Spatio-temporal dependence of the signaling response in immune-receptor trafficking networks regulated by cell density: a theoretical model

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) 'k I ' is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) 'k E ', the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) 'k R ', the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as α e lt + β e mt + γ with α , β , γ  = constant and l and m are roots of a quadratic equation,r 2 + k I + k E + k R r + k I k E + k I k R + k E k R = 0 . When l and m are realk I 2 + k E 2 + k R 2 > 2 k I k E + k E k R + k R k I , amounts of proteins in three states shows no oscillatory change but a monotonic change after a transient increase (or decrease); when l and m are complexk I 2 + k E 2 + k R 2 < 2 k I k E + k E k R + k R k I , amounts of proteins in three states are expressed as α e lt + β e mt + γ = 2 g 2 + h 2 sin b t + σ e at + γ ( α , β , l , m  = complex and γ , a , b , g , h , σ  = real: α , β  = conjugate each other; l , m  = 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,k I ,k E andk R . The present study indicates that complex numbers have an essentially important meaning in appearance of oscillatory phenomena in bodily and cellular function.

Mathematical modelling and computational study of two-dimensional and three-dimensional dynamics of receptor-ligand interactions in signalling response mechanisms

Cell signalling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signalling pathways. But the binding process still presents many enigmas and, by analogy with surface catalytic reactions, two different mechanisms can be conceived: the first mechanism is related to the Eley-Rideal (ER) mechanism, i.e. the bulk-dissolved ligand interacts directly by pure three-dimensional (3D) diffusion with the specific surface receptor; the second mechanism is similar to the Langmuir-Hinshelwood (LH) process, i.e. 3D diffusion of the ligand to the cell surface followed by reversible ligand adsorption and subsequent two-dimensional (2D) surface diffusion to the receptor. A situation where both mechanisms simultaneously contribute to the signalling process could also occur. The aim of this paper is to perform a computational study of the behavior of the signalling response when these different mechanisms for ligand-receptor interactions are integrated into a model for signal transduction and ligand transport. To this end, partial differential equations have been used to develop spatio-temporal models that show trafficking dynamics of ligands, cell surface components, and intracellular signalling molecules through the different domains of the system. The mathematical modeling developed for these mechanisms has been applied to the study of two situations frequently found in cell systems: (a) dependence of the signal response on cell density; and (b) enhancement of the signalling response in a synaptic environment.