The adaptive filter of the yeast galactose pathway

In the yeast Saccharomyces cerevisiae, the interplay between galactose, Gal3p, Gal80p and Gal4p determines the transcriptional status of the genes required for galactose utilization. After an increase in galactose concentration, galactose molecules bind onto Gal3p. This event leads via Gal80p to the activation of Gal4p, which then induces GAL3 and GAL80 gene transcription. Here we propose a qualitative dynamic model, whereby these molecular interaction events represent the first two stages of a functional feedback loop that closes with the capture of activated Gal4p by newly synthesized Gal3p and Gal80p, decreasing transcriptional activation and creating again the protein complex that can bind incoming galactose molecules. Based on the differential time scales of faster protein interactions versus slower biosynthetic steps, this feedback loop functions as a derivative filter where galactose is the input step signal, and released Gal4p is the output derivative signal. One advantage of such a derivative filter is that GAL genes are expressed in proportion to the cellular requirement. Furthermore, this filter adaptively protects the cellular receptors from saturation by galactose, allowing cells to remain sensitive to variations in galactose concentrations rather than to absolute concentrations. Finally, this feedback loop, by allowing phosphorylation of some active Gal4p, may be essential to initiate the subsequent long-term response.

is lacking. In this paper we propose a mathematical model of the early response to galactose and we analyze its dependence upon time delays, protein degradation rate and initial conditions. The model accounts for the above-mentioned sensitivity to galactose fluctuations. It also proposes a solution to the apparent paradox described above by showing that a feedback loop brings active Gal4p onto gene promoters, thus allowing its phosphorylation and consequent maintenance of transcriptional activation.

Assumptions
The present model deals with the early steps of galactose induction; it does not consider the events occurring after Gal4p phosphorylation. It does not emphasize the details of signal transmission from galactose to Gal4p (except in Appendix A). Thus, Gal4p appears in this model either bound to DNA, or bound to DNA and to Gal80p. Gal80p is either bound to DNA and to Gal4p,or bound to Gal3p,or unbound. An equilibrium between nuclear and cytoplasmic forms of Gal80p has been considered by other authors (Peng and Hopper, 2000;Peng and Hopper, 2002;Verma et al., 2003), but is not relevant here given the scope of our model. The order in which galactose, ATP, Gal3p and Gal80p bind together is not fully known but should have no effect on the conclusions reached with our model, which simply considers Gal80p consumption upon galactose addition.
The GAL1 gene is a paralogue of the GAL3 gene (Wolfe and Shields, 1997) that encodes a galactokinase, while Gal3p does not have galactokinase activity (Platt et al., 2000). Galactokinase activity is irrelevant to the present model which does not address galactose catabolism. Therefore, Gal1p and Gal3p are taken to play a similar role in GAL pathway activation, averaged over their respective abundances and inducing properties. In the model, they will be lumped together under the name of Gal3p. Figure 1 illustrates the different states of the dynamic system. Figure 2A illustrates the core regulatory mechanism. In the absence of galactose, Gal4p can bind to Gal80p and has no transcriptional activity. Following a step increase in galactose, Gal3p rapidly binds galactose, and Gal80p is consumed by being recruited in a complex with Gal3p. As the concentration of unbound Gal80p decreases, the Gal4p/80p complex is destabilized, which activates Gal4p. Activated Gal4p then initiates the slower biosynthetic reactions, transcription of the GAL genes including GAL3 and GAL80, followed by translation into their protein products.

Dynamic description of the GAL system
Following Gal4p activation, and consequent GAL gene expression, newly synthesized Gal3p and Gal80p shift the equilibrium back towards Gal4p inactivation. As a result, GAL transcriptional activity decreases back. Newly formed proteins can bind incoming galactose molecules, thus restoring sensitivity to any further galactose input. This effectively closes the feedback loop, the central point of this model.

Model simplification
The detailed model shown in Figure 2A is needlessly complex relative to the focus of our study: the role of the feedback loop. In this section, we show how the model could be simplified, yielding a reduced model (Fig. 2B) that features the feedback-loop and preserves the qualitative dynamics of the detailed model, while allowing deeper analysis and understanding.
The GAL3 and GAL80 genes are both transcriptionally regulated by Gal4p. However, the GAL3 gene is activated about five-fold stronger than GAL80 (Peng and Hopper, 2002). This fact may amplify or accelerate the response. Indeed, a relative increase of Gal3p with respect to Gal80p will shift complex formation towards additional Gal80p consumption, increasing further the concentration of activated Gal4p. Even though this may bring about changes in the exact response kinetics, it does not change the qualitative behavior of the system. Furthermore, concomitant overexpression of GAL3 and GAL80 was shown to suppress the constitutive GAL gene expression elicited by overexpression of GAL3 alone (Suzuki-Fujimoto et al., 1996), suggesting that their two products play a complementary role in the reaction cascade. Accordingly, Gal3p and Gal80p are lumped together in this model as the 'Gal3/80p' complex. This simplification is relaxed in a more complex model described in Appendix A. This model may be used for simulation. However, it is not amenable to analytical understanding, as it involves six equations and eight parameters. It is shown that the results on the qualitative behavior remain unchanged (Appendix A).
In the absence of galactose, Gal3p, Gal80p and Gal4p form an inactive complex Gal4/3/80p called 'receptor' ('R';Fig.2B). This simplifies the model, lumping a cascade of reactions into one. A 'bound receptor' ('BR') comprising Gal3p, Gal80p and galactose remains inactive or may be degraded. Finally, the total Gal4p concentration is assumed to be constant during the GAL response, as suggested by transcriptomics data (Ren et al., 2000;Ideker et al., 2001).
The three main transformations of the simplified model corresponding to the three reactions of Figure 2B are shown below (Eq. 1-3). We represent a gene, its encoded mRNA and protein as a single entity. 'G4' denotes an active Gal4p protein. The first equation pertains to the slow biosynthetic steps of transcription and translation, comprising the binding of Gal4p to the GAL3 and GAL80 gene promoters, and all subsequent actions until Gal3/80p molecules are newly synthesized one at a time, and Gal4p stays activated. The second equation represents the inactivation of Gal4p into its inactive form called the receptor 'R'. The third equation expresses the activation of Gal4p due to the binding of galactose ('Gal') to the receptor, yielding the bound receptor ('BR').
To facilitate analyses and simulations, this model can be further reduced to a two equation, two variable system readily amenable to phase plane analysis. It emphasizes the role played by the negative feedback loop in the GAL pathway dynamics. Combining transformations (1) and (2) into (2a), gives the following set of two equations, with three kinetic constants K.
'S 0 ' (for "Sugar") is the initial quantity of galactose, and 'R 0 ' (for "Receptor") the total amount of Gal4p. These values are constant, as galactose is initially provided in a finite amount, and total Gal4p has been assumed to be a constant amount (Johnston et al., 1994;Ren et al., 2000;Ideker et al., 2001)).
This leads to two non-linear differential equations, with K 1 normalized to 1. From assuming a homogeneous spatial distribution and a mean cell volume of 70 µm 3 (Ruhela et al., 2004), the order of magnitude of concentration of galactose and proteins involved is of the order of 10 -6 M. This led to S 0 and R 0 of the order of 10 -6 M and K 1 =K 2 =K 3 =1. One to 10 hours is the typical time response of the galactose switch. These values imply that the order of magnitude of time unit is 5x10 2 s.
In the special case where K 2 is set equal to K 3 , these equations can be simplified into linear equations:

Steady States and Phase Plane Portrait
The phase plane for equations (1c) and (2c) is the Cartesian coordinate system representing the two variables, receptor and free galactose concentrations (Fig. 3). Two nullclines are represented, that correspond to the situations where: 1) the receptor synthesis and consumption by galactose fixation are balanced, or 2) free galactose addition and galactose fixation are balanced. Thus, the nullclines correspond to steady states, where the derivatives of equations (1c) and (2c) are null (Fig. 3). The two nullclines cross each other at a stable point which corresponds to the situation where 1) and 2) simultaneously hold true. This stable point turns out to correspond to low free galactose and high receptor concentrations (Fig. 3). When K 1 is normalized to 1, and K 2 and K 3 are equal (equations (1d) and (2d)), the nullclines are linear, thus facilitating further analysis (Fig. 3 A-B). The two nullclines (plain lines) delimit three domains in the plane. From wherever of these three domains is the initial starting point, trajectories (Fig. 3B, broken line) return to the unique steady state. This provides robustness towards fluctuations of the receptor concentration.

Influence of parameters values
To assess the influence of the variations of parameters K 1 , K 2 and K 3 , we studied how the phase plane was modified by varying each of the parameters separately (Smolen et al., 2001;Morohashi et al., 2002;Sriram and Gopinathan, 2005). Three different sets of values for K 1 , K 2 and K 3 are explored in Figure 3, giving different pairs of nullclines. K 2 range has been explored over two decades ( Fig 3C) and only affects the concavity of the nullclines without modifying the qualitative dynamics. Then, keeping the ratio of K 2 and K 3 constant, we explored two decades of K 3 variations and this modifies the area between the two nullclines ( Fig 3D). This does not change qualitatively the dynamics of the system. K 1 variations were also studied. For K 1 greater than 1, the two nullclines cross each other at the stable point that corresponds to an equilibrium between the reactants ( Fig 3E) and does not alter the qualitative dynamics otherwise. This qualitative behavior looks like the one of the model with degradation (see below and Fig 5). For K 2 lesser than 1 (Fig 3F), the two nullclines cross each other outside of the reachable concentration area but still near the bottom right corner of the phase plane. As in all cases the stable point is kept in the same area, the trajectories are not qualitatively different and the dynamic behavior appears to be robust.

Functional interpretation as a 'derivative filter'
The 'chicken and egg' paradox can be readily explained by this feedback loop model. In cells expressing a mutated Gal4p A699S (where an alanine replaces the serine 699) that cannot be phosphorylated, GAL gene activation can initiate but cannot be maintained (Yano and Fukasawa, 1997). Figure 4A shows that in response to the addition of galactose (dashed line), free Gal4p (bold line) increases and initiates transcription of the GAL3/80 genes. As a consequence, Gal4p binds to new Gal3/80p to produce receptors (plain line) and its free concentration decreases, hence Gal4p cannot maintain transcriptional activity. During the early response phase modelled here, this system acts as a derivative filter (Lauffenburger, 2000;Basu et al., 2004), where galactose is the input step signal and Gal4p is the output peak signal, an approximate derivative of the step input. Furthermore, the model is consistent with the observation that transcription is permanently activated in gal80∆ mutant cells (Ideker et al., 2001). Indeed, within the present model, the absence of Gal80p breaks the loop; Gal80p can no longer sequester Gal4p which remains permanently active.

Influence of time delays
As transcriptional interactions generally occur at longer time scales than proteinprotein interactions, we experimented with the introduction of time delays (de Jong, 2002) in the differential equations. We introduced a delay in the transcription reaction that led to the equations (1e) and (2e).
In the simulations shown on figure 4A-C, ten units of galactose are initially introduced at time t=0 in the presence of ten units of receptor. These three simulations correspond respectively to no, short, or long biosynthetic delay. As protein concentrations shift away from equilibrium during the biosynthetic steps, potential chemical energy is stored in the form of Gal3/80p undergoing synthesis. In the absence of delay, this energy is immediately released by the consumption of Gal4p and of the remaining galactose. As the delay increases, more Gal3/80p are being synthesized simultaneously, and therefore more potential energy will suddenly be released at the end of the delay. Thus, the differential time scales of fast protein interactions and slower biosynthetic processes are responsible for the observed feedback acceleration and output signal sharpening.

Adaptation of receptor concentration
To investigate how the system adapts to various galactose inputs, galactose was initially introduced in larger quantity units (15 units) than the receptor (10 units) (Fig.  4D). Compared with an initial dose of ten units of galactose in an otherwise identical experiment (Fig. 4C), a three-step response is observed on figure 4D. Firstly, the receptor is entirely consumed and excess galactose remains as a plateau. Secondly, newly formed receptors rapidly bind the remaining galactose. Thirdly, the initial receptor and Gal4p concentrations are restored. Thus, the receptor concentration is maintained at a constant level, independent of the initial galactose concentration. The GAL genes are expressed in proportion to the cellular requirement, at each increase of galactose concentration.

Effect of protein degradation
To investigate the effect of protein degradation, Gal3/80p half-life was set to 1 arbitrary time unit in the presence of a long transcriptional delay of 2 units. It appears that Gal3/80p degradation elicits dampened oscillations of the receptor, Gal4p and Gal3/80p concentrations (Fig. 4E). These oscillations depend on the presence of a transcriptional delay, as shown on figure 4F where Gal4p concentration is monitored in the absence (dashed bold line) or in the presence (bold line) of a long transcriptional delay. As was already shown on panels 3A and 3C, introducing a delay sharpens the time evolution of Gal4p. However, Gal3/80p degradation makes the signal more realistic in that, for instance, Gal4p becomes in excess to Gal3/80p and never reaches zero values.
When Gal3p and Gal80p are distinguished according to the more complete model shown on figure 2A, while including protein degradation, the outcome is similar (Appendix A).

Discussion
Earlier attempts to model the galactose pathway did not emphasize the feedback loop involving Gal4p activation and induction of Gal3p and Gal80p synthesis. Here we propose that the crucial mechanism generating the ill-understood behavior of the system prior to Gal4p phosphorylation is this feedback loop. A similar approach was recently developed independently, that emphasizes the importance of autoregulation (Ruhela et al., 2004), but with no discussion of the underlying mechanism.
To emphasize the role that feedback may play in the GAL pathway, we simplified our model of the molecular machinery by lumping together Gal3p and Gal80p to the point where its behavior could be described by two differential equations. This simplification has been justified but aposteriori it can be also verified. We have shown that the simple model and simulations of the more detailed one have the same qualitative behaviors. In both case, the system is adaptive and acts as a derivative filter (Fig. 3-4 & Fig. 5-6). Degradation modifies moderately the dynamic preventing concentrations to go back down to zero but led to an equilibrium (Fig. 4 E-F & Fig.  5). Both models are robust to parameter variations (Fig. 3 & Fig. 6). The simple twoequation model has several advantages.
Firstly, it captures the key qualitative features of the system dynamics. For instance, phase plane analysis provides useful insight into the poorly understood mechanism of adaptation and high sensitivity to galactose fluctuations. A consequence of the existence of a feedback loop is that receptor concentration is maintained at a constant level, independent of the amount of galactose captured by the cell. Hence, the receptor cannot be saturated by galactose and cells remain sensitive to galactose variations, not to the absolute galactose concentration, without requiring a high number of receptors. Another advantage is that Gal4p-regulated genes are expressed in proportion to cellular requirements, i.e. the galactose flux or the time derivative of the galactose concentration.
Secondly, predictions obtained with this model are consistent with previous biological investigations, which were mostly based on the study of one branch of the loop, signal transduction from galactose to Gal4p (Ideker et al., 2001). Thirdly, the model is rich enough to allow predictions that could be tested at the bench. For instance, consider a mutant strain expressing a Gal4p A699S version to suppress the onlocking of the GAL activation, and Gal2p under the control of a constitutive promoter to avoid permease-dependence of the response to galactose. In these mutant cells, each successive galactose step increase should result in an increase of a Gal1p-GFP chimeric reporter. Furthermore, if the reporter protein is destabilized, its increase should be followed by a decrease that would constitute the signature of the feed-back loop.
In the future, the model could be refined by considering the relationship between the short-term adaptation described here and the long-term response achieved through Gal4p phosphorylation, or by immersing this system into the wider web of other sugar regulatory pathways and of other types of interaction.    (1d) and (2d). Simplified phase portrait and trajectory for equations (1d) and (2d). The nullclines (plain lines) are represented for K 1 =1 and K 2 =K 3 . Arrow pairs represent the slopes in each of the three domains defined by the two nullclines. (B) A typical trajectory is represented by a broken line. In the absence of galactose, the system is at the stable point in the lower right corner. When galactose is introduced, all galactose is in its free form and the state moves to the upper right corner. Then, galactose binds to the receptor and causes free galactose and receptor concentrations to decrease. This binding causes Gal4p to activate transcription, thereby producing new receptor whose concentration increases back to the stable point. (C-F) Phase portrait for equations (1c) and (2c). These curve pairs are represented for various values of K 1 , K 2 and K 3 . (C) K 1 =1, K 2 =10, K 3 =10 (bold line) , K 1 =1, K 2 =100, K 3 =10 (plain line) , K 1 =1, K 2 =1, K 3 =10 (dashed line). (D) K 1 =1, K 2 =3, K 3 =1 (bold line) , K 1 =1, K 2 =30, K 3 =10 (plain line) , K 1 =1, K 2 =0.3, K 3 =0.1 (dashed line). (E) K 1 =1.5, K 2 =1, K 3 =1. (F) K 1 =0.6, K 2 =1, K 3 =1. . Although no additional free Gal4p has been introduced, Gal3/80p decay results in excess Gal4p that is not sequestered in the receptor, hence the non-zero value of initial and final Gal4p. (F) Long transcriptional delay (bold line), compared to the absence of delay (dashed line), and Gal3/80p degradation as in (E). Only Gal4p is represented here (same remark as in E). Along the ordinate axis, the Gal4p scale has been amplified five-fold, and the galactose scale has been decreased two-fold. This simulation involves no delay in transformations. In this respect, the result may be compared with Figure 4A.  Fig.5) for 10-fold increase or decrease of parameter k1-k7 and ±50% variation in parameter c. The square represents the model with no parameter variation. Each vertical bar indicates how much the Gal4p response decreases after the maximal signal response until time t=6 (d m distance in Figure 5).