The stability conditions at the steady states for Hill coefficients <i>n</i>₁ = <i>n</i>₂ = 1.

<p>Remark: </p

Modeling and Dynamical Analysis of Virus-Triggered Innate Immune Signaling Pathways

<div><p>The investigation of the dynamics and regulation of virus-triggered innate immune signaling pathways at a system level will enable comprehensive analysis of the complex interactions that maintain the delicate balance between resistance to infection and viral disease. In this study, we developed a delayed mathematical model to describe the virus-induced interferon (IFN) signaling process by considering several key players in the innate immune response. Using dynamic analysis and numerical simulation, we evaluated the following predictions regarding the antiviral responses: (1) When the replication ratio of virus is less than 1, the infectious virus will be eliminated by the immune system’s defenses regardless of how the time delays are changed. (2) The IFN positive feedback regulation enhances the stability of the innate immune response and causes the immune system to present the bistability phenomenon. (3) The appropriate duration of viral replication and IFN feedback processes stabilizes the innate immune response. The predictions from the model were confirmed by monitoring the virus titer and IFN expression in infected cells. The results suggest that the balance between viral replication and IFN-induced feedback regulation coordinates the dynamical behavior of virus-triggered signaling and antiviral responses. This work will help clarify the mechanisms of the virus-induced innate immune response at a system level and provide instruction for further biological experiments.</p> </div

The definitions and values of all parameters in model (1).

<p>Note: The degradation rate of virus, IFN or AVP amounts to ln2/T_half-life<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone.0048114-Bazhan1" target="_blank">[30]</a> and its unit is h⁻¹. The half-life of virus is about 7 hour <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone.0048114-Arnheiter1" target="_blank">[22]</a>, the half-life of IFN is about 1∼7 hour <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone.0048114-Vitale1" target="_blank">[23]</a>, which the half-life of IFNβ is about 1∼3 hour and the half-life of IFNα is about 4∼7 hour, and the half-life of AVP is 2∼24 hour <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone.0048114-Janzen1" target="_blank">[24]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone.0048114-Haller2" target="_blank">[25]</a>, which is in fact the range of the half-life of Mx protein which is a kind of important anti-virus protein.</p

Stabilization of oscillation when Hill coefficients n₁ and n₂ are greater than one.

<p>(A). No delays, oscillation system. (B). <i>τ</i>₁ = 1, steady state. (C). <i>τ</i>₄ = 5, steady state. Other dimensionless parameters: <i>n</i>₁ = 4, <i>n</i>₂ = 3, <i>σ</i>₁ = 4, <i>σ</i>₂ = 0.3, <i>α</i>₂ = 2, <i>α</i>₄ = 4 and <i>K</i> = 2. The initial values are (10, 5, 2).</p

The influence of time delays on the stability of the system.

<p>The settings of the dimensionless parameters are <i>n</i>₁ = <i>n</i>₂ = 2, <i>σ</i>₁ = 0.5, <i>α</i>₂ = 5, <i>α</i>₄ = 4, <i>K</i> = 2, and the initial value is [10 5 2] for all. <i>σ</i>₂ = 1 for A and B. <i>σ</i>₂ = 5 for C and D. <i>τ</i>₁ = 5, <i>τ</i>₂ = 3, <i>τ</i>₃ = 2, <i>τ</i>₄ = 4, <i>τ</i>₅ = 6 for A and C, and <i>τ</i>₁ = 50, <i>τ</i>₂ = 30, <i>τ</i>₃ = 20, <i>τ</i>₄ = 40, <i>τ</i>₅ = 60 for B and D.</p

Initial concentration values of three components for the simulation in Figure 3.

<p>Initial concentration values of three components for the simulation in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048114#pone-0048114-g003" target="_blank">Figure 3</a>.</p

Schematic diagram of stability conditions without a synergistic effect.

<p>The first quadrant in the plane <i>σ</i>₁–<i>σ</i>₃ is divided into three regions , and by the lines <i>σ</i>₁ = 1, <i>σ</i>₃ = <i>1</i>, and </p

Schematic diagram of stability conditions with a synergistic effect.

<p>The first quadrant in the plane <i>σ</i>_1–<i>σ</i>₃ is divided into six regions , , , , and by the lines <i>σ</i>₁ = 1, <i>σ</i>₃ = 2, and the curves and </p

Bifurcation diagram about time delays τ₁ and τ₄.

<p>(A). System (2) undergoes a process from oscillation to stability and then to the oscillation again when <i>τ</i>₁ changes from 0 to 2. (B). System (2) undergoes a process from oscillation to stability and then to oscillation again when <i>τ</i>₄ changes from 0 to 4. The dimensionless parameters are <i>n</i>₁ = <i>n</i>₂ = 1, <i>σ</i>₁ = 4, <i>σ</i>₂ = 3, <i>α</i>₂ = 12 (>C = 10.2347), <i>α</i>₄ = 4 and <i>K</i> = 2. All other time delays are 0.</p

The stability conditions at the steady states for Hill coefficients <i>n</i>₁ = <i>n</i>₂ = 2.

<p>Remark: </p