Theoretical Systems Biology, Institute of Food Research, Norwich Research Park, Colney Lane, Norwich NR4 7UA, 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.

Background

Bistability is key for understanding basic phenomena of cellular functioning, such as decision-making processes in cell cycle progression, cell differentiation, and apoptosis

Bistable switches are typically enabled by positive feedback loops in signal transduction networks. Here a sufficiently strong (external) signal switches on a self-amplifying process leading to expression of the corresponding target genes. Due to the corresponding hysteresis effect this process can retain its activity without a persistent signal. Such switches are therefore called 'decision-making'. One often discussed example is the restriction point control for the regulation of G1-S transition of the mammalian cell cycle, where recently a detailed small ordinary differential equation (ODE) model was presented

Given its outstanding biological importance, it is clear that bistable switches also attracted the attention of theoretical biologists. A frequently discussed problem is the necessary (and/or sufficient) condition for bistability. The central result goes back to the work of Clarke

Another approach for identifying necessary structural conditions for any dynamic behaviour is the identification of the corresponding minimal systems. Such systems have the advantage of being "simple enough to understand at an intuitive level"

Distinguished minimal MAK systems

**System**

**Reaction scheme**

**MAK model (ODEs)**

**Ref.**

**Minimal bistable MAK system**

**Minimal bistable chemical system**

This paper

**Minimal oscillating MAK system**

**Minimal MAK system with limit cycle**

**Minimal chemical system with limit cycle **

Here we present and discuss the smallest bistable chemical reaction system. Application of our previously presented Instability Causing Structure Analysis (ICSA

Results

The smallest bistable chemical reaction system

We define the smallest chemical system (contains only mono- and bimolecular reactions, reversible reactions are considered as two irreversible ones) by the following criteria in decreasing order of importance:

According to this definition, the following bistable system is unique (Methods section contains the proof for this statement).

Assuming spatially homogeneous conditions, the system can be described by the two-component mass-action kinetic ODE system (_{1}):

Due to its simplicity, the mathematical analysis of the system is simple as well. The system has two elementary flux modes

One mode uses reactions 1-3, and the other reactions 1,2, and 4. Bistability can, of course, only arise if all reactions are active. Introducing dimensionless quantities (_{1}/(_{2}_{1}, _{3}/_{2 }→ _{3}, _{4}/(_{2}_{4}, _{2}_{2 }= 1, without restriction of generality (_{1}, _{3}, _{4 }> 0). The system has three steady states: _{1 }- 4_{3}_{4}. A saddle-node bifurcation occurs at

Generally, in two-component systems a steady state is locally stable if the trace ^{2}, focus otherwise). If the corresponding determinant is negative, the steady state is a saddle point. It can be seen from the Jacobian _{1}_{4}, ^{2 }at the first and third steady state, so these are always stable nodes.

For _{1 }= 8, _{3 }= 1, _{4 }= 1.5 the second and third steady state are _{bi }= 1, the concentration S is identical to the apparent rate constant _{1 }= _{bi}_{1 }small fluctuations in the concentrations would drive the system to the positive steady state (the 'on' state).

Signal-response curve (bifurcation diagram) of system (2) for the parameters _{1 }= 8, _{2 }= 1, _{3 }= 1, _{4 }= 1.5

**Signal-response curve (bifurcation diagram) of system (2) for the parameters **_{1 }= 8, _{2 }= 1, _{3 }= 1, _{4 }= 1.5. Solid lines indicate locally stable steady states, the dashed line locally unstable steady states. The inset shows the signal-response curve if an additional small constant influx into X (here 0.6) is assumed (enabling a positive 'off' state, leaving the 'on' state and bifurcation point nearly unchanged). This is the classical toggle switch (terminology of Tyson et al. (6), others use the term toggle switch to describe a double negative (i.e. positive) feedback loop (4)) picture enabling the hysteresis cycle: starting with low values and increasing the signal continuously increases the response, until the saddle-node bifurcation at about S = 1.7 is reached. Further increase of the signal leads to a sudden jump of the response to the upper steady state. Decreasing the signal now leads to a continuous decrease of the response, the systems stays in the upper steady state until the left bifurcation point is reached where the response jumps back to the lower steady state.

Figure

Rate curves _{1 }= 8, _{2 }= 1, _{3 }= 1, _{4 }= 1

**Rate curves **** of system (2) for the parameters **_{1 }= 8, _{2 }= 1, _{3 }= 1, _{4 }= 1.5. The thick solid line is the rate of the removal of reactant X (sum of the negative terms in _{2}^{2 }dashed, the effectively cubic term _{3 }_{4 }

The Instability Causing Structure Analysis (ICSA) of system (2)

Recently we presented a new method for topological network analysis of dynamical systems, the

ICSA needs no knowledge about kinetic details. It can be applied to (bio)chemical systems where just the stoichiometric matrix is known (or even just the signs of its elements) or to signal transduction/gene regulatory networks represented by interaction graphs

We apply ICSA for additional analysis of system (2). The stoichiometric matrix **S **is given in (2c). Multiplication of **S **with the reaction velocity substrates ector (contains the substrates for each reaction) (_{1 }(_{2 }(_{3 }(_{4 }(^{T }leads to **J**_{G }(39) of system (2):

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 _{2x }in **J**_{G21 }shows that x activates y by the second reaction, and equivalently for the two terms in **J**_{G12 }: _{1y }→ y activates x by the first reaction, - _{3y }→ y inhibits x by the third reaction. Figure

Interaction graph of system (2)

**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).

The system contains one positive and one negative feedback loop. The positive loop is the necessary structural condition for bistability

System (2) is the minimal chemical reaction system with bistability. Therefore, any ingredient is essential. That also means, without the negative loop (without reaction 3 being catalyzed by Y) this system cannot be bistable. In fact, the negative feedback prevents explosion of the system: without reaction 3 the system has just one locally stable off-state and one unstable positive steady state. Figure

Summarizing, system (2) contains the three different necessary conditions for bistability: (i) a positive feedback loop, (ii) a mechanism for filtering out small stimuli, and (iii) a mechanism for preventing explosion. Interestingly, the third condition lacks discussion in the literature

The first step in the general ICSA procedure is the analysis of feedback cycles resulting from the off-diagonal terms of the general Jacobian. The second step is the identification of the topological structures that actually cause instability. As mentioned above, standard local stability analysis of two-component systems needs consideration of the Jacobian's trace and determinant _{2x }_{3y }+ _{1y }(_{3x }+ _{4x }- _{2x}). The only negative term in

Discussion

The minimal bistable chemical reaction system (2) is based on definition (1). This definition is based on a chemical/physical point of view, but other definitions might also be possible. The definition for the smallest chemical reaction system with Hopf bifurcation _{1 }= _{2 }= 1,

Interestingly, system (2) is similar to a previously presented "minimal reaction network"

Bistable systems play important roles also beyond biology. They are usually depicted by the mechanical example of a ball rolling into two different valley basins. Bistable chemical systems, in particular, have been studied extensively to analyse relaxation kinetics

Positive feedback is clearly associated with bi/multistability. Negative feedback, in contrast, is often discussed in the context of oscillations **J**_{G }(for system (2) it is given in (3)) for a simple definition of feedback: if the **J**_{G }terms close any cycles, then feedback exists (e.g. the positive and negative feedback cycles of system (2)), otherwise not. Analysis of **J**_{G }guarantees a unique identification of all in a system contained feedback cycles. Goldbeter ^{2+}, and cAMP oscillations. However, a more detailed analysis of these systems shows that a negative feedback (according to our definition) is always contained (results unpublished). An example is the simplest model for glycolytic oscillations, the Higgins-Selkov oscillator (_{2y }- _{2x }- _{3y }and _{2x}_{3y}, respectively. Obviously, _{2y }in **J**_{G22 }is the only instability causing term (a positive feedback), so the second reaction is the only ICS in the system. However, inspection of the off-diagonal elements of **J**_{G }reveals a negative feedback as well: the larger x, the larger becomes y, but the larger y, the smaller becomes x. The same is realised in the bistable system (2), where the positive feedback is the only ICS and another negative feedback is contained. These examples show how the analysis of the general Jacobian helps clarifying the discussion of feedback loops.

We have shown that a mechanism for preventing explosions is a third necessary condition for bistability (complementing the previously discussed two other conditions positive feedback and filtering out of small stimuli). In system (2) this is achieved by a negative feedback. Other bistable systems contain negative feedbacks as well (e.g. ERK pathway

Conclusion

Bi/multistability and oscillations are the two most important dynamic phenomena in biology. Limit cycle oscillations are associated with biological clocks and cell signalling

Minimal systems are well suited for basic studies and for teaching purposes. This explains the great success of, for instance, the Lotka-Volterra

Methods

System (2) is the smallest bistable chemical reaction system according to definition (1) - proof (inductive proof systematically considering all possibilities):

1. The Schloegl system (Table ^{3 }+b x^{2}-c x+d (a, b, c>0) possesses two positive extrema as can simply be seen considering f'(x) = 0. Thus, the minimal bistable 1d system reads

2. Using our general quasi-steady-state-approximation procedure

3. The lemma of the index sum ^{3 }term (higher order polynomials would require more bimolecular reactions). In 2d MAK systems this can only be realized by one ODE with an xy term and the other with x^{2 }and y terms. Inserting the corresponding steady state expression y = x^{2}..., into the other ODE's xy term gives the cubic term. A direct x^{3 }term is forbidden in bimolecular systems. The symmetric case y^{2 }and x needs no extra consideration. These terms already correspond to at least three reactions. One can show that such three reactions are not sufficient to give a bistable system (the explicit proof is not necessary, because it turns out indirectly from the following analysis). So the minimal bistable system has at least four reactions.

4. To realize a -x^{3 }term in the corresponding steady state equation, one either needs the terms y and x^{2 }with different signs and -xy in the other ODE, or y and x^{2 }with the same signs and +xy in the other ODE. But the same signs variant cannot work: ^{2 }terms must have different signs in one ODE.

5. There are two variants for the different-sign-case: 1. ^{2 }term has to appear also in the ^{2 }term (still realizable by four reactions) would be added in the

6. However, the second system

The slightly modified system replacing reaction

Acknowledgements

I thank Klaus R. Schneider for referring to the lemma of the index sum. The work was supported by a BBSRC Core Strategic Grant for IFR.