Energy transfer and molecular switching. i. the nerve action potential

Time evolution of the action potential in plant cells

In this paper we extend and reconsider a solitonic model of the actionpotential in biological membranes for the case of plant cells. Aiming togive at least a qualitative description of the K(+),Cl(-) and Ca(2+) driven process of propagation ofthe action potential along plant cells we put forward the hypothesis ofthree scalar fields φ(i) (X), i = 1, 2, 3 which representK(+), Cl(-) and Ca(2+) ions,respectively. The modulus squared of these fields carries the usualquantum-mechanical (probabilistic) interpretation of the wave function. Onthe other hand, the fields are described themselves by the Lagrangiandensities ℒ[Formula: see text]. Moreover, the interaction and self-interaction term ℒ[Formula: see text] between thefields is considered. The Lagrangian densities ℒ[Formula: see text]include a double-well potential (which is proportional toσ(4) (i)) that leads to spontaneous symmetrybreaking which may produce structures with non-zero topological charge, e.g.longitudinal solitons. In order to describe the transversal motion of theions of concern we need to assume only non-uniform solutions of the system of equation of motion. Hence we seek for solutions (travelling waves) whichpreserve the shape and which move without dissipation and in this way wereconstruct the main dynamical features of the action potential in plants.

Catastrophe modelling in the biological sciences

Catastrophe Theory was developed in an attempt to provide a form of Mathematics particularly apt for applications in the biological sciences. It was claimed that while it could be applied in the more conventional "physical" way, it could also be applied in a new "metaphysical" way, derived from the Structuralism of Saussure in Linguistics and Lévi-Strauss in Anthropology. Since those early beginnings there have been many attempts to apply Catastrophe Theory to Biology, but these hopes cannot be said to have been fully realised. This paper will document and classify the work that has been done. It will be argued that, like other applied Mathematics, applied Catastrophe Theory works best where the underlying laws are securely known and precisely quantified, requiring those same guarantees as does any other branch of Mathematics when it confronts a real-life situation.