Evidence
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The approach is particularly well suited to systems where the functional forms of the underlying processes are uncertain but the network structure is known, such as ecological food webs, coupled laser systems, and socioeconomic models of dynastic cycles. In each case, the normalization procedure identifies natural parameters of the system, and the Jacobian at a steady state is derived as a function of these parameters, so that local bifurcations can be computed analytically.
Passages supplied to the Evidence version
Card 1 · Bifurcations and Multistability in a Model of Cytokine-Mediated Autoimmunity
This paper investigates the dynamics of immune response and autoimmunity with particular emphasis on the role of regulatory T cells (Tregs), T cells with different activation thresholds, and cytokines in mediating T cell activity. Analysis of the steady states yields parameter regions corresponding…
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This paper investigates the dynamics of immune response and autoimmunity with particular emphasis on the role of regulatory T cells (Tregs), T cells with different activation thresholds, and cytokines in mediating T cell activity. Analysis of the steady states yields parameter regions corresponding to regimes of normal clearance of viral infection, chronic infection, or autoimmune behavior, and the boundaries of stability and bifurcations of relevant steady states are found in terms of system parameters. Numerical simulations are performed to illustrate different dynamical scenarios, and to identify basins of attraction of different steady states and periodic solutions, highlighting the important role played by the initial conditions in determining the outcome of immune interactions.
Card 2 · Generalized models as a universal approach to the analysis of nonlinear dynamical systems
We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure…
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We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of …
Card 3 · Parameter regions that give rise to 2[n/2]+1 positive steady states in the n-site phosphorylation system
The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the react…
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The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[n/2]+1 positive steady states (that is, n+1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1/4 of the intermediates occur in the reaction mechanism. This result is based on the general framework developed by Bihan, Dickenstein, and Giaroli (2018), which can be applied to other networks. We also describe how to implement these tools to search for multistationarity regions in a …
Card 4 · An algebraic method to calculate parameter regions for constrained steady-state distribution in stochastic reaction networks
Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuat…
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Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuations. Although several methods have been proposed to detect and analyze the stability of steady states for deterministic models, these methods cannot be applied to stochastic reaction networks. In this paper, we propose an algorithm based on algebraic computations to calculate parameter regions for constrained steady-state distribution of stochastic reaction networks, in which the means and variances satisfy some given inequality constraints. To evaluate our proposed method, we perform computer simulations for three typical chemical reactions and demonstrate that the results obtained with our method are consistent with the simulation results.
Card 5 · Detecting parameter regions for bistability in reaction networks
Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to de…
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Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to determine, exclusively via symbolic computations, the stability of the steady states for unspecified parameter values. In particular, our approach fully determines the stability type of all steady states of a broad class of networks. To this end, we combine the Hurwitz criterion, reduction of the steady state equations to one univariate equation, and structural reductions of the reaction network. Using our method, we prove that bistability occurs in open regions in parameter space for many relevant motifs in cell signaling.
Model-reported card use
Card 2 · fact
Affected output: The approach is particularly well suited to systems where the functional forms of the underlying processes are uncertain but the network structure is known, such as ecological food webs, coupled laser systems, and socioeconomic models of dynastic cycles.
Model explanation: This row supplied the concrete fact that generalized models are not restricted to specific functional forms and can describe a class of systems sharing a similar structure, which grounds the claim about the approach's suitability for systems with uncertain functional forms.
Card 2 · fact
Affected output: In each case, the normalization procedure identifies natural parameters of the system, and the Jacobian at a steady state is derived as a function of these parameters, so that local bifurcations can be computed analytically.
Model explanation: This row supplied the specific methodological steps—normalization to identify natural parameters, derivation of the Jacobian as a function of these parameters, and analytical computation of local bifurcations—which were used to describe the approach's implementation.
Card 2 · terminology
Affected output: ecological food webs, coupled laser systems, and socioeconomic models of dynastic cycles
Model explanation: This row provided the specific example systems from the source paper, which were used as illustrative applications of the generalized modeling approach.