Figure 1.
Regulatory network of cortisol in the HPA axis.
Stress induces the secretion of corticotrophin release hormone (CRH) in the hypothalamus that diffuses to the pituitary gland to activate aceto-corticotrophin hormone (ACTH). ACTH in turn activates cortisol (CORT) in the adrenal gland. The secreted cortisol binds to glucocorticoid (G) receptor to form a complex GR followed by the dimerization reaction of GR complex. Cortisol down regulates its own production through GR complex that binds to both CRH and ACTH and form a closed loop. This closed loop gives rise to negative feedback in the circuit.
Figure 2.
Time series and power spectrum of PTSD subjects.
(A) The time series of three individual subjects, (B) concatenated time series of the three individual time series, and (C) amplitude spectrum of the time series. The height of dominant peak in the spectrum is denoted by h, and is the difference in the frequency of the time series corresponding to the full width at half maximum (FWHM, given as exp(−1/2)* h),
is the background noise. The fundamental frequency
of the concatenated time series is 0.0417 Hz and this corresponds to the frequency where maximum peak “h” occur in the spectrum.
Figure 3.
Experimental and fitted concatenated cortisol time series from the model.
(A) Normal (B) depression, and (C) PTSD. The parameter set used to simulate each of the category is given in Table 1.
Figure 4.
Simulated free glucocorticoid receptor time series of normal, depressed, and PTSD subjects.
The free glucocorticoid receptor (G) in PTSD is much higher than in normal and depressed subjects indicating a stronger negative feedback loop.
Table 1.
Estimated kinetic parameters of normal, PTSD, and depressed subjects used in the bifurcation analysis and numerical integration.
Figure 5.
Whisker plots for stress and inhibition strength parameters of the 50 phenotypes.
(A) The reestimated stress parameter of the 50 sets indicates that for PTSD (k) is much higher than for depressed (k
) and normal (k
) subjects. (B) The fifty parameters K
obtained for PTSD are lower than these for normal (K
) and depressed subjects (K
), indicating the presence of a strong feedback loop. On the other hand, the depressed patients have scattered, and larger K
values indicating a disruptive, weak negative feedback loop in comparison to the normal subjects.
Figure 6.
Whisker plots of the peak and nadir cortisol levels of the 50 phenotypes.
(A) Peak and (B) nadir levels of the cortisol for the normal (N), PTSD (P), and depressed (D) categories, for which significant group differences are observed (p-value0.05). For depression, there is a wide range of cortisol values observed at nadir, whereas in PTSD, the range is extremely small, indicating that in depression, due to the weak negative feedback, the fluctuations are found to be enormous that resulted in a wide range of cortisol values.
Figure 7.
Sensitivity Indices (SI) for the optimal set of parameters.
Sensitivity analysis shows that the sensitivity of cortisol with respect to K for depressed subjects is much higher than for normal subjects, while for PTSD is much lower. This difference is due to the difference in the sensitivity of the HPA axis for normal, depressed, and PTSD subjects. On the other hand, hardly any difference in the sensitivity to k
is seen. No sensitivity to the Hill coefficient n
is observed, which is the homodimerization of GR receptor, indicating that the circadian oscillations are due to the other Hill coefficient n
, that determines the nonlinearity and robustness of the oscillations. The most sensitive parameter is the total glucocorticoid receptor, which plays a strong role in cortisol feedback regulation. This is in tandem with the hypothesis. The insensitive parameters V
, K
, are the enzymatic degradation of the cortisol, while K
is the autonomous/dilution rate of the CRH. This indicates that in the model cortisol enzymatic degradation is not necessary, because of the strong binding rates (k
highly sensitive) of the GR receptors and, the autonomous degradation k
(again highly sensitive), that removes the cortisol much more efficiently than the enzymatic degradation.
Figure 8.
The correlation values between the dynamic sensitivities for the all the parameters are shown with the diagonal being self correlated. The levels of correlation are differently shaded as shown on the horizonal bar on the right that ranges from highly correlated (+1) to anti-correlated (−1). The sensitivities of the parameters of interest, namely k and K
are shown to be strongly and positively correlated, whereas the sensitivity of G
, the total glucocorticoid is strongly anti-correlated with respect K
, the strength of negative feedback loop in PTSD.
Figure 9.
One and two parameter bifurcation analyses.
Bifurcation analysis is carried out for (A) normal, (B) PTSD, and (C) depressed subjects with k as the bifurcation parameter. All the parameters were estimated, and kept constant throughout the simulation except for k
and K
values that are differed in the normal, depressed and PTSD subjects. The parameters are given in Table 1. Initially, for simulating the bifurcation diagram, k
was chosen as 0.001, a stable steady state. As k
increased, bistability was only observed in depressed patients (shown as inset in (C)), and Hopf bifurcation (indicated as HB) was observed in all the three subjects. Also, Hopf bifurcation began (E-G) and ended (H-J) at different k
values in all the three subjects. (D) Two parameter bifurcation diagram indicates the presence of oscillations for a wide range of k
-K
values, and N (circle), D (dotted circle) and P (star) indicates the estimated parameters for which normal, depressed and PTSD subjects lie in the parameter plane. SS–stable steady state, USS–unstable steady state, and HB–Hopf bifurcation which is supercritical in all the subjects.
Figure 10.
Variation of period to the corresponding changes in stress and inhibition strength.
The inhibition strength K is varied from the lowest (0.8) to the highest value (1.7) in steps of 0.1 along the direction indicated by the arrow, and the bifurcation parameter is k
. Inset shows the period of cortisol dynamics that varies between 23.5 and 24.5 for a very wide range of k
values indicating the robustness of the system.
Figure 11.
Distinction of normal, depressed and PTSD phenotypes.
(A) Normal, depressed and PTSD patients are distinguished in the stress-feedback-period parameter space. All three categories can be clearly distinguished with PTSD having a strong negative feedback (low K values). (B) Normal, depressed, and PTSD patients are distinguished in the stress-feedback-cortisol (nadir) concentration space. The cortisol concentration is found to be lower for PTSD, with low K
(strong feedback) values in comparison to the normal. In (A) and (B), clearly the transition from Normal
PTSD (moving towards low K
in (A), low cortisol values in (B)), Normal
Depression (moving towards high K
values, high cortisol values), and Depression
PTSD (moving towards a low K
, low cortisol values) can also be distinguished. PTSD
depression (moving towards a high K
, high cortisol values) can also be distinguished.