Oscillations in MAPK cascade triggered by two distinct designs of coupled positive and negative feedback loops

Sarma, Uddipan; Ghosh, Indira

doi:10.1186/1756-0500-5-287

BMC Research Notes

Table 2 Flux of signal flow in cytoplasmic MAPK cascades ‘S1’ and ‘S2’

From: Oscillations in MAPK cascade triggered by two distinct designs of coupled positive and negative feedback loops

Model reactions	Flux equations in model S1	Flux equations in model S2
1] M3K→M3K*	$v 1_{neg} = \frac{\frac{S i g . M 3 K}{K 1}}{(1 + \frac{M 3 k}{K 1}) . (1 + {(\frac{{M K}^{* *}}{K I})}^{n 1})}$	$v 1_{pos} = \frac{(S i g . \frac{M 3 K}{K 1}) . (1 + {(\frac{A . {M K}^{* }}{K a})}^{n}}{(1 + \frac{M 3 K}{K 1}) . (1 + {(\frac{{M K}^{ *}}{K a})}^{n 1}}$
2] M3K*→ M3K	$v 2 = \frac{k 2 . P 1 \frac{{M 3 K}^{}}{K 2}}{1 + \frac{{M 3 K}^{}}{K 2}}$	$v 2 = \frac{k 2 . P 1 . \frac{{M 3 K}^{}}{K 2}}{1 + \frac{{M 3 K}^{}}{K 2}}$
3] M2K→M2K*	$v 3_{pos} = \frac{(k 3 . {M 3 K}^{} . \frac{M 2 K}{K 3}) . (1 + {(\frac{A . {M K}^{ }}{K a})}^{n 3})}{(1 + \frac{{M 3 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K a})}^{n 3})}$	$v 3_{neg} = \frac{(k 3 . {M 3 K}^{} . \frac{M 2 K}{K 3})}{(1 + \frac{{M 2 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K I})}^{n 3})}$
4] M2K→M2K*	$v 4_{pos} = \frac{(k 4 . {M 3 K}^{} . \frac{M 2 K }{K 4}) . (1 + {(\frac{A . {M K}^{* }}{K a})}^{n 4})}{(1 + \frac{M 2 K }{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K a})}^{n 4})}$	$v 4_{neg} = \frac{(k 4 . {M 3 K}^{} . \frac{M 2 K }{K 4})}{(1 + \frac{{M 2 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{ *}}{K I})}^{n 4})}$
5] M2K*→ M2K	$v 5 = \frac{\frac{k 5 . P 2 . {M 2 K}^{* }}{K 5}}{1 + \frac{M 2 K }{K 5} + \frac{M 2 K }{K 6}}$	$v 5 = \frac{\frac{k 5 . P 2 . {M 2 K}^{* }}{K 5}}{1 + \frac{{M 2 K}^{ }}{K 5} + \frac{{M 2 K}^{}}{K 6}}$
6] M2K*→ M2K	$v 6 = \frac{\frac{k 6 . P 2 . {M 2 K}^{}}{K 6}}{1 + \frac{M 2 K }{K 6} + \frac{{M 2 K}^{* *}}{K 5}}$	$v 6 = \frac{\frac{k 6 . P 2 . M 2 K }{K 6}}{1 + \frac{{M 2 K}^{}}{K 6} + \frac{{M 2 K}^{* *}}{K 5}}$
7] MK→ MK*	$v 7 = \frac{\frac{k 7 . M 2 K * * . M K}{K 7}}{1 + \frac{M K}{K 7} + \frac{{M K}^{*}}{K 8}}$	$v 7 = \frac{\frac{k 7 . {M 2 K}^{* } . M K}{K 7}}{1 + \frac{M K}{K 7} + \frac{{M K}^{}}{K 8}}$
8] MK→ MK*	$v 8 = \frac{\frac{k 8 . M 2 K * * . {M K}^{}}{K 8}}{1 + \frac{M K }{K 8} + \frac{M K}{K 7}}$	$v 8 = \frac{\frac{k 8 . {M 2 K}^{* } . {M K}^{}}{K 8}}{1 + \frac{{M K}^{*}}{K 8} + \frac{M K}{K 7}}$
9] MK** → MK*	$v 9 = \frac{\frac{k 9 . P 3 . {M 3 K}^{* }}{K 9}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$	$v 9 = \frac{\frac{k 9 . P 3 . {M K}^{* }}{K 9}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$
10] MK* → MK	$v 10 = \frac{\frac{k 10 . P 3 . {M K}^{}}{K 10}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$	$v 10 = \frac{\frac{k 10 . P 3 . M K }{K 10}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$

The fluxes of signal flow and the values of kinetic parameters used for simulation of S1, S2, S1n and S2n are shown in Table2. In this table, Ki, i = 1–10 are the Km values of the reactions and ki, i = 2–10 are the kcat values of the reactions. The numerical value of ‘i’ corresponding to Ki and ki represents the reaction number, KI are the kinetic parameters associated with negative feedback. Ka and A are the kinetic constants associated with the positive feedback. The hill coefficient used in the equations 1, 3 and 4 are shown as n1, n3 and n4 respectively.

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Model reactions	Flux equations in model S1	Flux equations in model S2
1] M3K→M3K*	$v 1_{neg} = \frac{\frac{S i g . M 3 K}{K 1}}{(1 + \frac{M 3 k}{K 1}) . (1 + {(\frac{{M K}^{* *}}{K I})}^{n 1})}$	$v 1_{pos} = \frac{(S i g . \frac{M 3 K}{K 1}) . (1 + {(\frac{A . {M K}^{* }}{K a})}^{n}}{(1 + \frac{M 3 K}{K 1}) . (1 + {(\frac{{M K}^{ *}}{K a})}^{n 1}}$
2] M3K*→ M3K	$v 2 = \frac{k 2 . P 1 \frac{{M 3 K}^{}}{K 2}}{1 + \frac{{M 3 K}^{}}{K 2}}$	$v 2 = \frac{k 2 . P 1 . \frac{{M 3 K}^{}}{K 2}}{1 + \frac{{M 3 K}^{}}{K 2}}$
3] M2K→M2K*	$v 3_{pos} = \frac{(k 3 . {M 3 K}^{} . \frac{M 2 K}{K 3}) . (1 + {(\frac{A . {M K}^{ }}{K a})}^{n 3})}{(1 + \frac{{M 3 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K a})}^{n 3})}$	$v 3_{neg} = \frac{(k 3 . {M 3 K}^{} . \frac{M 2 K}{K 3})}{(1 + \frac{{M 2 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K I})}^{n 3})}$
4] M2K→M2K*	$v 4_{pos} = \frac{(k 4 . {M 3 K}^{} . \frac{M 2 K }{K 4}) . (1 + {(\frac{A . {M K}^{* }}{K a})}^{n 4})}{(1 + \frac{M 2 K }{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{* *}}{K a})}^{n 4})}$	$v 4_{neg} = \frac{(k 4 . {M 3 K}^{} . \frac{M 2 K }{K 4})}{(1 + \frac{{M 2 K}^{}}{K 4} + \frac{M 2 K}{K 3}) . (1 + {(\frac{{M K}^{ *}}{K I})}^{n 4})}$
5] M2K*→ M2K	$v 5 = \frac{\frac{k 5 . P 2 . {M 2 K}^{* }}{K 5}}{1 + \frac{M 2 K }{K 5} + \frac{M 2 K }{K 6}}$	$v 5 = \frac{\frac{k 5 . P 2 . {M 2 K}^{* }}{K 5}}{1 + \frac{{M 2 K}^{ }}{K 5} + \frac{{M 2 K}^{}}{K 6}}$
6] M2K*→ M2K	$v 6 = \frac{\frac{k 6 . P 2 . {M 2 K}^{}}{K 6}}{1 + \frac{M 2 K }{K 6} + \frac{{M 2 K}^{* *}}{K 5}}$	$v 6 = \frac{\frac{k 6 . P 2 . M 2 K }{K 6}}{1 + \frac{{M 2 K}^{}}{K 6} + \frac{{M 2 K}^{* *}}{K 5}}$
7] MK→ MK*	$v 7 = \frac{\frac{k 7 . M 2 K * * . M K}{K 7}}{1 + \frac{M K}{K 7} + \frac{{M K}^{*}}{K 8}}$	$v 7 = \frac{\frac{k 7 . {M 2 K}^{* } . M K}{K 7}}{1 + \frac{M K}{K 7} + \frac{{M K}^{}}{K 8}}$
8] MK→ MK*	$v 8 = \frac{\frac{k 8 . M 2 K * * . {M K}^{}}{K 8}}{1 + \frac{M K }{K 8} + \frac{M K}{K 7}}$	$v 8 = \frac{\frac{k 8 . {M 2 K}^{* } . {M K}^{}}{K 8}}{1 + \frac{{M K}^{*}}{K 8} + \frac{M K}{K 7}}$
9] MK** → MK*	$v 9 = \frac{\frac{k 9 . P 3 . {M 3 K}^{* }}{K 9}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$	$v 9 = \frac{\frac{k 9 . P 3 . {M K}^{* }}{K 9}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$
10] MK* → MK	$v 10 = \frac{\frac{k 10 . P 3 . {M K}^{}}{K 10}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$	$v 10 = \frac{\frac{k 10 . P 3 . M K }{K 10}}{1 + \frac{{M K}^{ }}{K 9} + \frac{{M K}^{}}{K 10}}$