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The smallest chemical reaction system with bistability.

Wilhelm T - BMC Syst Biol (2009)

Bottom Line: 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Theoretical Systems Biology, Institute of Food Research, Norwich Research Park, Colney Lane, Norwich NR4 7UA, UK. thomas.wilhelm@bbsrc.ac.uk

ABSTRACT

Background: 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.

Results: 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.

Conclusion: 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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Interaction graph of system (2). It follows directly from the off-diagonal elements of the general Jacobian (3). The positive feedback loop is the only instability causing structure (ICS) in the system, allowing for a locally unstable steady state (presupposition for bistability).
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Figure 3: Interaction graph of system (2). It follows directly from the off-diagonal elements of the general Jacobian (3). The positive feedback loop is the only instability causing structure (ICS) in the system, allowing for a locally unstable steady state (presupposition for bistability).

Mentions: where the indices x and y denote the corresponding partial differentiation. The off-diagonal elements represent the fundamental activating and inhibiting interactions in the system: the positive v2x in JG21 shows that x activates y by the second reaction, and equivalently for the two terms in JG12 : v1y → y activates x by the first reaction, - v3y → y inhibits x by the third reaction. Figure 3 shows the corresponding interaction (incidence) graph summarizing these interactions. Interaction graphs can often be found in the biological literature and corresponding databases (KEGG [42]; BIOBASE [43,44]; Dynamic Signaling Maps . ICSA [39] was developed for structural analyses of (bio)chemical systems (KEGG [42]; BRENDA [45]) AND such interaction graphs.


The smallest chemical reaction system with bistability.

Wilhelm T - BMC Syst Biol (2009)

Interaction graph of system (2). It follows directly from the off-diagonal elements of the general Jacobian (3). The positive feedback loop is the only instability causing structure (ICS) in the system, allowing for a locally unstable steady state (presupposition for bistability).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2749052&req=5

Figure 3: Interaction graph of system (2). It follows directly from the off-diagonal elements of the general Jacobian (3). The positive feedback loop is the only instability causing structure (ICS) in the system, allowing for a locally unstable steady state (presupposition for bistability).
Mentions: where the indices x and y denote the corresponding partial differentiation. The off-diagonal elements represent the fundamental activating and inhibiting interactions in the system: the positive v2x in JG21 shows that x activates y by the second reaction, and equivalently for the two terms in JG12 : v1y → y activates x by the first reaction, - v3y → y inhibits x by the third reaction. Figure 3 shows the corresponding interaction (incidence) graph summarizing these interactions. Interaction graphs can often be found in the biological literature and corresponding databases (KEGG [42]; BIOBASE [43,44]; Dynamic Signaling Maps . ICSA [39] was developed for structural analyses of (bio)chemical systems (KEGG [42]; BRENDA [45]) AND such interaction graphs.

Bottom Line: 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Theoretical Systems Biology, Institute of Food Research, Norwich Research Park, Colney Lane, Norwich NR4 7UA, UK. thomas.wilhelm@bbsrc.ac.uk

ABSTRACT

Background: 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.

Results: 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.

Conclusion: 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.

Show MeSH
Related in: MedlinePlus