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Signal transduction pathways allow cells to respond to environmental changes and stimuli and to adapt their gene expression programme to actual demands. Their function and temporal behaviour are in the focus of extensive experimental work. We develop a mathematical model for the description of the dynamics of the pheromone pathway after external stimulation.
The yeast Saccharomyces cerevisiae may be present in one of two haploid cell types, which are able to mate, MATa and MATα cells. Pheromones released by one type bind to a specific plasma membrane receptor of the respective other type. The pheromone receptor activates a heterotrimeric G protein, which transmits the signal from the cell surface receptor to intracellular effectors. It follows the activation of a scaffold protein-bound mitogen-activated protein (MAP) kinase (K) cascade and the phosphorylation and activation of nuclear proteins that control cell polarity and changes in cytoskeletal structure, transcription and progression through the cell cycle. The target of these changes in response to pheromone is preparation for cell fusion. Polarized cell growth is required to form the site of cell fusion and to ensure that the two cells grow towards each other. New gene transcription is necessary to produce new proteins, e.g. proteins that mediate cell adhesion. The arrest of cell cycle progression and growth arrest are required to synchronize the two mating partners.
A huge amount of data and information has been collected about the mechanisms of various signalling pathways, their functioning and their output. A challenging task in current bioscience is the conversion of this information into knowledge about phenotypic traits and understanding biological processes such as regulation and adaptation. Integration of the detailed investigation of particular biological processes, on the one hand, and consideration of network properties of complex biological systems, on the other hand, lead to a holistic approach called systems biology. An important tool of systems biology is the application of mathematical modelling and simulation, especially the development of models with predictive value. Mathematical modelling is a powerful method for the analysis of structural characteristics and dynamics of biological processes (e.g. Chen et al., 2000; Van Wuytswinkel et al., 2000; Heinrich and Schuster, 1996; and many others). Mathematical models have been employed to study metabolic networks (for examples, see Fell, 1997; Heinrich and Schuster, 1996; Cornish-Bowden and Cardenas, 2000). Mathematical models are increasingly used to study certain traits of signal transduction pathways, i.e. robustness against concentration variations (Batchelor and Goulian, 2003), threshold properties and bistability (Levchenko et al., 2000; Ferrell, 2002), characteristic times (Llorens et al., 1999; Heinrich et al., 2002), feedback effects (Asthagiri and Lauffenburger, 2001; Bhalla and Iyengar, 1999; Ferrell, 2002; Kholodenko, 2000) and ultrasensitivity (Huang and Ferrell, 1996; Kholodenko, 2000). Two models address specifically G protein activity in yeast (Yi et al., 2003; Hao et al., 2003).
Especially in the case of the yeast pheromone pathway, a lot of experimental information is available about the corresponding changes in gene expression and cell cycle, concerning both the interaction of the involved proteins, the sequence of events, temporal data, and even kinetics. However, there are still many open questions that cannot easily be investigated in an experiment, since not all compounds can be accessed with high accuracy. The mathematical model will be used to test whether current understanding of the mechanisms of cellular adaptation to pheromone stimulation may reflect the true processes. It allows predictions for the temporal performance of compounds that are experimentally not accessible, it can be used to simulate a range of experimental conditions, and it can point to open questions for further experiments.
In the following we describe the processes that are known in considerable detail from experimental investigation of the yeast pheromone response and will be considered in our model (for reviews, see e.g. Dohlman and Thorner, 2001; Elion, 2000; Gustin et al., 1998).
Materials and methods
For a biochemical reaction system, it is customary to use a set of ordinary differential equations (ODEs) to describe the changes in the concentration of a biochemical species. In a system of m biochemical species with the concentration ci (i = 1, .., m) and r biochemical reactions with the rates vj (j = 1, .., r), one may write:
where the quantities nij denote the stoichiometric coefficients (the number of dedicated reactants entering a reaction). The rate of a reaction is a function of the concentrations of substrates, products and probable effectors.
In the model presented below, we consider all unbound substances in their different phosphorylation states and all complexes formed of these substances as individual species with their respective concentrations. All complex formations and disintegrations, degradations and phosphorylations or dephosphorylations are considered as reactions with their respective rates.
The complete set of differential equations and expressions for the individual rates is analysed with respect to qualitative properties of the model. This concerns conservation relations (for a compound that is not newly produced or degraded in the frame of the model, the conserved moiety comprises all unbound species and all complexes in which it is involved) and flux analysis. The system description allows the calculation of steady states, i.e. states in which concentrations do not change with time. For simplicity, we assume that our system is in steady state before the treatment with the α-factor. Furthermore, the simulation of the system of differential equations yields the temporal behaviour of all considered compounds after the stimulation. For the simulations, we used the numerical integration routine offered by Mathematica, Wolfram Research, Version 4.1.
The mathematical model
A spatial diagram of the current understanding of the pheromone signalling pathway is given in Figure 1. As a basis for the mathematical model, we use the scheme of the considered reactions as depicted in Figure 2, which favours the temporal order of the processes rather than the spatial. It involves the following fundamental modules: (a) the receptor activation; (b) the G protein cycle; (c) the complex formation involving Ste5; (d) the MAPK cascade; and (e) the downstream effects of phosphorylated Fus3 and activated Far1. We have chosen to model a MATa cell, which is exposed to the pheromone α-factor.
We describe the dynamics of systems using a set of differential equations governing the concentration changes of individual components and of complexes over time. The equations are listed in Table 1. For simplicity, we assume that protein production and degradation are slow compared to the events of the signalling pathway. In this way, we consider only the activation or deactivation of compounds or their involvement in complexes or their release from complexes. The parameters of the model are taken from the literature as far as possible (see Table 2). For the remaining parameters, the models have been used to test the effect of parameter choice (see e.g. Figure 8). The initial concentrations are given in Table 3.
Table 1. Equations governing the dynamics of the pheromone pathway (mathematical model)
Ordinary differential equations
ν1 = α [t]· Bar1active[t]· k1
ν2 = Ste2[t]· α [t]· k2
ν3 = Ste2active[t]· k3
ν4 = Ste2active[t]· k4
ν5 = Ste2[t]· k5
ν6 = Ste2active[t]· G α β γ[t]· k6
ν7 = Gα GTP[t]· k7
ν8 = Gα GTP[t]· Sst2active[t]· k8
ν9 = Gα GDP[t]· Gβ γ[t]· k9
ν10 = Gβ γ[t]· C[t]· k10
ν11 = D[t]· k11
ν12 = Ste5[t]· Ste11[t]· k12
ν13 = A[t]· k13
ν14 = Ste7[t]· Fus3[t]· k14
ν15 = B[t]· k15
ν16 = A[t]· B[t]· k16
ν17 = C[t]· k17
ν18 = D[t]· Ste20[t]· k18
ν19 = E[t]· k19
ν20 = E[t]· k20
ν21 = E[t]· k21
ν22 = F[t]· k22
ν23 = F[t]· k23
ν24 = G[t]· k24
ν25 = G[t]· k25
ν26 = H[t]· k26
ν27 = H[t]· k27
ν28 = I[t]· k28
ν29 = L[t]· Fus3[t]· k29
ν30 = K[t]· k30
ν31 = K[t]· k31
ν32 = L[t]· k32
ν33 = Fus3PP[t]· k33
ν34 = Ste12[t]· Fus3PP[t]· k34
ν35 = Ste12active[t]· k35
ν36 = Ste12active[t]· Bar1[t]· k36
ν37 = Bar1active[t]· k37
ν38 = Bar1active[t]· k38
ν40 = Far1PP[t]· k40
ν41 = Far1[t]· Cdc28[t]· k41
ν42 = Gβ γ[t]· Far1PP[t]· k42
ν43 = M[t]· k43
ν44 = N[t]· k44
ν45 = Far1PP[t]· Cdc28[t]· k45
ν47 = Sst2active[t]· k47
Table 2. Parameter values for the mathematical model
The processes and individual reactions that are taken into account are described below. The respective reactions entering the mathematical model are denoted in the text.
Receptor activation by pheromone
The α-factor secreted by MATα cells approaches the cell surface. It may be degraded (reaction ν1) by the protease Bar1, the transcription of which is enhanced after pheromone stimulation. The α-factor binds to the specific receptor Ste2 (reactions ν2 and ν3), which is only expressed in MATa cells. This renders the receptor active. The receptors are the only components of this pathway that determine the specificity of the response with respect to the mating type; all the other components are identical in both cell types (Kurjan, 1992). The active receptor can be downregulated (reaction ν4) and the non-activated receptor may be degraded by an ubiquitin-dependent mechanism (reaction ν5) (Esch and Errede, 2002; Wang and Dohlman, 2002; Yi et al., 2003). We do not consider receptor synthesis in the time frame of the signalling process, as has been done by Yi et al. (2003).
The scaffold complex
Pheromone-induced signalling requires a scaffold protein–MAP kinase complex involving Ste5, Ste7, Ste11 and Fus3. The formation of this complex appears to be independent of the pheromone stimulus and is not necessary for signal transduction (Inouye et al., 1997b; Yablonski et al., 1996). Interactions of Ste5, Ste7 and Ste11 do not require signalling from either G protein or Ste20; the complex also exists in cells that are not exposed to pheromone stimulation (Marcus et al., 1994).
Ste5 is a scaffold protein (for review, see e.g. Elion, 2001), which is able to bind the kinases Ste7, Ste11 and Fus3 or Kss1. It seems to be crucial for signal and substrate specificity by preventing cross-talk between several pathways, accumulating the proteins in specific areas of the cell, bringing them into a special relationship to each other (Burack and Shaw, 2000), preventing the influence of negative regulators on the bound kinases, and suppressing auto-inhibitory conformations of the kinases (especially Ste11) (van Drogen et al., 2000).
Ste11 is the MAPKKK of the yeast pheromone response pathway (Yablonski et al., 1996). Ste11 is associated with Ste50 and auto-inhibits itself. Binding of Ste5 to Ste11 is essential for its activation by the mating pheromone (Elion, 2001).
The MAPKK of the pheromone pathway is Ste7. The unbound N-terminus is capable of interacting with the MAP kinase Fus3. Both proteins form a stable complex independently of Ste5 or a signal (Bardwell and Thorner, 1996). The C-terminal kinase domain interacts specifically with Ste5 (Choi et al., 1994; Marcus et al., 1994; Printen and Sprague, 1994).
Fus3 acts as the MAP kinase of the pheromone response pathway. It shows some similarities to Kss1, which is involved in the invasive growth pathway but probably has a different activating mechanism (Breitkreutz and Tyers, 2002; Madhani and Fink, 1998). Fus3 in its inactive form establishes a stable association with Ste5 (Kranz et al., 1994; Choi et al., 1994; Marcus et al., 1994; Printen and Sprague, 1994).
Theoretically, there are different possibilities for the MAP kinases Ste7, Ste11 and Fus3 to bind to Ste5. They can associate with the scaffold protein one after another in a special order or in a random mechanism, where it makes no difference whether the MAPKKK, the MAPKK or the MAPK binds first. But there is experimental evidence that the four components first form a Ste5–Ste11 complex and a Ste7–Fus3 complex, which associate afterwards (Choi et al., 1994; Sette et al., 2000). Theoretical studies show that the characteristic times for the formation of the Ste5–Ste7–Ste11–Fus3 complex and its steady-state concentration differ insignificantly for these various mechanisms. Based on experimental knowledge, we consider in the model the formation of the Ste5–Ste11 complex (complex A in Figure 2, reactions ν12 and ν13) and the Ste7–Fus3 complex (complex B, reactions ν14 and ν15), respectively, and their association to complex C (reaction ν16). The disaggregation of the scaffold protein–MAP kinase complex is included in reaction ν17.
The activity cycle of the G protein
The G protein binds to the intracellular domains of the receptor Ste2 (Conklin and Bourne, 1993). The G protein is a heterotrimer and consists of the subunits Gα (Gpa1), Gβ (Ste4) and Gγ (Ste18). Gβ and Gγ act as the heterodimer Gβ γ (Hirschman et al., 1997). Gα interacts with two regions of Gβ γ (Ford et al., 1998). For the full coupling with the receptor, all three subunits of the heterotrimer are required (Blumer and Thorner, 1990). The G protein cycle is schematically represented within Figure 2.
Upon activation, the interaction of the receptor with the Gα subunit leads to a series of conformational changes, allowing for the release of GDP from Gα and the association of GTP with the nucleotide binding site (Conklin and Bourne, 1993) (reaction ν6). The related conformational change leads to the release of Gβ γ, which is in turn capable of binding and activating components of the mating pheromone response pathway. Gβ γ also has a regulatory function for the G protein cycle. Gα is a negative regulator of Gβ γ. It regulates Gβ γ signal transmission to multiple effectors (Ford et al., 1998).
After hydrolysis from GαGTP to GαGDP (as a slow process reaction ν7, as process enhanced by action of Sst2 reaction ν8), Gα can reassociate with Gβ γ to the heterotrimeric Gα β γ (reaction ν9). In this way, the G protein cycle is closed.
Sst2 is a regulator of G protein signalling (RGS protein) and as a GTPase accelerating protein (GAP) it is able to accelerate the hydrolysis (reaction ν8) (Dohlman and Thorner, 1997; Lan et al., 2000). Through binding to Gα and subsequent conformational changes in the Gα subunit, a RGS protein shortens the lifetime of the active GTP-bound state and accelerates reassociation to the heterotrimer. This restrains the cellular response. Consequently, the RGS protein counteracts the receptor. The activity of Sst2 is dependent on phosphorylated Fus3 (reaction ν46). Note that in our model the activity of Sst2 is part of a feedback loop, while in the model of Yi et al. (2003) Sst2 is either active or inactive.
The intensity of the G protein signal action on the further components of the signalling cascade depends on the rate of nucleotide exchange, the rate of GTP hydrolysis and the rate of subunit reassociation (Dohlman, 2002).
The downstream effects of Gβ γ, which is not associated to Gα
Unbound Gβ γ is able to interact with a number of proteins including Cdc24, Cdc42, Ste20 and the scaffold protein Ste5. The Gβ subunit is responsible for interactions with other proteins and the Gγ subunit is responsible for tethering to the plasma membrane via dual lipid modification (Manahan et al., 2000). The binding of Gβ γ to Cdc24 and Cdc42 is responsible for bud formation and, in this pathway, for polarized growth and for forming a projection (so-called ‘shmoo tip’) towards the pheromone source (chemotropism). In addition, the scaffold protein Far1 and the proteins Ste20 and Bem1 are necessary (Butty et al., 1998; Elion, 2000; Nern and Arkowitz, 1999). The formation of the projection itself is not considered in our model.
Gβ γ binds to the C-terminus of the p21-activated protein (PAK) Ste20 (Leeuw et al., 1998; Song et al., 2001) (reactions ν18, ν19). This interaction is dependent on pheromone and essential for signal transduction (Leeuw et al., 1998). Gβ γ has no activating or inhibiting influence on Ste20 but promotes its interactions with other proteins (Pryciak and Huntress, 1998). Through Cdc42, this system is anchored to the plasma membrane, accumulated in the projection tips and stimulated (Zhao et al., 1995).
The scaffold protein Ste5 is tethered to the plasma membrane due to binding to the Gβ γ dimer (reactions ν10, ν11) (Inouye et al., 1997b; Pryciak and Huntress, 1998). This step is pheromone-dependent and essential for signal transmission to the MAPKKK Ste11. By localizing Ste5 at the plasma membrane (with bound Ste11), the scaffold protein is brought into proximity of the membrane-associated Ste20 (Pryciak and Huntress, 1998). This proximity alone is not sufficient to enable Ste20-dependent phosphorylation of Ste11; the interaction of Ste5 with Gβ γ is necessary to permit access of Ste20 to Ste11 (Sette et al., 2000). Autophosphorylation and activation of Ste20 makes it competent to initiate the following MAP kinase cascade by phosphorylating Ste11 (reaction ν20) (Dan et al., 2001; van Drogen et al., 2000; Whiteway et al., 1995). Therefore, Ste20 could be called a MAPKKKK of the MAPK cascade, but it is regarded as an upstream activator of the cascade.
The MAPK cascade
The phosphorylation of Ste11 by Ste20 activates its kinase function. Activated Ste11 (Ste11PPP) does not diffuse in the cell. Unbound Ste11PPP is unstable and could be rapidly degraded by an ubiquitin-dependent mechanism. A feedback phosphorylation of Ste11PPP by activated Fus3 promotes its rapid degradation and, thereby, decreases the possibility of cross-talk to other pathways (Esch and Errede, 2002; van Drogen et al., 2001). Ste11PPP, which is bound to the MAPK scaffold protein Ste5, activates Ste7 by phosphorylating two highly conserved residues in the activation loop of Ste7 (reaction ν22) (Neiman and Herskowitz, 1994). The MAPKK is competent to feedback phosphorylate Ste11. The phosphatases that are responsible for the deactivation of Ste7 and Ste11 are not yet known (Dohlman and Thorner, 2001; van Drogen et al., 2000).
The activation of Fus3 takes place by a Ste7-dependent phosphorylation of threonine and tyrosine residues in the activating loop (Bardwell and Thorner, 1996) (reaction ν24). Fus3PP is able to initiate some feedback phosphorylations (on Ste5, Ste7 and Ste11) (reaction ν26).
While Fus3PP dissociates rapidly from Ste5 (reaction ν28), the scaffold protein remains bound at the plasma membrane and forms a platform which allows an activation of many Fus3 molecules (reactions ν28, ν29, ν30, and ν31) before it dissociates from Gβ γ itself. This may lead to an amplification of the signal. The magnitude of this amplification is still unknown (van Drogen and Peter, 2001; van Drogen et al., 2001). In the model we also consider a possible disaggregation of the whole complex at several stages (reactions ν21, ν23, ν25, ν27, and ν32). The phosphatases responsible for the deactivation of Fus3 are Ptp2, Ptp3 and Msg5 (reaction ν33) (Zhan et al., 1997).
Activated Fus3 and its effects
Fus3PP phosphorylates and regulates various nuclear and cytoplasmatic proteins (Dohlman, 2002): Sst2 (necessary for G protein inactivation), Ste5 (the scaffold protein of the MAPK cascade; Kranz et al., 1994), Ste11 and Ste7 (Bardwell and Thorner, 1996), Far1 (required for morphological changes and for cell cycle arrest; Peter et al., 1993), Ste12 (involved in the transcriptional activation; Metodiev et al., 2002), Dig1/Rst1 and Dig2/Rst2 (necessary for transcriptional inhibition; Tedford et al., 1997) and others.
The transcriptional activator Ste12
The protein Ste12 is a transcriptional activator and responsible for the expression of pheromone-induced genes (Pi et al., 1997). The repressors Dig1/Rst1 and Dig2/Rst2 inhibit Ste12 by direct binding to the regulatory domain. Due to Fus3PP-dependent phosphorylation of these repressors, they dissociate from Ste12, rendering it activated (reactions ν34, ν35) (Bardwell et al., 1998; Pi et al., 1997; Tedford et al., 1997).
Via activation of Ste12 the transcription of the endoprotease Bar1 is induced (reactions ν36, ν37). Bar1 may be excreted from the cell (reaction ν38) and cleaves the α-factor between leucin6 and lysin7 (reaction ν1) and, in this way, inactivates the pheromone (Ballensiefen and Schmitt, 1997). This mechanism constitutes one negative feedback loop in the pheromone response pathway.
Far1—a cyclin-dependent kinase inhibitor
Activated Ste12 also increases the transcription of Far1 (Chang and Herskowitz, 1990; Oehlen et al., 1996) and Fus3PP is capable of phosphorylating, and hence stabilizing, Far1 (reactions ν39, ν40) (Breitkreutz et al., 2001; Breitkreutz and Tyers, 2002; Oehlen et al., 1996; Peter et al., 1993; Tyers and Futcher, 1993). Far1 is a bi-functional scaffold protein. While cytoplasmatic Far1 is involved in polarized growth (Butty et al., 1998; Nern and Arkowitz, 1999), nuclear Far1 has a key function in the control of the cell cycle by being involved in cell cycle arrest in the G1 phase in response to an external stimulus (Gartner et al., 1998; Peter et al., 1993).
Far1 is degraded in the nucleus in a ubiquitin- and proteasome-dependent manner (reaction ν41). This degradation is cell cycle-dependent; Far1 is stable only in the G1 phase of the cell cycle and during the other stages it is rapidly degraded (Henchoz et al., 1997).
The cell cycle is driven by the sequential activations and deactivations of cyclin-dependent kinases (CDK). For the G1- to S-phase transition, the CDK complexes formed of Cdc28 (kinase) and Cln (G1 cyclins: Cln1, Cln3 and especially Cln2) have to be active. The phosphorylation of Far1 by Cdc28-Cln (reactions ν44, ν45) leads to a recognition signal for a G1–S ubiquitination system. This ubiquitin-dependent degradation of the cell cycle proteins has emerged as a key mechanism for regulating the cell cycle transitions (Henchoz et al., 1997; Peter et al., 1993).
The complete model
In the simulation of the signalling processes after pheromone stimulation, we include scaffold formation, receptor activation, the G protein cycle, the MAPK cascade, repeated Fus3-phosphorylation, transcriptional activation and pathway downregulation via activation of Sst2 and Bar1. We do not consider the formation of the shmoo tip itself or further changes in the cell cycle.
The developed model is used to analyse the dynamics of the pheromone pathway. It is also applied to test the effect of parameter changes and to assess the influence of experimentally investigated mutations on the characteristic features of the pheromone response. The temporal behaviour of different components of the signalling process is illustrated in Figures 3–9 for wild-type cells and for a number of different mutants.
Before pheromone stimulation the system is in a steady state. The scaffold complex (complex C), consisting of Ste11, Ste7, Fus3 and Ste5, is in a dynamic equilibrium with its individual components and the intermediary complexes Ste5Ste11 (A) and Ste7Fus3 (B). Upon stimulation with the α-factor at time t = 0 min the concentration of active receptor Ste2 increases transiently (not shown). The consequences for the G protein are displayed in Figure 3. A small quantity of Gα β γ is cleaved into Gβ γ and GαGTP. Gβ γ is quickly involved in the formation of complex D, thus the concentration of free Gβ γ rises only for a short time period of about 48 s. GαGDP, produced by hydrolysis of GαGTP, rises to a maximal value within about 10 min and then decays slowly.
After release of Gβ γ, the subsequent complexes involving Gβ γ and the complex C are formed rapidly (Figure 4). This process may be divided into three phases: in the first phase, the complex is formed involving Gβ γ, Ste5, Ste11, Ste7, Fus3 (complex D) and Ste20 (complex E). In the second phase, the respective proteins are phosphorylated and thereby activated (complexes F–H). The various phosphorylated states are present in measurable concentrations only for a short time, since they pass into the next state quickly. Complexes I–L are present for a comparatively longer time. In this third, long-lasting phase, Fus3 molecules are phosphorylated repeatedly. Free Fus3PP assumes its maximal concentration after about 0.7 min and then decays slowly (Figures 5 and 9).
Fus3PP catalyses the activation of Far1 and of Ste12 (Figure 6). Hence, the concentration of Far1PP increases transiently, allowing for the binding to Gβ γ or Cdc28. The production of Far1PP–Gβ γ (complex M) and Far1PP–Cdc28 (complex N) is the signal for polarized growth and cell cycle arrest, respectively. These subsequent processes are not considered here. The activation of the transcription factor Ste12 is followed by the production of active Bar1 (Figure 6). Active Bar1 is an important part of a feedback regulation loop in the pheromone pathway. Bar1active is exported into the medium, where it may cleave the α-factor. In this way it downregulates the activation of the receptor—the termination of the signalling process starts.
Fus3PP also activates Sst2, which supports an accelerated hydrolysis of GαGTP to GαGDP. Since GαGDP may bind Gβ γ to the complete G protein, Gβ γ is less and less available for the formation of signalling complexes. This process constitutes a second feedback regulation loop and contributes to the termination of signalling.
Parameter dependence of the model performance
We performed intensive checks of parameters for which no experimentally determined value was found. In Figure 7, we present the dependence of the concentration of free Fus3PP on the rate of degradation of complex L (k32) as an example. Obviously, the ability to respond to an external signal and to switch the response off is dependent on the value of k32. For a low degradation rate, we find a strong response but no downregulation. For a high degradation rate, no pool of free Fus3PP can form, and hence the signal is not transmitted to downstream processes. For the calculations we have chosen a value of k32 = 5 min−1.
Temporal characterization of the signalling process
Several measures have been introduced to assess the time a certain signal needs to progress from the beginning of a pathway to the end, e.g. from the receptor to the transcription factor or from the activation of the MAP kinase kinase kinase kinase to the phosphorylation of the MAP kinase. Furthermore, the duration of the signal, δ, as well as the maximal value of the signal, Max, and the time, at which this maximal value is attained, tMax, can be of interest. We use the definition of the characteristic time introduced by Llorens et al. (1999) and a deduced definition of the signal duration according to Heinrich et al. (2002) (see Table 4).
Table 4. Temporal characterization of the signalling pathway
The duration of the signal passage is an important feature of a signalling pathway (how long does the information need to proceed from the receptor to the effector?). The assessment of this duration is not straightforward. Here, we present the values for different quantities applicable for temporal characterization of reaction systems.
Characteristic time, τ (min)
Signal duration, δ (min)
Maximal value, Max (nM)
Time of Max, tMax (min)
The characteristic quantities have been evaluated for concentration changes of Gβ γ, Fus3PP, Ste12active, Bar1active and the complex I (the complex which is completely phosphorylated, including the feedback phosphorylation of Ste5). The characteristic time, τ, and the time of maximal signal (tMax) for one compound may differ significantly. This effect is even more pronounced if the signal duration is long. This points to an important feature in signalling; the respective compound is active just as long as it is abundant, not only at its maximal concentration.
Graded response to varying stimulus strength
It has been observed that different concentrations of the α-factor are required to activate various phenotypic aspects of the pathway. Transcriptional activation requires low concentrations of α-factor (about 10−12M or 10−3 nM), whereas cell cycle arrest requires several orders of magnitude more α-factor (about 10−1 nM) and projection formation requires even more (10 nM) (Cole et al., 1990). This dependence can be well explained by our model. In Figure 8 the maximal values of different concentrations (relative values) during the process of pheromone response are shown for varying concentrations of α-factor. The abundance of Ste12active approaches its maximal value already for very low concentrations of α-factor. This means that the transcription of Ste12-dependent proteins may take place at low pheromone stimulation. The next species to respond at increasing concentration of α-factor is complex N, Far1PP–Cdc28, which is necessary for cell cycle arrest. Both the concentration of Ste12active and the concentration of Far1PP–Cdc28 rise notably, although increase in the concentration of Fus3PP is still moderate. The remarkable rise of Fus3PP is only achieved at higher concentrations of α-factor. The strongest stimulation is necessary for Far1PP–Gβ γ (complex M), for the following reason. The formation of Far1PP–Gβ γ necessitates free Gβ γ. Excess Gβ γ is only available if more Gα β γ is cleaved into Gα and Gβ γ than Gβ γ is included in formation of the complex D, i.e. at high stimulation by α-factor. The difference in the dose–response curves presented in Yi et al. (2003) may be based on the fact that they were measured after 1–3 min, but our model approaches maximal activation for low α-factor concentration only after longer stimulation (about 15 min).
Modelling the phenotype of mutants
The effect of mutations has been tested with this model in detail. A list of experimentally investigated mutants (with references), their phenotypes and the possible implementation of their defects into our model is given in Table 5. The respective time courses are given in Figure 3 (for the G protein) and in Figure 9 (Fus3PP, Far1PP-Cdc28, and Far1PP-Gβ γ). In detail:
1.Gα overexpression. A higher amount of total Gα leads to an imbalance in the ratio of Gα and Gβ γ at the instant of formation of the G protein and, therefore, to a higher quantity of free GαGDP. The free GαGDP may bind Gβ γ and the concentration increase of available Gβ γ for the downstream signalling is delayed (Figure 3E). This results in a shortened complex formation, in less Fus3 phosphorylation (Figure 9A), in a lower amount of Far1PP–Gβ γ (Figure 9C) and, eventually, in reduced pheromone sensitivity.
2.Gβ with defect in binding to Gα. The binding defect leads to a higher concentration of free Gβ γ (Figure 3E, C). The effect is a perturbed termination of signalling: Fus3PP and Far1PP–Gβ γ are available for much longer (Figures 9A, D). The cells have problems in leaving pheromone-induced cell cycle arrest.
3.Gβ γ overexpression. A higher amount of total Gβ γ also leads to an imbalance in the ratio of Gα and Gβ γ, and to a higher quantity of free Gβ γ (Figure 3E). The effect is similar to an everlasting stimulation or to a stronger stimulation with more α-factor. We find an increased Fus3 phosphorylation (Figure 9A), a higher amount of Far1PP–Gβ γ (Figure 9D) and a higher concentration of complex M (Figure 9E). The downregulation of the signalling pathway is disturbed.
4.Sst2 mutant. The hydrolysis of GαGTP to GαGDP may only occur in the slow mode of reaction ν7. Hence, the reassociation of GαGDP with Gβ γ is lagged (Figures 3B, D, F), and the termination of signalling is delayed (Figure 9B for Fus3PP; Figure 9D for Far1PP–Gβ γ). In the extreme case that no hydrolysis may occur, the result is a missing termination of signalling and the cells stay in cell cycle arrest.
5.Sst2 loss-of-function mutant. The hydrolysis of GαGTP to GαGDP is almost stopped. The reassociation of GαGDP with Gβ γ is more strongly delayed, as in the case of mutant (4) (Figures 3B, D, F). In the extreme case where almost no hydrolysis occurs, the termination of signalling fails (Figure 9B for Fus3PP; Figure 9D for Far1PP–Gβ γ) and the cells must stay in cell cycle arrest.
6.Sst2 gain-of-function mutant. An increase in the rate of hydrolysis of GαGTP has almost no effect. It leads to a slightly increased concentration of free Fus3PP, but not to qualitative changes (not shown).
7.ste5Δ cells. In the absence of Ste5, no MAPK complex including the scaffold protein Ste5 is formed. The proximity of Gβ γ and Ste20 as well as Ste20 and Ste11 as a prerequisite for activation is not facilitated. Hence, the subsequent processes are also missed. There is no Fus3 phosphorylation (Figures 9A) and no pheromone sensitivity in terms of Far1PP effects (Figures 9C and E). The cells are sterile.
8.Ste5 with defect in binding Gβ γ. The active MAPK complex is not formed, no further signalling occurs [cf. mutant (7); see Figures 9A, C and E].
9.Overexpression of Ste5. Higher abundance of Ste5 leads to slightly increased concentrations of all complexes involving Ste5. This results in a slightly increased concentration of Fus3PP (Figure 9A). The fact that overexpression of Ste5 shows less effect in the simulation than in the experiment may point to the critical role of the scaffold protein in real signalling, and may be due to the fact that other compounds of the signalling cascade become limiting.
10.Ste5-mutant with reduced binding to Ste7. The formation of MAPK complex is restricted and the transmission of the signal is disturbed. Eventually, the phosphorylation of Fus3 is diminished (Figure 9A). The severeness of the effect depends, of course, on the remaining strength of binding. In the extreme case of no binding, no further signalling would occur.
11.ste5Δ ste11Δ cells. These mutants show the same behaviour as mutant (7), the ste5Δ cells. These mutations have no influence on the formation of the Ste7–Fus3 complex (not shown).
12.Ste20 with defect in binding Gβ γ. The active MAPK complex is not accomplished. Ste20 (the MAPKKKK) cannot approach Ste11 (the MAPKKK) and no further signalling occurs (Figures 9A, C and E). Note that in vivo ste20Δ mutants are not sterile, since the protein Cla4 may fulfil a bypass function. This is not considered in our model.
13.msg5Δ cells. This mutant shows a lower or no phosphatase activity on Fus3PP. This results in an increased concentration of Fus3PP (Figure 9B). The signals for further cellular processes, Far1PP–Gβ γ and Far1PP–Cdc28, are overrepresented, corresponding to the experimentally observed increased pheromone sensitivity (Figure 9C,E).
14.msg5Δptp2Δ ptp3Δ cells. This mutant has no phosphatase activity on Fus3PP. The concentration of Fus3PP is very strongly increased for very long times (Figure 9B) and the return from cell cycle is inhibited.
15.Msg5 overexpression. The phosphatase activity on Fus3PP is increased. The turnover of Fus3 is enhanced and the amount of free Fus3PP is diminished (Figure 9A). Consequently, Far1PP–Gβ γ and Far1PP–Cdc28 are produced to a lower degree (Figure 9C,E), giving rise to the observed reduced mating efficiency.
16.Far1-mutant unable to bind Gβ γ. The complex Far1PP–Gβ γ is not formed (Figure 9C). No processes initiated by this complex may take place, and polarized growth is perturbed.
Table 5. Experimentally observed mutants, their phenotypes and the mode of implementation into the model. The numbering of mutants corresponds to the numbering in the text. In the column ‘implementation’, the respective parameter changes in the model are given, which are implied by the type of mutation. The corresponding outcome of the model is discussed in the text
We present for the first time a mathematical model for the dynamics of the complete yeast pheromone response pathway. The model comprises the activation of the membrane-bound pheromone receptor, the activation of the G protein, the formation and activation of the scaffold-bound MAP kinase cascade, the activation of the transcription factor Ste12, and connections to the downstream effects on gene expression alteration and preparation for mating, as well as the downregulation of the signalling process.
Despite the yet incomplete knowledge about the kinetics of the individual steps, and even, in some cases, about the mechanism, the model shows how to map the experimental knowledge to a mathematical description. The proposed set of differential equations presents a working-type model for the temporal behaviour of yeast cells after pheromone stimulation. On the other hand, the mathematical model can be used to predict in silico the phenotypic consequences of manipulations in the experimental system. We were able to reproduce the gradual response of yeast cells to varying pheromone concentrations. We can also reproduce the quantitative features of the G protein cycle model of Yi et al. (2003), although our model is different with respect to receptor synthesis and, more importantly, with respect to the feedback loops concerning degradation of α-factor and activation of Sst2. Furthermore, we were able to map the systemic changes in the various experimentally investigated mutants to the phenotypic consequences. To this end we simulated the model under consideration of the respective changes in protein concentration or rate constants.
The kinetic parameters (rate constants, total concentrations) used in the model rely partially on experimental measurements. The rest of the parameters were fitted to the observed phenotypic behaviour. We do not claim that the chosen model parameters reflect in every case the real values of binding constants, rate constants or concentrations. Even the assumptions about the kinetic types of the various reactions (mainly linear kinetics) are very simple and probably deserve improvement. Any further measurements of detailed kinetic types and rate constants may be used to refine the model and make it more adequate to the real cellular processes. However, with the current set of parameters we were quite able to reproduce typical phenotypes.
The model may be used to test different hypotheses about the structure of the pheromone pathway, the process flow and the reasoning for cellular decisions. For example, it can be investigated, whether the degradation of involved proteins is necessary for the downregulation of signal transduction and, if so, how fast it should proceed. Simulations of the current model do not support the hypothesis that degradation of proteins involved in the MAP kinase cascade is necessary for pathway downregulation (not shown).
Typical characteristics of signalling pathways may well be studied with the pheromone response model. The regulatory interactions embedded in the model contain several feedback loops. First, Fus3PP carries out the activation of Sst2, which in turn stimulates the hydrolysis of GαGTP. This supports the closure of the G protein cycle. Second, the transcription and activation of Bar1 is activated in the course of signalling. In the activated state Bar1 may be excreted from the cell and cleave the α-factor. In the absence of one or the other feedback loop (e.g. in deletion mutants) the downregulation of the pathway is disturbed. The response to pheromone is clearly prolonged compared to wild-type.
The characteristic times of the key events in the signal cascade can be determined with the model in order to estimate the time in which significant changes occur. This is especially important if we consider adjoining processes in a further stage of the modelling.
In the process of developing the model we have tested various scenarios for the interaction of the compounds. In some cases, there have been reported different mechanisms in the literature. For example, we investigated the influence of the order of protein aggregations in the MAPK scaffold complex. We found that the specific order (whether Ste5 binds the other proteins in a sequential or in a random manner, or if two pre-complexes form the final complex) does not matter with respect to the equilibrium concentrations of the complex and the free proteins before stimulation or with respect to the temporal characterization of the formation of complex D after stimulation. An important difference was found, if we consider different scenarios for the phosphorylation of Fus3 and the fate of the scaffold complex after a single phosphorylation event. If the whole complex (here complex I) disaggregates as soon as one molecule, Fus3, has been phosphorylated, much less Fus3 may be phosphorylated than if the complex remains intact and may phosphorylate several Fus3 molecules (cycle of complexes I, K and L). In the latter case, a remarkable amplification of the signal is achieved.
The pheromone response model can also be regarded as a building block in a more comprehensive modelling effort in order to get a deeper understanding of the yeast stress response. In a more extended model, further aspects of signalling pathways, such as the integration of different signals to an appropriate response or the prevention or benefit of cross-talk between various signalling pathways, should be investigated.
In the biological reality, the activation of pheromone signalling pathway causes a cell cycle stop and the preparation for mating. With minor appropriate adaptations our model can also operate together with mathematical models of the yeast cell cycle, e.g. with the model published by Chen et al. (2000). The simulation of addition of α-factor in this coupled model leads to a cell cycle stop in the G1 phase. A complete modelling of the pheromone response, of course, would necessitate consideration of the further cellular processes.
E. Klipp is supported by the Berlin Centre for Genome Based Bioinformatics, which is financed by a grant from the German Ministry for Education and Research. We would also like to thank an unknown reviewer for valuable suggestions.