Essential requirement of complex number for oscillatory phenomenon in intracellular trafficking process

Intracellular protein trafficking processes consisting of three intracellular states are described by three differential equations. To solve the equations, a quadratic equation is required, and its roots are generally real or complex. The purpose of the present study is to clarify the meanings of roots of real and complex numbers. To clarify the point, we define that: 1) ' $$ ' is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) ' $$ ', the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) ' $$ ', the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as $$ with $$  = constant and $$ and $$ are roots of a quadratic equation, $$ . When $$ and $$ are real $$ , amounts of proteins in three states shows no oscillatory change but a monotonic change after a transient increase (or decrease); when $$ and $$ are complex $$ , amounts of proteins in three states are expressed as $$ ( $$ , $$ , $$ , $$  = complex and $$ $$  = real: $$  = conjugate each other; $$  = conjugate each other), showing an oscillatory change with time. The frequency of oscillatory change appearance is evaluated to be 60% at random combinations of three trafficking rates, $$ , $$ and $$ . The present study indicates that complex numbers have an essentially important meaning in appearance of oscillatory phenomena in bodily and cellular function.