Model comparison via simplicial complexes and persistent homology

Vittadello, Sean T.; Stumpf, Michael

doi:10.1098/rsos.211361

Cited by 14 publications

(29 citation statements)

References 48 publications

(59 reference statements)

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“…In the following example we apply our model-comparison methodology to the two main categories of models for developmental pattern formation, namely Turing-pattern (TP) models and positional-information (PI) models. We have described models from these two categories previously [24]: four TP models, namely activatorinhibitor, substrate depletion, inhibition of an inhibition, and modulation; five PI models, namely linear gradient, synthesis-diffusion-degradation (SDD), opposing gradients, annihilation, and induction-contraction (active modulation). Interestingly, we found that the TP activator-inhibitor model is equivalent to the PI annihilation model from a significant conceptual perspective.…”

Section: Methodology For Determining Model Equivalencementioning

confidence: 99%

“…We now state Operations 1, 2, and 5. Further details are in Vittadello and Stumpf [24]. Recall that two distinct vertices in a simplicial complex are adjacent if they belong to the same simplex, and for a vertex u in a simplicial complex K we denote the set of all vertices adjacent to u as V K (u), noting that u / ∈ V K (u).…”

Section: Model Comparison By Equivalencementioning

confidence: 99%

“…In Vittadello and Stumpf [24] we present a novel framework for model comparison where models are represented as labelled simplicial complexes. Models are then compared in terms of their simplicial representations using two general methodologies: the first is comparison by distance, where the difference between models is measured in terms of the differences between the corresponding simplicial representations; the second is comparison by equivalence, where we identify any equivalences between the components of the simplicial representations of different models, and then employ particular operations on the complexes to transform one into the othersuch equivalence, when it exists, reveals common underlying characteristics of the models.…”

Section: Introductionmentioning

confidence: 99%

“…In this article we continue the development of our methodology for model comparison [24] in two ways.…”

Section: Introductionmentioning

confidence: 99%

“…In this section we provide a brief and informal description of the necessary background in simplicial algebraic topology; further details are available from standard references [27][28][29]. We also discuss our methodology for model comparison by equivalence as described in Vittadello and Stumpf [24], and developed further in this work.…”

Section: Introductionmentioning

confidence: 99%

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A group theoretic approach to model comparison with simplicial representations

Vittadello¹,

Stumpf²

2021

Preprint

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The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying structure is therefore required. In a previous work we developed an appropriate framework for model comparison where we represent models as labelled simplicial complexes and compare them with two general methodologies, namely comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we develop an automatable methodology for determining model equivalence using group actions on the simplicial complexes, which greatly simplify and expedite the process of determining model equivalence. Second, we develop an alternative framework for model comparison by representing models as groups, which allows for the application of group-theoretic techniques within our model comparison methodology.

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Section: Methodology For Determining Model Equivalencementioning

confidence: 99%

Section: Model Comparison By Equivalencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this article we continue the development of our methodology for model comparison [24] in two ways.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A group theoretic approach to model comparison with simplicial representations

Vittadello¹,

Stumpf²

2021

Preprint

Self Cite

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show abstract

A group theoretic approach to model comparison with simplicial representations

Vittadello

Stumpf

2022

J. Math. Biol.

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The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

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Algebra, Geometry and Topology of ERK Kinetics

et al. 2022

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The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death (Shaul and Seger in Biochim Biophys Acta (BBA)-Mol Cell Res 1773(8):1213–1226, 2007). Here we study a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. (Curr Biol 2019, https://doi.org/10.1016/j.cub.2019.12.052) with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. We demonstrate how methods from computational algebraic geometry, differential algebra, Bayesian statistics and computational algebraic topology can inform the model reduction, identification and parameter inference of MEK variants, respectively. Throughout, we show how this algebraic viewpoint offers a rigorous and systematic analysis of such models.

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Model comparison via simplicial complexes and persistent homology

Cited by 14 publications

References 48 publications

A group theoretic approach to model comparison with simplicial representations

A group theoretic approach to model comparison with simplicial representations

A group theoretic approach to model comparison with simplicial representations

Algebra, Geometry and Topology of ERK Kinetics

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