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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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Related in: MedlinePlus

Rate curves [6] of system (2) for the parameters k1 = 8, k2 = 1, k3 = 1, k4 = 1.5. The thick solid line is the rate of the removal of reactant X (sum of the negative terms in ) and the thick dashed line the rate of production (positive term in ). The three crossings indicate the three steady states . The thin lines show the contributions of the three degradation terms separately: quadratic term k2x2 dashed, the effectively cubic term k3 xy solid, and the linear term k4 x dotdashed. The inset shows a zoomed version for x < 2.1.
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Figure 2: Rate curves [6] of system (2) for the parameters k1 = 8, k2 = 1, k3 = 1, k4 = 1.5. The thick solid line is the rate of the removal of reactant X (sum of the negative terms in ) and the thick dashed line the rate of production (positive term in ). The three crossings indicate the three steady states . The thin lines show the contributions of the three degradation terms separately: quadratic term k2x2 dashed, the effectively cubic term k3 xy solid, and the linear term k4 x dotdashed. The inset shows a zoomed version for x < 2.1.

Mentions: Figure 2 shows rate curves of system (2). It can be seen that the three crossings of production and degradation rate (i.e. the three steady states) are due to the different contributions of the three degradation terms. This implies a simple general procedure for designing bi- or multistable systems: a bistable system can be created with one function for production and three different functions for degradation, e.g. a linear, a quadratic, and a cubic one as in our simple example system. Accordingly, tristable systems require 5 different functions to enable 5 crossings (three stable and two unstable steady states, cf. point 3 in Methods), and so forth for more steady states. This observation helps constructing minimal and/or realistic models of more complicated multistable systems. It can also be a starting point for the design of real bistable systems, for instance in synthetic biology. Note that all enzyme kinetic rate laws can be modelled with polynomial ODEs [40].


The smallest chemical reaction system with bistability.

Wilhelm T - BMC Syst Biol (2009)

Rate curves [6] of system (2) for the parameters k1 = 8, k2 = 1, k3 = 1, k4 = 1.5. The thick solid line is the rate of the removal of reactant X (sum of the negative terms in ) and the thick dashed line the rate of production (positive term in ). The three crossings indicate the three steady states . The thin lines show the contributions of the three degradation terms separately: quadratic term k2x2 dashed, the effectively cubic term k3 xy solid, and the linear term k4 x dotdashed. The inset shows a zoomed version for x < 2.1.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 2: Rate curves [6] of system (2) for the parameters k1 = 8, k2 = 1, k3 = 1, k4 = 1.5. The thick solid line is the rate of the removal of reactant X (sum of the negative terms in ) and the thick dashed line the rate of production (positive term in ). The three crossings indicate the three steady states . The thin lines show the contributions of the three degradation terms separately: quadratic term k2x2 dashed, the effectively cubic term k3 xy solid, and the linear term k4 x dotdashed. The inset shows a zoomed version for x < 2.1.
Mentions: Figure 2 shows rate curves of system (2). It can be seen that the three crossings of production and degradation rate (i.e. the three steady states) are due to the different contributions of the three degradation terms. This implies a simple general procedure for designing bi- or multistable systems: a bistable system can be created with one function for production and three different functions for degradation, e.g. a linear, a quadratic, and a cubic one as in our simple example system. Accordingly, tristable systems require 5 different functions to enable 5 crossings (three stable and two unstable steady states, cf. point 3 in Methods), and so forth for more steady states. This observation helps constructing minimal and/or realistic models of more complicated multistable systems. It can also be a starting point for the design of real bistable systems, for instance in synthetic biology. Note that all enzyme kinetic rate laws can be modelled with polynomial ODEs [40].

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