Spherical drop of cytoplasm with an effective surface tension influenced by oscillating enzymatic reactions. A simplified model for the biological clock, cell division, and chemotaxis

Abstract

A model suggested as a possible mechanism of a biological clock of the escapement type is constructed. A biological cell is treated as a spherical drop of cytoplasm with embedded contractile elements (microfilaments and microtubules). The contractile elements are thought to be positioned mainly in the thin layer of ectoplasm adjacent to the cell membrane. This layer is described as a mathematical surface with an “effective surface tension” and an “effective surface viscosity.” For simplicity, the synthesis of contractile elements is thought to be effectuated by one or more of the enzymes in a triangular enzyme cycle (Christiansen cycle). The enzyme concentrations in this model may perform damped oscillations around their stationary values, just as enzymes in real enzyme cycles. A fluid drop with surface tension is also able to perform damped oscillations. When the two systems are combined, calculations show that unstable, oscillatory solutions may be obtained. Those may generate limit cycles to be used in a mechanochemical clock. In a cell tissue we have the possibility of mechanical interaction between the cells and of nonlinear synchronization of the individual clocks. If there should not be a complete decoupling between the hydrodynamic and the chemical dispersion relations between complex growth constant and normal mode number, it is very important that the three surface diffusion coefficients of the enzymes are not all equal. Apart from that, calculations reveal several remarkable features of the model: (a) Stationary instabilities have loci which are independent of the absolute values of surface tension. The oscillatory modes have loci of marginal stability which depend on surface tension, except for the l = 1 surface harmonics. (b) For l = 1 (translation mode), stationary instability appears before oscillatory instability. This mode may therefore be used to model chemotaxis of unicellular organisms in the gradient of an external attractant. (c) The mode l = 2 (cell division mode) exhibits oscillatory instability before the appearance of stationary instability when the coupling parameter between enzyme kinetics and hydrodynamics is increased. The model may therefore be used to simulate the mitotic clock as well as the cell division process and various disturbances of the threshold values leading to cancer growth. (d) With the present model it is easy to obtain a perfect compensation of the temperature dependence of the chemical rate constants.