Nonself RNA rewires IFN-ß signaling: A mathematical model of the innate immune response.

Type I interferons (IFNs) are key coordinators of the innate immune response to viral infection, which, through activation of the transcriptional regulators STAT1 and STAT2 (STAT1/2) in bystander cells, induce the expression of IFN-stimulated genes (ISGs). Here, we showed that in cells transfected with poly(I:C), an analog of viral RNA, the transcriptional activity of STAT1/2 was terminated because of depletion of the interferon-ß (IFN-ß) receptor, IFNAR. Activation of RNase L and PKR, products of two ISGs, not only hindered the replenishment of IFNAR but also suppressed negative regulators of IRF3 and NF-κB, consequently promoting IFNB transcription. We incorporated these findings into a mathematical model of innate immunity. By coupling signaling through the IRF3-NF-κB and STAT1/2 pathways with the activities of RNase L and PKR, the model explains how poly(I:C) switches the transcriptional program from being STAT1/2 induced to being IRF3 and NF-κB induced, which converts IFN-ß-responding cells to IFN-ß-secreting cells.

A plausible identifiable model of the canonical NF-κB signaling pathway.

An overwhelming majority of mathematical models of regulatory pathways, including the intensively studied NF-κB pathway, remains non-identifiable, meaning that their parameters may not be determined by existing data. The existing NF-κB models that are capable of reproducing experimental data contain non-identifiable parameters, whereas simplified models with a smaller number of parameters exhibit dynamics that differs from that observed in experiments. Here, we reduced an existing model of the canonical NF-κB pathway by decreasing the number of equations from 15 to 6. The reduced model retains two negative feedback loops mediated by IκBα and A20, and in response to both tonic and pulsatile TNF stimulation exhibits dynamics that closely follow that of the original model. We carried out the sensitivity-based linear analysis and Monte Carlo-based analysis to demonstrate that the resulting model is both structurally and practically identifiable given measurements of 5 model variables from a simple TNF stimulation protocol. The reduced model is capable of reproducing different types of responses that are characteristic to regulatory motifs controlled by negative feedback loops: nearly-perfect adaptation as well as damped and sustained oscillations. It can serve as a building block of more comprehensive models of the immune response and cancer, where NF-κB plays a decisive role. Our approach, although may not be automatically generalized, suggests that models of other regulatory pathways can be transformed to identifiable, while retaining their dynamical features.

Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle.

A minimal model describing the embryonic cell division cycle at the molecular level in eukaryotes is analyzed mathematically. It is known from numerical simulations that the corresponding three-dimensional system of ODEs has periodic solutions in certain parameter regimes. We prove the existence of a stable limit cycle and provide a detailed description on how the limit cycle is generated. The limit cycle corresponds to a relaxation oscillation of an auxiliary system, which is singularly perturbed and has the same orbits as the original model. The singular perturbation character of the auxiliary problem is caused by the occurrence of small Michaelis constants in the model. Essential pieces of the limit cycle of the auxiliary problem consist of segments of slow motion close to several branches of a two dimensional critical manifold which are connected by fast jumps. In addition, a new phenomenon of exchange of stability occurs at lines, where the branches of the two-dimensional critical manifold intersect. This novel type of relaxation oscillations is studied by combining standard results from geometric singular perturbation with several suitable blow-up transformations.