- Research
- Open Access

# Differentiation between genomic and non-genomic feedback controls yields an HPA axis model featuring Hypercortisolism as an irreversible bistable switch

- Clemens A Zarzer
^{2}, - Martin G Puchinger
^{3}, - Gottfried Köhler
^{3}and - Philipp Kügler
^{1, 2}Email author

**10**:65

https://doi.org/10.1186/1742-4682-10-65

© Zarzer et al.; licensee BioMed Central Ltd. 2013

**Received:**17 June 2013**Accepted:**29 October 2013**Published:**9 November 2013

## Abstract

### Background

The hypothalamic-pituitary-adrenal axis (HPA axis) is a major part of the neuroendocrine system responsible for the regulation of the response to physical or mental stress and for the control of the synthesis of the stress hormone cortisol. Dysfunctions of the HPA axis characterized by either low (hypocortisolism) or increased (hypercortisolism) cortisol levels are implicated in various pathological conditions. Their understanding and therapeutic correction may be supported by mathematical modeling and simulation of the HPA axis.

### Methods

Mass action and Michaelis Menten enzyme kinetics were used to provide a mechanistic description of the feedback mechanisms within the pituitary gland cells by which cortisol inhibits its own production. A separation of the nucleus from the cytoplasm by compartments enabled a differentiation between slow genomic and fast non-genomic processes. The model in parts was trained against time resolved ACTH stress response data from an *in vitro* cell culture of murine AtT-20 pituitary tumor cells and analyzed by bifurcation discovery tools.

### Results

A recently found pituitary gland cell membrane receptor that mediates rapid non-genomic actions of glucocorticoids has been incorporated into our model of the HPA axis. As a consequence of the distinction between genomic and non-genomic feedback processes our model possesses an extended dynamic repertoire in comparison to existing HPA models. In particular, our model exhibits limit cycle oscillations and bistable behavior associated to hypocortisolism but also features a (second) bistable switch which captures irreversible transitions in hypercortisolism to elevated cortisol levels.

### Conclusions

Model predictive control and inverse bifurcation analysis have been previously applied in the simulation-based design of therapeutic strategies for the correction of hypocortisolism. Given the HPA model extension presented in this paper, these techniques may also be used in the study of hypercortisolism. As an example, we show how sparsity enforcing penalization may suggest network interventions that allow the return from elevated cortisol levels back to nominal ones.

## Keywords

- Cortisol
- Chronic Fatigue Syndrome
- Anterior Pituitary Gland Cell
- Bistable Behavior
- Elevated Cortisol Level

## Background

The HPA axis is a major part of the neuroendocrine system in mammals and particularly in humans. A main task of this hormonal network is the regulation of the response to physical or mental stress that threatens to disrupt the homeostatic balance of the organism. If a stressor is sensed by the nervous system the hypothalamus is stimulated to produce and secret the corticotropin-releasing hormone (CRH). The secretion of CRH causes the anterior pituitary to synthesize adrenocorticotropin (ACTH). ACTH then stimulates the adrenal glands to release cortisol, which down regulates the blood concentration of CRH and ACTH via different negative feedback mechanisms[1–4].

The HPA axis is subject of intensive research in endocrinology as HPA malfunctions are implicated in various pathological conditions. These are often characterized by either elevated or insufficient blood cortisol levels compared to the average healthy human. For instance, hypocortisolism (insufficient cortisol level) is reported in patients suffering from the chronic fatigue syndrome and post traumatic stress disorders (cf.[5–9]), whereas hypercortisolism (elevated cortisol level) is observed in depression, dementia or postoperative delirium (cf.[10–15]). Especially in the context of personalized medicine the use of modeling and simulation of biological systems for the rational design of treatments and drug intervention strategies is more and more recognized[16–20]. For such endeavors the integration of biological information of different type into computational, hence predictive, models is a prerequisite. The emphasis of earlier HPA modeling approaches with ordinary and delay differential equations has been put on self regulatory ultradian and circadian oscillatory behavior in[21–27], oscillations in response to an independent external pacemaker from the suprachiazmatic nucleus have been described in[28, 29]. In comparison, the article[30] stands out as it incorporates intracellular glucocorticoid receptor kinetics which mediate bistable behavior of the HPA axis. Despite its parsimonious character the four state ODE model of[30] offers an explanation for hypocortisolism as an irreversible biological switch and served in[31] as a basis for the design of a therapeutic corrections of the HPA dysfunction. In[19] it is shown that the model of[30] also exhibits stable limit cycle oscillations, in[32] the four state rate equations of[30] were modified in order to fit oscillatory data of patients suffering from post traumatic stress disorder.

### A parsimonious HPA model featuring hypocortisolism

*CRH*], [

*ACTH*], [

*GR*] and [

*COR*] denote the concentrations of CRH, ACTH, GR and cortisol. If GR is bound to cortisol, the resulting complex is denoted by [

*COR-GR*]. As this binding is very fast compared to the other dynamics it is assumed to be in equilibrium, i.e. [

*COR-GR*] = [

*COR*] · [

*GR*]. The GR dimerization may generate an irreversible bistable switch in the model by which the dysfunctional abidance of the system at an alternative stable steady state characterized by low cortisol level can be explained. The therapeutic strategy of[31] for the correction of hypocortisolism based on (1)-(4) is to force the HPA axis by a further suppression of cortisol to a point where the only stable condition in proximity is the original healthy state. An alternative for supporting the search for pharmaceutical therapies is the manipulation of the feedback network by means of inverse bifurcation techniques in order to transform the irreversible switch into a reversible one, see[19, 33]. Though the model of[30] sheds new light on possible causes for hypocortisolism, its dynamic repertoire could not be shown to feature the opposite scenario of hypercortisolism. Furthermore, while the model takes the intracellular glucocorticoid receptor GR into account, it does not consider feedback mechanisms known to act on faster time scales.

### Motivation of model extensions

The above mentioned GR mediated negative feedback on the production of ACTH acts at the genomic level. Specifically, cortisol COR binds to the glucocorticoid receptor GR and forms a homodimer (cf.[34, 35]). This homodimer is known to work as a transcription factor, enhances the expression of the GR gene and down regulates the expression of the ACTH related gene. A consequence of the transcriptional regulation is that the GR feedback control works rather slowly compared to other cellular processes such as a vesicle transport, cell signaling or extracellular events such as changes in the CRH blood concentration. For mammalian cells one can expect at least a delay of the down regulation in the range of 15 minutes up to 2 hours (c.f.[1]). However, it has been frequently reported that cortisol induced inhibitory effects on anterior pituitary gland cells can already occur after a few seconds (c.f.[36, 37]), a phenomenon that cannot be explained by means of the genomic feedback mechanism.

The common hypothesis for this observation is that besides the GR receptor a second receptor in the membrane may directly inhibit the release of ACTH when bound to cortisol (cf.[38–42]). In[43] the authors were able to provide evidence of a glucocorticoid receptor (GPCR) in the anterior pituitary cell membrane, possibly enabling the fast response related to cortisol induced inhibition. In the following we incorporate this fast and non-genomic feedback mechanism and for this purpose integrate a detailed pituitary gland cell model into the HPA network of[30]. As a consequence, the extended dynamic repertoire of the resulting HPA axis feedback network also features bistable behavior compatible with the dysfunctional state of hypercortisolism and the ability to simulate time resolved ACTH stress response data from *in vitro* AtT-20 pituitary tumor cells.

## Results

### Model comparison with experimental data

For training the model against the experimental data by adaptation of the model parameters we followed a variational regularization approach (cf.[33, 45]), see Methods for details. Sparsity enforcing penalization was used to eliminate model components less relevant for reproducing the observed kinetics. Figure4B shows that the experimental data can be fitted by the HPA model of Figure2 even if the autocatalytic loop for the GR gene regulation would be neglected and if the production of the TF (transcription factors for CRH) receptors would be considered to be independent of CRH.

In order to assess the relevance of the featured GPCR receptor for the model we knocked out the corresponding pathway by elimination of the respective ODE equations from our model and addressed the data fitting task with the reduced model. The result in the data space is shown in Figure4C and indicates that the consideration of the GPCR feedback pathway in the model is in fact vital for reproducing the experimentally observed dynamics. For comparison we finally run the data fitting procedure with the parsimonious model (1) - (4) from[30]. The model lacks a GPCR mediated feedback pathway for fast response and fails to reproduce the corresponding experimental observation, see Figure4D.

### Exploration of the dynamic repertoire of the model

*f*

_{ x }(

*x*,

*q*) (with

*f*denoting the right hand side of an ODE model) and search for regions in the parameter space that give rise to limit-point or Hopf bifurcations. Figure5B displays a resulting bifurcation diagram for the extended HPA model from Figure2 with the stress level of Equation (8) as the bifurcation parameter. It gives an explanation of the dysfunctional alteration from a steady state of normal cortisol level (upper branch of the equilibrium curve) to a second steady state with low cortisol level (lower branch of the equilibrium curve) in response to a temporary stressor. If the abscissa of the limit point

*LP*

_{2}lies below the basal stress level, say for instance 0.1, the return to the upper branch is hampered and the HPA axis is trapped at low cortisol as observed in hypocortisolism. Figure4A shows a corresponding cortisol concentration profile obtained by simulation of the model from Figure2.

Further explorations of the dynamic repertoire of the extended HPA model revealed a region in the parameter space which allows for an alternative bistable behavior that can be linked to the pathological condition of hypercortisolism. Figure5D provides an example of a corresponding bifurcation diagram in which the qualitative scenario of Figure5B is inverted. If a stressor drives the HPA axis from the lower cortisol branch to the upper one, it abides at elevated cortisol levels even after removal of the stressor if the abscissa of the limit point *LP*_{2} lies below the basal stress level, again say for instance 0.1. Figure5C shows a corresponding cortisol concentration profile obtained by simulation of the model depicted in Figure2. While the positive feedback mediated by the glucocorticoid receptor *GR* dimerization has been already suggested in[30] as the cause for the bistable behavior associated to hypocortisolism, our in-silico pathway knock out tests on the extended HPA model indicate that the positive feedback on the genomic level mediated by the CRH hormone is vital for the bistability associated to hypercortisolism. Without this component, we could not find a parameter region giving rise to a bifurcation diagram as displayed in Figure5D. Similarly, the results of our numerical analysis suggest that the dynamic repertoire of the parsimonious model of[30] excludes the bifurcation behavior of Figure5D, possibly due to its simplistic description of the CRH pathway.

Finally, we explored the parameter space of the extended HPA network model with objective functions for the detection of Hopf bifurcations and found parameter regions allowing for stable limit cycle oscillations, see Figure5E. The displayed temporal sequence in which the core species CRH, ACTH and cortisol reach their maximum value is in accordance with the established cascade in release of the HPA hormones. While oscillatory behavior for the parsimonious model was obtained in[30] in response to repeated stress pulses, limit cycle oscillation for (1)-(4) were found in[19] and for a modified version also in[32]. However, to our knowledge the HPA model shown in Figure2 is the first one able to capture two distinct types of bistability (as reflected in Figures5B and5D) as well as autonomous oscillatory behavior.

### Model based correction of Hypercortisolism

Based on the parsimonious model of Figure1 from[30] a therapeutic strategy for the correction of hypocortisolism has been suggested in[31]. This therapy aims at perturbing the HPA axis trapped at the low cortisol branch by a further suppression of cortisol in order to enable the return to a healthy cortisol level. An alternative idea for the treatment of hypocortisolism has been presented in[33] which aims at an intervention with the reaction network in order to turn the dysfunctional irreversible switch into a reversible one by shifting the critical limit point in the bifurcation diagram. As the extended HPA model of Figure2 also captures a switching scenario associated to hypercortisolism, the inverse bifurcation approach of[33, 48, 48, 49] can also be applied to study which parts of the HPA network need to be targeted in order to allow the return from the abnormal hypercortisol steady state back to a healthy cortisol level.

For the purpose of illustration we suppose that a representative *q*^{0} out of the parameter region compatible with the switching behavior of Figure5D has been chosen (for instance by fitting of corresponding patient data) in which the abscissa *S*_{2}(*q*^{0}) of the limit point *LP*_{2}(*q*^{0}) lies below the basal stress level *S* = 0.1 (on a fictive scale). Then, a short but strong stress signal may drive the system from a normal cortisol level to an abnormal elevated one where it abides even if the stress drops back to basal level, see Figure5C. A therapeutic qualitative change from an irreversible to a reversible steady state transition can be achieved by interfering with the HPA model such that the abscissa of the limit point *LP*_{2}(*q*^{0}) is shifted to a value
with
while, e.g., maintaining the abscissa value of the second limit point *LP*_{1}(*q*^{0}), see Figure5A.

*F*maps the correction

*x*onto the abscissas

*S*

_{1}(

*q*

^{0}+

*x*) and

*S*

_{2}(

*q*

^{0}+

*x*) of the limit points

*LP*

_{1}(

*q*

^{0}+

*x*) and

*LP*

_{2}(

*q*

^{0}+

*x*) defined by the HPA model with altered parameter vector

*q*

^{0}+

*x*. In order to keep the number of necessary interventions with the HPA network as low as possible a sparse solution of (5) is sought for by solving the optimization problem

with *φ* as in (35) and the sparsity promoting choice *p* < 1.

*q*

^{0}) giving rise to the re-engineered bifurcation diagram of Figure6A (green curve) and allowing the return to normal cortisol levels, see Figure6B. The parameters

*K*

_{6},

*v*

_{3}and

*d*

_{7}to be altered concern the feedback controls related to the hormone CRH. A decrease of

*v*

_{3}and an increase of

*d*

_{7}correspond to a reduction of the sensitivity of the cells with respect to extracellular CRH. Moreover, an increase of the dissociation/Michaelis-Menten constant

*K*

_{6}reduces the sensitivity of the TFs transcription factor regulation by the CRH-CRHR related pathways. Furthermore, the increase of

*k*

_{tl2}calls for an increase of the translation rate of GR mRNA in order to further diminish the effect of extracellular CRH due to higher availability of GR. The intervention strategy shown in Figure6C also suggests to alter the parameters

*K*

_{3},

*d*

_{3}and

*d*

_{2}in order to reduce the production of CRH and to increase the sensitivity with respect to the feedback via cortisol. As these parameters concern processes outside of the anterior pituitary gland cells, an extension of HPA model by a mechanistic description of the hypothalamus is necessary in order to enable a more detailed interpretation of those parameter changes.

## Conclusions

Motivated by the experimental detection of a glucocorticoid membrane receptor GPCR we have built a mathematical model of the anterior pituitary cell that differentiates between genomic and GPCR mediated non-genomic signaling pathways. After the integration of our model into a literature framework of the hypothalamus pituitary adrenal axis and the training against stress induced ACTH concentration profiles of murine AtT-20 pituitary tumor cells we obtained a dynamic repertoire capturing two types of bistable switches as well as stable limit cycle oscillations. While the first bistable switch has been previously associated to the diseased state of hypocortisolism and utilized in the development of therapeutic strategies, to our knowledge the second bistable switch is novel in the context of HPA axis modelling. Along similar lines it can be linked to another dysfunctionality of the HPA axis referred to as hypercortisolism in which a continued abidance of the axis at elevated cortisol levels is observed even after complete removal of the stressing factor. Our in-silico knockout studies in particular indicate that the positive feedback on the genomic level mediated by the CRH hormone is responsible for the generation of the hypercortisolism switch. Using inverse bifurcation analysis with sparsity enforcing penalization we finally detected key model parameters that need to be altered in order to turn the irreversible switch into a reversible one then featuring a healthy stress response characterized by a return to normal cortisol levels.

While systems biology strongly advertises the use of modeling and simulation in the design of future drugs and therapies, one current challenge is to appropriately picture the complexity of the biological system at hand in the defining mathematical equations and their parameters. Though with respect to HPA axis modeling we reached a certain level of detail by introducing both genomic and non-genomic signaling pathways in the pituitary cell, we only adopted the parsimonious description of the link between the pituitary and the hypothalamus as well as the adrenal from the literature. Furthermore, the information content of the available ACTH concentration did not allow to sufficiently constrain the model parameters such that we so far have to content ourselves with parameter space regions corresponding to different qualitative model behaviour. Nevertheless, even at its current stage our model offers to endocronologists a different view on hypercortisolism as an irreversible bistable switch while the outlined mathematical methodologies will also apply to more realistic models and improved data sets. Finally, the model may also serve as a tool for the design of a model predictive control based treatment of hypercortisolism in which the HPA axis is driven by a - though at first glance counter-intuitive - *additional administration* of cortisol to a point where the only stable condition in proximity is the original healthy state, compare with[31].

## Methods

### Model development

In order to distinguish between slow genomic and fast non-genomic processes we use the modeling concept of compartments and separate the nucleus from the cytoplasm of the pituitary gland cell. Moreover, we base our model on the principles of Michaelis-Menten enzyme kinetics ([50–52]) in order to deal with physical meaningful parameters.

We refrain from a detailed description of gene regulation, RNA processing or cleaving and rather concentrate on the rate limiting processes such as transport/diffusion in and out of the nucleus. We apply this modeling strategy not only to the glucocorticoid mediated feedbacks but also to the stimulating pathways involving the hypothalamic hormone CRH. CRH does not enter the pituitary gland cells but interacts with a specific membrane receptor (CRHR) only. Still, also the binding of CRH induces both a genomic and a non-genomic feedback (cf.[53–55]). As this positive feedback via CRH counteracts the negative feedback via cortisol on the genomic as well as on the non-genomic level, also the catalyzing mechanisms related to CRH have to be modeled in more detail than in[30] for the purpose of differentiating between genomic and non-genomic effects in the anterior pituitary gland cells. Concerning the molecular mechanisms of the anterior pituitary cells, our model involves three different pathways which are associated with the three main receptors, namely the intracellular glucocorticoid receptor GR and the two membrane receptors (GPCR and CRHR) with respect to cortisol and CRH, see Figure2 and Figure3.

#### Extracellular species and the controls related to the adrenals and the hypothalamus

*k*

_{diff1}) of cortisol into the anterior pituitary cells:

*V*

_{ex}and

*V*

_{in}are the volumes of the extracellular compartment and the intracellular compartment. Analogously to[30] we model the temporal development of the CRH concentration by

#### The GR mediated feedback control

_{in}) is subject to the transport of the dimerized GR-COR complex into the nucleus. This is described by

which follows from the assumption of the fast binding of cortisol and GR and the quick formation of the homodimer. A detailed derivation is provided in the Additional file1.

which then are transported to the cytoplasm and translated to the GR protein.

#### The GPCR mediated feedback control

The second pathway we consider in our model is the negative feedback on the release of ACTH via the glucocorticoid membrane receptor (GPCR), see Figure3B. Concerning the production of the GPCR receptor we assume no regulation by means of other compounds and hence disregard the processes on the genomic level. With respect to the complex formation (COR-GPCR) we suppose a (quasi) equilibrium as it has been experimentally observed that two ligands bind to the receptor and exhibit positive cooperativity (cf.[43]).

#### The CRHR mediated feedback control

Concerning the fast response to CRH stimulation it is the common hypothesis that the release of ACTH is regulated via a signaling cascade which is induced by the binding of CRH to the CRHR receptor in the cell membrane. Signaling cascades are very fast compared to other cellular processes. In fact the response to cortisol or CRH has been observed already after a few micro seconds (cf.[40, 54, 58, 59]). Thus we assume an immediate effect of the membrane receptor complex formation on the release of ACTH, as modeled in the ODE (13).

#### Model parameters

We now comment on the role of the parameters in our ODE model. We have three subsidiary parameters describing the average volumes of the three compartments *V*_{in}, *V*_{ex} and *V*_{nu}. These parameters are phenomenological and are fixed in the subsequent analysis.

Another class of parameters are the constants related to diffusion and transport of the compounds. In our model *k*_{diff1}, *k*_{1ex}, *k*_{2in}, *k*_{2ex}, *k*_{3ex} and *k*_{3in} comprise the list of these parameters. We do not explicitly model active transport, as this would involve additional regulatory species and mechanisms and consequently would lead to an even more complex model. Subsequently, we assume that the rate of diffusion/transport is constant on average.

The largest portion of model parameters is related to degradation processes whose rates are assumed to be proportional to the concentration of the respective species. The associated rate constants are denoted by *d*_{1}, *d*_{2}, …, *d*_{14} and *d*_{15}.

As a consequence of the considered enzymatic reactions and ligand-receptor complexes the model includes several dissociation constants. Moreover, we have some constants arising due to the Michaelis-Menten approximations, i.e. the maximum reaction rates and the Michaelis-Menten constants. They are closely related to the dissociation constant of the respective complex and approximate them if the rate of the product formation dominates the degradation of the enzyme-substrate complex. *k*_{GRdim}, *k*_{GC2}, *k*_{CRC} and *k*_{GRC} denote the dissociation constants of the considered ligand-receptor complexes. *K*_{1}, *K*_{3}, …, *K*_{9} denote the Michaelis-Menten constants of the respective enzyme-substrate complexes. *v*_{1}, *k*_{s}, *k*_{s}*k*_{sb} and *v*_{4} denote the corresponding limiting rates of the Michaelis-Menten approximations.

Our description of the genomic mechanisms involves rates for transcription (*v*_{5}, *v*_{6}, *v*_{7} and *v*_{8}), translation (*k*_{tl1} and *k*_{tl2}) and transport (*k*_{trs1} and *k*_{trs2}) of mRNA. We neglect the post-processing of mRNA as our focus is on the respective timescales and as the time delay is mainly due to the translocation. The constants *v*_{2} and *v*_{3} denote the production rate of the two membrane receptors. As we have not considered their regulatory mechanism we lumped the transcription and translation rate together.

### Model training against data

*φ*that guarantees the continuous dependence of the parameter solution on the data

*q*of length 46.

*ρ*

_{1}and

*ρ*

_{2}in (33) denote the initial conditions corresponding to the two different experimental settings as described above and the two resulting data sets and displayed in Figure5A. While the initial concentration of some species are known from the experimental setup and can be taken into account by the constraints (34), the remaining components of

*ρ*

_{1},

*ρ*

_{2}have to be estimated from the data as well. As the available amount of data is not sufficient to uniquely determine the model parameters, we have chosen a two step approach for finding the model components most relevant for reproducing the data. First, we applied classical Tikhonov regularization[45] and solved (31) with

*p*= 2 in the penalty term

*q*

^{ tik }then served as an initial guess for solving (31) with a

*p*-value less than one. Sparsity enforcing

*ℓ*

_{ p }functionals with the choice 0 <

*p*≤ 1 aim at solutions whose less important components are driven to zero[60–62] and have been used in the context of parameter identification in[63, 64]. For nonlinear inverse problems, the regularization properties of (35) with 0 <

*p*≤ 1 have been analyzed in[65, 66]. Figure7 displays the parameter vector

*q*

^{∗}obtained for

*p*= 0.9 in relation to

*q*

^{ tik }. The near zero components of

*q*

^{∗}correspond to reactions and pathways that can rather be neglected in fitting the experimental data, see Figure5B. In particular, the autocatalytic loop for the GR gene regulation, represented by the parameters

*k*

_{trs},

*v*

_{7},

*v*

_{8},

*v*

_{9},

*d*

_{15},

*d*

_{11}, seems negligible with respect to the fast dynamics. Furthermore, the near zero value of

*K*

_{6}indicates that the production of the TFs receptors is independent of CRH, i.e. the genomic part of the CRH related feedback.

### Origin of experimental data

Murine anterior pituitary AtT-20 cells (passage 12-24) were seeded and maintained as previously described[44]. After 92 hours of cell growth AtT-20 cells were exposed to doses of 10nM CRH and in addition 100nM cortisol. The supernatant was carefully removed from the cell layer after 1 minute to 1 hour of administration and centrifuged for 10 minutes with 800xg at 37°C. The concentration of ACTH-molecules in the cell-free supernatant was detected using Fluorescence Correlation Spectroscopy Immunoassay[44].

The AtT-20 cells (ATCC no. CCL-89) were purchased from the American Type Culture Collection (ATCC, Manassas, USA). Cell culture media and reagents were from Sigma (Sigma-Aldrich Inc., St. Louis, MI).

## Declarations

### Acknowledgements

Financial support by the Viennese Science and Technology Funds WWTF, project grant MA07–30, is kindly acknowledged.

## Authors’ Affiliations

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