal changes of compound concentrations that are due to production, degradation, modification or transport. The rate of change of the concentration *c*_{i} of the *i*th compound reads:

(1)

where *r* is the number of reactions in the network, *v*_{j} the rate of the *j*th reaction and *n*_{ij} the stoichiometric coefficient of c_{i} in reaction *j*. For convenience, one may put all concentrations, rates and coefficients in vectors or matrices, such as:

(2)

The rates are functions of the concentrations **c** of compounds or modifiers, such as enzymes, and of parameters **p** (including kinetic constants and fixed concentrations of external compounds):

(3)

These rates can be described by different types of kinetic laws. Those rate laws are in turn the result of a modelling effort and are based on a set of assumptions, as indicated for the specific rate laws discussed below. Most of the rate laws are based on the mass action law35, stating that the rate of a reaction is proportional to the probability of collision of its reactants, which in turn is proportional to their concentrations *c*_{i} to the power of molecularity *m*_{i}:

(4)

where means the product overall for *i* = 1, .., *n*. Here, *m*_{i} is an integer. It is zero if c_{i} is not a substrate of v_{j}, and 1, 2, … otherwise.

In experiments, rates often do not show the linear dependence of rate on concentrations, as assumed in mass action laws. This fact is acknowledged in power law kinetics used in the S-systems approach90, 91. Here, the rate reads:

(5)

where the concentrations and rates are normalized to some standard value denoted by superscript 0 and *g*_{i, j} is a real number instead of an integer, as in equation (4). The normalized rates and concentrations are dimensionless quantities. The power law kinetics can be considered as a generalization of the mass action rate law. The exponent *g*_{i, j} is equal to the concentration elasticities ε_{j, i},42 i.e. the scaled derivatives of rates with respect to substrate concentrations:

(6)

In traditional enzyme kinetics, one considers separate steps of reactant binding to and release from the enzyme and even isomerization of enzyme–reactant complexes, which leads to the classical Michaelis–Menten expression3, 68. For one reactant and irreversible conversion, this reads:

(7)

where *V*_{max, j} is the maximal rate of reaction *j* and *K*_{m, (i, j)} is the Michaelis constant of reactant *i* in reaction *j*, i.e. the concentration of reactant *i* leading to half-maximal rate. A basic assumption is that the enzyme concentration *E*_{j} is much lower than the reactant concentrations, thereby allowing for a quasi-steady state of the enzyme-bound complexes. Here, the rate is proportional to the enzyme concentration, which is hidden in the maximal rate (*V*_{max, j} = *E*_{j}·*k*_{cat}, with *k*_{cat} turnover number of enzyme *j*). This approach can be extended to cases with several reactants, reversible conversion of substrate into products, various sequences of reactant binding, or the effect of modifiers (see textbooks such as Cornish-Bowden14 and Klipp *et al.*55). Simplifications of complicated rate laws for multiple reactants with a high number of parameters lead to convenience kinetics67.

In metabolic modelling studies, approximative kinetic formats are used (for a recent review, see Heijnen38). One assumes that the rate is proportional to the enzyme concentration. The rates [equation (3)] are normalized with respect to a reference state, which is usually a steady state. This leads to the general expression:

(8)

where the superscript 0 denotes the values in the reference state and **E**_{c} is the matrix of concentration elasticities given in equation (6). One example is so-called ‘linear-logarithmic kinetics’:

(9)

where **I** is the *r* × *r* identity matrix. Another example is an approximation of the power law kinetics:

(10)

Sigmoid dependence of rates on reactant concentrations can sometimes be observed, e.g. if the enzyme has several subunits. They are described by Hill kinetics

(11)

(where *n* is called the Hill coefficient) or by more detailed kinetic laws, such as those suggested by Monod and colleagues69 and Koshland and colleagues60.

Most of these kinetic expressions have been developed for metabolic reactions catalysed by enzymes and they are subject of specific assumptions. Currently they are used to describe various types of reactions and protein–protein or protein–nucleic acid interactions. In some cases, their appropriateness remains to be proved. This can be done as soon as their respective kinetics can be assessed experimentally.

The stoichiometric matrix **N** introduced in equation (2) contains the information about the network structure in the sense of producing and degrading reactions. It does not describe activating or inhibiting interactions by compounds that are not reactants of the considered reaction. Such information is stored in the rate laws and hence in the elasticity coefficients (equation 6). Analysis of the matrix **N** provides two additional results. First, solving the steady state equation in terms of the rates *v*_{j} is mathematically equivalent to solving the matrix equation **N**·**K** = **0**. Every admissible set of fluxes, i.e. rates in steady state, is a linear combination of the columns vectors of **K**, which is a representation of the Kernel space of **N**, i.e. the space of solutions of the matrix equation. Solving the matrix equation **G**·**N** = **0** provides all conservation relations of the system at any state expressed as **G**·**c** = *const*.

ODE systems for biochemical networks can be simulated by using general purpose mathematical tools, but there exist also a number of specific tools that simplify this task (for an overview, see Klipp *et al.*56). Note that some of these tools allow switching from a quasi-continuous and deterministic simulation to a discrete and stochastic simulation of one and the same model, provided that all rates follow mass action kinetics.

Metabolism is necessary to provide living cells with the energy and materials that they require for all processes, such as growth, replication and repair of DNA, building of membranes, movement and many others. Metabolism is the means by which cells reproduce and survive. It comprises two major processes: catabolism, i.e. breakdown of complex compounds to obtain building blocks and energy, and anabolism, i.e. synthesis of compounds that are needed for cell functioning.

Metabolic processes are balanced in exponentially growing cells in the absence of perturbation, leading to homeostasis. This observation is the basis for the frequent assumption of the systems being in steady state: by assuming that we investigate cells in a time window, where fast processes, such as small external perturbations, quickly level out and slow processes, such as protein synthesis and degradation, are too slow to disturb or are sensibly balanced; net rates of metabolite concentration changes are negligible in respective approaches.

As compared to other biochemical networks, metabolic networks are well defined by the presence or absence of the enzymes that catalyse the individual reactions. This facilitates stoichiometric analysis of metabolic networks in steady state. The approaches to solving steady-state equations (see Methods) formed the basis for flux balance analysis93. Together with the large-scale reconstruction of metabolic networks26, 61, it enables prediction of steady-state fluxes in yeast metabolism and provides the starting point for examination of the function and capacity of yeast's metabolic machinery24 and for testing hypotheses about the effect of gene deletions2, 36. An overview of recent developments in network-based approaches is given by Papin *et al.*83 A dynamic version of flux balance analysis was presented recently for fed-batch *S. cerevisiae* fermentation, which couples a detailed steady-state description of primary carbon metabolism with dynamic mass balances on key extracellular species44.

As mentioned above, dynamic modelling of metabolism in yeast is usually restricted to networks of limited size. There is a long tradition in modelling of metabolic networks, and especially of glycolysis, in *S. cerevisiae* (Figure 3). One major focus was to describe the dynamics and steady-state properties of this experimentally intensively studied pathway. Another focus was to detect under which conditions metabolic oscillations, which have been found in experiments, occur.

A model of the fermentation pathway from glucose to ethanol, glycerol and polysaccharides18, 100 has been used to systematically compare the dynamic description with traditional biochemical notation and with biochemical systems theory90, 91, first of all for a steady state. Moreover, by performing dynamic simulations following a perturbation pulse in the NADH : NAD ratio the simulated behaviour was first compared with the experimental data. Second, the predictive power of the sensitivity analysis performed in steady state was tested and it was found that the dynamic results reinforce the conclusions of the sensitivity analysis.

A seminal model describing the central carbon metabolism of yeast cells was presented by the Reuss group87, 108. It includes glycolysis from glucose down to pyruvate, the TCA cycle, branches to biosynthetic pathways, glycerol and ethanol, as well as balances for ATP/ADP and for NAD/NADH. This model was based on previous kinetic modelling of individual reactions and comprehensive new measurements of metabolite concentration time courses in glucose pulse experiments. It comprises 21 ordinary differential equations for concentration changes, 25 rates for transport and reaction steps, three algebraic equations for mass balances, as well as all the kinetic parameters. Kinetic expressions are mainly of the Michaelis–Menten type. Model simulations confirm that the model is able to reproduce the included time course information, a fact that is not trivial for such a large ODE system.

Employing a model of almost the same complexity (except for the TCA cycle), Teusink and colleagues106 investigated the question of whether a model that uses *in vitro* kinetics for the individual reactions is capable of reproducing *in vivo* kinetics. They found profound discrepancies and discussed, in detail, the causes related to individual enzymes.

An interesting question about why the structure of metabolic networks is the way we find it today was tackled by studying the impact of the phosphorylation in the upper part of glycolysis107. On the one hand, the coupling of an early reaction in the pathway to the conversion of ATP into ADP leads to a strong initial drop in free energy and thereby to an enhancement of the glycolytic flux39. The initial ATP investment is more than repaid by a surplus in ATP production downstream. On the other hand, such a turbo design can be dangerous and cause substrate-accelerated death. This effect has been observed when mutant cells are subjected to an excess of a substrate that previously limited growth in continuous culture. Using a metabolic model, it was shown that metabolic pathways with turbo design require special types of regulation in environments that have rapidly changing substrate availability.

A series of models for yeast glycolysis has been developed in order to study the distribution of flux and concentration control41, 51 within this network. Cortassa and Aon15 found that for the three different scenarios of carbon, nitrogen and phosphate limitation, sugar uptake always exerts the most control. Pritchard and Kell84 tested the flux control regimes for glucose-derepressed yeast glycolysis using the Teusink model106 and also found dominant control by hexose transport. Wang and Hatzimanikatis117 constructed a comprehensive steady-state model and showed by statistical analysis that yeast cells growing in batch culture condition exhibit dramatically different control schemes from those growing in a chemostat. They concluded that the difference is mainly due to the feedback introduced by the constraints of the chemostat.

Metabolic models with linear-logarithmic kinetics (see Methods) have been employed for the estimation of kinetic parameters from metabolite data obtained from dynamic as well as steady-state perturbations75. Moreover, the question of parameter identifiability from dynamic perturbation data in the presence of noise was addressed.

An interesting question in the modelling of metabolic pathways is, what circumstances cause or prevent oscillations?5, 30 Glycolytic oscillations can be observed experimentally by monitoring the absorption of light by NADH at 340 nm, using the absorption at 400 nm as a reference. It had been realized early on that main reason for oscillation is that autocatalytic processes are involved, e.g. the phosphofructokinase (PFK) reaction in glycolysis43, 97. This problem was also addressed in terms of non-linear thermodynamcis16, resulting in the conclusion that the presence of an autocatalytic feed-back loop with a specific type of kinetic non-linearity was sufficient to account for the oscillatory behaviour.

Several contributions have explored the effect of coupling of oscillators in populations of yeast cells19, 118–121. These investigations were stimulated by experimental results showing that oscillations of NADH or ATP in yeast can be induced by the addition of KCN85, leading to the synchronization of metabolic oscillations of the individual cells, which otherwise simply levelled out. The authors concluded that acetaldehyde synchronizes the oscillations of the individual cells. In an exploratory study, Wolf and Heinrich118 considered the coupling of oscillators described with a minimal model of two components. In a model that included more glycolytic reactions120 the effect of the various processes on synchronization is characterized quantitatively. Hypotheses for the causes of autonomous oscillations, as experimentally observed in aerobic continuous cultures, were tested computationally121. For a more detailed model based on comprehensive experimental data50, 74, it was determined which set of parameters associated with the models allows for oscillations of glycolysis.

Dynamic modelling is not restricted to just mirroring experimentally measured system dynamics. Since modelling is abstraction, one can always question whether the current representation is appropriate or if another representation would better serve a certain purpose. It is always possible to include more components into a model in order to represent their influence on the system of interest. Another interesting aspect is the reduction of models in order to make them easier to comprehend and less computationally expensive. Liebermeister *et al.*66 presented a model reduction approach that allowed them to focus on a selected system while preserving major modes of dynamics for the reduced system. Model reduction with preservation of its basic dynamic properties was also the aim of a study to reduce an oscillatory model of yeast glycolysis from 20 to eight, six or three variables20.

Signalling pathways enable cells to sense changes in their environment, to integrate external or internal signals, and to respond to them by changes in transcriptional activity, metabolism or other regulatory measures. The proper functioning of these pathways is crucial for adaptation and survival under varying conditions, but also for differentiation and cell fate. Since the environmental changes induce dynamic responses within the cell, those processes are especially attractive for experimental investigation and dynamic modelling. Signal transduction in yeast has been intensively investigated for years; an overview is given in Hohmann45.

In order to understand the complex behaviour of signalling networks, researchers have adopted computational modelling approaches, ranging from abstract models that emphasize some key features of signalling pathways40, 82 to detailed models that describe the dynamics of specific pathways in specific organisms94, 105, 114, 123. Several models from both categories have been developed and experimentally tested, which has revealed interesting behaviour. Overviews on structural properties and dynamic features of signalling pathway models are given in Kholodenko52, Papin *et al.*81 and Tyson *et al.*110.

The response of yeast cells to external stimuli, environmental changes, nutrient supply or availability of a mating partner is ensured by a variety of signalling pathways that partly overlap by the use of common proteins (Figure 4). Despite their diversity in function and design, many signalling pathways use the same essential components, which are often highly conserved through evolution and between species. Signalling pathways frequently consist of ubiquitous building blocks, such as receptors, MAP kinase cascades, G proteins and small G proteins23, and their design seems to be conserved throughout all kingdoms of life. For example, proteins in yeast pathways have homologues in human pathways and G proteins or MAP kinases are conserved throughout the kingdoms.

The pathway that has attracted most attention is the pheromone pathway of budding yeast. Yeast cells have a life cycle that involves diploid and haploid stages. Haploid cells belong to different mating types, MAT**a** and MATα, and they secrete specific pheromones. The occurrence of the pheromone α-factor indicates for a MAT**a** cell the presence of a potential mating partner, which induces cell cycle arrest and formation of a mating projection to enable mating. The pathway, which is employed to transfer the signal from the receptor to the targets has been subject of intensive experimental studies22, 23, 111, 112.

A first model of the receptor activation and G protein regulation within the pheromone pathway has been presented by Yi and colleagues123. This model takes into account G protein activities that have been measured using fluorescence resonance energy transfer (FRET). It comprises the production, degradation and activation of the G protein-coupled α-receptor (Ste2p), the activity cycle of the G protein and its regulation by the regulator of G protein (RGS) Sst2p. This model has been adapted and incorporated into a more comprehensive model of the pheromone pathway59, which includes downstream processes of the activation of Gβγ, such as binding of the components of the MAP kinase cascade to the scaffold protein and their subsequent activation. The phosphorylated MAPK (Fus3p) triggers the following events, including the activation of the transcription factor Ste12p, the activation of the cell cycle regulator Far1p and the activation of the RGS Sst2p.

The regulation of signal transduction within the pheromone pathway is ensured by several feedback loops that also contribute to pathway downregulation after successful signalling, including active Fus3p-induced repeated Fus3p phosphorylation, as well as Sst2p regulation. Yi *et al.*123 studied strains with either constitutively active or inactive Sst2p. Moreover, the transcription factor Ste12p enhances the expression of the protease Bar1p, which is exported and cleaves the α-factor, and thereby counteracts the input signal. Hence, the pathway design ensures the long-term downregulation of the pathway after successful activation of target processes.

Recently, a pheromone pathway model has been presented98 which does not take into account the feedback via the protease Bar1p by considering a *bar1*Δ strain but, compared to the model described above59, provides an extended representation of the complex formation and phosphorylation processes at the scaffold protein Ste5p.

A Petri net model of the pheromone pathway and adjacent events88 allows qualitative aspects of the network, such as reachability (which system states can be reached starting from a set of initial conditions), liveness (which transitions are possible in certain states) and boundedness (is there an accumulation of tokens at a place?), to be tested.

Another intensely studied pathway is the high osmolarity glycerol (HOG) pathway, which is involved in osmoregulation and specifically induces the increased production of glycerol. Osmoregulation is a fundamental and highly conserved process by which cells control their water balance. The response to osmotic stress has been described by a model58 that comprises the HOG pathway, transcriptional regulation, the effect on metabolism and the change in the production of glycerol, and the regulation of volume and osmotic pressure. The HOG pathway has two input branches, the Sln1 branch and the Sho1 branch; the latter is not considered in the model. The Sln1 branch is a phospho-relay system under the control of the transmembrane receptor Sln1p. Under non-stress conditions, it is continuously phosphorylated and transmits its phosphate group via Ypd1p to Ssk1p, thereby keeping Ssk1p phosphorylated and inactive. Osmotic stress interrupts phosphorylation of Sln1p; thus, Ssk1p switches to a non-phosphorylated active state. In this form, it triggers the HOG MAP kinase cascade, which involves the redundant proteins Ssk2p and Ssk22p as well as Pbs2p and Hog1p. Phosphorylated Hog1p can enter the nucleus and regulate the transcription of a series of genes. Some of these genes code for metabolic enzymes involved in the production of glycerol. Another player contributing to the accumulation of the osmolyte glycerol is the glycerol transporter Fps1p, which normally allows glycerol flow out of the cell and which closes upon osmotic stress, thereby ensuring initial fast accumulation of glycerol. The model helped to reveal import aspects of yeast osmoregulation: (a) the contribution of osmotic and turgor pressure changes to the regulation of biochemical processes; (b) the role of the glycerol channel Fps1p for the immediate cell response; and (c) the function of the induced changes of gene expression as long-term contributions to the upregulation of glycerol.

The regulatory effect of the nuclear translocation of the active MAPK of the HOG pathway and of the phosphorylation and activation of nuclear targets on the expression of specific genes has been investigated in a theoretical approach73 which, in turn, applied the principles of extensive model analysis work concerning the properties of MAP kinase cascades48, 89.

For heat-shock response, the nuclear events have been described in a mathematical model86. The heat-shock response is a ubiquitous molecular response to proteotoxicity resulting from the appearance of non-native and damaged proteins70, 71. The central elements of this process are the heat-shock proteins (HSPs) that function as molecular chaperones. Upon sensing a stress signal, such as elevated temperatures, small toxic molecules, oxidants or heavy metals, cells transiently overexpress chaperones to high levels to meet the stress demand. Heat-shock transcription factor-1 (HSF1) regulates the expression of the major HSPs. The mathematical model considers the nuclear events during the heat-shock response. It includes nuclear translocation and phosphorylation of the transcription factor HSF1 upon reception of a stress signal. The phosphorylation of HSF1 results in elevated transcription of hsp mRNAs and subsequent translation of HSPs, i.e. inducible molecular chaperones which are capable of participating in protein refolding and/or regulation of their own expression. Three regulatory events are included: HSPs binding to, and inactivation of, active HSF; HSP-dependent sequestration of free HSF and initiation of transcriptional activation; and HSP-independent increase of mRNA transcript stability due to stress. Using this model, the contribution of different regulatory steps to the determination of HSP levels was tested, especially the role of mRNA stabilization compared to increased translation.

For the performance of a signalling pathway, it is not only the wiring or the kinetics of individual reactions that are important, but also the profile of the applied stress. It has been demonstrated that repeated activation leads to different effects, e.g. a repeated activation of the HOG pathway by successive addition of salt results in a repeated pathway activation, while the pheromone pathway becomes desensitized, such that a second application of pheromone has only minor effects57.

At first glance, signalling can be seen as a linear connection between input elements (the receptors) and output elements (such as regulators of gene expression). A closer inspection reveals that signalling pathways interact with each other, forming a network. Schwartz and Baron96 introduced the notion of cross-talk, referring to the case that two inputs (here, growth factors from different families) work through distinct signalling pathways but cooperate to regulate cell growth. Intensive experimental work has revealed numerous potential paths for cross-talk.

The integration of signals by pathways that share compounds has been exemplarily studied for the pheromone and the filamentous growth pathways92. The impact of the activation of one or the other or both pathways on their targets was quantified by their intrinsic and extrinsic specificity, i.e. the relative effect of the activation of one pathway compared to the simultaneous activation of both pathways. It was demonstrated that the pheromone pathway experiences a cross-activation by the filamentous growth pathway and mutual signal amplification, while the filamentous growth pathway is inhibited by an activated pheromone pathway (mainly due to the degradation of Tec1p triggered by phosphorylated Fus3p).

The cell cycle is the series of events accompanying cell growth and division (Figure 5). It comprises the phases G_{1}, S, G_{2} and M. DNA replication and early bud formation occur in S phase. In M phase, chromosomes become separated and division is prepared. G_{1} and G_{2} are growth or gap phases and, especially during the G_{1} phase, the cell must grow sufficiently to replicate its building blocks to give rise to a new organism. Cell cycle progression has been the subject of a number of dynamic modelling approaches. Early modelling attempts tried to understand the oscillatory nature of the cell cycle32, 109. Later models from the Tyson and Novak groups paved the way to include more and more molecular details and to understand the contribution of individual components of the cell cycle machinery. The major components of these models are cyclins, i.e. proteins that are periodically produced and degraded during the cell cycle, and cyclin-dependent kinases (CDK) as well as CDK inhibitors (CKI). A series of interactions of these components, combined with transcriptional regulators and protein degradation, form the backbone of cell cycle progression. There are models of increasing complexity for fission yeast77–79, 102–104, and for budding yeast7, 8 as well as a generic model for eukaryotic cells17. These workers tackled important questions such as: (a) the timing of DNA replication to prevent repeated replication without mitosis and ensure full DNA replication before onset of mitosis79; and (b) the performance of checkpoints ensuring completion of relevant molecular interactions before further cell cycle progression77. Sveiczer *et al.*103 proposed a stochastic version of those models, considering as origins of stochasticity: (a) the known asymmetry of cell division, which produces daughter cells of slightly different sizes; and (b) the difference in nuclear volumes between the two newborn cells. By assuming that the accumulation of cyclins in the nucleus is proportional to the ratio of cytoplasmic to nuclear volumes, they investigated the role of the nucleocytoplasmic ratio in cycle-time regulation.

Comprehensive phase plane analysis and bifurcation analysis demonstrated that these models are able to account for cell cycle oscillations. For this analysis, the accumulation of mass and the ratio of mass to DNA content are used as the driving force of the relevant protein production processes and for the periodic cell division. With respect to kinetics, these models employ mostly Michaelis–Menten kinetics with a special choice of parameters (the so-called Goldbeter–Koshland function33), which ensure sharp switches of protein activity. The models are based on a huge amount of experimental evidence in the literature and the major test for their validity is to predict the phenotypes of a large series of mutants correctly. While qualitative predictions of the model for wild-type cells agree with available data, time-course data proving the correctness of the predicted dynamics for various proteins are not yet comprehensively available.

A detailed model of the G_{1}–S transition of the yeast cell cycle1 with simple mass action kinetics takes into account the fact that the cell cycle network is compartmentalized into cytoplasm and nucleus, and it describes cellular growth as a gain in volume. In this way, this model can explain how cells can determine the size at which they undergo the transition to S phase, a hitherto open question.

Cell cycle regulation is closely coupled to cell growth and division. Dynamic models of cell size distribution in growing yeast populations11, 37, 113 give insight into the underlying control mechanisms, but also into the different timing of the unbudded cell cycle phases in mother and daughter cells.

Modelling of the dynamic regulation of gene expression became relevant after microarray techniques started to provide comprehensive quantitative datasets. The following examples show that the existence of an appropriate dataset can be stimulating and influential for the development of dynamic models. Focus of a number of approaches was on the regulation of cell cycle genes, all inspired by the same sets of quantitative data10, 101. The prevailing question was: can one extract the transcriptional network driving (or accompanying) cell cycle progression from gene expression data?

A dynamic model for quantification of transcriptional regulation116 uses an ODE approach, which relates gene expression with the set of potential regulators to target genes. The best combination of target genes and regulators is extracted by fitting the model to the experimental data.

In another approach, the dynamics of the cell cycle transcriptional network is represented by first assuming that the changes in the expression of a specific gene depends on the product of binding constants of transcription factors weighted by an exponent, α. Second, the dynamics of these α-coefficients is determined by a time-translation matrix that relates successive measurement points12.

The dynamics of cell cycle related gene expression have also be presented with stochastic methods6, 9. Another approach4 combined Boolean logic to describe genetic regulatory interactions and Bayesian methodology to learn model structure and parameters from gene expression data.

All these models have in common that they try to extract the network structure and the relevant dynamic parameters from the available set of expression data, but not from previous knowledge from other sources about the gene regulatory interactions.

Network component analysis (NCA) was also applied to cell cycle regulation65. This approach is related to principal components analysis (PCA) but, different to PCA and to the above-described methods, this approach takes into account the present and probably incomplete knowledge about the regulatory links in the network. Based on the outcome, the relation between transcription factors and the expression profile of cell cycle-regulated genes was dynamically modelled and fitted to experimental data64, 101.

In order to understand certain aspects of the dynamics of yeast cells upon response to stress or just during their normal cell cycle, it is necessary not only to consider either metabolism or signalling pathways or gene expression regulation separately, but also to study how these networks are integrated and influence each other. Most cell cycle models7, 8 integrate protein–protein interactions and protein production via transcription and translation. For various signalling models, the protein activation cascade and the resulting regulation of gene expression has been analysed58, 59, 73. The regulation of metabolism, on one side, via a signalling cascade and resulting gene expression changes and, on the other side, by direct modulation of metabolic fluxes (by shutting down of a glycerol transporter) was studied for the specific case of the osmotic stress response58.

Gat-Viks and colleagues28, 29 have presented a mixed Boolean and Bayesian approach to derive the comprehensive structure of such integrated networks from diverse types of qualitative and quantitative data. They have exemplified their strategy using the yeast HOG pathway and by modelling the interaction of this pathway with other stress response pathways.

In order to understand the response of cells to external changes, such as stress and, especially, changes in the availability of nutrients, models are required that comprise both the metabolic pathways that convert the nutrients and the expression of the enzymes as regulated in response to sensed changes of the nutrient availability.

Here, models were presented that use optimality considerations to predict the changes in gene expression. There are two basic underlying assumptions. First, it is considered that cells have adapted during evolution to a changing environment and that they have evolved response programs for typical and frequent types of stress, and they might even be able to ‘anticipate’ certain external changes. Second, in order to keep models simple and manageable, and also due to a lack of more detailed data, it is assumed that the changes in protein concentrations (or, more specifically, in enzyme concentrations) can be represented in a good approximation by the changes in gene expression measured on the mRNA level by microarrays.

A dynamic model of central carbon metabolism with mass action kinetics54 was used to predict the optimal temporal changes of enzyme concentrations under the condition of glucose depletion. It was assumed that the cells are optimized during evolution to maintain the homeostasis of ATP (to keep an energy charge) and NADH (to ensure balancing of the redox state of the cell) and to keep their level above a certain threshold value for as long as possible. In the optimal scenario, cells first activated enzymes necessary to use up all available glucose to produce ethanol, i.e. the lower part of glycolysis and ethanol production. Activation of lower glycolysis also ensured that no carbohydrates were ‘wasted’ for synthetic pathways. Only in a second phase, when glucose was depleted, were glycolytic enzymes downregulated and enzymes necessary for respiration upregulated. Simulation results are close to the experimental data of DeRisi *et al.*21, who measured expression changes of all 6000 or so genes of *S. cerevisiae* upon glucose starvation.

Vilaprinyo and colleagues115 investigated quantitative design principles for gene expression following heat shock. To this end, they established a small model of energy metabolism considered as important for providing metabolites necessary for mediating appropriate heat stress responses, i.e. ATP and NADPH. The model employs the generalized mass action formalism. Moreover, they produced a large number of gene expression profiles related to the enzymes involved in the metabolic network, assuming that gene expression changes are indicative of enzyme concentration changes. By formulating a set of criteria, they selected those gene expression profiles that allow the necessary metabolic switch associated with heat stress response. Those criteria include: (a) flux-based conditions (increased production of ATP, NADPH, and trehalose); (b) avoiding unnecessary accumulation of other metabolites; (c) economic adaptation, i.e. minimal gene expression changes, as well as maximal glycerol accumulation.

Expectations towards systems biology range from scepticism to the hope for a whole-cell model in the near future. The examples presented show that between those two extremes, mathematical representations of cellular networks of limited size can be very useful for understanding specific aspects of regulation, such as the cause of glycolytic oscillations or the optimal regulation of gene expression programmes.

The numerous reviews about systems biology (maybe more than serious original papers in the field) have quickly extracted the basic principles of this field. The first principle is the ‘iterative cycle’ denoting the prospective successful approach of first experimental evidence and data → initial mathematical model → new experiments and data → updated model, and so forth. The second principle is that a model must be predictive and that the prediction should be experimentally verified. Unfortunately, as soon as this confirmation is available, it is hard for the reader to decide which was first–the evidence or the prediction. There is only a very limited set of cases in which prediction and verification were provided independently, that is not by the same group of scientists or within the same project95. A third principle matures only slowly, i.e. the apprehension that not only experiments but also modelling results must be reproducible. This demand induces a set of quality conditions on models that are rarely fulfilled. Those conditions include publishing not only all model information, such as equations, parameters and initial conditions, but also all assumptions that have been used to arrive at the level of abstraction presented by the model. A more extensive discussion of this issue is given in Le Novere *et al.*63.

Dynamic modelling of yeast cellular processes has focused, up to now, on central carbon metabolism, a selected number of signalling pathways and cell cycle progression. Only a few models attempt to connect these different types of regulatory networks. It is clearly an upcoming task for further modelling to tackle questions such as: What is the impact of gene regulation on metabolic fluxes (besides the drastic effect of a knock-out mutation)? Or, What is the effect of the activation of a signalling pathway on its targets depending on the current cell cycle state? Successful dynamic models have been developed in areas where appropriate data are available. Metabolic models have been based on a long tradition of studying enzymes and pathways in energy metabolism. Therefore, a critical amount of information was available to construct sensible models, which can be used to test hypotheses about the system's behaviour. Time-resolved data on yeast gene expression related to cell cycle regulation10, 101 have inspired a series of modelling approaches or have been used as a test set for existing models4, 6, 8, 12, 116. These examples show clearly that the development of theoretical and computational approaches needs appropriate and reliable time-resolved datasets. This aspect was also emphasized in responses to a questionnaire on standards in computational systems biology56.

One can conclude that the development of predictive models requires appropriate datasets that allow sensible definition of the network structure (which compounds and interactions to include) and that allow estimation of the relevant parameters. To this end, it will be helpful if modelling and experimental planning are tightly linked. Upcoming single-cell measurements, such as those using single-cell imaging techniques (e.g. Gordon *et al.*34) will provide useful information, e.g. about the effect of a perturbation on cells in different cell cycle stages. The investigation of the pheromone response of yeast cells13 yielded, for example, clues about the origin of noise within the pathway.

The purpose of modelling is to provide an abstract description of a biological process or structure that fosters the understanding/representation of specific aspects of this process or structure. Such a model must neglect other aspects for the sake of simplicity. The choice of included features will change with a change in the specific question to be answered by the model. Therefore, one cannot establish fixed rules for a model that are valid once and forever. On the other hand, the growing interest in dynamic models and the need for communication between modellers and experimental researchers make it necessary to establish some rules how specific aspects should be expressed in a model of a certain type. A prominent approach for the development of such a standard is the Systems Biology Markup Language (SBML)49, which serves as a unified exchange language for the description of biochemical network models. Another standard is the Minimal Requirements in the Annotation of Models (MIRIAM)80 for the description and documentation of models in a publication. Model repositories such as JWSonline99 or Biomodels62 make developed and curated models available to public. Dedicated simulation software such as Copasi47 or CellDesigner53 can also enable interested biologists with limited preparatory training to analyse or even develop dynamic models.

This work was supported by yeast systems biology grants of the European Community: the Yeast Systems Biology Network (YSBN, EU Project No. FP6-2004-LSH-5 # 018942) and QUASI (EU Project No. LSHG-CT2003-530203). I thank Dr Wolfram Liebermeister for critical reading of the manuscript.

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