### Mathematical Biosciences and Engineering

2019, Issue 6: 7589-7615. doi: 10.3934/mbe.2019381
Research article Special Issues

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

• 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

### Related Papers:

• 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.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3

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