Persistent Subspaces of Reaction-Based Dynamical Systems

Ibrahim, Bashar; Stephan, Peter

doi:10.46793/match.90-2.471i

Cited by 4 publications

(4 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As shown in 21 , Theorem 4 does not only hold true for RDS but can also be transferred to the special cases of ODE and patch-like systems. Note that the concept of persistence used here is a new one that was defined in 20 .…”

Section: Resultsmentioning

confidence: 93%

“…The previously defined organizations were generalized towards so-called distributed organizations (DOs), which are introduced and broadly discussed in 20 , 21 .…”

Section: Preliminariesmentioning

confidence: 99%

“…To incorporate such spatial and temporal separation in chemical organization theory, Peter et al 20 recently introduced the concept of a distributed organization (DO). A distributed organization is a subset of species that achieves overall self-maintenance by separation into suitable compartments 20 , 21 . As a generalization of the above-mentioned relation of fixed points and organizations, it was also shown that every persistent subspace of a PDE system is a DO 20 , 21 .…”

Section: Introductionmentioning

confidence: 99%

“…A distributed organization is a subset of species that achieves overall self-maintenance by separation into suitable compartments 20 , 21 . As a generalization of the above-mentioned relation of fixed points and organizations, it was also shown that every persistent subspace of a PDE system is a DO 20 , 21 . This represents a method for analyzing dynamical systems that are comparable to commonly used fixed point analyses 22 .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing all persistent subspaces of a reaction-diffusion system

Peter,

Woitke,

Dittrich

et al. 2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

An algorithm is presented for computing a reaction-diffusion partial differential equation (PDE) system for all possible subspaces that can hold a persistent solution of the equation. For this, all possible sub-networks of the underlying reaction network that are distributed organizations (DOs) are identified. Recently it has been shown that a persistent subspace must be a DO. The algorithm computes the hierarchy of DOs starting from the largest by a linear programming approach using integer cuts. The underlying constraints use elementary reaction closures as minimal building blocks to guarantee local closedness and global self-maintenance, required for a DO. Additionally, the algorithm delivers for each subspace an affiliated set of organizational reactions and minimal compartmentalization that is necessary for this subspace to persist. It is proved that all sets of organizational reactions of a reaction network, as already DOs, form a lattice. This lattice contains all potentially persistent sets of reactions of all constrained solutions of reaction-diffusion PDEs. This provides a hierarchical structure of all persistent subspaces with regard to the species and also to the reactions of the reaction-diffusion PDE system. Here, the algorithm is described and the corresponding Python source code is provided. Furthermore, an analysis of its run time is performed and all models from the BioModels database as well as further examples are examined. Apart from the practical implications of the algorithm the results also give insights into the complexity of solving reaction-diffusion PDEs.

show abstract

Section: Resultsmentioning

confidence: 93%

“…The previously defined organizations were generalized towards so-called distributed organizations (DOs), which are introduced and broadly discussed in 20 , 21 .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing all persistent subspaces of a reaction-diffusion system

Peter,

Woitke,

Dittrich

et al. 2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

show abstract

Revealing the hierarchical structure of microbial communities

Ruth,

Peter,

Ibrahim

et al. 2024

Sci Rep

View full text Add to dashboard Cite

Measuring the dynamics of microbial communities results in high-dimensional measurements of taxa abundances over time and space, which is difficult to analyze due to complex changes in taxonomic compositions. This paper presents a new method to investigate and visualize the intrinsic hierarchical community structure implied by the measurements. The basic idea is to identify significant intersection sets, which can be seen as sub-communities making up the measured communities. Using the subset relationship, the intersection sets together with the measurements form a hierarchical structure visualized as a Hasse diagram. Chemical organization theory (COT) is used to relate the hierarchy of the sets of taxa to potential taxa interactions and to their potential dynamical persistence. The approach is demonstrated on a data set of community data obtained from bacterial 16S rRNA gene sequencing for samples collected monthly from four groundwater wells over a nearly 3-year period (n = 114) along a hillslope area. The significance of the hierarchies derived from the data is evaluated by showing that they significantly deviate from a random model. Furthermore, it is demonstrated how the hierarchy is related to temporal and spatial factors; and how the idea of a core microbiome can be extended to a set of interrelated core microbiomes. Together the results suggest that the approach can support developing models of taxa interactions in the future.

show abstract

Cell Cycle Complexity: Exploring the Structure of Persistent Subsystems in 414 Models

Peter,

Josephraj,

Ibrahim

2024

Biomedicines

View full text Add to dashboard Cite

Background: The regulation of cellular proliferation and genomic integrity is controlled by complex surveillance mechanisms known as cell cycle checkpoints. Disruptions in these checkpoints can lead to developmental defects and tumorigenesis. Methods: To better understand these mechanisms, computational modeling has been employed, resulting in a dataset of 414 mathematical models in the BioModels database. These models vary significantly in detail and simulated processes, necessitating a robust analytical approach. Results: In this study, we apply the chemical organization theory (COT) to these models to gain insights into their dynamic behaviors. COT, which handles both ordinary and partial differential equations (ODEs and PDEs), is utilized to analyze the compartmentalized structures of these models. COT’s framework allows for the examination of persistent subsystems within these models, even when detailed kinetic parameters are unavailable. By computing and analyzing the lattice of organizations, we can compare and rank models based on their structural features and dynamic behavior. Conclusions: Our application of the COT reveals that models with compartmentalized organizations exhibit distinctive structural features that facilitate the understanding of phenomena such as periodicity in the cell cycle. This approach provides valuable insights into the dynamics of cell cycle control mechanisms, refining existing models and potentially guiding future research in this area.

show abstract

Persistent Subspaces of Reaction-Based Dynamical Systems

Cited by 4 publications

References 30 publications

Computing all persistent subspaces of a reaction-diffusion system

Computing all persistent subspaces of a reaction-diffusion system

Revealing the hierarchical structure of microbial communities

Cell Cycle Complexity: Exploring the Structure of Persistent Subsystems in 414 Models

Contact Info

Product

Resources

About