Asymptotic Analysis On Current-voltage Relationship Between Two Ions Through An Ion Channel

Ion channels are a special type of protein in cells.Their main function is to help water-soluble substances in living cells to enter and leave cells,enabling living cells to perform metabolism,and thus complete a series of behaviors in life activities.Understanding the inherent mechanism of ion channel itself is great significance to biology and clinical medicine,for example,the control of heart rate is related to various ions such as Na+,K+,Ca2+.When a certain channel is diseased,it can cause a series of diseases such as arrhythmia.If intracellular Ca2+or Na+exceeds the normal level for a long time,it will cause abnormal contraction of vascular smooth muscle,leading to brain ischemia and so on.Many scholars at home and abroad have proposed many mathematical models for studying the characteristics of ion channels.The Poisson-Nernst-Planck(PNP)has commonly used for ion channel research,because of considering the actual shape of the ion channel,the ion concentration and the potential difference between the two ends of the channel.One can describe the selectivity of the ion channel by the PNP equation,which can be characterized by the current-voltage(I-V)relationship within the cell membrane.Up to now,using the PNP model to study the current-voltage relationship of ion channels is still one of the hot topics of scholars at home and abroad in recent years.In this paper,we study the current-voltage relationship of two oppositely charged ions under non-electrical neutral conditions.That is,assume the relationship between the two ion concentrations at the left and right ends of the ion channel is L1??L2=?L,R1=?R2?pR,the relationship the current-voltage of the system.For the PNP system,this paper firstly does variable transformation and asymptotic expansion,and transform it into two equivalent subsystems,inner system and outer system.Secondly,the potential and concentration of the zero-order inner system and outer system are expanded asymp-totically with respect to the parameter ? and ?.Similarly,the potential and concentration of the first-order inner system and outer system are also expanded asymptotically with respect to the parameters ? and ?.Then the potential and concentration of second-order inner system and outer system with respect to the parameters ? and ? are derived via the results of the zero-order and first-order.Further,the asymptotic matching expansion method is used to connect the potential and concentration of the inner system and outer system to derive asymptotic expressions of the second-order current-voltage relationship.Finally,we give the current-voltage relationship of the system when the parameters ? and p take different ranges.