A Case Study on the Parametric Occurrence of Multiple Steady States

Bradford, Russell J.; Davenport, James H.; England, Matthew; Errami, Hassan; Gerdt, Vladimir P.; Grigoriev, Dima; Hoyt, Charles Tapley; Košta, Marek; Radulescu, Ovidiu; Sturm, Thomas; Weber, Andreas

doi:10.1145/3087604.3087622

Cited by 30 publications

(47 citation statements)

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“…New applications are found regularly, such as the derivation of optimal numerical schemes [31], and the validatation of economic hypotheses [46], [45]. CAD has also found application independent of QE, such as reasoning with multi-valued functions [21] (where we decompose to see where simplification rules are valid); and biological networks [7], [29] where we decompose to identify regions in parameter space where multi-stationarity can occur.…”

Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

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Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

England

Florescu

2019

Lecture Notes in Computer Science

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There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. Here we extend this result by training ML models to select the variable ordering directly, and by trying out a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human-made heuristics for the problem: the Brown heuristic and sotd. On this dataset all of the ML approaches outperformed the human-made heuristics, some by a large margin.

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Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

“…6 Proportion of x0 occuring in polynomials. 7 Proportion of x1 occuring in polynomials. 8 Proportion of x2 occuring in polynomials.…”

Section: Featuresmentioning

confidence: 99%

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

England

Florescu

2019

Lecture Notes in Computer Science

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“…Finally we summarise and give final thoughts in Section 11. This journal article follows published conference works at ISSAC 2017 (Bradford et al, 2017) and CASC 2017 . The present article reproduces this material clarifying, correcting and extending in places.…”

Section: Outline and New Contributionsmentioning

confidence: 99%

“…Software Remark 1. In the authors' ISSAC 2017 paper (Bradford et al, 2017) the description of the evaluated solution component ended here. However, following the publication of that paper a bug was uncovered by one of the authors in the simplifier of the Regular Chains Library when working with a different MAPK model to the one considered presently.…”

Section: The Main Solution Component From Lrtmentioning

confidence: 99%

Identifying the parametric occurrence of multiple steady states for some biological networks

Bradford

Davenport

England

et al. 2020

Journal of Symbolic Computation

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We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods.The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters.We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory.We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

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“…Many kinds of approaches have been taken to quantitatively assess robustness in this 69 way, such as by parametric sensitivity [37][38][39][40], or by estimating volume and 70 shape [41][42][43][44][45][46][47]. Algebraic methods can sometimes provide an analytical description of 71 parametric regions [46,[48][49][50][51], but these methods tend to scale poorly with the 72 complexity of the system. For systems arising from networks of biochemical reactions, 73 methods also exist which give parametric conditions under which bistability 74 occurs [52-60] and some of these apply to PTM systems [55,[59][60][61][62][63][64].…”

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confidence: 99%

Robustness and parameter geography in post-translational modification systems

Nam

Gyori

Amethyst

et al. 2019

Preprint

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AbstractBiological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this “parameter geography” for bistability in post-translational modification (PTM) systems. We use the previously developed “linear framework” for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼109 parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.Author SummaryBiological organisms are often said to have robust properties but it is difficult to understand how such robustness arises from molecular interactions. Here, we use a mathematical model to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making. The robustness of this property is revealed by the size and shape, or “geography,” of the parametric region in which the property holds. We use advances in reducing model complexity and in rapidly solving the underlying equations, to extensively sample parameter points in an 8-dimensional space. We find that under realistic molecular assumptions the size of the region is surprisingly small, suggesting that generating multiple internal states with such a mechanism is much harder than expected. While the shape of the region appears straightforward, we find surprising complexity in how the region grows with increasing amounts of the modified substrate. Our approach uses statistical analysis of data generated from a model, rather than from experiments, but leads to precise mathematical conjectures about parameter geography and biological robustness.

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A Case Study on the Parametric Occurrence of Multiple Steady States

Cited by 30 publications

References 30 publications

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Identifying the parametric occurrence of multiple steady states for some biological networks

Robustness and parameter geography in post-translational modification systems

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