Local relative sensitivity analysis for the estimated parameters.

<p>The bars show in blue the sensitivity due to the effect of the parameters on HSF13S and in red the corresponding to HHp. As expected, there are several parameters having very little influence on the measured states, therefore, the parameter values obtained from the global optimization have to be taken with caution. Most of the parameters that appear insensitive in this analysis are sensitive to other species, thus experimental data of these intermediate states would help to better estimate these values. The numbers on the abscissa corresponds to the kinetic constants given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042958#pone-0042958-t001" target="_blank">Table 1</a>.</p

Fitted and experimental time series of the phosphorylated HH complex.

<p>Simulated mono-phosphorylated (shown in continuous line) for value of 0.9557, and the experimental data (shown in red circles) is taken from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042958#pone.0042958-Kline1" target="_blank">[25]</a>. The percent conversion of the time series is calculated as , for all the dynamical variables, and Y is the simulated time series.</p

Correlation matrix.

<p>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.</p

A Detailed Modular Analysis of Heat-Shock Protein Dynamics under Acute and Chronic Stress and Its Implication in Anxiety Disorders

<div><p>Physiological and psychological stresses cause anxiety disorders such as depression and post-traumatic stress disorder (PTSD) and induce drastic changes at a molecular level in the brain. To counteract this stress, the heat-shock protein (HSP) network plays a vital role in restoring the homeostasis of the system. To study the stress-induced dynamics of heat-shock network, we analyzed three modules of the HSP90 network—namely trimerization reactions, phosphorylation–dephosphorylation reactions, and the conversion of HSP90 from an open to a closed conformation—and constructed a corresponding nonlinear differential equation model based on mass action kinetics laws. The kinetic parameters of the model were obtained through global optimization, and sensitivity analyses revealed that the most sensitive parameters are the kinase and phosphatase that drive the phosphorylation–dephosphorylation reactions. Bifurcation analysis carried out with the estimated kinetic parameters of the model with stress as bifurcation parameter revealed the occurrence of “mushroom”, a type of complex dynamics in which S-shaped and Z-shaped hysteretic bistable forms are present together. We mapped the molecular events responsible for generating the mushroom dynamics under stress and interpreted the occurrence of the S-shaped hysteresis to a normal level of stress, and the Z-shaped hysteresis to the HSP90 variations under acute and chronic stress in the fear conditioned system, and further, we hypothesized that this can be extended to stress-related disorders such as depression and PTSD in humans. Finally, we studied the effect of parameter variations on the mushroom dynamics to get insight about the role of phosphorylation–dephosphorylation parameters in HSP90 network in bringing about complex dynamics such as isolas, where the stable steady states in a bistable system are isolated and separated from each other and not connected by an unstable steady state.</p></div

Observation of different dynamical scenario for the parameter set obtained from fitting the time series data by global optimization.

<p>Loss of mushroom-like dynamics in the full ordinary differential equation (ODE) model with as the bifurcation parameter. (A) The Z-shaped bistability is lost, but ultrasensitivity is retained. A mushroom-like dynamics is possible without any memory effect at high chronic stress due to loss of Z-shaped bistability. (B) Irreversible transition in the S-shaped bistability is observed. Z-shaped bistability is lost and it is replaced by Hopf bifurcation with a very low period. The parameters used to simulate the dynamics for each of the cases are given in a separate file (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042958#pone.0042958.s006" target="_blank">Model file S2</a>).</p

Bifurcation diagram in the absence of the negative feedback loop.

<p>One-parameter bifurcation diagram is constructed in the absence of negative feedback loop by silencing the kinetic constants and and to zero. Only the S-shaped part of the mushroom is seen, and the Z-shaped part is absent. This indicates that to generate Z-shaped bistable part, a strong negative feedback loop is important, and this strong feedback loop may be responsible for the occurrence of depression and post-traumatic stress disorder (PTSD) (see also <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042958#pone-0042958-g007" target="_blank">Fig. 7</a>).</p

Whisker plots of the peak and nadir cortisol levels of the 50 phenotypes.

<p>(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.</p

Modeling Cortisol Dynamics in the Neuro-endocrine Axis Distinguishes Normal, Depression, and Post-traumatic Stress Disorder (PTSD) in Humans

<div><p>Cortisol, secreted in the adrenal cortex in response to stress, is an informative biomarker that distinguishes anxiety disorders such as major depression and post-traumatic stress disorder (PTSD) from normal subjects. Yehuda <em>et al.</em> proposed a hypothesis that, in humans, the hypersensitive hypothalamus-pituitary-adrenal (HPA) axis is responsible for the occurrence of differing levels of cortisol in anxiety disorders. Specifically, PTSD subjects have lower cortisol levels during the late subjective night in comparison to normal subjects, and this was assumed to occur due to strong negative feedback loops in the HPA axis. In the present work, to address this hypothesis, we modeled the cortisol dynamics using nonlinear ordinary differential equations and estimated the kinetic parameters of the model to fit the experimental data of three categories, namely, normal, depressed, and PTSD human subjects. We concatenated the subjects (n = 3) in each category and created a model subject (n = 1) without considering the patient-to-patient variability in each case. The parameters of the model for the three categories were simultaneously obtained through global optimization. Bifurcation analysis carried out with the optimized parameters exhibited two supercritical Hopf points and, for the choice of parameters, the oscillations were found to be circadian in nature. The fitted kinetic parameters indicate that PTSD subjects have a strong negative feedback loop and, as a result, the predicted oscillating cortisol levels are extremely low at the nadir in contrast to normal subjects, albeit within the endocrinologic range. We also simulated the phenotypes for each of the categories and, as observed in the clinical data of PTSD patients, the simulated cortisol levels are consistently low at the nadir, and correspondingly the negative feedback was found to be extremely strong. These results from the model support the hypothesis that high stress intensity and strong negative feedback loop may cause hypersensitive neuro-endocrine axis that results in hypocortisolemia in PTSD.</p> </div

Simulated and experimental time series of heat-shock factor 1 (HSF1) trimer.

<p>In the experiments <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042958#pone.0042958-Abravaya1" target="_blank">[24]</a>, the time courses of HSF levels were obtained for three different temperatures, namely, , , and , shown in red circles in (A), (B), and (C), respectively. For a very high temperature (), HSF levels reach a new steady state. Fitted time series of HSF13S levels (continuous line) are shown for three different that correspond to the stress values of (A) 0.4136, (B) 0.9859, and (C) 1.2066. The percent conversion of the time series is calculated as , for all the dynamical variables, and Y is both experimental and simulated time series.</p

Time series and power spectrum of PTSD subjects.

<p>(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)* <i>h</i>), 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.</p