Protein folding and protein maturation largely occur in the controlled environment of the Endoplasmic Reticulum (ER). Perturbation to the correct functioning of this organelle leads to altered proteostasis and accumulation of misfolded proteins in the ER lumen. This condition is commonly known as ER stress and is appearing as an important contributor in the pathogenesis of several human diseases. Monitoring of the quality control processes is mediated by the Unfolded Protein Response (UPR). This response consists in a complex network of signaling pathways that aim to restore protein folding and ER homeostasis. Conditions in which UPR is not able to overcome ER stress lead to a switch of the UPR signaling program from an adaptive to a pro-apoptotic one, revealing a key role of UPR in modulating cell fate decisions. Because of its high complexity and its involvement in the regulation of different cellular outcomes, UPR has been the center of the development of computational models, which tried to better dissect the role of UPR or of its specific components in several contexts. In this review, we go through the existing mathematical models of UPR. We emphasize how their study contributed to an improved characterization of the role of this intricate response in the modulation of cellular functions.

Inositol 1,4,5-trisphosphate (IP3) plays a key role in calcium signaling. After stimulation, it diffuses from the plasma membrane where it is produced to the endoplasmic reticulum where its receptors are localized. Based on in vitro measurements, IP3 was long thought to be a global messenger characterized by a diffusion coefficient of ~280 micron2s-1. However, in vivo observations revealed that this value does not match with the timing of localized Ca2+ increases induced by the confined release of a non-metabolizable IP3 analog. A theoretical analysis of these data concluded that in intact cells diffusion of IP3 is strongly hindered, leading to a 30-fold reduction of the diffusion coefficient. Here, we re-investigated the characteristics of IP3 diffusion using a stochastic model of Ca2+ puffs. Our simulations concluded to a moderate decrease of the effective IP3 diffusion coefficient, which quantitatively agrees with a buffering effect by non-fully bound, hence inactive, IP3 receptors. The value of ~100 micron2s-1 predicted by the model is in average not much affected by the presence of the endoplasmic reticulum, which represents an obstacle to the free displacement of molecules, but can be significantly increased when cells display an elongated, 1-dimensional like geometry.