Introduction

Hormones are regulatory molecules secreted by endocrine cells into the circulation, where they affect target tissues1. Hormones impact physiological functions including growth, metabolism, reproduction, stress response, immunity, mood and behavior1,2,3,4,5,6. Dysregulation of hormones underlies a wide range of pathologies including diabetes, thyroid disorders, fertility disorders, growth delay, Cushing’s syndrome, anemia and mood disorders1,7.

Hormones share a common aim—to regulate physiological function. However, they differ in their regulation circuitry1. Some hormones are secreted under the control of neuronal stimulation, such as antidiuretic hormone (ADH)8. Some are secreted under the control of other hormones, such as thyroxine, which is controlled by thyroid-stimulating hormone (TSH)9. Others are secreted in response to metabolic signals, such as insulin, which is secreted in response to elevated blood glucose10. Yet other hormones are regulated in complex cascades. For example, several key hormones are regulated by the hypothalamus by means of an intermediary gland, the pituitary, that controls multiple effector glands11.

We ask what determines the regulation circuitry for a given hormone. In order to do so, it is useful to define recurring patterns and concepts12. This approach was successful in deriving principles in gene regulation networks and other biochemical systems such as network motifs with defined information-processing functions13. Well-known principles in endocrinology likewise have been instrumental in guiding experimental and clinical studies. For example, feedback regulation is a hallmark of homeostatic endocrine systems such as insulin control of glucose14.

The discovery of such feedback loops led to mathematical models that resulted in formulas that are useful for clinical research. One example is the estimate for insulin resistance based on insulin and glucose blood tests, known as the Homeostasis Model Assessment-estimated Insulin Resistance (HOMA-IR), developed by Matthews et al. based on fasting insulin and glucose measurements15,16. Clinical practice in diabetes also benefits from the Disposition Index (DI), an estimate of beta-cell function relative to insulin sensitivity. The Disposition Index was developed by Bergman et al. (1981) based on a minimal mathematical model analysis of the glucose tolerance test17.

Principles help to form analogies between systems, allowing researchers to carry ideas from one system to another. Examples include the development of HOMA-like formula for the thyroid18,19 and the discovery of long feedback loops in the hypothalamic-pituitary hormone axes20,21.

Recent work has defined regulatory principles related to changes in endocrine gland mass due to the trophic effect of hormones in the same pathway22,23,24. This allows gland mass to grow or shrink, and thus to produce more or less hormone. For example, the thyroid grows when iodine is scarce in order to produce more thyroxine. Mass changes on the scale of weeks compensate for physiological changes such as insulin resistance25, pregnancy26 and addiction27. They govern responses over months to chronic stress22, hormone seasonality24 and the transition between subclinical and clinical autoimmune diseases in which upstream glands can partially compensate for destruction of tissue23.

There remain open questions which we aim to address here. What determines the regulation circuitry of each hormone system? How does the regulation circuitry affect the function of the circuit? Why doesn’t the hypothalamus directly communicate with target organs like the adrenal, and instead goes through an intermediate gland, the pituitary? The answers to these questions can deepen our understanding of the endocrine system and contribute to the field of systems endocrinology28,29,30,31,32.

In this study we identify unifying design principles in the human endocrine system. In order to do this, we studied 43 hormone systems, seeking to categorize them based on regulation circuitry. We find that diverse systems can be organized into 5 classes of circuits according to shared regulatory motifs—regardless of biochemical detail. These classes are only a tiny fraction of the possible circuits. Using dynamical mathematical models of circuit function, we show that each class offers specific regulatory functions. We show how the pituitary, an intermediate gland that is a key element of several endocrine circuits can offer functional advantages over alternative designs that lack such an intermediate gland. These include amplification of hormone secretion, buffering against hormone-secreting tumors and providing speedup of response to chronic stress. These principles enhance our understanding of human hormone systems.

Results

Hormone circuits can be categorized into five distinct classes

There are 63 known hormone systems in humans1,4. We analyzed the regulatory interactions between cells via secreted hormones as described in the primary textbooks in clinical endocrinology1 and complemented this by a literature search. For each hormone, we identified the regulatory interactions that control its secretion, the gland or tissue that secretes the hormone and the major physiological variable (metabolite or other hormone) that is affected by the hormone. We call this set of interactions a hormone circuit. These circuits represent baseline biological consensus under normal physiological conditions. We found sufficient information to systematically categorize the regulation of 43 hormone systems, which make up 30 hormone circuits (e.g. TRH, TSH and thyroid hormones are all in one circuit) (SI Table S2). Examples of hormone systems not included due to lack of information include apelin and pituitary adenylate cyclase activating polypeptide (PACAP) (full list in SI Table S1).

We focus on regulatory interactions between cells, and not on the regulation inside the cell. For example, insulin control by glucose occurs through an elaborate circuit of interactions inside beta-cells, including glycolysis, energy sensors, calcium channels and secretion machinery33. We do not explore the potential motifs on this intracellular level.

We found that the hormone systems that we studied can be organized into five classes of circuits, which we term classes 1-5 (Fig. 1). The various hormone circuits within each class obey the same regulation circuitry design. By analogy with gene-regulatory and other networks, we describe these classes as circuit motifs13. Note that these circuits differ from classical gene-regulatory motifs that act within the cell because they describe cell-cell signaling and cellular growth. Class 1 circuits are the simplest and class 5 are the most complex.

Fig. 1: The five classes of endocrine circuits and their dynamic response to brief and prolonged input stimuli.
figure 1

Diagrams are generic and stimulation and inhibition may be exchanged. a−e Class 1−5, respectively. Metabolite, hormone and gland mass dynamics from simulations with a brief input pulse of 9 days (dashed blue line) or a prolonged pulse of 150 days and a second prolonged pulse with twice the amplitude at day 350 (orange line). Constant tissue in class 2 indicates that the secreting cell type growth is not regulated by elements in the circuit. The functions of the circuits are summarized. Simulation parameters are \(q={q}_{3}=16\,{da}{y}^{-1}\), \({q}_{1}=288\,{da}{y}^{-1}\), \({q}_{2}=48\,{da}{y}^{-1}\), \({r}_{{Classes}1-3}={r}_{3}=16\,{da}{y}^{-1}\), \({r}_{1}=288\,{da}{y}^{-1}\), \({r}_{{Class}4}={r}_{2}=48\,{da}{y}^{-1}\), \(c={c}_{1}={c}_{2}=d={d}_{1}={d}_{2}=1/15\,{da}{y}^{-1}\), \(s=48\,{da}{y}^{-1}\), \(a=16\,{da}{y}^{-1}\) (Box 1, Methods).

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We note that the number of possible regulatory circuits is much larger than 5. For example, if one enumerates all possible connected circuits analogous to class 5 circuits (circuits with three hormones secreted by three cell types) one obtains 512 possible circuits (6 interactions between the cell types and 3 possible autocrine regulations providing a total of 9 interactions, each of which can be present or absent totaling \({2}^{9}=512\) circuits). If one includes all combinations of possible regulatory signs on the arrows (each either negative or positive), this number grows to \({3}^{9}=19683\) possibilities. Only a tiny fraction (0.025% or 1%) of the possibilities are observed in physiology, suggesting a selected, meaningful function for the 5 classes.

For each class of circuits we consider a simple mathematical model (Box 1), aimed to describe the interactions between cells and the changes in functional gland mass. These models describe the dynamics of the hormone concentration, metabolite concentration and the gland functional mass. The models are based on previous approaches—the Topp model for the insulin system34 and the model for the Hypothalamic-Pituitary-Adrenal (HPA) axis by Karin et al.22. The models have a fast (hours) timescale of hormone production and removal, and a slow (weeks) timescale of gland mass changes. We emphasize that for different research questions, different models are required: for analysis of ultradian and diurnal timescales, for example, more detailed models of the intracellular circuitry are required than, for example, in studying the HPA axis31 and other systems32.

The models describe the essential mechanisms with a minimal number of parameters (production rates, removal rates, growth rates). We provide simulations of the dynamics of the hormones and metabolites for each type of circuit for both brief and prolonged input pulses (Fig. 1). For a brief input pulse we chose a 9 day pulse, much shorter than the typical timescale of gland-mass changes, and for a prolonged input pulse we used 150 days, much longer than the gland change timescale. We then provided a second input pulse of twice the amplitude. The response time of the circuits to rapid stimuli is determined by the hormone removal rates on the scale of minutes to hours. The response time of changes on the scale of weeks of circuit classes 3-5 is determined by the turnover rates of the gland functional masses. These rates were previously calibrated for the thyroid23, HPA22 and other hormone systems24. For each class, the simulations indicated distinct behavior related to the circuit topology. We discuss each of these behaviors and their sensitivity to parameters.

Class 1 and 2 circuits function as fast input-output devices

Class 1 and class 2 circuits, the simplest classes, are both input-output devices that convert a neuronal input into a secretion rate of a hormone. Class 1 circuits (Fig. 1a) are simply neurons that directly secrete a hormone. An example is the system of hypothalamic neurons that secrete Anti-Diuretic Hormone (ADH) and oxytocin8. Class 2 circuits (Fig. 1b) have a neuronal input to an endocrine cell. Examples include sympathetic control of adrenal medulla cells that secrete adrenaline and noradrenaline35,36. Simulations of class 1 and 2 circuits responded to an input pulse on the fast timescale of minutes to hours (SI Fig. S2). The response was elicited for as long as the input pulse was present, indicating that they are both input-output devices that convert a neuronal input into a secretion rate of a hormone. These forms of regulation enable immediate hormonal response to a specific signal. In this respect it is not surprising that included in these two classes are the fight or flight hormones adrenaline and noradrenaline, oxytocin, and melatonin, which regulates the circadian rhythm in response to darkness (for additional examples and references see Supplementary Table S2).

Class 3 circuits enable robust homeostasis of metabolite levels

In class 3 circuits, endocrine cells secrete a hormone in response to a metabolic signal (Fig. 1c). The hormone acts to restore the metabolite to a homeostatic level. The metabolite also regulates endocrine cell growth (hyperplasia or hypertrophy) on the timescale of weeks to months. Examples are beta-cells which secrete insulin under control of blood glucose. Glucose is a beta-cell growth signal, acting primarily via hypertrophy in humans after childhood37,38 and via hyperplasia in rodents39. Another example is the parathyroid chief cells, which secrete parathyroid hormone (PTH) under control of blood-free calcium ions. Free calcium ions also act to regulate parathyroid cell proliferation and survival40.

Whereas class 1 and 2 circuits provide differing output levels according to inputs, simulations of class 3 circuits achieve a fixed target concentration of a metabolite, as seen in Fig. 1c. Class 3 circuits control metabolites with well-defined set points such as 5 mM glucose or 1 mM calcium ions41,42. Deviations from these concentrations can be lethal43,44.

Class 3 circuits are able to act robustly, in the sense that the specific homeostatic concentration of the metabolite is achieved in the face of wide variation in the physiological parameters of the circuit. This is seen in Fig. 2a−c, which shows the effect of changing each of the parameters in the circuit. For example, various levels of hormone-sensitivity parameter s (such as insulin resistance) lead to the same steady-state metabolite level.

Fig. 2: Step changes in parameters cause transient responses and a return to baseline over weeks.
figure 2

Simulations began at steady state with nominal parameter values, and then at \(t=0\) one parameter was changed as indicated, causing a rapid change followed by recovery to the new steady state level, which in all cases is identical to the pre-change level. Blue horizontal line is no change from the nominal parameter (see schematic in panel a). a−c Class 3 circuit metabolite and hormone dynamics upon changes in: a hormone sensitivity, \(s\). b hormone specific production rate (maximal secretion rate per unit biomass of secreting cell type), \(q\). c rate constant for hormone removal, \(r\). d−f Class 4 hormone or metabolite dynamics upon changes in: d rate constant for hormone removal, \(r\). e metabolite specific production rate (maximal secretion rate per unit biomass of secreting cell type), \(q\). f rate constant for metabolite removal, \(a\). g-l Class 5 circuit dynamics of the three hormones: g hormone 1 dynamics with different values of hormone-specific production rate, \({q}_{1}\). h hormone 1 dynamics with different values of rate constant for hormone removal, \({r}_{1}\). i hormone 2 dynamics with different values of hormone-specific production rate, \({q}_{2}\). j hormone 2 dynamics with different values of rate constant for hormone removal, \({r}_{2}\). k hormone 3 dynamics with different values of hormone-specific production rate, \({q}_{3}\). l hormone 3 dynamics with different values of rate constant for hormone removal, \({r}_{3}\). Parameters not varied in this figure are as in Fig. 1. See Box 1 for model equations.

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Robust homeostasis is achieved by a combination of fast responses and slow gland mass changes13. The cells respond within minutes to changes in metabolite, such as postprandial insulin secretion. They also respond within weeks by changing their gland functional mass to compensate for physiological changes (Fig. 2a−c). As long as the metabolite is away from its set point the cell mass grows or shrinks until the set point is achieved. An example is the hypertrophy of beta-cells seen in individuals with insulin resistance45. The change in gland mass can compensate precisely for changes in physiological parameters, as long as the gland is not limited by a maximal size known as a carrying capacity23,46,47. When the gland mass approaches carrying capacity, deviations from homeostasis can occur, such as prediabetes and type-2 diabetes in response to elevated insulin resistance25.

Mathematically, the reason for robustness is the integral feedback loop that controls the mass of the secreting cells13,25 — see the model in Box 1. The equation for the change in gland mass \(G\) regulated by the metabolite \(m\) is \({dG}/{dt}=G({cm}-d)\), where \(c\) and \(d\) are production and removal parameters of gland mass. Since \(G\) is made of cells, it multiplies both production and removal terms (all cells come from cells). This equation ensures that as long as there are secreting cells \((G\,\ne\, 0)\), the only possible steady-state of the metabolite is \({m}_{{st}}=d/c.\) This steady-state solution for the metabolite \({m}_{{st}}\) does not depend on any of the other model parameters, including the metabolite, input \(u\), its sensitivity \(s\), the hormone specific production rate \(q\) or the hormone removal rate \(r\) (Fig. 2a−c).

Class 3 circuits include, in addition to glucose and calcium control, hormones that regulate electrolytes in the blood that must be kept within a tight range such as potassium and sodium. For more examples and references see Supplementary Table S2.

Class 3 circuits show an undershoot of metabolites after a prolonged (weeks or more) period of high input is stopped (Fig. 1c). This undershoot is an intrinsic behavior of this circuit in response to such a change, because the gland functional mass has grown over weeks during the high input period, and it takes it weeks then to adjust back to baseline. In this transient period, hormone levels are higher than in steady state, causing an undershoot of the metabolite. In the body, however, such undershoots would be eliminated by counterregulatory hormones. For example, an undershoot in glucose would cause glucagon (as well as cortisol and other hormones) to increase endogenous glucose production and ensure baseline glucose. Similarly, an overshoot is predicted upon a change between prolonged low input and high input. Such a glucose overshoot may occur in refeeding syndrome after prolonged fast48.

Class 4 circuits enable allostatic control with adjustable set points

Class 4 circuits (Fig. 1d) describe secretory cells whose input signal is a hormone, rather than a metabolite as in class 3. These cells secrete a metabolite under control of the input hormone. The input hormone is also a growth factor for the cells on the slow timescale (weeks-months). The input hormone is itself secreted by another cell type. For example, stomach parietal cells secrete acid under control of the input hormone gastrin, which is secreted by other cells in the digestive tract called G-cells that sense neuronal and nutritional signals associated with meals49. Gastrin is also a growth factor for the gastric parietal cells50.

Simulations show that this circuit can provide allostasis—the output metabolite has a steady-state set point that can be tuned to physiological needs (Fig. 1d). This tuning is achieved by changing the secretion rate of the input hormone and other parameters.

Class 4 circuits lock the input hormone to a homeostatic value on the slow timescale. In a certain sense, the class 4 circuit is the “inverse” of class 3. In class 3, a metabolite achieves homeostasis by allostasis of a hormone. In class 4, a metabolite achieves allostasis by homeostasis of a hormone.

The difference in behavior between class 3 and 4 is due to the position of the hormone in the circuit (see also equations in Box 1). The input signal in class 3 is a metabolite, and in class 4 it is a hormone. When a signal controls the cell growth rate, it participates in an integral feedback loop that locks the input signal to a fixed point on the scale of weeks. In class 3 circuits, metabolites are thus locked to a constant value (e.g., 5 mM in the case of glucose41). In class 4 circuits, the input hormone is locked, but the output, such as stomach acid, depends on the gland mass, which can adjust to varying parameters on the slow timescale. This steady-state solutions do not depend on any of the other model parameters, including hormone removal rate \(r\), the metabolite specific production rate \(q\), or the metabolite removal rate \(a\) (Fig. 2d−f).

We found that enteroendocrine hormones, such as gastrin and secretin, often have a class 4 circuit design. This makes sense, as their corresponding metabolites have adjustable set points according to different digestive needs49. For additional examples see Supplementary Table S2.

Class 5 circuits have several functional advantages

Class 5 circuits (Fig. 1e) are the most complex. They involve three cell types, a top hypothalamic neuronal cells and two downstream glands—a pituitary cell type and an effector gland (effector glands can come in pairs as in the right and left adrenals, which we consider as a single gland for our purposes). The pituitary and effector glands can change their functional mass according to the upstream and downstream hormones that act as growth factors. The mass changes are on the slow timescale of weeks-months. This circuit resembles two instances of a class 4 circuit stacked on top of each other in series.

The size of the cell type structures shows a hierarchy where the top neuronal cell type is the smallest in cell number, the middle (pituitary) gland cell type is of intermediate cell number and the effector gland is the largest. This follows the observation that gland mass (cell number) is proportional to its target tissue mass51.

Class 5 circuits are found in the hypothalamic-pituitary axes. These axes control major functions in vertebrates. The hypothalamic-pituitary-adrenal (HPA) axis controls stress response via the hormone cortisol20. The thyroid axis controls metabolic rate via thyroid hormones52. Similarly, the sex hormone and growth hormone pathways share a class 5 design.

There are minor variations in design between these axes, which can be considered as several subtypes of the class 5 circuit. In the thyroid axis, the growth factor for the pituitary is not its upstream hormone but instead the effector hormone (see Supplementary Table S2 for details and references)23.

Simulations of the generic class 5 circuit showed complex dynamical properties. The end product of the circuit (the effector hormone 3 in Fig. 1e) shows allostasis and overshoots. The overshoot speeds up the response to changing levels of input hypothalamic signal. The upstream hormones 1 and 2 both show homeostasis. The positions of hormone 1 and hormone 2 in circuit 5 are analogous to the position of the hormone in circuit 4 and the metabolite in circuit 3, and as the simulations showed, their secretion rate is maintained under homeostatic regulation on the timescale of weeks, though it can vary on the timescale of minutes to hours.

Class 5 circuit also shows robustness properties with respect to changes in its parameters (Fig. 2g−l). The steady state levels of \({h}_{1}\) and \({h}_{2}\) hormones averaged over weeks (the slow timescale steady state) are independent of most of the model parameters. On the short timescale, they often show pulsatile ultradian and circadian dynamics. The effector hormone \({h}_{3}\) depends linearly on the input \(u\), as appropriate for axes that transduce a brain input to a proportional physiological output13,22.

Due to the complexity of class 5 circuits and centrality in hormone regulation, we next consider their functions in more detail in the remainder of this study.

The pituitary serves as an endocrine amplifier

Next, we explore the structure-function relationship of class 5 circuits. One question is what advantage this design, with a pituitary gland in the middle, might have compared to simpler circuits that lack a middle gland. The anterior pituitary secretes certain effector hormones (e.g., POMC products like beta-endorphins53) and generates pulsatile dynamics and band-pass filtering on the fast timescale11. In addition to these, we consider here possible advantages on the timescale of weeks.

The hormones that are secreted from the pituitary in class 5 circuits, such as ACTH, TSH, LH and FSH, do not have a major physiological role except to serve the next step in the circuit, inducing the effector gland to grow and to secrete the effector hormone (thyroid hormones in the case of TSH, cortisol in the case of ACTH and sex hormones in the case of LH and FSH)1,11. This can be seen from the expression pattern of the receptors for these hormones in the human body54,55,56,57,58,59. In the next few sections, we examine the role of the pituitary and offer several functional advantages that are absent from designs that lack a middle gland.

We propose that one function of the pituitary is to act as an amplifier of the hypothalamic hormones. We showed in previous work that a single endocrine cell serves about 2000 target cells on average, regardless of the hormone system in question51. In the Hypothalamic-Pituitary (HP) class 5 circuits, a small brain region, the hypothalamus, secretes hormones. If there were no pituitary, the hypothalamus would need to secrete enough hormones for the effector gland, such as the adrenal, thyroid and female and male gonads. These glands in turn serve nearly the entire body, \(5\times 1{0}^{12}\) cells60, and thus have about \(1{0}^{9}\) cells51. Without a pituitary, the hypothalamic regions that secrete each hormone would thus need to have a mass of about \(1{0}^{6}\) cells, which is 2 orders of magnitude larger than observed for the neuronal cells secreting CRH, TRH and GnRH (we could not find estimates for the number of GHRH secreting neurons)51,61,62. It may be implausible to host such a large number of cells in the hypothalamus. The pituitary, which lies external to the skull, can more easily accommodate a large number of endocrine cells. It may thus have an amplification role, allowing \(1{0}^{4}\) hypothalamic cells to produce enough hormone for the \(1{0}^{7}\) pituitary cells11,51,63, which then provides enough hormones for the \(1{0}^{9}\)-\(1{0}^{10}\) cells of the effector gland. For scale, \(1{0}^{9}\) cells typically weigh about 1 gram64 (Bionumber ID 111609).

The pituitary can compensate for toxic adenomas up to a threshold

Beyond this amplification role, we asked whether the pituitary also has dynamical functions. Another way to pose this question is to ask why the pituitary is regulated to change its functional mass in class 5 circuits, rather than having a constant mass.

We begin by noting that changes in the pituitary mass can buffer physiological and pathological variations. An example has been described previously in the context of thyroid disease23. We add to this previous work an exploration of how class 5 circuits might compensate for tumors that hypersecrete hormones.

As a concrete example we consider the HPA axis, in which tumors arise quite frequently, in up to a few percent of the population (often called incidentalomas)65. They occur both in the pituitary cells that secrete ACTH and in the adrenal cells that secrete cortisol66,67. When the tumors occur, the corresponding non-tumorous gland mass shrinks68,69. The tumors often have no physiological consequence — the levels of the target hormone cortisol are normal70,71. When tumors that secrete hormones exceed a threshold size, they dysregulate the hormone levels, causing overt hypercortisolism called Cushing’s syndrome. The tumors thus transition from a subclinical to a clinical disease.

We asked what sets the threshold between subclinical and clinical disease. We also asked whether there are qualitative differences in the dynamics between tumors in the pituitary and tumors in the adrenal that have the same net effect of increasing cortisol levels.

To address this, we used the mathematical model for the HPA axis from Karin et al.22, which represents a special case of the class 5 equations in Box 1 (Methods). We updated the equations to include the growth of the tumor and its effect on cortisol production. We modeled the growth of the tumor C with an additional equation \({dC}/{dt}={rC}(1-C/K)\) describing logistic growth at rate \(r\) with carrying capacity \(K\). We make the simplifying assumption that the carrying capacity depends only on constant features like anatomy and not on dynamic variables like cortisol. We modeled cortisol production \(\gamma\) per unit tumor mass by adding a term \(\gamma C\) to the equation describing cortisol dynamics \(d{h}_{3}/{dt}\). We analytically solved the updated HPA class 5 mathematical model at steady state (Methods).

We begin with an adrenal tumor that secretes cortisol (Fig. 3a). We assumed, as commonly observed, that the tumor secretion rate is not regulated by ACTH72. We find that, as long as the normalized tumor secretion rate \(\beta\) is below a critical threshold, the non-tumorous adrenal mass shrinks to precisely compensate for the extra hormone secreted by the tumor (Fig. 3a−d).

Fig. 3: Transition between subclinical and clinical Cushing’s syndrome cases for pituitary and adrenal tumors.
figure 3

a Schematic showing how an adrenal cortisol-secreting tumor causes the native (non-neoplastic) adrenal cortex cortisol-secreting mass to shrink. Clinical disease appears in cases in which the adrenal cortex cortisol-secreting mass shrinks to zero. b CRH, c ACTH and d cortisol as a function of adrenal tumor cortisol secretion rate \(\beta\). e Schematic showing how a pituitary ACTH-secreting tumor causes the native pituitary corticotroph mass to shrink. Clinical disease appears in cases in which the pituitary ACTH-secreting mass shrinks to zero. f CRH, g ACTH and h cortisol as a function of pituitary tumor ACTH secretion rate \(\alpha\). We simulated using \({q}_{1}={r}_{1}=288\,{da}{y}^{-1}\), \({q}_{2}={r}_{2}=48\,{da}{y}^{-1}\), \({q}_{3}={r}_{3}=16\,{da}{y}^{-1},{b}_{P}={a}_{P}={b}_{A}={a}_{A}=\frac{1}{15}\,{da}{y}^{-1},\) \(r=1\,{da}{y}^{-1}\), \({K}_{{gr}}=4\), \(n=3\), \(K=1\),\(\,u=1\), and \(\zeta=0.1\). See Methods for model Eq. 7−12 for the simulations in (b−d) and Eq. 31−36 for the simulations in (f-h). Panels (a) and (e) were created using Procreate software.

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However, at a critical secretion rate, the native (non-tumorous) adrenal cortex functional mass shrinks to zero at steady state. This critical rate occurs when the tumor produces exactly the same amount of cortisol as the native pre-tumor adrenal glands. Thereafter, as the tumor secretion grows, cortisol levels exceed normal levels (Fig. 3a−d) and clinical symptoms of hypercortisolism occur, called Cushing’s syndrome. Since tumor growth typically evolves over years, whereas glands adjust over months, we assume quasi-steady state and simulate the circuit with a constant value of total cortisol secretion \(\beta\). This represents a very slow change in \(\beta\) or a non-temporal distinction between individuals with subclinical and clinical tumors.

We compared this adrenal tumor to a different form of Cushing’s pathology in which a pituitary tumor secretes ACTH. To address this, we used the mathematical model from Karin et al.22 and updated the equations to include the growth of the tumor and its effect on ACTH production. We modeled the growth of the tumor M with an additional equation \({dM}/{dt}={rM}(1-M/K)\) describing logistic growth at rate \(r\) with carrying capacity \(K\). We again make the simplifying assumption that the carrying capacity depends only on constant features like anatomy and not on dynamic variables like cortisol. We modeled ACTH production \(\eta\) per unit tumor mass that is suppressed by cortisol by adding a term to the equation describing ACTH dynamics \(d{h}_{2}/{dt}\). We find that at a secretion rate \(\alpha\) below a critical value, the native (non-tumor) pituitary corticotroph mass shrinks to compensate and maintain a normal level of ACTH and cortisol due to the negative feedback of cortisol secreted from the adrenal, as known from the literature1,73 (Fig. 3e−h). However, beyond a critical tumor secretion rate, the non-tumor pituitary corticotroph functional mass shrinks to zero at steady state. The critical secretion rate occurs when the tumor secretes the same amount of ACTH as the pre-tumor pituitary corticotrophs. Thereafter, higher tumor secretion causes elevated levels of cortisol, resulting in Cushing’s disease (Fig. 3e−h). As above, we assume a quasi-steady state of very slow tumor growth compared to gland mass changes and hence constant \(\alpha\).

In both ACTH- and cortisol-secreting tumors, the system undergoes a transition which, in the language of dynamical systems, is a transcritical bifurcation74. Beyond the transition, the compensating gland functional mass (the non-tumor part of the gland) drops to zero. We conclude that changes in gland mass can compensate for toxic adenomas until they reach a critical mass.

The pituitary can speed responses on the scale of weeks compared to simpler circuits

We also asked whether the pituitary can provide dynamical benefits to class 5 circuits as compared to simpler class 1-4 circuits in non-pathological situations. For this purpose we studied the HPA model, using the equations from Karin et al.22 from above (Methods). We tested this model with a prolonged stress input that rises and remains high for months and then falls, to explore the onset and recovery from such prolonged stress. The reason we chose to analyze the dynamics of the HPA axis during stress is because stress and high cortisol secretion are expected to occur during health (stressful work periods, emigration, lack of proper sleep due to childbearing, etc.) and therefore studying the system before, during and after such prolonged stress periods can reveal dynamical properties of its function.

We compared the class 5 HPA circuit with two hypothetical simpler designs for cortisol control. One has a pituitary but the pituitary cannot change its mass. We thus set \({dP}/{dt}=0\) and \(P={P}_{0}\). The second alternative circuit is a class 4 circuit without a pituitary. To model this we dropped the \(d{h}_{2}/{dt}\) and \({dP}/{dt}\) equations and replaced \({h}_{2}\) with \({h}_{1}\). Here the adrenal is directly activated by a hormone from the hypothalamus. To allow a ‘mathematically controlled comparison’13,75,76 we set all hormone half lives and specific production rates to be equal between the circuits.

We simulated these three circuits and their responses to a prolonged (500-day) stress input (Fig. 4, Methods). In the class 5 HPA circuit (Fig. 4a, d, e), the onset of stress causes a rapid response on the scale of hours, and then a slower increase on the scale of weeks as gland masses change22. Similarly, at the end of the stress input pulse, there is a rapid reduction on the scale of hours, followed by a slower adjustment due to gland mass changes on the scale of weeks.

Fig. 4: Comparison of alternative HPA designs indicates that the natural design overshoots and has the fastest rise time.
figure 4

a Natural class 5 design of the HPA axis in which the pituitary functional mass is regulated, b alternative design with a pituitary that has a constant functional mass (c) alternative design without a pituitary (no middle gland). d Dynamics of cortisol in response to a prolonged stress input which rises to 3 times the normal input and drops back 500 days later. e Zoom-in on the rise phase shows that class 5 circuit reaches 90% (dashed black line) of steady-state response fastest. We simulated with parameters \({q}_{1}={r}_{1}=288\,{da}{y}^{-1}\), \({q}_{2}={r}_{2}=48\,{da}{y}^{-1}\), \({q}_{3}={r}_{3}=16\,{da}{y}^{-1},{b}_{P}={a}_{P}={b}_{A}={a}_{A}=\frac{1}{15}\,{da}{y}^{-1},\) \({K}_{{gr}}=4\), \(n=3\), \(u=1\). See Methods for the model equations used in the simulations.

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The alternative class 5 circuit with constant-mass pituitary (Fig. 4b, d, e) has a 35% slower rise time, defined as the time to first reach 90% of the hormone steady state value. The alternative class 4 circuit without a pituitary (Fig. 4c, d, e) has the slowest rise time (75% slower). The same effect is found upon recovery from the stress pulse — the class 5 circuit reaches within 10% of the baseline faster than the other circuits. The class 5 circuit, unlike the alternative ones, also shows an overshoot due to the adjustment of pituitary mass absent from the other two simpler circuits. We conclude that the pituitary with changing mass can provide a speedup on the scale of weeks when stress conditions change.

Discussion

We analyzed literature data on 43 human hormone systems which make up 30 hormone circuits. We found that hormone circuits can be organized into 5 classes according to recurring regulation circuitry motifs and modeled them mathematically. The design of each class of circuits serves a specific function such as acute quantitative response, endocrine amplification, strict homeostasis or adjustable set points, as we demonstrate using simulations and analytical solutions. Regulation has two timescales, a fast secretion timescale of minutes to hours, and a slow adjustment of gland functional mass on the scale of weeks, which allows robustness to physiological variations and the possibility of different set points.

The five classes of circuits defined here have characteristic functional roles. Class 1 and 2 circuits can serve as fast input-output devices. In class 1, specialized neurons directly produce hormones, whereas in class 2, neurons activate hormone-secreting glands. We speculate that class 2 may allow for a more nuanced or quantitative response compared to class 1, which may primarily provide all-or-none output. Class 3 circuits can lock a metabolite to a tight range around a specific concentration. An example is control of glucose tightly around 5 mM and control of blood-free calcium ions around 1mM41,42. Class 4 circuits lock the input hormone level and can offer allostatic control of their output metabolite. Examples are intestinal hormones secretin and gastrin that control bicarbonate secretion and stomach acid secretion, respectively50,77,78. Finally, class 5 circuits are the most complex. They can amplify hormones to levels that serve the entire body, speed responses and compensate for small toxic tumors and for variations in physiological parameters such as specific hormone production rates.

The mass of endocrine glands can expand by hypertrophy or hyperplasia, depending on the gland and on age23,25. It can also grow, in principle, by differentiation of progenitor cells79. One basic function of such mass growth occurs when high levels of hormones are needed for long times. The circuits of classes 3-5 can sense this need and signal the endocrine cells to grow in total mass and thus in their hormone secretion potential. A well-known example is endemic goiter, in which the thyroid expands when iodine, essential for thyroid hormone production, is very low80. The thyroid mass can grow by a factor of ten or more. The thyroid also grows in pregnancy to meet the needs of the fetus81. Similarly, the adrenal cortex grows under chronic stress or depression82,83.

Another consequence of the gland mass regulation by these circuits is that they offer a solution to the problem of organ size control. Since cells expand exponentially, their growth and removal rates need to be precisely matched to avoid excess mass growth or shrinkage47. The circuits ensure that the gland mass production and removal rates balance precisely when the signal reaches a functional level25. Thus, the same circuits solve two problems: expansion of gland mass when more hormone is needed, and organ size control.

One question in class 5 circuits is the purpose of a middle gland—the pituitary in hypothalamic-pituitary (HP) axes. Why would an alternative design without a middle gland, like class 4 circuits, not be chosen by natural selection instead? We propose several answers. First, the middle gland acts as an amplifier. In the hypothalamic-pituitary (HP) axes, a small brain region in the hypothalamus secretes a hormone, and the effector gland needs to serve the entire body, and is thus on the order of \(1{0}^{9}\) cells as found in the mass law of51 where each endocrine cell serves about 2000 target cells. In order to supply such a large effector gland, it is useful to have a middle gland (eg \(1{0}^{7}-1{0}^{8}\) for each secretory cell type) so that the hypothalamic region can be small. A tiny amount of hypothalamic hormone thus regulates a sub-pea-sized pituitary, which regulates a large effector gland. In the HP axes, there is about a thousand-fold ratio between the top hypothalamic cell type and the middle gland, and a similar ratio between the middle and effector gland.

A second functional advantage of the middle gland is speedup of responses to inputs that change on the scale of many weeks. Alternative designs that lack a middle gland, or with a middle gland that can’t change its mass, have slower response on the scale of weeks. The middle gland in class 5 circuits achieves speedup by causing a mild overshoot or undershoot of several weeks in the hormone dynamics. This overshoot can cause mild dysregulation when entering or exiting prolonged periods of high excitation, as in prolonged stress22, postpartum26 or after addiction27.

Finally, the middle gland can participate in compensation of physiological or pathological perturbations. We demonstrate this by analyzing the effects of hormone-secreting tumors in the HPA axis that cause Cushing’s syndrome. A cortisol-secreting tumor in the adrenal can be fully compensated by reduction of the rest of the adrenal cortisol-secreting mass (caused by a reduction in the adrenal growth factor ACTH due to feedback regulation), provided that the tumor secretion rate is below a threshold value. This avoids clinical consequences of mildly cortisol-secreting tumors, which may account for 15% of tumors (incidentalomas) found in the adrenal67. These mildly secreting tumors are thus quite common subclinical events given that incidentalomas are found in about 4% of individuals undergoing high-resolution abdominal imaging84. When the tumor secretion rate crosses a threshold value, however, the effective adrenal mass shrinks to zero, and compensation cannot continue. Thereafter, high cortisol with clinical symptoms can result, a condition known as Cushing’s syndrome. The same phenomenon can also be found for ACTH-secreting tumors in the pituitary, which can be compensated by reduction of pituitary functional mass until the native pituitary shrinks to zero and clinical signs of hypercortisolism occur.

The present study reflects a generalist understanding of endocrine systems by focusing on interactions between cells (e.g. TSH control of thyroid cells), rather than on intracellular circuits. The latter can be quite complex, as in the thyroid synthesis pathway, which involves multiple import, export and enzymatic steps. Thus, when we state that some hormone systems are regulated in complex cascades whereas others are not, we would like to emphasize that we do not mean that the intracellular circuitry is not complex.

It would be interesting to compare these modelling predictions against data from humans and other organisms. Hormone regulation circuitry tends to be conserved in mammals and vertebrates, with the main differences being switches between the dominant chemical form of the hormone (e.g. cortisol in humans, corticosterone in mice) and sometimes in its biological functions on target tissues. It is thus plausible that the same five circuit classes will occur across vertebrates despite variation in biochemistry. The question of the role of gland-mass changes in other species also requires further research. Recent advances signal the availability of large-scale data in other species in the near future, such as a study on the hormone network of a primate, the mouse lemur85.

It would be instructive to ask about convergent evolution of these circuits by comparing them to invertebrates and plants. For example, the corpora allata in insects serves a similar role as the anterior pituitary in mammals and other vertebrates as an “intermediate gland”86. This raises the possibility that it plays a comparable role in amplifying neuroendocrine signals and speeding up the response time.

The present hormone circuit classes have implications for understanding disease states. Cell types in class 3 and 5 circuits are prone to CD8 T-cell-based autoimmune diseases such as type 1 diabetes (T cells attack beta-cells) and Hashimoto’s thyroiditis (T cells attack thyroid cells). A recent theory connects the circuit structure to the autoimmune attack by hypothesizing that physiological removal of mutant cells that hypersense the growth signal can prevent toxic adenomas that hyper-secrete the hormone87. Furthermore, these cell types can have a carrying capacity that limits their ability to compensate for physiological changes. When the carrying capacity is approached, subclinical disease can occur, followed by clinical disease. For example, carrying capacity in beta-cells (class 3) can lead to prediabetes when insulin resistance rises46. Similarly, carrying capacity of pituitary thyrotropes can lead to subclinical Hashimoto’s (high TSH but normal thyroid hormones), since TSH can compensate for immune killing until the carrying capacity is approached23.

We grouped all interactions in each hormone system as defined in textbooks (e.g., thyroid system, HPA axis and so on). We call each such set of interactions a hormone circuit. The circuits represent baseline biological consensus under normal physiological conditions. Some hormones affect cells outside of their circuit as defined here. For example, cortisol affects the thyroid axis88,89. This forms a combined, more complex circuit that connects the HPA circuit and the HP-thyroid circuit. Similarly, the HPA axis impacts the HP-growth and HP-gonad axes5,90,91. Future work can seek motifs on this higher level of organization, which have recently been called hyper-motifs92.

Future research can refine and expand the present framework. It would be valuable to incorporate regulatory insights at the cellular and molecular level, such as circuitry specialized for steroid versus peptide hormones. It would be interesting to address individual physiological differences, and to use comparative biology to identify additional regulatory classes or nuances. One question is whether similar circuit motifs appear in plants and invertebrates, which may indicate convergent evolution of circuit motifs similar to that seen in gene regulatory motifs93.

Limitations

The exclusion of certain hormone systems due to insufficient data and reliance on existing literature may affect the comprehensiveness and accuracy of the categorization. This study focused on timescales of weeks; additional categorization can be gained by considering shorter timescales such as ultradian and circadian timescales, and by including intracellular regulatory circuitry.

We excluded 20 hormone systems due to lack of data, the main missing information being the lack of knowledge regarding the growth factors of the secreting cell type. It is thus not possible to know whether these hormone systems will fall into classes 1 and 2 or classes 3 and 4, but they are likely to fall within the present classification.

In summary, endocrine regulatory mechanisms fall into 5 classes of recurring circuit motifs with specific dynamical functions. These circuits cause gland masses to grow or shrink to compensate for physiological and pathological changes. It would be interesting to explore whether other design principles can be found to deepen our understanding of systems endocrinology.

Methods

Hormone circuit inference

We searched the literature for all known human hormone systems, primarily using an endocrinology textbook1 and a medical database4, and found 63 hormone systems. Using the literature, we focused on each hormone system to understand its regulation circuitry and gland mass control. For each of the secreting cell types, we sought the major growth factor signals, and the major signals for hormone secretion. We excluded 20 hormone systems due to lack of data, the main missing information being the lack of knowledge regarding the growth factors of the secreting cell type. (excluded hormone systems are: adiponectin, alpha-melanocyte stimulating hormone (alpha-MSH), amylin, angiotensin, apelin, c-type natriuretic peptide (CNP), calcitonin-gene related peptide, galanin, glucagon-like peptide 1 (GLP-1), glucagon-like peptide 2 (GLP-2), leptin, motilin, osteocalcin, parathyroid-hormone related protein (PTHrP), pancreatic polypeptide (PP), peptide histidine-isoleucine (PHI), peptide YY (PYY), pituitary adenylate cyclase activating polypeptide (PACAP), renin, vasoactive intestinal polypeptide (VIP)).

Mathematical modeling

We used the principles in ref. 47 to model the different systems: We used ordinary differential equations with production and first-order removal equations for each hormone and metabolite. We modeled positive and negative regulatory effects of a signal on these rates by multiplication or division respectively. We used equations for cell mass by having cell production and removal be both proportional to mass (all biomass comes from biomass), with a growth rate regulated by signals. Simulations used Python 3.9.12. We modelled input signals in two ways. The first is piecewise constant input signals that rise and fall in a step-like manner with pulses on the order of several days to hundreds of days (Fig. 1). The second approach is to use pulse trains of about 3-minute, step-like input pulses that occur stochastically with a frequency that is piecewise constant over intervals of days or hundreds of days (SI Fig. S1).

Steady state equations for class 5 circuit with tumor growth

Adrenal adenoma

We modeled the tumor using the Karin et al. model for the HPA axis22 and updated the class 5 equations from Box 1 to include the growth of the tumor and its effects on cortisol secretion, yielding the following equations:

$$\frac{d{h}_{1}}{{dt}}={q}_{1}\frac{u}{{h}_{3}(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}-{r}_{1}{h}_{1}$$
(1)
$$\frac{d{h}_{2}}{{dt}}={q}_{2}P\frac{{h}_{1}}{(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}-{r}_{2}{h}_{2}$$
(2)
$$\frac{d{h}_{3}}{{dt}}={q}_{3}A{h}_{2}-{r}_{3}{h}_{3}+\gamma C$$
(3)
$$\frac{{dP}}{{dt}}=P({b}_{P}{h}_{1}-{a}_{P})$$
(4)
$$\frac{{dA}}{{dt}}=A({b}_{A}{h}_{2}-{a}_{A})$$
(5)
$$\frac{{dC}}{{dt}}={rC}\left(1-\frac{C}{K}\right)$$
(6)

where \({h}_{1}\) is CRH, \({h}_{2}\) is ACTH, and \({h}_{3}\) is cortisol. The parameters \({q}_{1},{q}_{2},{q}_{3}\) represent the corresponding hormone specific production rate parameters (maximal secretion capacity per unit biomass of secreting cell type) and \({r}_{1},{r}_{2},{r}_{3}\) the corresponding rate constant for hormone removal. The gland parameters \({b}_{P},{b}_{A}\) are the cell proliferation/growth rates of the pituitary and adrenal, respectively; \({a}_{p},{a}_{A}\) are the corresponding cell removal/death rates. For the tumor, \(r\) is the growth rate and \(\gamma\) is the concentration of cortisol secreted per unit mass of tumor \(C\). \({K}_{{gr}}\) is the receptor affinity for cortisol, \(n\) is the receptor cooperativity for cortisol and \(K\) is the tumor carrying capacity. Note that the important variable is the total cortisol secretion by the tumor at its final size \(\beta=\gamma K\), where we define \(\beta\) as the rate of cortisol secretion \(\gamma\) from the tumor at its carrying capacity \(K\), which is the maximal cortisol secretion rate. One need not assume that \(\gamma\) or \(r\) is constant between individuals or over time, or that each unit mass of the tumor secretes cortisol equally.

We normalized the equations by substituting \(C\to {CK}\), \({h}_{1}\to {h}_{1}{a}_{P}/{b}_{P}\), \({h}_{2}\to {h}_{2}{a}_{A}/{b}_{A}\), \({h}_{3}\to {h}_{3}{K}_{{GR}}\), \(P\to P\frac{{a}_{A}{r}_{2}{b}_{P}}{{b}_{A}{q}_{2}{a}_{P}}\), \(A\to A\frac{{b}_{A}{r}_{3}{K}_{{GR}}}{{a}_{A}{q}_{3}}\), \(u\to u\frac{{a}_{P}{r}_{1}{K}_{{GR}}}{{b}_{P}{q}_{1}}\), \(\gamma \to \gamma \frac{1}{{r}_{3}{K}_{{GR}}}\) to obtain:

$$\left(\frac{1}{{r}_{1}}\right)\frac{d{h}_{1}}{{dt}}=\frac{u}{{h}_{3}+{{h}_{3}}^{1+n}}-{h}_{1}$$
(7)
$$\left(\frac{1}{{r}_{2}}\right)\frac{d{h}_{2}}{{dt}}=\frac{{h}_{1}P}{1+{{h}_{3}}^{n}}-{h}_{2}$$
(8)
$$\left(\frac{1}{{r}_{3}}\right)\frac{d{h}_{3}}{{dt}}={h}_{2}A+\beta C-{h}_{3}$$
(9)
$$\left(\frac{1}{{a}_{P}}\right)\frac{{dP}}{{dt}}=P({h}_{1}-1)$$
(10)
$$\left(\frac{1}{{a}_{A}}\right)\frac{{dA}}{{dt}}=A\left({h}_{2}-1\right)$$
(11)
$$\left(\frac{1}{r}\right)\frac{{dC}}{{dt}}=C(1-C)$$
(12)

At steady state, Eqs. 7 - 9 have a single solution, whereas Eqs. 10 - 12 can each be satisfied by two solutions (\(P=0\) or \({h}_{1}=1\), \(A=0\) or \({h}_{2}=1\), and \(C=0\) or \(C=1\), respectively). We are not interested in situations with \(C=0\), which lack the tumor.

Steady state solutions in normalized form with \(\beta\) below the bifurcation -- defined implicitly by \(u=\beta (1+{\beta }^{n})\) -- are:

$${h}_{1}=1$$
(13)
$${h}_{2}=1$$
(14)
$$C=1$$
(15)
$${h}_{3}=A+\beta$$
(16)
$$P=\frac{u}{A+\beta }$$
(17)
$$u={h}_{3}+{{h}_{3}}^{n+1}=(A+\beta )(1+{(A+\beta )}^{n})$$
(18)

Here the native adrenal functional mass \(A > 0\) shrinks to compensate for the tumor, and the tumor reaches its carrying capacity \(C=K\).

Steady state solutions in normalized form with \(\beta\) above the bifurcation -- defined implicitly by \(u=\beta (1+{\beta }^{n})\) -- are:

$${h}_{1}=\frac{u}{\beta+{\beta }^{1+n}}$$
(19)
$${h}_{3}=\beta$$
(20)
$$A=0$$
(21)
$$C=1$$
(22)
$$\frac{{dP}}{{dt}}={a}_{P}\left(\frac{u}{\beta+{\beta }^{1+n}}-1\right)P$$
(23)
$$\frac{d{h}_{2}}{{dt}}={r}_{2}\left(\frac{u}{(1+{\beta }^{n})(\beta+{\beta }^{1+n})}P-{h}_{2}\right)$$
(24)

Here native adrenal functional mass = 0, and the tumor reaches its carrying capacity \(C=K\).

While Equations \((19-22)\) are in steady state, we leave Equations \((23),(24)\) as differential Equations to show the time dependence on the external input \(u\) and the rate of cortisol concentration secreted from the tumor at carrying capacity \(\beta\). If external input \(u > \beta+{\beta }^{1+n}\), then \(P\to \infty\) and \({h}_{2}\to \infty\). If external input \(u < \beta+{\beta }^{1+n}\), then \(P\to 0\) and \({h}_{2}\to 0\).

Pituitary adenoma

We modeled the tumor using Karin et al. model for the HPA axis22 and updated the equations to include the growth of the tumor and its effects on cortisol secretion, to yield

the following equations:

$$\frac{d{h}_{1}}{{dt}}={q}_{1}\frac{u}{{h}_{3}(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}-{r}_{1}{h}_{1}$$
(25)
$$\frac{d{h}_{2}}{{dt}}={q}_{2}P\frac{{h}_{1}}{(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}+\frac{\eta M}{1+\zeta {h}_{3}}{-r}_{2}{h}_{2}$$
(26)
$$\frac{d{h}_{3}}{{dt}}={q}_{3}A{h}_{2}-{r}_{3}{h}_{3}$$
(27)
$$\frac{{dP}}{{dt}}=P({b}_{P}{h}_{1}-{a}_{P})$$
(28)
$$\frac{{dA}}{{dt}}=A({b}_{A}{h}_{2}-{a}_{A})$$
(29)
$$\frac{{dM}}{{dt}}={rM}(1-\frac{M}{K})$$
(30)

Here \(M\) is the tumor mass, \(\eta\) is the specific production rate of \({h}_{2}\) secreted per one unit mass of tumor \(M\) and \(\frac{1}{\zeta }\) is the half-maximal inhibitory concentration of \({h}_{3}\) on \(\eta M\). Other parameters are as above.

We normalized the equations by substituting \(M\to {MK}\), \({h}_{1}\to {h}_{1}{a}_{P}/{b}_{P}\), \({h}_{2}\to {h}_{2}{a}_{A}/{b}_{A}\), \({h}_{3}\to {h}_{3}{K}_{{GR}}\), \(P\to P\frac{{a}_{A}{r}_{2}{b}_{P}}{{b}_{A}{q}_{2}{a}_{P}}\), \(A\to A\frac{{b}_{A}{r}_{3}{K}_{{GR}}}{{a}_{A}{q}_{3}}\), \(u\to u\frac{{a}_{P}{r}_{1}{K}_{{GR}}}{{b}_{P}{q}_{1}}\), \(\zeta \to \zeta /{K}_{{GR}}\), \(\eta \to \eta \frac{{r}_{2}{a}_{A}}{{b}_{A}}\) to obtain:

$$\left(\frac{1}{{r}_{1}}\right)\frac{d{h}_{1}}{{dt}}=\frac{u}{{h}_{3}(1+{{h}_{3}}^{n})}-{h}_{1}$$
(31)
$$\left(\frac{1}{{r}_{2}}\right)\frac{d{h}_{2}}{{dt}}=\frac{{h}_{1}P}{1+{{h}_{3}}^{n}}+\frac{\alpha M}{1+\zeta {h}_{3}}-{h}_{2}$$
(32)
$$\left(\frac{1}{{r}_{3}}\right)\frac{d{h}_{3}}{{dt}}={h}_{2}A-{h}_{3}$$
(33)
$$\left(\frac{1}{{a}_{P}}\right)\frac{{dP}}{{dt}}=P({h}_{1}-1)$$
(34)
$$\left(\frac{1}{{a}_{A}}\right)\frac{{dA}}{{dt}}=A({h}_{2}-1)$$
(35)
$$\left(\frac{1}{r}\right)\frac{{dM}}{{dt}}=M(1-M)$$
(36)

where we define \(\alpha=\eta K\) as the rate of ACTH secretion from the tumor at its carrying capacity, which is the maximal ACTH secretion rate.

Steady state solutions below the bifurcation (\(\alpha < 1+\zeta A\), where \(A\) is defined implicitly by \(u=A+{A}^{n+1}\), and where native pituitary functional mass \(P > 0\) shrinks to compensate for the tumor, and the tumor reaches its carrying capacity \(M=K\)):

$${h}_{1}=1$$
(37)
$${h}_{2}=1$$
(38)
$${h}_{3}=A$$
(39)
$$M=1$$
(40)
$$P=\frac{u}{A}(1-\frac{\alpha }{1+\zeta A})$$
(41)
$$u=A+{A}^{n+1}$$
(42)

Steady state solutions above the bifurcation (\(\alpha > 1+\zeta A\), where \(A\) is defined implicitly by \(u=A+{A}^{n+1}\), native pituitary functional mass = 0, tumor in its carrying capacity \(M=K\)):

$${h}_{1}=\frac{u}{\frac{\eta M-1}{\zeta }(1+{(\frac{\eta M-1}{\zeta })}^{n})}$$
(43)
$${h}_{2}=1$$
(44)
$$M=1$$
(45)
$${h}_{3}=\frac{\alpha -1}{\zeta }$$
(46)
$$P=0$$
(47)
$$A=\frac{\alpha -1}{\zeta }$$
(48)

Model equations for comparison of alternative HPA designs

To model alternative HPA designs, we started from the following equations from Karin et al.22 for the full HPA axis:

$$\frac{d{h}_{1}}{{dt}}={q}_{1}\frac{u}{{h}_{3}(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}-{r}_{1}{h}_{1}$$
(49)
$$\frac{d{h}_{2}}{{dt}}={q}_{2}P\frac{{h}_{1}}{(1+{(\frac{{h}_{3}}{{K}_{{GR}}})}^{n})}-{r}_{2}{h}_{2}$$
(50)
$$\frac{d{h}_{3}}{{dt}}={q}_{3}A{h}_{2}-{r}_{3}{h}_{3}$$
(51)
$$\frac{{dP}}{{dt}}=P({b}_{P}{h}_{1}-{a}_{P})$$
(52)
$$\frac{{dA}}{{dt}}=A({b}_{A}{h}_{2}-{a}_{A})$$
(53)

For the alternate circuit with a constant functional mass of the pituitary, we set \({dP}/{dt}=0\) and \(P={P}_{0}\), dropping Eq. 52. For the circuit without a pituitary (equivalent to class 4 circuit design), we dropped Eqs. 50 and 52, and we replaced \({h}_{2}\) with \({h}_{1}\) in Eq. 51.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.