Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system

Magalí Giaroli; Rick Rischter; Mercedes P. Millán; Alicia Dickenstein; Magalí Giaroli; Rick Rischter; Mercedes P. Millán; Alicia Dickenstein

doi:10.3934/mbe.2019381

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7589-7615. doi: 10.3934/mbe.2019381

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Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system

1.
Dto. de Matemática, FCEN, Universidad de Buenos Aires, and IMAS (UBA-CONICET), Ciudad Universitaria, Pab. I, C1428EGA Buenos Aires, Argentina
2.
Instituto de Matemática e Computação, IMC, Universidade Federal de Itajubá (UNIFEI), Av. BPS 1303, Bairro Pinheirinho, 37500-903, Itajubá, Minas Gerais, Brazil

Received: 25 April 2019 Accepted: 13 August 2019 Published: 20 August 2019

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 $2 \lfloor \frac{n}{2} \rfloor+1$ 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 $\frac 1 4$ 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.
Citation: Magalí Giaroli, Rick Rischter, Mercedes P. Millán, Alicia Dickenstein. Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7589-7615. doi: 10.3934/mbe.2019381

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Abstract

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 $2 \lfloor \frac{n}{2} \rfloor+1$ 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 $\frac 1 4$ 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.

References

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