Lower bounds for positive roots and regions of multistationarity in chemical reaction networks

Bihan, Frédéric; Dickenstein, Alicia; Giaroli, Magalí

doi:10.1016/j.jalgebra.2019.10.002

Cited by 24 publications

(58 citation statements)

References 30 publications

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“…In the previous paper [1], parameter regions on all the parameters are given for the occurrence of multistationarity for the n-site sequential phosphorylation system, but no more than three positive steady states are ensured. These conditions are based on a general framework to obtain multistationary regions jointly in the reaction rate constants and the linear conservation constants.…”

Section: Introductionmentioning

confidence: 99%

“…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 computer algebra system and present some computer aided results.…”

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confidence: 99%

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Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

Giaroli

Rischter

Millán

et al. 2019

Mathematical Biosciences and Engineering

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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 computer algebra system and present some computer aided results.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

Giaroli

Rischter

Millán

et al. 2019

Mathematical Biosciences and Engineering

Self Cite

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“…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]. Bistable parametric 75 regions have thereby been demarcated in various contexts [54,58,60,62,64].…”

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confidence: 99%

“…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]. Bistable parametric 75 regions have thereby been demarcated in various contexts [54,58,60,62,64]. However, 76 the relevant conditions for bistability are typically sufficient, but not always necessary, 77 making it difficult to exactly determine bistable regions.…”

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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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Families of Polynomials in the Study of Biochemical Reaction Networks

Dickenstein

2021

Computer Algebra in Scientific Computing

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Lower bounds for positive roots and regions of multistationarity in chemical reaction networks

Cited by 24 publications

References 30 publications

Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

Robustness and parameter geography in post-translational modification systems

Families of Polynomials in the Study of Biochemical Reaction Networks

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