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BIOMD0000000233 - Wilhelm2009_BistableReaction


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Reference Publication
Publication ID: 19737387
Wilhelm T.
The smallest chemical reaction system with bistability.
BMC Syst Biol 2009; 3: 90
Theoretical Systems Biology, Institute of Food Research, Norwich Research Park, Colney Lane, Norwich NR4 7UA, UK.  [more]
Original Model: BIOMD0000000233.xml.origin
Submitter: Thomas Wilhelm
Submission ID: MODEL2425356597
Submission Date: 08 Sep 2009 16:56:04 UTC
Last Modification Date: 20 Apr 2012 22:19:37 UTC
Creation Date: 30 Jun 2009 17:26:59 UTC
Encoders:  Thomas Wilhelm
set #1
bqbiol:isVersionOf Gene Ontology response to chemical
set #2
bqbiol:occursIn Taxonomy cellular organisms

This a model from the article:
The smallest chemical reaction system with bistability
Thomas Wilhelm BMC Systems Biology 2009;Sep 8;3:90. 19737387 ,
Bistability underlies basic biological phenomena, such as cell division, differentiation, cancer onset, and apoptosis. So far biologists identified two necessary conditions for bistability: positive feedback and ultrasensitivity.
Biological systems are based upon elementary mono- and bimolecular chemical reactions. In order to definitely clarify all necessary conditions for bistability we here present the corresponding minimal system. According to our definition, it contains the minimal number of (i) reactants, (ii) reactions, and (iii) terms in the corresponding ordinary differential equations (decreasing importance from i-iii). The minimal bistable system contains two reactants and four irreversible reactions (three bimolecular, one monomolecular). We discuss the roles of the reactions with respect to the necessary conditions for bistability: two reactions comprise the positive feedback loop, a third reaction filters out small stimuli thus enabling a stable 'off' state, and the fourth reaction prevents explosions. We argue that prevention of explosion is a third general necessary condition for bistability, which is so far lacking discussion in the literature. Moreover, in addition to proving that in two-component systems three steady states are necessary for bistability (five for tristability, etc.), we also present a simple general method to design such systems: one just needs one production and three different degradation mechanisms (one production, five degradations for tristability, etc.). This helps modelling multistable systems and it is important for corresponding synthetic biology projects.
The presented minimal bistable system finally clarifies the often discussed question for the necessary conditions for bistability. The three necessary conditions are: positive feedback, a mechanism to filter out small stimuli and a mechanism to prevent explosions. This is important for modelling bistability with simple systems and for synthetically designing new bistable systems. Our simple model system is also well suited for corresponding teaching purposes.

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This model originates from BioModels Database: A Database of Annotated Published Models ( It is copyright (c) 2005-2012 The Team.
For more information see the terms of use .
To cite BioModels Database, please use: Li C, Donizelli M, Rodriguez N, Dharuri H, Endler L, Chelliah V, Li L, He E, Henry A, Stefan MI, Snoep JL, Hucka M, Le Novère N, Laibe C (2010) BioModels Database: An enhanced, curated and annotated resource for published quantitative kinetic models. BMC Syst Biol., 4:92.

Publication ID: 19737387 Submission Date: 08 Sep 2009 16:56:04 UTC Last Modification Date: 20 Apr 2012 22:19:37 UTC Creation Date: 30 Jun 2009 17:26:59 UTC
Mathematical expressions
r1 r2 r3 r4
Physical entities
Compartments Species
batch S P X
Reactions (4)
 r1 [S] + [Y] → 2.0 × [X];  
 r2 2.0 × [X] → [X] + [Y];  
 r3 [X] + [Y] → [P] + [Y];  
 r4 [X] → [P];  
 batch Spatial dimensions: 3.0  Compartment size: 1.0  (Units: Predefined unit volume)
Compartment: batch
Initial concentration: 1.0  (Units: substance)
Compartment: batch
Initial concentration: 1.0  (Units: substance)
Compartment: batch
Initial concentration: 1.0  (Units: substance)
Compartment: batch
Initial concentration: 1.0  (Units: substance)
r1 (1)
Value: 8.0
r2 (1)
Value: 1.0
r3 (1)
Value: 1.0
r4 (1)
Value: 1.5
Representative curation result(s)
Representative curation result(s) of BIOMD0000000233

Curator's comment: (updated: 12 Oct 2009 15:56:19 BST)

The reproduces figure 1 of the reference publication.The bifurcation analysis was performed using the AUTO2000 front end and SBW 2.7.10. Both branches were calculated independently, the off state with X[t=0]=1 and the on state with X[t=0]=4. Continuation was performed from S=0 to 2 in negative direction.