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Wilhelm T. (2009). The smallest chemical reaction system with bistability.

September 2013, model of the month by Vladimir Kiselev
Original model: BIOMD0000000233


Bistability is a fundamental phenomenon in nature. Something that is bistable can be resting in either of two states. The defining characteristic of bistability is simply that two stable states (minima) are separated by a peak (maximum). This is illustrated in Figure 1.

Originally bistability found a lot of applications in physics and engineering. However, in the last several decades it has been widely used in describing biological and chemical systems. In biology, bistability is involved in large variety of processes ranging from decision-making in cell cycle progression to apoptosis. It has also been shown to occur in various diseases, like cancer.

Despite the fact that it is easy to understand the basic principal of bistability, it has always been a challenge for scientists and engineers to find its necessary and sufficient conditions. However, recently one of the necessary condition in chemical and biological systems was distinctly defined as the presence of positive feedback loops [1, 2]. Another necessary condition was the presence of "some type of non-linearity" or "ultrasensitivity" in the feedback loop. While a positive feedback loop can be naturally introduced in the system under consideration, the ultrasensitivity condition is difficult to satisfy.

Since current biological systems usually involve hundreds of species, it can be sometimes impossible to find their bistability conditions. However, in many cases large systems can be reasonably split into small subsystems and one could analyse the behaviour of the subsystems first and then incorporate it back to the original system. An extreme case of such an approach is the identification of the bistability minimal system. In this paper, Wilhelm [3, BIOMD0000000233] describes how this can be done.

Figure 2

Figure 2 Signal-response curve (bifurcation diagram) of system (1) for the parameters k1=8, k2=1, k3=1, k4=1.5, (note dimensionless units here). Solid lines indicate locally stable steady states, the dashed line indicate locally unstable steady states. The inset shows the signal-response curve if an additional small constant influx into X (0.6) is assumed. Figure taken from 1].

Generalization of the minimal system (1)

Of high importance, based on system (1) the author shows a simple general procedure of designing bi- or multistable systems. Looking at the plot of rate curves of system (1) (Figure 4), one can see that the three crossings between the total production and total degradation rates are due to the shape of the degradation rate curve. This shape is the result of summation of three degradation rates (thin lines in Figure 4 corresponding to equations 2-4 in system (1)). Note that all three degradation rate lines are three different functions of X: linear (k4X), quadratic (k2X2) and effectively cubic (k3XY )

The author argues this observation implies that there exists a simple general way to design multistable systems: a bistable system can be created with one function for production and three different functions for degradation, like system (1)). Accordingly, tristable systems require 5 different functions to enable 5 crossings (three stable and two unstable) and so forth. This procedure allows one to construct realistic models of more complicated multistable systems or even to design real bistable systems.

Figure 4

Figure 4 Rate curves for the parameters k1=8, k2=1, k3=1, k4=1.5, (note dimensionless units here). The thick solid line is the total rate of the removal of reactant X (sum of all thin lines-three removal rates). The thick dashed line is the rate of production of X. The three crossings indicate the three steady states (0,2,6). The inset shows a zoomed version for X < 2.1. Figure taken from [1].

The Instability Causing Structure Analysis (ICSA) of system (1)

ICSA is a new method for topological network analysis developed by the author [4]. Application of this method to system (1) provides some insights, which complement the analysis described above. The resulting interaction graph of system (1) is presented in Figure 5.

System (1) contains one positive and one negative feedback loop, which are represented by reactions 1-3. As described above the positive feedback loop (reaction 1-2) is the necessary condition for bistability. In agreement with the second condition of bistability reaction 4 in system (1) is the simplest equivalent of ultrasensitivity (or the mechanism of filtering out small stimuli). Figure 4 shows that without this reaction (linear degradation term) the second unstable steady-state would merge with the first stable steady-state making it unstable. Finally, it appears that the negative feedback loop (reactions 2-3) is also essential for bistability. Again, Figure 4 shows that without cubic degradation term there are just two steady states, one of which is unstable. Thus, the negative feedback loop is necessary for the bistability of system (1) (it prevents explosion of the system).

Figure 1

Figure 1 Bistable system. Balls marked "1" and "3" are in the two stable positions (stable steady states). Ball marked "2" can go to any of the two steady states. This figure was taken from the Wikipedia article for bistability.

Minimal system definition and analysis

Wilhelm provides his criteria of the smallest chemical system (in decreasing order of importance):

  1. Minimal number of reactants
  2. Minimal number of reactions
  3. Minimal number of terms in the ODEs

He also proves that according to the definition above, the following bistable system is unique: Equation 1

The author shows that system (1) has three steady states, two of which are always stable and the third one is always unstable. The system behaviour can be represented by signal-response curve (see Figure 2). It shows steady state values of X depending on the signal amplitude S. The saddle-node bifurcation occurs at S = 0.75. Beyond this point the system has two stable steady states and the system's dynamic behaviour becomes that of a toggle- switch: small fluctuations in the concentrations would drive the system to the positive steady state.

By simulating system (1) in COPASI software, one can clearly see how a very small fluctuation of S switches X to the second steady state (Figure 3).

Figure 3

Figure 3 Toggle-switch like behaviour of X with a small fluctuation of S..

Figure 5

Figure 5 Interaction graph of system (1). The positive feedback loop is the only instability causing structure in the system. Figure take from [1].


The author conclude that the mechanism for preventing explosions (negative feedback loop in system (1)) is a third necessary condition for bistability (in complement with the first two: positive feedback loop and filtering out of small stimuli). It has also been shown in some other bistable systems (e.g. ERK pathway). Since system (1) is proved to be minimal, the negative feedback loop is probably indeed a typical feature of bistable systems.

Interestingly, the author mentions that oscillating systems besides the necessary negative feedback loop often contain a positive feedback loop. This makes bistable and oscillatory systems to be based on the same set of feedback cycles.

Bibliographic references

  1. Clarke BL. Stability of Complex Reaction Networks. Wiley Online Library. (2007).
  2. Thomas R. The role of feedback circuits: positive feedback circuits are a necessary condition for positive real eigenvalues of the jacobian matrix. Wiley Online Library. (2010.)
  3. Wilhelm T. The smallest chemical reaction system with bistability. BMC Syst Biol. (2009). Sep 8;3:90.
  4. Wilhelm T. Analysis of structures causing instabilities Phys. Rev. (2007), E 76, 011911.