Taking the interaction between a DNA damage repair module, an ATM module, and a P53--MDM2 oscillation module into account, this paper presents a mathematical model of a P53 oscillation network triggered by a DNA damage signal in individual cells. The effects of the DNA damage signal and the delay time of P53-induced MDM2 expression on the behaviours of the P53 oscillation network are studied. In the oscillatory state of the P53--MDM2 oscillator, it is found that the pulse number of P53--P oscillation increases with the increase of the initial DNA damage signal, whereas the amplitude and the period of P53--P oscillation are fixed for different initial DNA damage signals, and the period numbers of P53--P oscillations decrease with the increase of time delay of MDM2 expression induced by P53. These theoretical predictions are consistent with previous experimental results. The combined negative feedback of P53--MDM2 with the time delay of P53-induced MDM2 expression causes oscillation behaviour in the P53 network.
The robustness and breakup of spiral wave in a two-dimensional lattice networks of neurons are investigated. The effect of small- world type connection is often simplified with local regular connection and the long-range connection with certain probability. The network effect on the development of spiral wave can be better described by local regular connection and changeable long-range connection probability than fixed long-range connection probability because the long-range probability could be changeable in realistic biological system. The effect from the changeable probability for long-range connection is simplified by multiplicative noise. At first, a stable rotating spiral wave is developed by using appropriate initial values, parameters and no-flux boundary conditions, and then the effect of networks is investigated. Extensive numerical studies show that spiral wave keeps its alive and robust when the intensity of multiplicative noise is below a certain threshold, otherwise, the breakup of spiral wave occurs. A statistical factor of synchronization in two-dimensional array is defined to study the phase transition of spiral wave by checking the membrane potentials of all neurons corresponding to the critical parameters(the intensity of noise or forcing current)in the curve for factor of synchronization. The Hindmarsh-Rose model is investigated, the Hodgkin-Huxley neuron model in the presence of the channel noise is also studied to check the model independence of our conclusions. And it is found that breakup of spiral wave is easier to be induced by the multiplicative noise in presence of channel noise.
MA Jun 1,2, TANG Jun 2 , ZHANG AiHua 3 & JIA Ya 2 1 Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
