A mathematical study of the adrenocorticotropic hormone as a regulator of human gene expression in adrenal glands

A. Manickam; A. Benevatho Jaison; D. Lakshmi; Ram Singh; C. T. Dora Pravina

doi:10.1515/cmb-2023-0122

Open Access Published by De Gruyter Open Access March 9, 2024

A mathematical study of the adrenocorticotropic hormone as a regulator of human gene expression in adrenal glands

A. Manickam , A. Benevatho Jaison , D. Lakshmi , Ram Singh and C. T. Dora Pravina

From the journal Computational and Mathematical Biophysics

https://doi.org/10.1515/cmb-2023-0122

Abstract

In this research, we have introduced compartments for asymptomatic and symptomatic individuals, along with reduced susceptibility, as key factors defining our investigation. The study is carried out in diverse scenarios, considering them as crucial for the essential generation number of the model, set at 3.18( r 0 > 1 ). The persistent reproduction differential method was used to explore the impact of continuous adrenocorticotropic hormone (ACTH) administration on the global gene expression in primary cultures of both fetal and adult adrenal cells. The study also investigates ACTH’s genetic effects on both adult and fetal human adrenal cells. The conclusion of this study is demonstrated through relevant and correct medical applications.

Keywords: ACTH; asymptomatic; cortisol; estimation of parameter; enzymes

MSC 2010: 92B05; 92B10; 92B25

1 Introduction

The primary biomarker for adrenal steroidogenesis is adrenocorticotropic hormone (ACTH), known for evaluating the production of aldosterone, cortisol, and dehydroepiandrosterone-sulfate (DHEA-S). Besides steroidogenic enzymes, additional gene targets have been identified. Utilizing the microarray technique in a mouse adrenal tumor cell model, a series of ACTH-sensitive genes has been identified. However, there has been a lack of research on the impact of ACTH on gene expression in human adrenal cells. This study addresses that gap by using primary cultures of human adrenocortical cells as models to illustrate how ACTH alters genomes. Through extensive research on fetal and adult adrenal cells across various species, it is found that the chronic response to ACTH involves the activation of genes responsible for encoding steroidogenic enzymes, a fact well-established in contemporary adult discussions [4,19].

ACTH therapy led to the elevation of all crucial steroidogenic enzymes required for cortisol release. During the 24 h administration of ACTH to adrenal cells in this study, a significant increase in 518 genes was observed. Among these genes, the gonadotropin hormone-releasing hormone (GnRH) gene exhibited notable growth in the human fetal adrenal. Additionally, the GnRH transcript showed increased stimulation in the fetal adrenal following ACTH, suggesting its regulatory influence on the hormone, as discussed in previous debates [11,13,14].

Examination of the adult and fetal adrenal microarray data revealed 20 genes that were elevated in both cell cultures under the influence of ACTH. The comparison of various models led to the identification of a comprehensive set of ACTH-responsive genes. However, due to the extended duration of treatment (48 h) discussed earlier, the microarray techniques used were unable to detect an increase in the rapid response gene. Interestingly, ACTH was found to decrease gene expression by a factor of four. Notably, only the home domain of protein X was identified in both adult and fetal adrenal cells, and it has been previously recognized in lung tumors as a potential tumor suppressor gene. Moreover, a recently discovered home domain-only protein has been linked to the development of heart disease [5,8,10,15].

The rest of the article is organized as follows: The mathematical model with governing equations is formulated in Section 2. In Section 3, the mathematical analysis with stability analysis is provided. Results and discussion is given in Section 4, and finally conclusion is drawn in Section 5.

2 Mathematical model

In this study, we used factors, such as asymptomatic, symptomatic, and decreased insusceptibility, to establish a representation of ACTH transmission in adrenal cells. The overall populace is partitioned into the overall gene cells in ACTH cortisol secretion ( V 0 ), uncovered gene cells ( F ), asymptomatic gene cells ( D P ), gene cells contamination ( C p ), and recuperated overall gene cells ( R ), vulnerable cells already tainted ( R 0 ), and isolated gene cells ( I ). The overall number of populaces at time t is given by

S ( t ) = V 0 ( t ) + F ( t ) + D P ( t ) + C P ( t ) + R ( t ) + R 0 ( t ) + I ( t ) .

The effects of long-term ACTH treatment on global gene expression in primary stage of fetal and adult adrenal cells are investigated in this work. The microarray analysis approach was utilized to show that 48 h of ACTH therapy increased the expression of 30 adults and 84 fetal adrenal genes by more than fourfold, with 20 genes shared between the two cell cultures.

We accept that the v 1 parameter found in compartments E , I a , and C P may be passing due to gene expression in primary cultures of fetal and adult adrenal cells [3]. As a result, individual containment will cause negative effects and will require hospitalization. The required parameters and their description are listed in Table 1 [1,2,6,9,16].

(1) d v 0 d t = Γ − ( χ 1 D P V 0 + χ 2 C P V 0 ) − v V 0 ,

(2) d E d t = ( χ 1 D P V 0 + χ 2 C P V 0 ) − σ F − v 1 F ,

(3) d D P d t = ε σ F − η D P − β 1 D P − v 1 D P ,

(4) d C P d t = ( 1 − ε ) σ F − ω C P − β 2 C P + χ 1 D P R 0 + χ 2 C P R 0 + η D P − v 1 C P ,

(5) d R d t = β 1 D P + β 2 C P + Ψ I − τ R − v R ,

(6) d R 0 d t = τ R − χ 1 D P R 0 − χ 2 C P R 0 − v R 0 ,

(7) d I d t = ω C P − Ψ I − v 1 Q ,

with V 0 ( 0 ) ≥ 0 , F ( 0 ) ≥ 0 , D P ( 0 ) ≥ 0 , C P ( 0 ) ≥ 0 , R ( 0 ) ≥ 0 , R 0 ( 0 ) ≥ 0 , and I ( 0 ) ≥ 0 as the initial condition for the aforementioned Mathematical model (Table 1 and Figure 1).

Table 1

Model parameters and their description

Parameter	Description
V 0 ( t )	The overall gene cells in ACTH cortisol secretion in adult adrenal cells
F ( t )	Uncovered gene cells
D P ( t )	Asymptomatic gene cells
C P ( t )	Gene cell contamination
R ( t )	Recuperated over all gene cells
R 0 ( t )	Vulnerable cells already tainted
I ( t )	Isolated gene cells

Figure 1

Transition diagram of population dynamics.

3 Mathematical analysis

Lemma 3.1

If the initial values V 0 ( 0 ) ≥ 0 , F ( 0 ) ≥ 0 , D P ( 0 ) ≥ 0 , C P ( 0 ) ≥ 0 , R ( 0 ) ≥ 0 , R 0 ( 0 ) ≥ 0 , and I ( 0 ) ≥ 0 , the solutions of V 0 ( t ) , F ( t ) , D P ( t ) , C P ( t ) , R ( t ) , R 0 ( t ) , and Q ( t ) of system are positive for all t > 0 .

Proof

Assume that ξ ( t ) = min { V 0 ( t ) , F ( t ) , D P ( t ) , C P ( t ) , R ( t ) , R 0 ( t ) , I ( t ) } , ∀ t > 0 .

Clearly ξ ( 0 ) = 0 .

Assuming that there exists a t 1 > 0 such that

ξ ( t 1 ) = 0 and ξ ( t ) > 0 , for all t ∈ [ 0 , t 1 ) .

If ξ ( t 1 ) = V 0 ( t 1 ) , then F ( t ) ≥ 0 , D P ( t ) ≥ 0 , C P ( t ) ≥ 0 , R ( t ) ≥ 0 , and R 0 ( t ) ≥ 0 for all t ∈ [ 0 , t 1 ] .

From the equation of model [11], we can obtain

d V 0 d t ≥ − ( χ 1 D P V 0 + χ 2 C P V 0 ) − μ V 0 , t ∈ [ 0 , t 1 ] .

Thus, we have

V 0 ( t ) > V 0 ( 0 ) exp − ∫ 0 t 1 [ χ 1 D P V 0 + χ 2 C P V 0 + μ V 0 ] d t .

The result will be positive, because both the exponential functions and initial solutions S 0 ( 0 ) are non-negative. Thus, V 0 ( t ) > 0 for all t ≥ 0 .

Similarly, we can also prove that F ( t ) > 0 , D P ( t ) > 0 , C P ( t ) > 0 , R ( t ) > 0 , R 0 ( t ) > 0 , and I ( t ) > 0 .

F = 0 Γ χ 1 ν Γ χ 2 ν 0 0 0 0 0 0 , V = − α − ν 0 0 σ ε − η − β 1 − ν 0 ( 1 − ε ) σ η − ω − β 2 − ν 1

and

F V − 1 = − σ ( ( ε − 1 ) χ 2 − ε χ 1 ) υ 1 + ( β 1 ε − η − β 1 ) χ 2 − ε χ 1 ( β 2 + ω ) Γ υ ( υ 1 + σ ) ( ν 1 + η + β 1 ) ( β 2 + ω + ν 1 ) Γ ( χ 1 ( β 2 + ω + υ 1 ) + χ 2 η ) υ ( β 2 + ω + υ 1 ) ( υ 1 + η + β 1 ) Γ χ 2 υ ( β 2 + ω + υ 1 ) 0 0 0 0 0 0 .

The Eigen values of F V − 1 are as follows:

λ 1 = − σ ( ( ε − 1 ) χ 2 − ε χ 1 ) ν 1 + ( β 1 ε − η − γ 1 ) χ 2 − ε χ 1 ( β 2 + ω ) Γ ν ( ν 1 + σ ) ( ν 1 + η + β 1 ) ( β 2 + ω + ν 1 ) , λ 2 , 3 = 0 .

According to the approach presented by [17,18,20,21],

r 0 = ξ ( F V − 1 ) = − σ ( ( ε − 1 ) χ 2 − ε χ 1 ) ν 1 + ( β 1 ε − η − β 1 ) χ 2 − ε χ 1 ( β 2 + ω ) Γ ν ( ν 1 + σ ) ( ν 1 + η + β 1 ) ( β 2 + ω + ν 1 ) .

Considering specific conditions, the innate recovery rate of infected individuals of both of the categories namely asymptomatic and symptomatic is considered as γ 1 = γ 2 = γ and transfer from infected individuals is just as likely as transmission from asymptomatic infected ones as β 1 = β 2 .

It was successful in obtaining the reproduction number for the situation denoted by R 0 β , where

□ r 0 β = Γ σ χ ( ε ω + ν 1 + η + γ ) ν ( ν 1 + η + γ ) ( ν 1 + σ ) ( ν 1 + ω + γ ) .

Lemma 3.2

All solutions of the system [4, 5,7, 11,13, 14,19] are bounded for all t ∈ [ 0 , t 0 ] .

Proof

Since S ( t ) = V 0 ( t ) + F ( t ) + D P ( t ) + C P ( t ) + R ( t ) + R 0 ( t ) + I ( t ) ,

we obtain d S d t = Γ − v ( V 0 + R + R 0 ) − v 1 ( F + D P + C P + I ) .

Assume that v = v 1 , to simplify the analysis process.

Then d S d t = Γ − v N .

Thus, we have 0 ≤ sup S ( t ) ≤ Γ v .

So all solutions of the system of equations (1)–(7) are ultimately bounded for all t ∈ [ 0 , t 0 ] .□

3.1 Non-endemic equilibrium point

The non-endemic equilibrium point of the COVID-19 disease model is obtained by setting D P = 0 , F = 0 , C P = 0 and substituting it into the system of equations (1)–(7) to obtain:

(8) P 0 = ( V 0 0 , F 0 , D P 0 , C P 0 , R 0 , R 0 0 , I 0 ) = Γ v , 0 , 0 , 0 , 0 , 0 , 0 .

3.2 Stability of non-endemic equilibrium point

Theorem 3.3

The non-endemic equilibrium points of equations (1)–(7) are locally asymptotically stable whenever it exists.

Proof

Substituting P 0 from equation (8) into the Jacobian matrix for the non-endemic equilibrium point we obtain:

J ( P 0 ) = − v 0 − χ Γ v − χ Γ v 0 0 0 0 − σ − v χ Γ v χ Γ v 0 0 0 0 ε σ − η − β 1 − v 1 0 0 0 0 0 ( 1 − ε ) σ η − ω − β 2 − v 1 0 0 0 0 0 γ 1 γ 2 − τ − v 0 δ 0 0 0 0 τ − v 0 0 0 0 q 0 0 − Ψ − v 1 .

The characteristic of the polynomial is

p ( Λ ) = 1 v ( Λ + v ) p 1 ( Λ ) = 0 , p 1 ( Λ ) = c 0 Λ 6 + c 1 Λ 5 + c 2 Λ 4 + c 3 Λ 3 + c 4 Λ 2 + c 5 Λ + c 6 .

From the polynomial p ( Λ ) we obtain Λ 1 = − v and for Λ i with i = 2 , 3 , … , 7 will be negative if c j > 0 , where j = 0 , 1 , 2 , … , 6 , R 0 < 1 , c 1 c 2 > c 0 c 3 , c 1 ( c 2 c 3 + c 0 c 5 ) > c 1 2 c 4 + c 0 c 2 , and c 1 c 2 c 4 > c 0 ( c 1 c 6 + c 2 c 5 ) . Since the coefficients in the characteristic equation p 1 ( Λ ) are complex, we proceed to analyze the coefficient values numerically with β 1 = β 2 . The results of the numerical analysis obtained show that for Λ i with i = 2 , 3 , … , 7 negative. Because λ j with j = 1 , 2 , … , 7 is negative, it can be concluded that the nonendemic equilibrium point of the system (1)–(7) is locally stable, so Theorem 3.3 is proven [17].□

Theorem 3.4

The non-endemic equilibrium point P 0 is globally asymptotically stable if Γ χ 1 v < v 1 and Γ χ 2 v < v 2 .

Proof

Let P 0 = ( V 0 0 , F 0 , D P 0 , C P 0 , R 0 , R 0 0 , I 0 ) = Γ v , 0 , 0 , 0 , 0 , 0 , 0 be the non-endemic equilibrium point of equations (1)–(7).

Define the Lyapunov function

V ( t ) = V 0 − V 0 * − V 0 * ln V 0 V 0 * + F + D P + C P + R + R 0 + I .

Differentiating with respect to time yields

d V d t = ( V 0 − V 0 * ) d v 0 d t V 0 * + d F d t + d D P d t + d C P d t + d R d t + d R 0 d t + d I d t d V d t = ( V 0 − V 0 * ) ( Γ − v V 0 ) V 0 * − ( V 0 − V 0 * ) ( χ 1 D P + χ 2 C P ) + χ 1 D P V 0 + χ 2 C P V 0 − σ F − v 1 F + ε σ F − η D P − γ D P − v 1 D P + ( 1 − ε ) σ F − ω C P − γ C P + χ 1 D P R 0 + χ 2 C P R 0 + η D P − v 1 C P + γ D P + γ C P + Ψ I − τ R − v R + τ R − χ 1 D P R 0 .

The value of d V d t will be negative if Γ χ 1 v < v 1 and Γ χ 2 v < v 1 .

By following Lasalle’s extension on Lyapunov’s method, disease-free equilibrium P 0 is globally asymptotically stable.

This concludes the proof.□

3.3 Endemic equilibrium points

Theorem 3.5

An endemic equilibrium point of the system P 1 = ( S 0 * , F * , D P * , C P * , R * , R 0 * , I * ) will exist if G > 0 and H > 0 or G < 0 and H < 0 .

Proof

The endemic point of this disease is endemic in certain areas for a certain period, which releases the COVID-19 in the population. It is indicated by the presence of compartments exposed to virus transmission F * , I α * , C P * at steady state. By calculating model of equations (1)–(7) and setting the right-hand side (RHS) zero we obtained:

V 0 * = Γ χ ( D P * + C P * ) + v , I A * = σ F P v 1 + η + γ , Y S * = ( 1 − ε ) σ F * + χ D P * R 0 * ω + γ − χ R 0 * , R * = γ ( D P * + C P * ) + δ I * v + τ , R 0 * = ξ R * χ ( I A * + C P * ) + v , I * = ω C P * Ψ + v 1 .

By substituting V 0 * , D P * , C P * , R * , R 0 * , I * into equation (2.2) and setting the RHS equal to zero,

(9) A 2 F 2 + A 1 F + A 0 = 0 .

This polynomial has two roots, F = 0 and F = F * , which can be expressed as F * = G H ′ .

Because the denominator of H is always positive, the steady state F * will exist if r 0 β > 1 and ε ω > 0 . If G > 0 and H > 0 , the system of equations (1)–(7) will have an endemic equilibrium point. The system of equations (1)–(7) has a single endemic equilibrium point under these conditions.□

4 Results and discussion

In this section, various profiles are visually presented. Figure 2 illustrates a profile depicting cortisol production in human fetal cells, revealing an increase in the human fetal cell population corresponding to elevated cortisol production. Moving on to Figure 3, it displays medical outcomes related to ACTH synthesis over a specific period, while Figures 4, 5, 6, 7, and 8 present mathematical results. These include Eigenvalue and reproduction (RD) graphs for cortisol production, Eigenvalue and reproduction (RD) graphs for cortisol and DHEA-S production, as well as asymptomatic and symptomatic relationships for cortisol and DHEA-S productions after ACTH activation.

Figure 2

ACTH results on cortisol synthesis in human fetal cells terminated time.

Figure 3

ACTH synthesis for a specific time continued by qualification of medium cortisol and DHEA-S.

Figure 4

Eigen value of RD for cortisol production.

Figure 5

Reproduction RD for cortisol production.

Figure 6

Eigen value of PRD for cortisol and DHEA-S production.

Figure 7

Reproduction of PRD for cortisol and DHEA-S production.

Figure 8

Asymptomatic and symptomatic relations.

Figure 2 illustrates the identification of corresponding medical outcomes regarding ACTH’s impact on cortisol synthesis in human fetal cells at a terminated time.

Moving on to Figure 3, it demonstrates the discovery of corresponding medical results for ACTH synthesis at a specific time, followed by an elaboration on the levels of medium cortisol and DHEA-S.

4.1 ACTH vitalizes cortisol secretion in adult adrenal cells

Injecting ACTH into mature adrenal cells resulted in a significant increase in cortisol production. The cortisol level exhibited a consistent rise during a specified time frame following ACTH treatment. The gradual rise of cortisol level was observed within 6 h of ACTH injection, and by the 48th hour, cortisol levels had surged to over 30 times their usual concentration.

4.2 Microarray data of adult adrenal 48-hour treatment with ACTH

Microarray technology was employed to compare samples treated with ACTH to untreated samples, aiming to assess the overall gene alterations induced by ACTH treatment in adult adrenal cells. Adrenal cells from three distinct donors, each from a different gland, were utilized in the tests. The analysis revealed that ACTH elevated 30 genes by more than fourfold compared to normal, while one gene experienced a reduction of more than fourfold compared to normal levels. This increase highlights the effectiveness of ACTH as a steroidogenic activator, boosting the expression of steroidogenic enzymes in adrenal cells and thereby enhancing their capacity for releasing steroid hormones over time [18,20,21].

4.3 ACTH causes cortisol and DHEA-S synthesis

Similar to adult adrenal cells, fetal adrenal cells depend on ACTH for enhancing steroid production. We isolated cells from three distinct fetal adrenal sources and conducted three independent experiments to explore the impact of ACTH on these cells. The application of ACTH resulted in a gradual increase in cortisol and DHEA-S levels over time. In comparison to untreated adrenal cells, the administration of ACTH amplified DHEA-S production by eightfold and cortisol production by 300. Following 48 h of ACTH treatment, 84 out of 18,390 genes exhibited an increase of more than fourfold, while five genes experienced a reduction to one-fourth of their usual levels. Notably, ACTH influenced the expression of four genes associated with steroidogenic enzymes [7].

4.4 Common genes shared by adult and fetal adrenal cells

Upon comparing the supragenomic effects of ACTH on adult and fetal cells, we identified a series of analogous genes that exhibited enhancement in both cell types. When applying a threshold four times higher, there were 30 genes regulated by ACTH in adult adrenal cells compared to 84 genes in fetal adrenal cells. Notably, 20 genes were found to be common between them, suggesting a potential shared set of human adrenal ACTH targets. To validate the microarray findings, quantitative PCR was employed for eight genes that were most enhanced by ACTH and one gene that experienced reduction in both cell types. The results from quantitative PCR and microarray analysis confirmed that all nine genes were responsive to ADH in both adult and fetal adrenal cells [12].

Figure 4 shows that medical results provide mathematical results of Eigen value of RD for cortisol production. Figure 5 shows that medical results provide mathematical results of Reproduction RD for cortisol production. Figure 6 shows that medical results provide mathematical results of Eigen value of Persistent Reproduction Differential (PRD) for cortisol and DHEA-S production. Figure 7 shows that medical results provide mathematical results of Reproduction PRD for cortisol and DHEA-S production. Figure 8 shows that medical results provide mathematical results of symptomatic and symptomatic relations for cortisol and DHEA-S productions after ACTH activation.

5 Conclusion

We conducted a scientific study examining the genetic effects of ACTH on adult and fetal human adrenal cell transmission. Our analysis considered symptomatic and asymptomatic individuals, and those with decreased susceptibility, using medical data. The compartment-based model categorizes the population into various gene cell conditions related to ACTH cortisol secretion ( V 0 ) , exposed gene cells ( F ), asymptomatic gene cells (DP), gene cell infection (CP), recovered overall gene cells ( R ), previously infected vulnerable cells ( R 0 ) , and isolated gene cells ( I ). Upon scientific analysis, the results reveal two equilibrium points: a disease-free equilibrium point and an endemic equilibrium point. Additionally, using the generation matrix method the fundamental propagation number ( R 0 ) for mathematical outcomes is determined to be 3.180126127. As immunity increases, the infected population also increases, and reinfection has no significant impact on the number of exposed and infected asymptomatic populations. The isolation period proves effective in slowing the spread of medical outcomes. In conclusion, our findings suggest the application’s relevance and monotony in mathematical outcomes, providing valuable insights for the medical profession.

Acknowledgement

The authors would like to extend their gratitude to the National Organization of Science for using the ANSYS application at VIT Bhopal University.

Funding information: This research received no specific grant from any funding agency, commercial or nonprofit sectors.
Conflict of interest: The authors have no conflicts of interest to disclose.
Ethical approval: This research did not required ethical approval.

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Received: 2023-09-14

Revised: 2024-01-10

Accepted: 2024-01-24

Published Online: 2024-03-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

A mathematical study of the adrenocorticotropic hormone as a regulator of human gene expression in adrenal glands

Abstract

1 Introduction

2 Mathematical model

3 Mathematical analysis

Lemma 3.1

Proof

Lemma 3.2

Proof

3.1 Non-endemic equilibrium point

3.2 Stability of non-endemic equilibrium point

Theorem 3.3

Proof

Theorem 3.4

Proof

3.3 Endemic equilibrium points

Theorem 3.5

Proof

4 Results and discussion

4.1 ACTH vitalizes cortisol secretion in adult adrenal cells

4.2 Microarray data of adult adrenal 48-hour treatment with ACTH

4.3 ACTH causes cortisol and DHEA-S synthesis

4.4 Common genes shared by adult and fetal adrenal cells

5 Conclusion

Acknowledgement

References

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