- Research article
- Open Access
Deterministic mathematical models of the cAMP pathway in Saccharomyces cerevisiae
© Williamson et al; licensee BioMed Central Ltd. 2009
Received: 09 March 2009
Accepted: 16 July 2009
Published: 16 July 2009
Cyclic adenosine monophosphate (cAMP) has a key signaling role in all eukaryotic organisms. In Saccharomyces cerevisiae, it is the second messenger in the Ras/PKA pathway which regulates nutrient sensing, stress responses, growth, cell cycle progression, morphogenesis, and cell wall biosynthesis. A stochastic model of the pathway has been reported.
We have created deterministic mathematical models of the PKA module of the pathway, as well as the complete cAMP pathway. First, a simplified conceptual model was created which reproduced the dynamics of changes in cAMP levels in response to glucose addition in wild-type as well as cAMP phosphodiesterase deletion mutants. This model was used to investigate the role of the regulatory Krh proteins that had not been included previously. The Krh-containing conceptual model reproduced very well the experimental evidence supporting the role of Krh as a direct inhibitor of PKA. These results were used to develop the Complete cAMP Model. Upon simulation it illustrated several important features of the yeast cAMP pathway: Pde1p is more important than is Pde2p for controlling the cAMP levels following glucose pulses; the proportion of active PKA is not directly proportional to the cAMP level, allowing PKA to exert negative feedback; negative feedback mechanisms include activating Pde1p and deactivating Ras2 via phosphorylation of Cdc25. The Complete cAMP model is easier to simulate, and although significantly simpler than the existing stochastic one, it recreates cAMP levels and patterns of changes in cAMP levels observed experimentally in vivo in response to glucose addition in wild-type as well as representative mutant strains such as pde1Δ, pde2Δ, cyr1Δ, and others. The complete model is made available in SBML format.
We suggest that the lower number of reactions and parameters makes these models suitable for integrating them with models of metabolism or of the cell cycle in S. cerevisiae. Similar models could be also useful for studies in the human pathogen Candida albicans as well as other less well-characterized fungal species.
The cAMP level is modulated by two phosphodiesterases: Pde2p has higher affinity for cAMP (around 1 × 10-3 mM)  compared to Pde1p which has a lower affinity for cAMP in crude extracts (around 0.1 mM) [16, 17]. Yeast cells previously starved for glucose exhibit a characteristic "spike" of cAMP following addition of glucose to the growth media. In wild-type cells, this spike reaches a peak at around 60 seconds before reaching a steady level after around 120 seconds
In the yeast cell, the only known function of cAMP is to activate protein kinase A (PKA). A molecule of PKA consists of two regulatory (R) and two catalytic (C) subunits. Under low cAMP concentrations, the R and C subunits are bound together to form a catalytically inactive heterotetramer. The complex is activated when two molecules of cAMP bind to each R subunit, causing their dissociation from the catalytic subunits. Following dissociation, the free C subunits can phosphorylate their targets. In yeast, the R subunit is encoded by BCY1, while the C subunits are encoded by the partially redundant genes TPK1, TPK2 and TPK3. Recently specific as well as common phosphorylation targets of the Tpk isoforms have been identified .
PKA exerts feedback on the system in several ways. First, it has been shown that the low affinity cAMP phosphodiesterase Pde1p is phosphorylated following a glucose pulse and Pde1p can be phosphorylated by bovine PKA . Phosphorylation of Pde1p leads to increased phosphodiesterase activity, which plays a part in reducing the cAMP level following a glucose induced spike. Secondly, PKA can phosphorylate Cdc25p, leading to its dissociation from Ras2p . This results in a decrease in adenylate cyclase activity. Finally, PKA may be able to regulate itself, as it has been demonstrated that Tpk1p is phosphorylated following a glucose pulse .
The roles of certain components of the cAMP pathway are still disputed. One of them is that of the Kelch Repeat Homologue proteins Krh1 and Krh2, also known as Gpb1 and Gpb2, as they are believed to function as beta subunits of Gpa2p. According to Harashima and Heitman  the Krh proteins stabilize the Ira proteins, the GTPases of the Ras proteins. Deletion of the Krh proteins leads to a loss of the Ira proteins, and therefore cAMP signalling is increased. However, there is evidence that shows that the Krh proteins enhance the association between the regulatory and catalytic subunits of PKA, and this enhancement is removed when the Krh proteins form a complex with activated Gpa2 [7, 23]. Further evidence for the role of the Krh proteins comes from studies of adenylate cyclase (cyr1Δ) mutants . Yeast cyr1Δ pde2Δ mutants can survive on YPD supplemented with 5 mM cAMP. However, the quadruple cyr1Δpde2Δkrh1Δkrh2Δ mutants survive in the presence of 1 mM cAMP, suggesting that the Krh proteins directly inhibit PKA activity, as PKA activity is necessary for yeast survival. In addition a cyr1Δpde2ΔGPA2Q 300Lmutant (with Gpa2 locked in its constitutively active GTP bound state) requires 1 mM cAMP for survival. This gives further support to the theory that Krh is recruited to active Gpa2.
The reductionist approach  has taught us much about individual elements of the cAMP pathway; however a quantitative and integrated mathematical representation is needed to fully understand its dynamics. Models of two broad categories can be used for this purpose: deterministic and stochastic [25–27]. Deterministic models which usually consist of a series of ordinary differential equations (ODEs) to describe the system in respect to time, have been used to study yeast systems such as glycolysis , the pheromone pathway [29–31] and the cell cycle . Stochastic models on the other hand are used when intrinsic noise is important to the system, such as when low species numbers are involved . However, stochastic models can be computationally expensive to simulate .
A stochastic model has been developed to examine the effects of altering the intracellular GTP levels on the Ras/cAMP/PKA pathway . However, in yeast the components of the cAMP pathway are present in high numbers (proteins in thousands, nucleotides in millions) making a deterministic model more appropriate. Moreover, this stochastic model did not include the Krh proteins. In this study we present a deterministic mathematical model of the yeast Ras/PKA/cAMP pathway, with components such as the Krh proteins that have not been included before. Our model has been fitted to experimental data. It is much easier to simulate than is the previously reported stochastic model, yet it can faithfully replicate intracellular species concentrations observed at steady state, and following a perturbation of the system with glucose.
Summary of the models generated in this study.
No. of parameters
No. of variables
PKA Model A
Deterministic model of the PKA module based on Cazzaniga et al 
PKA Model B
PKA Model A with optimized parameter values
PKA Model C
Simplified PKA module with mass action kinetics
PKA Model D
Simplified PKA module with Michaelis-Menten kinetics
Simplified cAMP Model A
Conceptual model of the entire cAMP pathway
Simplified cAMP Model B
Simplified Model A modified to include Krh proteins
Complete cAMP Model
Complete model of the cAMP pathway with estimated parameters
where S ij is the sensitivity of species i in relation to parameter j, Xss i is the steady state level of species i, p j is the value of parameter j, and Δp j is a perturbation of parameter j (equal to 1% of the parameter value).
where C(nM) is the nanomolar concentration of cAMP, C(nmolesgww-1) is the cAMP concentration in nanomoles per gram of wet weight reported in the literature, Cw is the conversion factor from grams wet weight to grams dry weight (0.15) and Vc is the volume of 1 × 107 cells in litres (2.68 × 10-6, there are approximately 1 × 107 cells in 1 gram of dry weight).
We recognise that ODE models of this type assume that all cells are identical, which may well not be the case .
The values of system parameters which were not experimentally derived, were fitted to experimental cAMP time course data using simulated annealing [40, 41], an estimation method that is very efficient in finding a close approximation of the global minimum of an optimization problem. It is based on a probabilistic search, in which every iteration of the algorithm replaces the current solution by a random nearby solution, using a probability distribution that tends to move the solution towards the global minimum. The simulated annealing algorithms found in SBToolbox in Matlab  with the SBToolbox function SBparameterEstimation were used for parameter estimation in the current study.
The Protein Kinase A module
The only known biochemical role of cAMP is to activate PKA. This process has a complicated reaction scheme, which is challenging to model. A general guiding principle when building models is to make the model as simple as possible, while capturing realistic behaviour . The expected behaviour of any PKA model must be consistent with the currently available experimental evidence. Firstly, a degree of PKA activity is required for cell viability . If no cAMP is present, the cell is nonviable ; therefore all catalytic subunits must be contained within the inactive tetramer in the absence of cAMP. The level of free catalytic subunits must be sensitive to the level of cAMP. The cAMP level can range from 0.015 mM in glucose starved cells, to approximately 0.05 mM (a peak of cAMP induced by a glucose pulse) .
The stochastic PKA module reported by Cazzaniga et al  makes several assumptions. The binding constants for the association of a cAMP molecule with the PKA tetramer are the same for all cAMP bound states of PKA, as well as the dissociation constants. The underlying assumption is that cAMP binds to PKA in a non-cooperative manner, i.e. the binding of a molecule of cAMP to PKA does not affect the binding/dissociation of further cAMP molecules. In addition, the dissociation of the cAMP-bound PKA holoenzyme, and the subsequent dissociation of cAMP from the free R subunit is considered to be very fast, as is the reassociation of the PKA holoenzyme. We have adopted the same assumptions for our deterministic model.
Reactions of PKA Model A
cAMP + PKA.x*cAMP ⇒ PKA.(x+1)*cAMP
k cAMPgain [PKA·x*cAMP][cAMP]
PKA.x*cAMP ⇒ PKA.(x-1)*cAMP + cAMP
k cAMPloss [PKA·x*cAMP]
PKA*4cAMP ⇒ 2R*2cAMP + 2 C
k PKAdiss [PKA*4cAMP]
R*2cAMP ⇒ R + 2 cAMP
k RcAMPdiss [R*2cAMP]
2 R + 2 C ⇒ PKA
k PKAass [R]2[C]2
These models are defined by the following ODEs.
The results of simulating these models show that it is possible to simplify the PKA module greatly without loss of performance. It is preferable to use the mass action based module, as it has just three state variables and two parameters. This compares favourably to the complex PKA module which has nine state variables and four parameters. Therefore we adopted the mass action based module to construct the model of the entire cAMP pathway.
Development and simulation of a conceptual model of the complete cAMP pathway
Parameters of Simplified cAMP Model A
GPR k F
GPR k R
PKA k F
PKA k R
V max AC
K m AC
K i AC
V max Pde1
K m Pde1
V max Pde2
K m Pde2
where GPi and GPa are the numbers of inactive and active G proteins, respectively. PKAi and PKAa are the number of inactive and active PKA molecules, respectively.
To test if the model would also accurately reproduce phenotypic cAMP profiles of pde1Δ and pde2Δ mutants, the cAMP ODE (equation 10 defined above) was modified to remove the Pde1 and Pde2 reactions. The resultant "mutant" models were simulated as before, and as shown in Figure 5 (panel B), the simulations accurately reproduce the experimental data  (again with the exception of the slight dip in cAMP profile seen in the wild type and pde2Δ model mutants). We therefore conclude that this greatly simplified conceptual model is capable of reproducing the essential dynamics of changes in cAMP levels observed in response to glucose addition in wild-type as well as in the cAMP phosphodiesterase deletion mutants.
We tested Simplified cAMP Model B to see if it could reproduce the results from studies on adenylate cyclase mutants by Peeters et al. . For this purpose, adenylate cyclase was removed from the model. The adenylate cyclase deletion model (cyr1Δ) was simulated with cAMP concentration set to 1. The GPA2Q 300L(Gpa2 constitutively active) mutant was modeled by setting the concentration of GPa to 1 and the parameter V maxGPdeact to 0. The pde2Δ mutant model was simulated as described earlier. The krh mutant was simulated fixing GPa levels to 1.
We attempted to make a model of Krh activity as proposed by Harashima and Heitman . In the Simplified cAMP Model B, Krh is quickly reassociated with the G proteins, allowing the system to exert negative feedback. However, any feedback in the mechanism proposed by Harashima and Heitman  is impossible because the Ira proteins are degraded, and re-synthesis of these proteins could not be fast enough to allow the Ira proteins to inhibit the Ras proteins. Therefore in all further developments of the complete cAMP pathway models Krh was retained as a direct inhibitor of PKA.
Modelling the complete cAMP pathway's response to glucose
Glucose import and metabolism
where v tr is the rate of transport (in mM per second), s is the extracellular glucose concentration, p is the intracellular glucose concentration, K M is the Michaelis constant (in mM) and K i is the interaction constant.
The metabolism of glucose via glycolysis was summarized with mass action kinetics, so that the intracellular glucose concentration did not exceed 1.5 mM during simulation, as described .
Gpa2 and Krh
As described earlier, Gpa2 is activated by Gpr1, and Gpr1 is activated by extracellular glucose. The activation of Gpr1 is modelled with mass action kinetics, whereby Gpr1 forms a complex with extracellular glucose. The activation of Gpa2 is based on mass action kinetics, with activated Gpr1 as an essential activator. Deactivation of Gpa2 is modelled using a basal rate of deactivation (representing the intrinsic GTPase activity of Gpa2), which can be enhanced by Rgs2. The binding of Gpa2 to Krh to form a complex is represented with simple mass action kinetics.
Ras2 is very challenging to model because a large number of molecular species are involved in its regulation. It is directly activated by Cdc25, but it is activated indirectly by glucose. We chose to model the activation of Ras2 using general hyperbolic modifier kinetics. In this reaction, glucose acts as a modifier which increases the rate of the reaction, but the reaction is dependent on Cdc25. The deactivation of Ras2 was modelled using modified mass action kinetics with Ira as an activator. This captured the intrinsic GTPase activity of Ras2.
Adenylate cyclase is represented as a Michaelis-Menten enzyme, with the following modifications. Activated Gpa2 and activated Ras increase the k cat of adenylate cyclase, increasing the maximum activity of the enzyme. In order to simplify the model, the substrate for adenylate cyclase (ATP) is not included, as the intracellular concentration of ATP is always greatly in excess of the cAMP concentration.
PKA is modelled using the mass action kinetics module with the addition of the actions of the Krh proteins described earlier. The forward reaction (PKA dissociation) is inhibited by Krh, and the backward reaction (PKA association) is activated by Krh.
Parameters of the complete cAMP pathway Model
Glucose transport K M
Glucose transport V
Glucose transport K i
Glucose Utilisation k F
Gpr1 Glucose association k1
Gpr1 Glucose dissociation k1
Gpa2 activation k A
Gpa2 deactivation k F
Gpa2 deactivation k A
Gpa2-Krh association k F
Gpa2-Krh dissociation k F
Ras2 activation k cat
Ras2 activation K M
1.38 × 10-3
Ras2 activation K d
Ras2 activation a
Ras2 activation b
Ras2 deactivation k F
Ras2 deactivation k A
cAMP synthesis k cat Gpa2
cAMP synthesis k cat Ras2
cAMP synthesis K M
4 × 10-3
PKA activation k F
7.6 × 108
PKA activation k I
PKA deactivation k F
PKA deactivation k A
2.2 × 104
Cdc25 phosphorylation k cat
Cdc25 phosphorylation K M
5.2 × 10-3
Cdc25 dephosphorylation k cat
Cdc25 dephosphorylation K M
1.6 × 10-2
Pde1 phosphorylation k cat
Pde1 phosphorylation K M
8.6 × 10-3
Pde1 dephosphorylation k cat
Pde1 dephosphorylation K M
1.07 × 10-3
cAMP hydrolysis (Pde1) k cat
cAMP hydrolysis (Pde1) K M
cAMP hydrolysis (Pde1p) k cat
cAMP hydrolysis (Pde1p)K M
6 × 10-7
cAMP hydrolysis (Pde2)k cat
cAMP hydrolysis (Pde2) K M
Reactions of the Complete cAMP pathway Model
Glucose transport (reversible)
k f [Glucin]
k f [Gpr1] [Glucout]
k f [Gpr1Glucout]
k A [Glucout] [Gpa2i]
(k A [Rgs2] + k f ) [Gpa2a]
k f [Gpa2a] [Krh]
k f [Gpa2aKrh]
(k A [Ira]+k f )·[Ras2a]
k f [C]2·(1 + (k A [Krh]))
cAMP hydrolysis (Pde1)
cAMP hydrolysis (Pde1P)
cAMP hydrolysis (Pde2)
We have successfully created a series of deterministic mathematical models to investigate the cAMP pathway in S. cerevisiae. These range from simplified, conceptual models of the pathway, to an extensive model that fits experimental data. We were able to build a simplified model of the PKA module, containing only two variables and two parameters, without compromising the behaviour of the system. The simplification of the PKA module demonstrates the power of deterministic models. The components of this pathway are present in high abundance (proteins in thousands, nucleotides in millions per cell), making a deterministic model better suited than a stochastic one (we note also that we are not seeking to model potentially hundreds of kinds of protein molecule with different post-translational modifications).
In our PKA Model, the activation of PKA is worthy of particular attention. In previously published models, PKA activity was directly proportional to the cAMP level . However, it has been proposed that PKA autophosphorylation provides a feed-forward mechanism for PKA activation , as Tpk1p is phosphorylated following a glucose pulse . Alternatively, it is proposed that Krh inhibits PKA, and this inhibition is removed when Krh is recruited to activated Gpa2 . Our Simplified cAMP Model B shows that the latter scenario is more likely, as this model corresponds well with observable phenotypes.
Our Simplified cAMP Model shows that the basic dynamics of the pathway in response to glucose can be explained with a relatively straightforward feedback mechanism. The activation of PKA by cAMP, followed by the activation of Pde1 and the inhibition of adenylate cyclase is sufficient to produce a characteristic "spike" of cAMP, followed by the emergence of a new steady state level of cAMP and PKA. This model has been tested by creating phosphodiesterase deletion mutant models (Figure 5, panel B). Deleting Pde2 in the model results in a higher steady state level of cAMP, but it does not significantly affect the cAMP spike. This phenotype is indeed found in yeast pde2Δ mutants . However, removing Pde1 from the model results in a cAMP spike with increased peak height and duration, which is comparable to that experimentally determined in pde1Δ mutant .
In the Simplified cAMP model A, a slight dip in the level of cAMP can be seen before the cAMP level reaches a steady state after a pulse of glucose. Although this slight oscillation is not widely noted in the literature, it is possible to observe it in some experiments . The presence of the slight oscillation in the model is dependent on the parameters of the model and the glucose concentration. It remains to be seen whether this oscillation is truly present in all or any circumstances.
The Simplified cAMP Model B (which incorporates the Krh proteins) demonstrates the significance of the negative feedback. Furthermore, it shows that this feedback is possible if the Krh proteins were acting as direct inhibitors of PKA as proposed by Peeters et al. [7, 45] rather than stabilising the Ira proteins as proposed by Harashima and Heitman . At the same time, it predicts that cAMP levels should decrease more rapidly in the cyr1Δpde2Δkrh1Δkrh2Δ mutant than in the cyr1Δpde2Δ mutant. It will be interesting to see if these mutants behave in the way predicted by our models.
Although the Simplified cAMP Model could account for the majority of the behaviour of the cAMP pathway, there were exceptions. Most notably, in simulations of the pde1Δ mutant model, the steady state level of cAMP became significantly higher after a glucose pulse than it was before (Figure 5, panel B). This is not seen experimentally, where there is little difference between the post-glucose cAMP levels seen in a wild type and pde1Δ mutant . This feature of the Simplified cAMP Model prompted us to develop the Complete Model. Our Complete cAMP Model represents the first effort to consolidate all the known elements of the cAMP pathway into one deterministic mathematical model. In addition to this, we have fitted the parameters of our model to experimental data. The fact that the complete cAMP pathway model can reproduce cAMP levels found in the literature indicates that the model is a reliable in silico approximation of the in vivo system. Furthermore, our model has other advantages. Firstly, as a deterministic model, it is computationally inexpensive to simulate and easy to analyze. Secondly, it represents a physiologically realistic steady state before glucose is introduced, in that the cAMP level is not zero. This contrasts with the model found in , in which the cAMP level is set to zero before glucose addition, which is biologically impossible, as cAMP is required for cell viability. After glucose addition, the model correctly represents the dynamical changes in cAMP level, until the cAMP level reaches a new steady state.
The models of the cAMP pathway described in this study make a number of predictions that could be tested experimentally. As a matter of further investigations in our lab, different species would be characterized following a pulse of glucose in terms of phosphorylation (Pde1p, Cdc25p), GTP loading (for Gpa2p), changes in cAMP levels (in cyr1Δpde2Δkrh1Δkrh2Δ in comparison to cyr1Δpde2Δ). Our Complete cAMP Model will no doubt be improved and tested further in the future. As more parameters are derived through experimentation, they can be included into the model to replace currently estimated parameters. We provide this model in SBML (Additional file 1), so that it can be easily expanded as scientific knowledge increases. For example, details on the mechanism of glucose activation of Ras2 could be incorporated when this mechanism is elucidated.
This model could be integrated with models of other pathways, a good example being that of the cell cycle, given the fact that cell cycle progression is controlled partly by the cAMP pathway . It could also be integrated with a metabolic model such as the community consensus version recently published  via known PKA targets. Furthermore it could be adapted to other organisms such as the human fungal pathogen Candida albicans, as it is well documented that the cAMP pathway plays a key role in regulating virulence .
We report a deterministic mathematical model of the cAMP-mediated signal transduction pathway in S. cerevisiae. The model is easier to compute and simulate as it has a reduced number of variables and parameters in comparison to previously reported stochastic model of this pathway. Furthermore, our model contains components such as the regulatory Krh proteins that have not been included before. It is able to simulate accurately experimentally derived patterns of cAMP changes observed in different pathway mutants in response to glucose addition. We suggest that it is suitable for integration with other models such as that of the cell cycle or metabolism and that it could be adapted to medically important yeast species such as the human fungal opportunistic pathogen C. albicans.
The authors acknowledge the helpful recommendations of the anonymous reviewers.
TW is a grateful recipient of a BBSRC-funded PhD studentship. JMS, LS and TW thank the BBSRC and EPSRC for support of the Manchester Centre for Integrative Systems Biology http://www.mcisb.org/.
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