The Study On The Kinetics Of Intracellular Ca²⁺ Spiral Wave

Ca²⁺ is one of the most important messengers. It transmits intracellular signals and takes part in intercellular coordination. The kinetics of the Ca²⁺ concentration involves a transition from locally stochastic release (e.g. Ca²⁺ spark) to intracellular global oscillations and waves, even waves spreading across cells. Ca²⁺ spiral wave is one of the most intriguing Ca²⁺ pattern. In order to explain the dynamics of Ca²⁺ concentration found in experiments, a number of mathematical models are presented.Many important factors in the cell, for example IP₃ concentration, are experimentallycontrollable. Investigating the effect of these controllable factors on Ca²⁺ spiral waves can help us understanding the mechanism of Ca²⁺ exchanging in the cell, and Ca²⁺ signalling. Thus, in this thesis, we have studied the factors effecting the Ca²⁺ spiral waves and the control of Ca²⁺ spiral waves. The main works are as follow:First, based on a spatial extended Tang-Othmer Ca²⁺ model, the dependence of spiral dynamics on IP₃ concentration is studied. we find: (i)The period of Ca²⁺ spiral wave changes un-monotonously with IP₃ concentration, and the increasing of periods corresponds to instability of spiral waves. (ii)Changing IP₃ concentration, the spiral dynamics undergoes fruitful transitions between rigidly rotating and meanderingspiral waves. The transitions are similar to that found in other systems (e.g. BZ reactions). (iii) Understanding the IP₃-dependent Ca²⁺ spiral dynamics, intuitively, some methods of controlling spirals through the control of IP₃ concentrationare introduced. (iv) Our results are experimentally accessable. Then, Based on previous work in BZ reaction, the electric fields are used to control Ca²⁺ spiral waves, and the controlling effects are exhibited by spiral tips. we find: (i) Under the influence of dc electric field, the spiral tip gradually drifts from center to edge of the system along a straight line; (ii) When the applied electric field is periodic, the system resonates at a frequencyω= 2ω₀ and the spiral tip drift along a straight line; (iii) These numerical results can be explained by an analytical method based on the weak deformation approximation.Finally, based on the model presented by Bugrim et. al., we have studied the effect of spatially discrete and random distribution of sites for Ca²⁺ releasing onCa²⁺ spiral waves. It is found that: (i) Only when the random distributions are considered, the spiral waves can be observed in reasonable parameters, vise verse, no spiral waves can be observed; (ii) the model considering random distribution of ion channel clusters can simulation the naturally initiation of Ca²⁺ spiral wave; (iii) when the random distributions are considered, the numerical results accord to experiments.