A network thermodynamic investigation of stationary and non-stationary proton transport through proteins

A model for the biological transport of protons in linear hydrogen-bonded chains formed from the amino acid side groups of membrane proteins has been investigated indetail.The description assumes first-order kinetics for transitions between all possible proton distributions in the hydrogen-bridged chain. The corresponding master equationis solved numerically and in some representative cases also analytically. The followingtime-dependent observables have been evaluated proton current at the conductor ends charge displacement within the conductor free energy decrement, and state of protonation of the conductor groups. It is shown which observable conductionproperties reveal features of the internal dynamics and structure of proton conductors. In particular, the following observations are considered:titration of the stationary, appliedvoltage-induced proton currents; coupling of the protontransport to alternating electric fields or to electric field jumps; measurement of therelaxation ofthe above four observables following injection or ejection of a proton. Wealso demonstrate thepossibility of constructing heterogeneous conductors with a diodic voltage-current characteristic.Allowing the interaction between the proton conductors and injecting or ejecting group to betime-dependent, we investigated the refractory phase that exists after an initial proton currentpulse and demonstrated the buffering capacity of the conductors, a function that we associatewith the “blue light effect” of bacteriorhodopsin. Among the theoretical developments are analgorithm to obtain the graph of the main kinetic pathways fromthe solution of a highdimensionalmaster equation, an expression for the proton resistance ofa conductor derived in the framework of linear irreversible thermodynamics, a reduction of the kinetic pathways by condensing fast processes to yield analytical expressions for the observables, andfinally theanalytical evaluation of relaxation times by the theory of first-passage times.