Mathematical modeling of the heat-shock response in HeLa cells

Abstract

The heat-shock response is a key factor in diverse stress scenarios, ranging from hyperthermia to protein folding diseases. However, the complex dynamics of this physiological response have eluded mathematical modeling efforts. Although several computational models have attempted to characterize the heat-shock response, they were unable to model its dynamics across diverse experimental datasets. To address this limitation, we mined the literature to obtain a compendium of in vitro hyperthermia experiments investigating the heat-shock response in HeLa cells. We identified mechanisms previously discussed in the experimental literature, such as temperature-dependent transcription, translation, and heat-shock factor (HSF) oligomerization, as well as the role of heat-shock protein mRNA, and constructed an expanded mathematical model to explain the temperature-varying DNA-binding dynamics, the presence of free HSF during homeostasis and the initial phase of the heat-shock response, and heat-shock protein dynamics in the long-term heat-shock response. In addition, our model was able to consistently predict the extent of damage produced by different combinations of exposure temperatures and durations, which were validated against known cellular-response patterns. Our model was also in agreement with experiments showing that the number of HSF molecules in a HeLa cell is roughly 100 times greater than the number of stress-activated heat-shock element sites, further confirming the model's ability to reproduce experimental results not used in model calibration. Finally, a sensitivity analysis revealed that altering the homeostatic concentration of HSF can lead to large changes in the stress response without significantly impacting the homeostatic levels of other model components, making it an attractive target for intervention. Overall, this model represents a step forward in the quantitative understanding of the dynamics of the heat-shock response.

Figure 1

Network diagram of the heat-shock response model. Highlighted reactions (red or gray) are those that have explicit temperature dependences, as described by Eqs. 15, 19, and 20. (HSE, heat-shock element; HSF, heat-shock factor; HSP, heat-shock protein; MFP, misfolded protein; mRNA, heat-shock protein messenger RNA; and Prot, healthy protein.) To see this figure in color, go online.

Figure 2

Comparison of model simulations with HSF DNA-binding data. Each plot shows the heat-stress temperature at the top, experimental data (circles), our model simulation (solid line), and model simulation of Petre et al. (16) (dashed line). Each box enclosing one or more plots is denoted by a letter and represents either one experiment or experiments from the same study plotted on the same scale: (A) (23), (B) (10), (C) (22), and (D) (10). To see this figure in color, go online.

Figure 3

Comparison of model simulations with HSP/HSF dynamics data. Each plot shows the heat-stress temperature at the top, experimental data (circles), our model simulation (solid line), and the model simulation of Petre et al. (16) (dashed line). Each box enclosing multiple plots is denoted by a letter and represents experiments from the same study plotted on the same scale: (A) (22) and (B) (25). To see this figure in color, go online.

Figure 4

Comparison of model simulations with HSP transcription and translation data. Each heat-stress temperature at the top, experimental data (circles), our model simulation (solid line), and the model simulation of Petre et al. (16) (dashed line). Results from the model of Petre et al. (16) are only shown in (E) because (A)–(D) correspond to the mRNA variable, which was not included in their model. Each box enclosing one or more plots is denoted by a letter and represents either one experiment or experiments from the same study plotted on the same scale: A (19), B (21), C (24), D (21), and E (26). To see this figure in color, go online.

Figure 5

Analysis of protein misfolding as a function of temperature and duration of exposure. (A) MFP versus time for three different temperatures. (B) AUC for 800 simulations similar to those in (A) for a variety of heating duration and temperature combinations. (Blue curve) Ratio of MFP AUC at a particular temperature to that at 43°C for the same duration. (Red curve) Ratio of MFP AUC at a particular temperature and duration to that at 43°C for the corresponding cumulative equivalent minutes at 43°C (CEM₄₃) duration. (Shaded regions) Standard deviation over multiple different heating durations.

Figure 6

Sensitivities of steady-state values and stress responsiveness of the heat-shock response relative to all parameters.