1School of Engineering Sciences and
2Department of
Mathematics
University of Southampton, Southampton SO17 1BJ
The photosynthetic reaction centre of purple bacteria consists of a protein complex which binds cofactors of the active branch of the electron transport chain: the primary electron donor (a bacteriochlorophyll dimer), a bacteriochlorophyll molecule (denoted by B), bacteriopheophytin H and a quinone Q. The quinone QB of the inactive branch forms the final electron acceptor in the chain: after reduction to QBH2 it diffuses away from the reaction centre, the ground state of the primary donor is replenished and the electron transfer cycle can start again. The electron transport occurs across a photosynthetic membrane (bacterial wall) and drives two energy sources for the bacterium: an electrostatic potential across the membrane, and a proton gradient. These are utilised by other enzymes embedded in the membrane which are considered as ‘users’ - indicated by the meter symbol. The overall electron transfer from P* to Q occurs with a high quantum yield (>90%), indicating that the electron transitions between the reduced states are much faster than the ‘recombination’ reaction from the reduced states to the ground state P. This is also confirmed by direct measurements for all the transitions, with the exception perhaps of the reverse transition from Q to H.
This paper will show that the electron transfer among the redox sites can be conveniently modelled using tools employed in the description of semiconductor solar cells (see Fig. 1; some of these ideas, including a review of the light harvesting processes, can be found in [1]). In equilibrium, the occupancy of all levels is described using the electrochemical potential m o which is identical for all levels. Away from equilibrium one can, by analogy with electron transfer across the p-n junction, introduce quasi-Fermi levels m red and m ox for the reduced and oxidised species. A common approximation in solid-state electronics is to assume that the charge separation process is thermodynamically reversible – in other words, that the quasi-Fermi levels remain parallel across the p-n junction. In photosynthesis, a similar approximation (sometimes called energy conservation) was used by Crofts et al to discuss the effects of the reverse reaction such as delayed luminescence [2].
Fig. 1 Electron transfer in bacterial photosynthesis
Using this model one can show that the difference of electrochemical potentials across the photosynthetic membrane (D m = m red - m ox) is related to the electron transfer rate K by a formula reminiscent of the solar cell equation
&nb
sp; &nb
sp;
(1)
where K plays the role of electrical current produced by the cell, and the parameters KL and Ko are analogous to the photogenerated and dark saturation currents. A similar equation follows from the work of Ross & Calvin [3] some 30 years ago.
The bacterium uses the energy D m to pump protons across the membrane and create a proton gradient, effectively storing the energy in a chemiosmotic form. This chemiosmotic energy (usually called protonmotive force) is then used to drive photophosphorylation: production of ATP from ADP. It is interesting to note that, unlike in solar cells, the free energy D m generated by the process contains two terms: the electrostatic potential difference D y across the membrane, and a term due to the shift of the redox couple Q-/Q towards the reduced moiety or, in other words, the electron occupation of the site Q. Assuming that the system remains electrically neutral overall, the latter process implies a shift of the electron donor P towards the oxidised form P+:
&nb
sp; (2)
&nb
sp;
where p and q are the probabilities that the redox couples Q-/Q and P/P+ are in the reduced and oxidised form, respectively, the subscript o refers to the equilibrium values, and other symbols have their usual meaning. One would expect that under normal operation p» q.
The properties of Eq. (1) are well known from the solar cell theory. They imply, in particular, that the rate of energy conversion KD m is zero when the production rate K and the energy produced D m are equal to their maximum values (since D m and K, respectively, are then zero). The maximum is achieved at some intermediate point D m max (or Vmax for a solar cell), and one aim of the photovoltaic system design is to ensure that the solar cells operate near this maximum power output. This is achieved by connecting the correct number of cells in series or, for optimum operation, by employing a maximum power point tracker. Photosynthetic organisms, however, do not have this option and it is therefore interesting to enquire how the photosynthetic apparatus delivers energy with a good conversion efficiency. It is suggested that one option available to the organism for maintaining high photosynthetic efficiency is to ensure an optimum value of the electric field across the membrane.
The electric field can be controlled, for example, by
controlling the
dielectric constant of the non-aqueous part of the membrane or by controlling the ion
flux
through the membrane although the time constant for latter process might be slow.
For a
fixed value of D m ,
a large
value
of D y implies low p
and q, and
vice
versa. Since the energy usage involves transfer of electrons from Q to P, the rate is
proportional to the product pq and depends on the value of D
y . Thus, by writing 
one
can show that
where
(see Fig. 2). These formulae can be understood as follows. High electric field increases the energy content of the product but retards the reaction rate since the electric field opposes the electron flow. Low field, on the other hand, leads to a relatively low energy product which, however, is produced at a rapid rate. An optimum value of the potential difference D y sets a compromise between these two effects and maximises the conversion efficiency.
Fig. 2 The reaction rate K, free energy D m , and the energy production rate KD m as functions of the electric field
References
1. T. Markvart, Light harvesting for quantum solar energy conversion, to appear in Progress in Quantum Electronics.
2. A.R. Crofts, C.A. Wraight and D.E. Fleischman, FEBS Lett. 15, 89 (1971)
3. R.T. Ross and M. Calvin, Biophys. J. 7, 595 (1967)
