The smallest chemical reaction system with bistability Literature seminar Vladimir Kiselev 21/10/13 Monday, 21 October 13 

Model of the month Monday, 21 October 13 

Bistability Balls marked "1" and "3" are in the two stable positions. Ball marked "2" can go to any of the two steady states. * Applications in physics and engineering. * Description of biological and chemical systems * Involved in large variety of bioprocesses and diseases, like cancer. Monday, 21 October 13 

Bistability Monday, 21 October 13 

Bistability conditions Necessary: * Positive feedback loops. * Presence of "some type of non-linearity" or "ultrasensitivity" in the feedback loop. Properties: * A positive feedback loop can be naturally introduced in the system under consideration * The ultrasensitivity condition is difficult to satisfy. Balls marked "1" and "3" are in the two stable positions. Ball marked "2" can go to any of the two steady states. Monday, 21 October 13 

Bistability conditions * Current biological systems usually involve hundreds of species -> it can be sometimes impossible to find their bistability conditions. * In many cases large systems can be reasonably split into small subsystems * An extreme case of such an approach is the identification of the bistability minimal system. (Wilhelm T. The smallest chemical reaction system with bistability. BMC Syst Biol. (2009). Sep 8;3:90) Balls marked "1" and "3" are in the two stable positions. Ball marked "2" can go to any of the two steady states. Monday, 21 October 13 

Minimal bistable system Criteria (in decreasing order of importance): 1. Minimal number of reactants 2. Minimal number of reactions 3. Minimal number of terms in the ODEs S is incorporated in k1 Monday, 21 October 13 

Minimal bistable system * Three steady states (two of which are always stable and the third one is always unstable) * 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 is a toggle-switch: small fluctuations in the concentrations would drive the system to the positive steady state. Solid lines - locally stable steady states Dashed line - locally unstable steady states. Monday, 21 October 13 

Minimal bistable system COPASI simulation Monday, 21 October 13 

Generalisation * The three crossings between the total production and total degradation rates are due to the shape of the degradation rate curve. * The shape is the result of summation of three degradation rates (thin lines). * Note that all three degradation rate lines are three different functions of X: linear (k4X), quadratic (k2X2) and effectively cubic (k3XY ) Thick solid line - total rate of the removal of reactant X (sum of all thin lines-three removal rates) Thick dashed line - rate of production of X. The three crossings indicate the three steady states (0,2,6). Monday, 21 October 13 

Generalisation “A bistable system can be created with one function for production and three different functions for degradation” “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 Monday, 21 October 13 

The Instability Causing Structure Analysis * ICSA is a new method for topological network analysis developed by the author * Application of this method to the minimal system provides some insights, which complement the analysis described above * The system contains one positive and one negative feedback loop, which are represented by reactions 1-3 * Positive feedback loop (reaction 1-2) is the necessary condition for bistability Monday, 21 October 13 

* The negative feedback loop (reactions 2-3) is also essential for bistability. * Without cubic degradation term there are just two steady states, one of which is unstable. * A negative feedback loop is necessary for the bistability of system! (it prevents explosion of the system). The Instability Causing Structure Analysis Monday, 21 October 13 

* In agreement with the second necessary condition of bistability reaction 4 in the system is the simplest equivalent of ultrasensitivity (or the mechanism of filtering out small stimuli). * Without this reaction (linear degradation term) the second unstable steady-state would merge with the first stable steady-state making it unstable. The Instability Causing Structure Analysis Monday, 21 October 13 

Paper Example Monday, 21 October 13 

1) A mechanism for preventing explosions (negative feedback loop) 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). 2) Since the system is proved to be minimal, the negative feedback loop is probably indeed a typical feature of bistable systems. 3) Interestingly, 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. Conclusions Monday, 21 October 13