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Bistability and bifurcation phenomena. (A). Bistability occurs when the value of σ1 changes from 0 to 10, which successively pass through domains , , and (σ2 = 4.1) in Figure 6. (B). Bifurcation occurs when the value of σ1 changes from 0 to 6, which successively pass through domains , , and (σ2 = 3.9) in Figure 6. The other parameters are fixed: α2 = 0.5, α4 = 4, K = 2 and n1 = n2 = 2.
Source publication
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-ind...
Citations
... Data-driven multi-scale mathematical modeling of SARS-CoV-2 infection assembly, which results in a large number of newly synthesized viruses. Upon viral replication, the interferon signaling pathway, one of the virus-mediated innate immune signaling pathways, is activated and engages type I interferons (IFNs) and antiviral proteins (AVPs), and to restrict the process of viral replication [29]. The newly synthesized virus leaves the target cell by means of budding, and other susceptible cells are further infected, forming a cascade of cell infection [30] (Fig 2 Intracellular). ...
... Here λ 1 , τ 1 , b 1 , m 1 , δ 1 are constants (details shown in Table A in S1 Text). With respect to interferons (IFNs), they are activated by viral RNA and exhibit positive autoregulation [29]. So, the influx of IFNs pure synthesis is ...
... The mammalian cell volume (v cell ) is 100 � 10000μm 3 [68] and the density of naïve T cell is approximately 4 × 10 −3 g/cm 3 [69] so that the counts of naïve T cell ([T 0 ]) are about 0.4 � 4.0 × 10 5 cells/ml. The half-life of IFNs, AVPs and SARS-CoV-2 were determined from published studies; the half-life of IFNs ranged from 1.3 to 4.7 hours [70], AVPs is 2�24 hours [29], and the half-life of SARS-CoV-2 is about 6.8 hours [71]. By the natural depletion rate δ = ln2/t 1/2 (t 1/2 is the half-life), the degradation of IFNs, AVPs and SARS-CoV-2 were estimated as δ 1 = 0.1h −1 , δ 2 = 0.4h −1 , δ 3 = 0.12h −1 , respectively. ...
Patients with coronavirus disease 2019 (COVID-19) often exhibit diverse disease progressions associated with various infectious ability, symptoms, and clinical treatments. To systematically and thoroughly understand the heterogeneous progression of COVID-19, we developed a multi-scale computational model to quantitatively understand the heterogeneous progression of COVID-19 patients infected with severe acute respiratory syndrome (SARS)-like coronavirus (SARS-CoV-2). The model consists of intracellular viral dynamics, multicellular infection process, and immune responses, and was formulated using a combination of differential equations and stochastic modeling. By integrating multi-source clinical data with model analysis, we quantified individual heterogeneity using two indexes, i.e., the ratio of infected cells and incubation period. Specifically, our simulations revealed that increasing the host antiviral state or virus induced type I interferon (IFN) production rate can prolong the incubation period and postpone the transition from asymptomatic to symptomatic outcomes. We further identified the threshold dynamics of T cell exhaustion in the transition between mild-moderate and severe symptoms, and that patients with severe symptoms exhibited a lack of naïve T cells at a late stage. In addition, we quantified the efficacy of treating COVID-19 patients and investigated the effects of various therapeutic strategies. Simulations results suggested that single antiviral therapy is sufficient for moderate patients, while combination therapies and prevention of T cell exhaustion are needed for severe patients. These results highlight the critical roles of IFN and T cell responses in regulating the stage transition during COVID-19 progression. Our study reveals a quantitative relationship underpinning the heterogeneity of transition stage during COVID-19 progression and can provide a potential guidance for personalized therapy in COVID-19 patients.
... Tan et al. developed a mathematical model describing the virus-induced interferon (IFN) signaling process. Dynamic analysis and numerical simulations led to the suggestion that a balance between viral replication and IFN-induced regulation is responsible for the dynamic behavior of virus-triggered signaling and also for antiviral responses (Tan et al., 2012). Dynamic modeling of infections with coronaviruses, especially with SARS-CoV-2, could broaden the understanding of its effects on the host and give further insight into potential targets for antiviral therapies. ...
Motivation
The novel coronavirus (SARS-CoV-2) currently spreads worldwide, causing the disease COVID-19. The number of infections increases daily, without any approved antiviral therapy. The recently released viral nucleotide sequence enables the identification of therapeutic targets, e.g. by analyzing integrated human-virus metabolic models. Investigations of changed metabolic processes after virus infections and the effect of knock-outs on the host and the virus can reveal new potential targets.
Results
We generated an integrated host–virus genome-scale metabolic model of human alveolar macrophages and SARS-CoV-2. Analyses of stoichiometric and metabolic changes between uninfected and infected host cells using flux balance analysis (FBA) highlighted the different requirements of host and virus. Consequently, alterations in the metabolism can have different effects on host and virus, leading to potential antiviral targets. One of these potential targets is guanylate kinase (GK1). In FBA analyses, the knock-out of the GK1 decreased the growth of the virus to zero, while not affecting the host. As GK1 inhibitors are described in the literature, its potential therapeutic effect for SARS-CoV-2 infections needs to be verified in in-vitro experiments.
Availability and implementation
The computational model is accessible at https://identifiers.org/biomodels.db/MODEL2003020001.
... Also, based on our observations we conclude that our model seems to be sensitive to infective rates even though any explicit sensitivity analysis has not been carried out in this paper. The future scope of this article lies in introducing the idea of multi group models with random perturbation in the papers (Ji, Jang, & Shi, 2011;Tan, Pan, Qiao, Zou, & Pan, 2012;Yu, Jiang, & Shi, 2009) by Ji. et al., J. Yu et al. and J. Tan et al. respectively, which will form a new frame work in the literature to analyze the mathematical models with more precision and accuracy. ...
Virus population disease dynamics in various species of ecosystem keep the research interests alive for many centuries. In this research article, an attempt has been made to understand the qualitative behavior of a virus infection model with Lytic and Non-Lytic Immune Responses by perturbing with randomness (white noise) via Lyapunov technique. The conditions for the extinction and permanence of the viral infection in the interacting populations has been found, analyzed and supported with numerical simulations. An application to HIV infection model has also been presented for drawing a comparative study of the model under various modeling methods. The research findings of this paper reveal that a study that includes random fluctuations of the environment prove to be the ideal way to bring out the qualitative analysis of a mathematical model that will depict the real world scenario.
... Then interaction between the IFNs and their receptors causes activation of JAK-STAT signaling pathway. Phosphorylated STAT proteins translocate to the nucleus and combine with interferon regulatory proteins to promote an abundant expression of a wide array of genes, including IFNstimulated genes (ISGs) (Takeuchi and Akira, 2010;Tan et al., 2012;Bailey et al., 2014). These ISGs encode distinct antiviral proteins with diverse biological effects that block multiple stages of the viral lifecycle including viral entry, translation, replication, assembly, and spread (Diamond and Farzan, 2013). ...
Avian Tembusu virus (ATMUV) is a highly pathogenic flavivirus that causes significant
economic losses to the Chinese poultry industry. Our previous experiments
demonstrated that ATMUV infection effectively triggered host innate immune response
through MDA5 and TLR3-dependent signaling pathways. However, little information is
available on the role of interferon-stimulated genes (ISGs) in defending against ATMUV
infection. In this study, we found that ATMUV infection induced robust expression of type
I and type III interferon (IFNs) in duck tissues. Furthermore, we observed that expression
of interferon-inducible transmembrane proteins (IFITMs) was significantly upregulated in
DEF and DF-1 cells after infection with ATMUV. Similar results were obtained from in vivo
studies using ATMUV-infected ducklings. Importantly, we showed that knockdown of
endogenous IFITM1 or IFITM3 by specific shRNA markedly enhanced ATMUV replication
in DF-1 cells. However, disruption of IFITM2 expression had no obvious effect on the
ATMUV replication. In addition, overexpression of chicken or duck IFITM1 and IFITM3 in
DF-1 cells impaired the replication of ATMUV. Taken together, these results reveal that
induced expression of avian IFITM1 and IFITM3 in response to ATMUV infection can
effectively restrict the virus replication, and suggest that increasing IFITM proteins in host
may be a useful strategy for control of ATMUV infection.
... Commonly, the inhibitory effects of the innate immune response are directly or indirectly antagonized by HIV proteins (Duggal and Emerman, 2012). Recent reports have theoretically studied the mutual negative interaction between the innate immune system and viral proteins describing the outcome of viral infection by deterministic dynamical behaviors that include monostability, bistability and oscillations (Tan et al., 2012;Zou et al., 2010). ...
... IIC was chosen to represent this negative interaction because of its simplicity and low non-linearity. The parameter k dV denotes a first-order decay rate of V (Tan et al., 2012). The parameter k C denotes the basal production rate of C. The second term represents the negative interaction of V with C, which was generated following the same logic of considering the negative interactions in the form of an IIC. ...
... The parameter k C denotes the basal production rate of C. The second term represents the negative interaction of V with C, which was generated following the same logic of considering the negative interactions in the form of an IIC. The parameter k dC represents the first-order decay of C (Tan et al., 2012). All the parameters in the equations have positive ranges defined by experimental reports and are tabulated in Table 1. ...
... At the early phase of viral infection, phosphorylated IRF3 and IRF7 translocate to the nucleus and trigger the expression of small amounts of early IFN-β and IFN-α. In the second phase of infection, robust transcription of IFN genes is induced and newly synthesized IFN bind to the type I IFN receptor (IFNAR) and activate the JAK/STAT pathway, leading to the up-regulation of hundreds of ISG [27][28][29]. These antiviral components inhibit viral replication and cause apoptosis of infected cells, subsequently resulting in the clearance of the infectious pathogens [30]. ...
Avian Tembusu virus (ATMUV) is a newly emerged flavivirus that belongs to the Ntaya virus group. ATMUV is a highly pathogenic virus causing significant economic loss to the Chinese poultry industry. However, little is known about the role of host innate immune mechanism in defending against ATMUV infection. In this study, we found that ATMUV infection significantly up-regulated the expression of type I and type III interferons (IFN) and some critical IFN-stimulated genes (ISG) in vivo and in vitro. This innate immune response was induced by genomic RNA of ATMUV. Furthermore, we observed that ATMUV infection triggered IFN response mainly through MDA5 and TLR3-dependent signaling pathways. Strikingly, shRNA-based disruption of IPS-1, IRF3 or IRF7 expression significantly reduced the production of IFN in the 293T cell model. Moreover, NF-κB was shown to be activated in both chicken and human cells during the ATMUV infection. Inhibition of NF-κB signaling also resulted in a clear decrease in expression of IFN. Importantly, experiments revealed that treatment with IFN significantly impaired ATMUV replication in the chicken cell. Consistently, type I IFN also exhibited promising antiviral activity against ATMUV replication in the human cell. Together, these data indicate that ATMUV infection triggers host innate immune response through MDA5 and TLR3-dependent signaling that controls IFN production, and thereby induces an effective antiviral immunity.
Electronic supplementary material
The online version of this article (doi:10.1186/s13567-016-0358-5) contains supplementary material, which is available to authorized users.
... For a more detailed review of models for signalling pathways activated during inflammation, see [240]. Also in the context of innate immunity we mention the existence of models for signalling pathways activated following infections with different pathogens (e.g., Francisella tularensis [140]), models for signalling pathways (e.g., PI3K) that control migration and polarisation of neutrophils [181], models that try to elucidate the pathways involved in the crosstalk between various cytokines that regulate immune responses, such as IFN-γ and IL-6 [198], models for gene regulatory networks that control genetic switching between cell fates, such as the GATA genes in hematopoietic stem cells [234], or models for the regulation of signalling pathways in innate immune cells following viral infections [229] and the optimal control of the innate response [230]. ...
The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.
... While state-structured population models deal with at least two independent dynamic variables, DDE models only consider temporal dynamics (cf. Bazhan and Belova 1999, Pawelek et al. 2012, Tan et al. 2012, for example). ...
The interferon system functions as a first line of defense against viral infections. The cellular recognition of viruses leads to the production of interferons (IFN) by the infected cells. Secreted IFN stimulates an antiviral response in an autocrine and paracrine manner: While autocrine IFN action inhibits virus production in infected cells, paracrine IFN signaling induces an antiviral protective state in naïve cells. Although central molecular components of the IFN system have been characterized, our quantitative understanding of its dynamics remains limited. In particular, it is not precisely known which molecular processes are decisive for the outcome of virus-host interactions.
Together with our experimental cooperation partners, we have studied virus-induced IFN signaling at single-cell resolution in communicating cell populations after infection with a non-spreading and a spreading virus. On this basis, we established two complementary mathematical models. First, we developed a stochastic multi-scale model accounting for the intracellular dynamics in individual cells and the cell-to-cell communication via secreted IFN. Second, we constructed a delay differential equation model to analyze the competition between viral spread and IFN-induced antiviral defense in a cell population. Both models were parameterized on the basis of original experimental data and the numerical analyses of the models aimed at deriving testable predictions for new experiments.
By live-cell imaging, we showed that key steps of the IFN pathway including virus-induced signaling, IFN expression, and induction of IFN-stimulated genes are stochastic events in individual cells. To relate the single-cell data after primary infection to antiviral protection at the cell-population level, we established a stochastic model which combines the heterogeneous IFN signaling in single cells with the intercellular communication through released IFN. The parameters describing the virus-induced activation of the transcription factors of the IFN gene were estimated from a distribution of observed single-cell activation times using as objective function Neyman’s chi-squares statistic. The minimization of this objective function by simulated annealing revealed that virus-induced signaling is cooperative. Moreover, fitting of the measured time delays between transcription factor activation and IFN gene induction in individual cells with a gamma distribution by applying the maximum-likelihood method implies that IFN gene induction downstream of transcription factor activation is a slow multi-step process. Notably, mathematical modeling and experimental validation indicate that reliable antiviral protection in the face of multi-layered cellular stochasticity can be achieved by paracrine propagation of the IFN signal. Therefore, a few IFN-producing cells are able to protect a large number of naïve cells (Rand, Rinas et al. 2012).
To investigate the competition between viral spread and IFN-induced antiviral defense, we examined virus-host interactions after infection with spreading Dengue virus. For this purpose, our experimental partners generated data showing the antiviral response dynamics of fluorescent reporter cells after infection with a fluorescently labeled Dengue virus. Based on these kinetic data, we established a delay differential equation model with time delays for virus replication, virus production and IFN secretion. Using data-driven least-squares fitting and profile likelihood analysis, we identified the model parameters within narrow confidence bounds and found that the timing of virus production and IFN secretion after infection are almost identical. This direct competition together with the highly heterogeneous IFN response in single cells fosters the coexistence of IFN-induced protection of naïve cells and viral spread in non-protected cells. To analyze which components of the antiviral IFN system have the greatest influence on viral spread, we compared the infection dynamics of the wild-type Dengue virus with the attenuated spread of the vaccine candidate Dengue virus E217A mutant. We quantified the differences between wild-type and mutant Dengue virus infections using data-driven parameter optimization constrained by the determined wild-type virus related parameter values. In this way, we identified two mutant virus specific parameters, which explain the attenuation of the mutant through a reduced virus production rate and an accelerated IFN secretion taking place much earlier than virus production. By mathematical modeling and validation experiments, we predict that rapid IFN action curbing virus production in infected cells is critical for the attenuation of the Dengue virus E217A mutant. Thus, a fast acting autocrine IFN signal could limit viral spread in such a way that accelerated paracrine IFN response has only a minor impact on the spread of Dengue virus (Schmid, Rinas et al. submitted).
In conclusion, our work demonstrates that mathematical modeling is an essential tool to integrate data and mechanisms from the molecular to the cell-population level. The research on understanding which molecular mechanisms shape virus-host interactions might inform the development of new antiviral therapies and vaccines.
... Mathematical modeling of viral infections has a rich tradition [56,68], but models that include the immune response explicitly have primarily focused on the adaptive immune response [69]. By comparison, the initial phase during which an invading virus must overcome the innate immune response to spread has rarely been addressed [70][71][72][73] and experimental data for the parameterization of such models have been scarce. Here we have developed a mathematical model for the dynamic interaction of viral spread and the IFN response and identified all model parameters from a consistent set of experimental data based on DENV infection of IFN-competent human cells. ...
Dengue virus (DENV) is the most common mosquito-transmitted virus infecting ~390 million people worldwide. In spite of this high medical relevance, neither a vaccine nor antiviral therapy is currently available. DENV elicits a strong interferon (IFN) response in infected cells, but at the same time actively counteracts IFN production and signaling. Although the kinetics of activation of this innate antiviral defense and the timing of viral counteraction critically determine the magnitude of infection and thus disease, quantitative and kinetic analyses are lacking and it remains poorly understood how DENV spreads in IFN-competent cell systems. To dissect the dynamics of replication versus antiviral defense at the single cell level, we generated a fully viable reporter DENV and host cells with authentic reporters for IFN-stimulated antiviral genes. We find that IFN controls DENV infection in a kinetically determined manner that at the single cell level is highly heterogeneous and stochastic. Even at high-dose, IFN does not fully protect all cells in the culture and, therefore, viral spread occurs even in the face of antiviral protection of naïve cells by IFN. By contrast, a vaccine candidate DENV mutant, which lacks 2'-O-methylation of viral RNA is profoundly attenuated in IFN-competent cells. Through mathematical modeling of time-resolved data and validation experiments we show that the primary determinant for attenuation is the accelerated kinetics of IFN production. This rapid induction triggered by mutant DENV precedes establishment of IFN-resistance in infected cells, thus causing a massive reduction of virus production rate. In contrast, accelerated protection of naïve cells by paracrine IFN action has negligible impact. In conclusion, these results show that attenuation of the 2'-O-methylation DENV mutant is primarily determined by kinetics of autocrine IFN action on infected cells.
... The topology of molecular networks can reveal essential principles of most cellular processes and biological functions [34], [35]. The dynamical analysis based on exploring topological organizations and biological modules in PPI networks will be helpful in understanding cellular processes [36], [37]. Fig. 4. The three core measurements as a function of penalty term p under the collins dataset. ...
The identification of protein complexes in protein-protein interaction (PPI) networks is fundamental for understanding biological processes and cellular molecular mechanisms. Many graph computational algorithms have been proposed to identify protein complexes from PPI networks by detecting densely connected groups of proteins. These algorithms assess the density of subgraphs through evaluation of the sum of individual edges or nodes; thus, incomplete and inaccurate measures may miss meaningful biological protein complexes with functional significance. In this study, we propose a novel method for assessing the compactness of local subnetworks by measuring the number of three node cliques. The present method detects each optimal cluster by growing a seed and maximizing the compactness function. To demonstrate the efficacy of the new proposed method, we evaluate its performance using five PPI networks on three reference sets of yeast protein complexes with five different measurements and compare the performance of the proposed method with four state-of-the-art methods. The results show that the protein complexes generated by the proposed method are of better quality than those generated by four classic methods. Therefore, the new proposed method is effective and useful for detecting protein complexes in protein-protein interaction networks.
