Q-bio:Designing a directed stochastic oscillator

Brief description

Biological networks are full of clocks and other oscillators. A typical way to organize these is by a limit cycle in a deterministic dynamical system. Stochastic dynamics offers other solutions. For example, noise can eject a particle from a fixed point onto a trajectory that leads back to the point, creating a noise-driven oscillator. Another option is to have two fixed points and to switch stochastically among them. Then the characteristic switching times define a period of a stochastic oscillator. In high-dimensional (>1) stochastic dynamical systems, this picture gets even more intriguing. In particular, the dynamics may be non-potential. That is, for example, for a system with three fixed points, the rates of 1-2, 2-3, and 3-1 transitions may be larger than 2-1, 3-2, and 1-3, breaking the symmetry and the detailed balance and creating a net drift along a directed cyclic path in the phase space. In this project, we will

1. design a simple example of this behavior (using, e.g., three mutually-suppressive genes)
2. numerically simulate the system to observe the drift
3. try to calculate the period of the oscillations using either either a variety of standard techniques, or the generating functional path-integral formalism, to be introduced in a lecture by Nikolai Sinitsyn

Contact instructors

Ilya Nemenman and Nikolai Sinitsyn

References

1. E Aurell and K Sneppen. Epigenetics as a first exit problem. Phys Rev Lett 88, 048101, 2002. PDF.
2. W Bialek. Stability and noise in biochemical switches. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in Neural Information Processing Systems 13, pages 103-109. MIT Press, 2001. PDF
3. D Gillespie. Stochastic Simulation of Chemical Kinetics. Ann Rev Phys Chem 58, 35-55, 2007. PDF.
4. N. A. Sinitsyn, and I. Nemenman. EPL 77 (2007) 58001. Abstract.

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