Predictions of thermodynamic efficiency in a pumped biochemical reaction

We propose and analyze a possible experimental system for the investigation of the thermodynamic efficiency of generating biochemical gradients. We investigate the efficiency of a model pump that uses 6-phosphofructokinase (EC 2.7.1.11, an enzyme that exhibits highly nonlinear kinetics), chromatophores from Rhodobacter sphaeroides, and light to generate a biochemical gradient. We analyze the experimental system and an equivalent alternative configuration and show that the establishment and maintenance of a concentration gradient across a membrane is thermodynamically equivalent to the establishment and maintenance of a stationary state in a single-phase, isothermal, open, homogeneous reaction system. With a constant input of light, the system can exist in a stable node (a stable steady state), a stable focus (upon perturbation from its steady state, the system returns to its steady state with an oscillatory component), and a stable limit cycle (at steady state the system exhibits stable oscillations). We investigate the efficiency of the system with both steady and oscillatory light input and observe efficiency changes that depend upon the autonomous state of the system and the frequency and amplitude of the periodic light input. When the system is in a stable focus, an efficiency maximum is seen when the system is perturbed at its resonant frequency. When the system is in a stable limit cycle, efficiency increases are seen near the 1:3, 1:2, and 2:1 entrainment regions and an efficiency decrease is seen near the 1:1 entrainment region. We further calculate various contributions to the efficiency: the phase shift of the force and flux, the magnitude of the response to the perturbation, and changes in the average values of the force and flux during a perturbation. We show that all three changes contribute to the overall changes in efficiency, but increases and decreases in the average force make the largest contributions.