### 4.1 Introduction to biological energy transfer

In this second lecture on photosynthesis, we shall look again at the possible decay pathways for excitation once a photon is captured by the photosynthetic antenna molecules. Then we shall focus on the theory behind the mechanism whereby energy is transferred from one chromophore to another, on its way towards the reaction centre. The image in Slide 1 shows the a-helices and b-sheets of the protein, with the embedded chlorophyll units highlighted in green. (For more about protein structures, see the topic ‘Biological molecules: Lecture 3 ‘Protein structures’.)

Slide 1 Energy transfer is a crucial part of photosynthesis. Analysing the fundamental mechanisms enables us to understand the nature and dynamics of chromophore coupling and the directionality of energy transfer

We shall lay the theoretical foundations, and towards the end of the lecture have a brief look at how this energy-transfer process can be understood from a deeper, quantum electrodynamical point of view (see Slide 2).

As we saw in Lecture 3, when an antenna molecule enters the excited state, several different processes can occur: there may be direct relaxation, energy may transfer to another chromophore or the energy associated with the excited state mayn start a charge transfer reaction (see Slide 3, which is a repeat of Slide 9 from Lecture 3).

### 4.2 Energy transfer theory

#### 4.2.1 Resonance energy transfer

In Lecture 3 we looked at the detailed molecular structure of some of the most important pigment molecules, notably chlorophyll and some carotenoid species. In Slide 4 (which was Slide 13 in Lecture 3) we can see how these molecules fit into much larger supramolecular structures, in host material made out of protein molecules, forming arrays of astonishing complexity and sometimes very high symmetry. These are colour-coded images processed from the results of X-ray crystallographic studies; the examples shown in Slide 4 come from plants, green sulphur bacteria and purple bacteria. Notice the very high (nine-fold) rotational symmetry of the array in the purple bacterial system. This was the first system of such astonishing symmetry to be discovered in the whole realm of biology.

We have already seen that the energy of photons, once acquired in absorption by the pigment molecules, needs to be transferred onwards, towards the reaction centre. This is achieved by an energy-transfer process that can involve several steps, as illustrated for the PSII supercomplex, one of the most commonly found photosynthetic complexes. The top picture of Slide 4 shows a typical trajectory for the energy-transfer process.

Let us first consider in general terms what is involved in each transfer of energy from one molecule to another one nearby, which is shown schematically in Slide 5. We shall call the molecule that starts off with the electronic excitation the donor, *D*, and the molecule that picks up the energy the acceptor, *A*. The process of energy transfer, because it conserves energy, is often called resonance energy transfer (RET). In Slide 5, S_{0} signifies the (singlet) ground state and S*n* represents an excited state (e.g. *n* = 1, first excited state).

Initially the donor is in its electronically excited state and the acceptor in its ground state; after the transfer this state of affairs is interchanged. In the diagram we indicate, by the blue background shading, the dense manifold of closely spaced and overlapping vibrational levels associated with the ground and excited singlet states for each of the two molecules, and the paired and unpaired electrons involved in the associated transitions – the donor decay and the acceptor excitation.

#### 4.2.2 The system Hamiltonian

The pictures that we have seen of photosynthetic arrays (Slide 4) show us that the donor and acceptor will generally be parts of a molecular aggregate consisting of a large number of such chromophores, held together in a biologically determined array geometry.

To develop the quantum mechanics we start with the Hamiltonian for the aggregate, a sum of terms that can be separated into those which involve individual molecules, and others which effect coupling between them (Slide 6).

The aggregate Hamiltonian, which is a function of the nuclear degrees of freedom, R, can be constructed in a similar way to the usual molecular Hamiltonian, but now we separate into intra- and intermolecular contributions:

The important point here is that individual chromophores, *m* and *n*, are separated in space by a distance *R _{nm}* (see Slide 7).

The intramolecular contributions, *H*_{m}, describe the individual molecules, and the latter ‘pair’ coupling terms represent the Coulomb interaction (which is a function of the separation between chromophores *m* and *n*); it is this term that is responsible for the mechanism of energy transfer between the chromophores.

#### 4.2.3 Chromophore couplings

Since the molecular structure of each chromophore comprises a set of positively charged nuclei and negatively charged electrons, the intermolecular interactions comprise three kinds of term: electron–electron interactions, internuclear interactions and electron–nuclear couplings (Slide 7).

For simplicity, as is common in molecular photophysics, we restrict our model to the valence electrons only, which essentially means that ‘nuclei’ denotes the full set of nuclear charges together with the core electrons – those not involved in bonding.

Then, the electronic coupling formally separates into a short-range contribution, where there is overlap between the donor and acceptor orbitals, and an electrodynamic contribution which operates at all separations. Usually the former contribution yields a wavefunction overlap integral which can be neglected in photosynthetic systems, where the chromophores are held apart from each other by the support structure, and each has a distinct electronic integrity. The bracketed equations represent the two transition dipole moments, being the electric dipole operator and k the orientation factor, as defined on the next slide.

In the commonly applied Förster picture developed by Theodore Förster in the 1950s, it is assumed that the coupling between molecules can be approximated as an interaction between the transition dipoles for the donor decay and acceptor excitation:

The exact form of this coupling (Slide 8) displays an overall inverse cubic dependence upon the distance between the donor and the acceptor: the first term entails the scalar product between the two transition dipoles; the second entails scalar products between each of these transition moments and the pair displacement vector~~.~~

Using the implied summation convention, where repeated subscript indices denote implied summation over all three Cartesian directions, the result can be more simply cast:

One can also factorise out an orientation factor κ:

It is worth noting (because it is quite commonly misreported) that to achieve effective coupling, the transition dipoles not only do not have to be parallel but also can even be orthogonal to each other, provided that they are not also perpendicular to the displacement vector. Generally the angular disposition determines whether energy transfer is favoured or inhibited.

#### 4.2.4 Thermal fluctuations and directionality

Within each chromophore the motions of the internal nuclear framework (which number 3*N*-6 independent vibrational modes, where *N* is the number of atoms) produce a stochastic broadening of the electronic levels – an effect that is further increased by the thermal motions of the surrounding host matter (Slide 9).

In consequence there is a distribution of possible energies for both the donor and the acceptor electronic states, as indicated by the continuous density of states in the upper and lower energy levels in the diagram shown in Slide 5. Moreover, the energy distribution in the ground electronic state arises from thermal fluctuations within the molecular structure – that is, vibrations of the nuclear framework. These effects can be directly observed in measurements of the donor fluorescence and acceptor absorption spectra. Moreover, for each chromophore, the fluorescence spectrum is usually red-shifted with respect to the absorption spectrum (a Stokes shift) because of intramolecular vibrational losses in the electronic excited state. (Further detail can be found in the text by Atkins & Friedman listed at the start of this lecture; see sections 10.7–10.13.)

The rate of energy transfer, Γ, is given by what is often called Fermi’s Golden Rule (Slide 10), which depends on the coupling between the donor and the acceptor, and also the density of states:

where ρ is the density of states. Because the coupling operates over relatively short distances, transfer processes typically involve a host of individual transfer steps within each photosynthetic complex, each one having a different rate constant.

#### 4.2.5 Dynamics of energy transfer

Putting all of this information together (Slide 11), we can secure a result for the rate of each energy-transfer step. The density of states can be cast in terms of a correlation between the emission characteristics of the donor and the absorption characteristics of the acceptor. Similarly, the coupling *V _{mn}* is a function of the relative orientation of the corresponding transition moments and the displacement distance between the coupled pair; it is also necessary to account for the refractive index of the medium in which energy transfer occurs.

After a significant amount of algebra we thus obtain:

Equation 8

Equation 8 involves a number of important, physically identifiable features: the integral is a frequency-weighted spectral overlap of the donor fluorescence spectrum and the acceptor absorption spectrum; in front of the integral there is a factor which involves the square of the orientation factor, the donor decay lifetime, τ_{D}, the refractive index of the medium, *n*, and most notably the inverse sixth power of the displacement.

The same formula can be cast in simpler form by defining a Förster distance,* R*_{0}, which is the distance of separation at which the probabilities of a donor releasing its energy (i) by spontaneous decay and (ii) by transfer to the acceptor are equal. Here *c* is the speed of light in a vacuum.

Equation 9

The definition of *R*_{0} can be written as:

Equation 10

where *c*' = *c*/*n* is the speed of light within the system, and *S* is the overlap integral in Equation 9.

Because of the inverse sixth power dependence of the rate on distance, the migration of energy beyond the Förster radius will usually involve a series of short hops, in preference to one long hop. It is for this reason that once energy is harvested from sunlight, it generally undergoes repeated hops towards the reaction centre.

What we now have to understand is why it is important that at each stage there is a small loss in energy, as the lower figure on Slide 11 indicates.

Because of the Stokes shift, which involves intramolecular vibrational redistribution (IVR), and also because the acceptor will often have an excited state a little lower in energy than the donor excited state, the overlap integral for forward transfer, donor to acceptor, is generally much larger in value than the corresponding overlap integral for back transfer, acceptor to donor (see Slide 12).

In antenna complexes, each transfer step is therefore subject to this spectroscopic gradient, in consequence of which, energy can be directed towards the reaction centre at each stage with a high degree of efficiency. This prevents undirected (essentially random walk) hopping that would otherwise greatly reduce the rate of delivering energy to the reaction centre.

If we look at what is going on in the well-studied case of purple photobacteria (Slide 13), we see that there are two kinds of ring-shaped light-harvesting array, denoted LH1 and LH2, adjacent to each other (there are in fact several LH2 units around each LH1); the reaction centre is in the middle of LH1.

Light is first captured by the pigments of LH2, and then repeated hops take the excitation around the ring before it hops over to the inner ring, from which transfer to the reaction centre occurs. What we see here is typical: there are several different chromophore types, and notice the successive differences in absorption maxima in the chromophores traversed in progress towards the reaction centre.

This example affords a clear illustration of a spectroscopic gradient. However, some hops are clearly taking place between chromophores of exactly the same type, held in electronically equivalent environments; for these, no spectroscopic gradient can apply. To tackle this issue, we need a further refinement of the theory.

### 4.3 Theoretical fundamentals

In this section we use a state-vector representation of quantum mechanics, which we hope will be familiar at undergraduate level. If further discussion is needed, see the text by Chester listed at the start of this lecture.

#### 4.3.1 Describing wavefunctions using an exciton basis

To consider the consequences of coupling between essentially equivalent chromophores (Slide 14), we must return to the aggregate Hamiltonian:

We now observe that when there are many chromophores all of the same chemical type, they would, if they acted in isolation, all have identical energies and electronic states. However, we know there is significant coupling between these chromophores. In consequence, any electronic excitation will usually be delocalised over closely neighbouring units, and these shared excitations are called excitons.

For a true description of the stable states of the coupled system we therefore have to diagonalise the Hamiltonian – in other words, find its eigenvector states and eigenvalue energies. Here *N* corresponds to the number of chromophore units, and T (T’) corresponds to the mathematical transformation (back transformation) between the site and exciton representation.

Equation 11

Another well-known example of an antenna complex is the Fenna-Matthews-Olson (FMO) complex (Slide 15), in which the different disposition of chemically similar chlorophylls gives each of them a slightly different energy level for its electronic excited state, and the coupling between them all becomes complicated by the formation of delocalised excitons.

#### 4.3.2 Localised vs delocalised states

To develop the quantum mechanics for such a system involves taking the individual chromophore wavefunctions on a local site basis, diagonalising the Hamiltonian and determining new exciton basis wavefunctions (Equation 11). The corresponding energies and transition moments are markedly different from the local basis set (see Slide 16).

Then to ascertain the dynamics of energy transfer it is necessary to include further dissipation terms in the Hamiltonian, accounting for vibrational relaxation and trapping. This is most readily achieved not in the Schrödinger wave equation (SWE) approach but in a density matrix formulation (Slide 17).

In the density matrix representation, diagonal terms designate state populations and off-diagonal terms signify quantum interferences between the basis states:

The evolution of the density matrix in time (Slide 18) is determined using the Liouville von Neumann (LVN) equation, based on the time-independent system Hamiltonian:

Equation 14

The advantage of this formalism (compared with the SWE method) is the incorporation of relaxation and trapping terms on a phenomenological basis. This introduces further directionality into the equations of motion, so Equation 14 can be modified to read:

Equation 15

Here, terms in γ are ‘relaxation terms’ and terms in Γ are ‘trapping terms’.

Recent years have seen the development of a new experimental technique called two-dimensional Fourier transform electronic spectroscopy. This allows experimental scientists to monitor the energy transfer process in real time (over about a picosecond with ‘snapshots’being taken every 20 femtoseconds or so). One of the most important observations was that quantum coherence between the electronic states occurs for much longer than anticipated (G. S. Engel, *Nature*, 446, p782, 2007). This was quite unexpected, as people assumed that the random motion of the protein and surrounding water molecules broke down the coherence. This has sparked great debate about whether quantum coherence facilitates the energy-transfer process. Theoreticians are now developing models that can be used to try to understand the mechanism behind this, and other, experimental observations.

Some recently performed calculations based on numerical implementation of the LVN equations give insight into the complex energy-transfer processes in the case of the FMO complex. These simulations (Slide 19) show how the exciton moves from an initially excited antenna at site 1 to the trap at site 8. It is particularly interesting to note the quantum beating seen at early times and how these beating effects disappear at longer times because of coupling to the environment, which is built into the models. These features, which some interpret as uniquely quantum mechanical effects at the heart of photobiology, are the subject of highly active current research.

### 4.4 Advanced topic: quantum electrodynamics and energy transfer

Deeper insights into the detailed mechanism for energy transfer can be gained from a quantum electrodynamics (QED) perspective. Here account is taken of the quantum uncertainty associated with not only the wavefunctions for the donor and acceptor species but also the location of the excitation itself.

Resonance energy transfer in the QED picture is determined by the propagation of virtual photons between the chromophores (Slide 20). A virtual photon is one that is not observed by any measurement, and which in consequence requires summation over every possible direction, frequency and polarisation.

When we consider energy transfer on this basis, two possible state sequences arise. In the diagram on the left we see how an initial state, in which the donor is excited and the acceptor is in its ground state, progresses by one of two routes to a final state, where the donor is in its ground state and the acceptor is excited. The upper route has an intermediate stage where both chromophores are in their ground states and energy is propagating between them. However, quantum theory also requires us to take into consideration an alternative, apparently less logical, route in which the donor and acceptor are simultaneously excited, while energy flows between them.

An alternative representation is given by the two Feynman diagrams shown on the right of Slide 20, in which time progresses upwards. What we see here are different representations of an interplay between retardation and quantum uncertainty. The distance at which the coupling photon essentially loses virtual character is determined by the uncertainty principle: if Δ*E* is the energy mismatch, this distance runs to ~~h~~c/Δ*E*. Energy conservation thereafter imposes increasingly tight constraints, and the photon becomes essentially real.

What we then find is that the coupling emerges in a different form, more complicated than the one shown in Slide 8. In Slide 21, the terms shaded in blue correspond to corrections to the short range formula seen in Slide 8.

Equation 16

Accordingly the transfer rate (Equation 7, Fermi’s Golden Rule) takes a form in which there are contributions not only from the inverse sixth power of distance but also terms involving inverse quartic and quadratic dependences:

Equation 17

The result is most effectively seen on a log-log plot of the rate against distance as shown in Slide 22. For a closely placed donor and acceptor, virtual photon behaviour is reflected in the inverse sixth power (Förster) distance dependence.

As distance increases, the behaviour changes to the emission of a real photon, captured by the acceptor with an inverse square law dependence. In consequence we can recognise that there can be no competition between so-called ‘radiative’ and ‘radiationless’ mechanisms; they are simply different asymptotes of one unified mechanism that operates over all distances.