Temperature affects the production of proteins at the level of translation. Yeast cells temporarily cease growing by reducing ribosome and tRNA synthesis, and they also generally reduce transcription [5]. Ribosomal RNAs account for 80% of the total RNA in a growing cell [6]. Therefore, reducing the expression of rDNA should be expected to free some of the resources needed for a faster expression of proteins that are required to overcome heat-induced problems. Conversely, cells under heat stress up-regulate genes that code for proteins and processes of immediate pertinence, including: energy production through carbohydrate and lipid metabolism; metabolite transport; respiration; redox balance and ROS detoxification; cell wall modification; DNA damage repair; as well as protein chaperones that are used for refolding and degradation.

In addition to affecting the translation rate, heat can alter protein synthesis through changes in the stability of mRNAs. Importantly, some mRNAs become more stable during heat stress. Of particular pertinence are mRNAs of genes that are associated with the transcription factors MSN2 and MSN4 [7], which are involved in stress responses (see below). By contrast, the half-lives of mRNAs whose production depends on heat shock factor HSF1 do not change with or during heat stress [7], whereas the mRNA levels of translation-related genes tend to decrease with heat stress [5].

Sufficiently high temperatures induce unfolding, denaturation and aggregation of proteins. This change in structure may be reversible or irreversible. Reversible denaturation is corrected by chaperonins, which refold un- or misfolded proteins, whereas irreversibly denatured and aggregated proteins need to be solubilized and are subsequently removed by proteolysis in the proteasome. The processes of complete unfolding, denaturation, and aggregation are typical for higher temperatures.

At lower temperatures, such as 35–40 °C, the first step in the heat response is a passive, partial unfolding of proteins. The resulting changes in protein structure are much milder than for higher temperatures and, in fact, they are used by the cell for regulatory purposes. In particular, proteins of the important sub-group that possesses chemical activity experience changes in their enzymatic activity. This combination of increased catalytic activity, due to the Arrhenius effect, and decreased activity, due to partial unfolding, leads to something like a bell-shaped response curve of enzyme activity as a function of temperature, with the maximum of activity located at the optimal temperature for the specific enzyme.

The slight unfolding of proteins triggers distinct, secondary cellular responses, which target: (1) transcription factor activation leading to altered physiological processes; (2) gene expression leading to metabolic adjustments; (3) rapid production of protective metabolites; or (4) signaling systems triggering tertiary responses.

(1) Effects of protein unfolding on transcription factor activation leading to altered physiological processes. Of particular prominence, the pathway associated with heat shock factor 1 (HSF1) is activated in response to heat (Figure 1). Hsf1p is a transcription factor that recognizes and binds to the heat shock element HSE (5'-NGAAN-3') [8]. Under normal conditions, Hsf1p exists in two states, namely free or bound to HSEs. In both states, Hsf1p is kept repressed through the association with repressor proteins like Cpr7p, Hsc82p, or Sse1p. Heat-unfolded proteins affect the response system through sequestration of these repressor proteins, which thereby permit the activation of Hsf1p [9]. Thus, unfolded proteins free a regulatory protein, HSF1, whose pathway is responsible for the production of protein chaperones, such as HSP82, SSAs and SSBs. HSF1 is also involved in cell cycle regulation and in protein turnover by regulating the expression of the genes UBC4 and CUP1.

(2) Effects of protein unfolding on gene expression leading to metabolic adjustments. Heat-induced protein unfolding also targets the zinc-finger transcription factors MSN2/MSN4, which control a large number of genes that appear to be associated with metabolic stress responses (Figure 2). MSN2 and MSN4, collectively called MSN, respond to heat stress and protein kinase A (PKA) in an antagonistic fashion [10]. As long as PKA is active, MSN is kept in the cytosol, where it is inactive. Sufficient heat induction inactivates the PKA pathway and causes MSN to relocate to the nucleus, where it becomes active [11]. Once in the nucleus, the transcription factors bind to specific Stress Response Elements (STRE; 5′-CCCCT-3′) and thereby activate the transcription of downstream genes [12,13]. Genes under the control of MSN code for protein chaperons, proteins involved in protective metabolic pathways (Hxk1p, Tps1p, Nth1p, Gpd1p) and proteins involved in antioxidant defenses (Ctt1p, Sod2p) [9].

Thus, the HSF1 and MSN pathways play different, complementary roles, indicating a division of labor between the two transcription factors. HSF1 controls physiological processes that are temporarily dispensable, such as cell cycle activities, and is essential for the cell’s recovery from short, high-intensity heat shock. By contrast, MSN seems to be primarily in charge of long-term survival at high, but tolerable temperatures [14]. A good review, although not recent, can be found in this reference [9].

One should note that heat affects the regulation of a number of genes that code for enzymes involved in central carbon metabolism. Two modes of action seem to play a role: Some steps are catalyzed by more than one protein paralog, in which case some of the paralogs are heat-inducible while the others are not (Table 1). Additionally, all genes coding for producing and degrading enzymes in some metabolic cycles (e.g., trehalose or glycogen) are up-regulated, which hints at the existence of downstream regulatory processes.

(3) Effects of protein unfolding on the rapid production of protective metabolites. Heat-induced protein unfolding, directly affects events at the metabolic level. In particular, temperature alters the activity of several enzymes of the trehalose pathway, thereby leading to the accumulation of the disaccharide trehalose, which protects proteins, membranes and DNA from damage. Intriguingly, heat stress induces a simultaneous increase in the expression of genes coding for both the synthesis and degradation of trehalose, glycogen and fructose-2,6-biphosphate [1]. This increased capacity for production and degradation of intermediates is at first puzzling, and one might be tempted to conclude that it constitutes a futile cycle. However, it rather appears to be evidence of a downstream regulatory mechanism. Such a mechanism can be inferred very nicely from the case of trehalose. Here, the producing enzymes (trehalose 6-phosphate synthase and phosphatase; Tps1p and Tps2p) have activity optima at temperatures of 35–45 °C, whereas the degrading enzyme (trehalase; Nth1p) has its optimum temperature at 30 °C [15]. With this discrepancy in optimal temperatures, very little trehalose is produced at 30 °C, and because trehalase is at its maximum activity, no trehalose accumulates. However, at 40 °C, trehalose production is high and the trehalase activity is reduced by a factor of ~2.4, which causes trehalose to accumulate. Once the temperature returns to normal values, the direct temperature dependence of these enzyme activities allows the cell immediately to degrade all trehalose accumulated at the higher temperature. Not to be wasteful, this degraded trehalose enters glycolysis in the form of two molecules of glucose.

In a slightly different mechanism, glycogen production and degradation are regulated by cAMP dependent phosphorylation: in the presence of glucose, glycogen synthase is activated and glycogen phosphorylase is inactivated; at higher temperatures (35 °C), the glucose effects on these enzymes are amplified [16].

(4) Effects of protein unfolding on signaling systems triggering tertiary responses. Heat-induced protein unfolding is expected to influence numerous signal cascades. For instance, an initial cascade of the cAMP-PKA system (see above and Figure 2) is directly affected by heat stress [17].

Another intriguing example is the effect of heat on the activity of enzymes involved in the sphingolipid pathway [18]. We will discuss this system later in more detail but, briefly summarized, it is known that sphingolipids like ceramide and sphingosine-1-phosphate play direct signaling roles in a variety of cell programs [19]. Specifically within the context of stress responses, heat induces changes in the enzyme profile of the biosynthetic pathway, which can lead to a significant alteration in the concentration profile of these lipids. This altered profile, in turn, evokes secondary changes in gene expression. It furthermore causes indirect ripple effects that initially affect the concentrations of other lipids, which again may have their own signaling functions. As a particular example, it was recently shown that heat stress induces an increase in the concentration of phytosphingosine-1-phosphate, which peaks about 10 to 20 min into the stress. The increase in this sphingolipid, in turn, has an effect on numerous other sphingolipid species and also regulates genes associated with cellular respiration, by affecting the HAP transcription factor complex [20].

As indicated in the previous paragraphs, heat induces a number of direct and mediated responses. While these commence more or less immediately when the temperature rises, their dynamics is quite different. As a case in point, the unfolding of proteins is very rapid, and if the protein is an enzyme, the corresponding change in catalytic activity is just as fast. By contrast, alterations in gene expression lead to physiological effects that are delayed by fifteen minutes or more, due to the time it takes to execute transcription and translation.

The human mind tends to have difficulties integrating diverse quantitative information, arising at different time scales, into numerical or even semi-quantitative mental constructs, and this shortcoming suggests the application of computational modeling. Modeling approaches in these situations are challenging as well, again because of differing time scales and because of the heterogeneity of the biological components contributing to the response.

Two generic, successful strategies in such a situation are the separation of time scales and the representation of processes in the format of a canonical model. The separation of time scales consists of focusing on a single time scale while keeping processes at distinctly different time scales constant. For instance, for the short time period where protein unfolding alters the activity of an enzyme, one assumes that changes in gene expression are inconsequential. Their effects will be seen later, but not during the first few minutes of changes in enzyme activities.

The use of canonical representations facilitates the initial model design. These representations, including uni- or multi-variate linear or power-law functions, permit the immediate translation of a dynamic interaction diagram into a symbolic, mathematical construct, which even at this early state allows certain diagnoses and analyses [21,22]. We demonstrate these strategies in the following section, starting with the main transcriptional regulators, MSN2 and MSN4. These are partially redundant, although MSN4, which is inducible by heat stress, is only mildly affected by it [5].

The development of a comprehensive mechanistic model of the transcriptional and translational processes is infeasible with our current modeling technologies, because the detailed physical and chemical events leading to the formation of an intact protein are exceedingly complex. Even within the realm of metabolism, which is much better understood, the choice of a mechanistic model is not without problems. As a case in point, the Michaelis-Menten approximation is often chosen as a default model for enzyme catalyzed reactions, but this rate law is in truth somewhat problematic because its underlying assumptions are not satisfied in vivo [23,24]. For instance, the intracellular milieu is certainly not homogeneous and well mixed; the total amount of enzyme is likely to change as a function of time, and a substrate may not exist in much higher concentrations than its enzyme. Thus, one must question whether the Michaelis-Menten representation can be validly used to capture the dynamics of enzymatic processes in vivo. Similarly, mass action kinetics is frequently used, but approximating the interactions between several proteins and RNAs in a crowded intracellular environment with an elementary reaction is probably not truly appropriate.

At a very coarse level, the biological complexity and the need for relatively unbiased representations can be tamed to some degree by the use of canonical modeling representations, such as power-law functions, which time and again have been shown to work well for the formalization of complex networks or systems. In particular, these functions are well suited as initial default representations for different types of interactions that are a priori ill characterized [25]. The use of power-law functions in such situations is a good compromise that does not impose linearity between components, is mathematically guaranteed to be correct at some nominal operating point, and often provides a reasonable approximation within an acceptable range of concentrations [26]. Due to these features, power-law functions are the central component of Biochemical Systems Theory (BST) [21,24,26,27], which provides a rigorous theoretical framework for modeling and analyzing biological systems.

One great advantage of power-law representations is that the model design step is in principle straightforward: Suppose a process P is directly affected only by a substrate S and a modulator M. Then we know immediately that this process is represented as a function of the type

(1)

Here γ is a positive rate constant, and the exponents g₁ and g₂ are real-valued kinetic orders, the first of which is positive, because S is the substrate, and the second of which is negative if M is an inhibitor or positive if it is an activator. The magnitude of each kinetic order reflects the strength of the effect of the variable, with which it is associated, on the process. In fact, if the modulator in Equation (1) has a negligible effect on P, its kinetic order g₂ is close to 0, M raised to this number is close to 1, and the influence of M essentially disappears from the equation.

In the case of heat stress in yeast, power-law functions may be used to represent the overall synthesis of transcripts as well as their degradation. To represent the specific case of a gene under the control of MSN, such as TPS1/2 or NTH1, the nuclear form of the Msn protein is included in the power-law function for gene expression, because it exerts a positive, activating effect (see Figure 2). The dynamics of proteins are formulated in canonical models in a similar manner, namely through overall production and degradation terms. For example, the power-law term for protein synthesis is formulated to depend directly on the abundance of its corresponding transcript.

As a more complex example, but again of the same mathematical format, Equation (2) shows how different factors can be included in a power-law representation (see [28]). In this case, we model a reaction step F_i, in which the enzyme activity depends explicitly on the temperature in the milieu. As before, we include in the representation the substrates (S_j) and modulators (M_k), and account for their respective roles with kinetic orders h_i,j and hm_i,k. We also specify a rate constant α_i and explicitly account for the amount of enzyme, P_i. If we are justified to assume a direct proportionality between enzyme amount and activity, its kinetic order is 1; otherwise a different, more appropriate kinetic order would be included. Finally, Q_i is the direct effect of temperature (T) on this enzyme (with reference to 30 °C). It is usually not included in metabolic models, but obviously becomes important for heat-stress studies. Therefore, the power-law formulation of the reaction step reads

(2)

Further details can be found in [28]. Thus, setting up a dynamic model in a symbolic canonical format is straightforward, because it is clear how different pieces of information are to be converted into components of the mathematical model. The real difficulties arise later, namely in the determination of appropriate parameter values, which are seldom known. However, as a default, experiential values and educated guesses can be employed, at least initially [29]. Upon completion, a canonical model of the transcriptional and translational aspects is expected to simulate the effects of heat stress on the concentrations of mRNAs and their corresponding proteins, at least in a coarse-grained manner.

While the proper translation of a biological phenomenon into a computable structure continues to be an unsolved challenge, it is relatively straightforward to set up initial canonical models in symbolic form, as described before. Yet, achieving the construction of such a symbolic model is only the first step of quantitative model design. A second challenge to be addressed is the identification of appropriate parameter values, and thus the mining of data and kinetic information. Depending on the level of modeling, different types of data and different methods have been proposed, but none of them so far is truly satisfactory [30].

For aspects of heat stress associated with transcription, Gasch et al. [5] published a seminal paper that describes numerous transcriptional responses of yeast to environmental changes. The paper is based on data that were made publically available [31] and, among other scenarios, quantifies how most of the transcriptome responds to a temperature jump from 25 °C to 37 °C. Indeed transcript levels are presented for a period of 80 min after the initiation of heat stress. Two further studies [32,33] also induced gene expression patterns under heat stress. Other authors [34,35,36] published complementary datasets for transcript abundances, transcriptional rates and transcript half lives. More recently, Castells-Roca et al. [7] published a genome-wide dataset containing mRNA amounts, as well as transcription and decay rates of each mRNA, obtained in a growing culture of yeast cells that were heat stressed by a temperature shift from 25 °C to 37 °C; the data were presented for several time points up to 40 min.

Some data are also available at the proteome level, although these are often not as precise and reliable as for transcripts. For instance, the literature contains accounts of protein amounts, translation rates and protein half-lives, albeit only under control conditions [34,37,38]. Also, more recently, a proteome-wide study characterized the changes triggered by shifting a yeast culture from 24 °C to 37 °C, but this study contains results for only two time points (0 and 30 min) [39]. In principle, these types of datasets render it possible to parameterize the aspects of a multi-level model that are related to transcripts or proteins.

To refine and extend the parameterization of the metabolic aspects of the model, additional data are needed. Often these are collected from different sources, which sometimes causes problems, due to variations in experimental conditions. Of course, further metabolic studies could be performed under essentially the same conditions that were used to study the transcriptome and proteome, so that a consistent dataset would characterize the three levels of transcripts, proteins, and metabolites. Some kind of standardization has already started to occur in this direction [40], at least for enzyme kinetics.

We demonstrate the generic modeling approach by beginning at the gene expression level. Of particular importance for heat stress responses are MSN2/4, as discussed before. For simplicity, it is useful to model these two transcription regulators as just one MSN gene or protein. This simplification seems to be supported by their structural and functional similarity. Associated with this transcription factor are a basal level of expression and the provision that heat might slightly increase this expression. As discussed previously, the activity of MSN also depends on protein kinase A (PKA), which itself is affected by cAMP and stress. A recent model [17] integrates these phenomena. It describes the PKA system in great detail and leads to the conclusion that cAMP-PKA and stress may cause an oscillatory shuttling of Msn2p between nucleus and cytoplasm. However, the model does not describe mechanistically or operationally how heat stress changes the localization of the MSN protein. Thus, by adjusting the main concepts of this model to our purposes, one might propose to model the change in localization according to the scheme in Figure 2, where heat stress promotes nuclear localization, whereas activation of PKA favors cytosolic localization. In this approach, PKA is modeled in one of two states, namely, activated (PKA_C) or inactivated (PKA_RC). The conversion to the activated state depends on glucose, whereas heat stress inactivates PKA. Once in the nucleus, the Msn protein activates the expression of genes coding for some of the enzymes associated with heat stress (TPS1,2; HXT5; ZWF1; HXK1; GLK1; PGM2; GPM2; GSY2; GLG1; NTH1) and with generic chaperonins that possess refolding functionality (see later and Figure 3). In a canonical model, the qualitative description of the various influences is straightforwardly translated into power-law terms that contain each contributing factor as a variable with an exponent [21,25].

The expression of HSF1 does not seem to change much with heat stress [5], and it is therefore not necessary to model its gene expression. Instead, one considers the total amount of protein as constant and partitions this amount into different activity states. Specifically, HSF1p can exist in three states: free, bound to HSE, or bound to repressor proteins (Figure 1). Hsf1p is kept inactivated by binding to a number of proteins with similar function. Again for the simplicity of an initial, coarse model, one representative for these proteins may be chosen, and a good candidate is Cpr7p. Activation of Hsf1p secondarily induces the expression of chaperonins and inhibits the cell cycle. A mathematical model of these aspects is found in this reference [41].

In order to model Hsf1p activation, we consider a pool of yeast proteins that is prone to heat denaturation and serves the purpose of providing the signal input to the heat stress response (Figure 1). These proteins might be enzymes or structural proteins that tend to unfold at non-optimal temperatures, or they might be intrinsically disordered proteins that are known to have signaling functions [42]. These types of heat-induced effects can be converted into a canonical model where a folded protein controls a heat signaling pathway and where its unfolding triggers—or at least contributes to—a stress response.

As is typical in nature, the ultimate response to a stress situation is the result of a balance between opposing forces. We already discussed the counteracting effects of cAMP-PKA and heat on the localization of MSN2/4 (Figure 2). Another example of the balance of opposing forces is the folding, unfolding, and refolding dynamics of proteins (Figure 3). The disaccharide trehalose protects proteins from unfolding, but interferes with the refolding and degradation of the unfolded protein [43]. By contrast, chaperonins (as representatives of heat shock proteins) promote refolding and facilitate the degradation of unfolded proteins. If these forces are entered into a model, the degradation of unfolded forms has to be balanced with the production of proteins, so that the model may eventually reach a steady state. This production term may be made heat stress sensitive, which is in line with the observation that many transcripts are simply down-regulated under heat stress [5]. At the same time, protein degradation is known to be affected by heat, and inclusion of this effect in the model might improve the functioning of this hypothetical signaling pathway under stress.

Events at the metabolic level are typically easier to model than at other levels, because specific kinetic information is often available and phenomena like the conservation of mass in reactions provide very valuable constraints that aid the parameter estimation process. As a consequence, several models have been proposed to analyze heat stress and its metabolic effects in yeast and other organisms (e.g., [44,45,46,47,48,49,50,51]). For example, Voit and Radivoyevitch [48] used a canonical modeling approach to study the metabolic consequences of changes in gene expression following heat stress. Although quite simplistic, the model suggested an explanation for the observed heat-induced expression profile, which is quite counterintuitive, with some genes up-regulated many fold, and other genes, even those coding for neighboring reactions, not changed in expression at all. Another study demonstrated the design features of the trehalose pathway with controlled comparisons that identified the role of every regulatory signal at the metabolic level, as well as the observed gene expression patterns [44]. Sorribas and his group refined these types of analyses with sophisticated optimization methods that explained why the observed gene expression patterns are metabolically superior to a priori imaginable alternatives [45,46,47,52]. These types of studies have shown that it is indeed possible to infer, with a fairly good degree of confidence, the changes in metabolic states from gene expression or, conversely, the changes in expression profiles from a metabolic model and a set of established physiological criteria based on experimental information.

Earlier studies relied on a possibly significant simplifying assumption, namely that there is a linear correlation between the changes in transcriptomic and proteomic profiles. Maybe more importantly, these approaches ignored the direct temperature effects on enzyme catalysis. A more recent model [28] takes these aspects into account. In particular, this work joins two dynamic sub-models that represent different time scales and shows that canonical models, using power-law functions (as in Equations (1) and (2)), can be constructed from experimental data in a top-down manner. The first sub-model simulates the time-dependent protein profiles from the network of interactions between transcripts and proteins, while the second sub-model is a metabolic model that is capable of simulating time-dependent metabolic profiles based on the amounts of enzymes catalyzing each step, which are supplied from the first sub-model.

The main focus of this joint model is the enormous accumulation of trehalose in response to elevated temperature. Interestingly, targeted experimental analyses demonstrated that naïve and heat-adapted cells respond in a qualitatively similar, but quantitatively very different manner. In particular, when cells are exposed to heat during their early exponential growth phase, later heat stress leads to almost ten times the amount of accumulated trehalose in comparison to naïve cells [28]. To analyze this phenomenon, we set up a model in the following fashion. We allowed the naïve and heat adapted cells to express different amounts of the enzymes that catalyze each metabolic step in the trehalose pathway. This strategy accounted for the fact that cells exposed to heat during growth had the opportunity to increase gene expression and thereby the abundance of pertinent mRNAs and proteins. Our experimental time series data even allowed us to quantify these changes numerically. Additionally, because these data were obtained under two different temperatures (30 °C and 39 °C), we were able to quantify some of the direct effects of temperature on the enzymatic activities, while the temperature dependence of the activities of the trehalose related enzymes was obtained from literature [15]. By combining this type of a metabolic model with a model capable of predicting changes in enzymes, brought about by the heat stress response (see Equation (2)), it is possible to obtain a model that predicts metabolic responses and the cell’s adaptation to heat stress exposure quite well. Also, by combining the two sub-models, we are in a position to close the trehalose loop, since trehalose production is accounted for in the metabolic sub-model, while its effects as protein stabilizer are modeled in the transcriptomic/proteomic sub-model. Thus, this modeling strategy allows us to grasp how the various components interact synergistically to elicit, regulate, and sustain an appropriate heat stress response.

Sphingolipids form a specific class of lipids that are crucial components of membrane microdomains, called lipid rafts, and furthermore play distinct, important roles in the regulation of cellular stress responses, differentiation, proliferation, apoptosis and other fundamental cell functions [19]. Interestingly, evidence has implicated sphingolipids in the coordination of the heat induced expression of genes under the control of the MSN proteins [53]. Sphingolipids have also been shown to be necessary for efficient translation initiation during heat stress [54]. Although the pathways of sphingolipid biosynthesis and degradation have been characterized over the past decades in some detail (see Figure 4), the collective functioning of the pathway is still puzzling, mainly because the unaided human mind is unable to integrate its many components and their regulation in a quantitative manner. Therefore, with the aid of modeling and computational techniques, we set out to characterize the pattern(s) that control the yeast sphingolipid pathway under heat stress.

We began by developing, over the span of a decade, a series of computational models [55,56,57,58]. Formulated within BST, the core model contains about 65 variables. The model was tested and validated, and it appears that it captures many dynamic features of the sphingolopid pathway quite well.

An ongoing goal of relevance here is the identification of enzymes that are affected in the pathway during a response to heat stress. The analysis, whose technical details will be presented elsewhere, is based on time course data of six key sphingolipids, measured at 0, 5, 10, 15, 20, and 30 min after the beginning of heat stress (39 °C). We smoothed these data with splines and then applied a piecewise optimization approach to estimate the dynamically changing profiles of all enzymes within the sphingolipid pathway, by minimizing the distance between the smoothed sphingolipid data and the simulation results at each time point from 0 to 30. Using a randomization scheme for the initial algorithmic settings, we generated 100 sets of dynamic adjustments in enzyme activities that led to metabolite concentration trends consistent with observations. The overall result thus consisted of a band for each enzyme activity, within which about 90% of all solutions laid, as well as the average trend in each enzyme activity (Figure 5). Details of this analysis will be shown elsewhere.

The results are quite intriguing in detail, because they reveal the balance of three forces acting, on the enzymes, induced by heat: Increased activity according to an enzyme’s Q₁₀ value, as alluded to in Equation (2); diminished activity due to partial protein unfolding, an altered half-life of the corresponding protein and/or mRNA, and/or a reduced production; and change in enzyme activity due to gene expression. As an example for the first category, the activity of phosphoserine phosphatase increases about three-fold and remains at this activity level for at least 30 min (Figure 5a). An example of the second category is diacylglycerol (DAG) ethanolamine phosphotransferase, whose activity was inferred to decrease, after a brief initial increase according to its Q₁₀ (Figure 5b). Sphingoid-1-phosphate phosphatase falls into the third category (Figure 5c). Initially its activity drops quickly, but after about 25 min not only recovers but increases well over its baseline activity. Of note is that these results were extracted from the concentration time series data and the dynamic model strictly by computational means and without additional information.