# Rate Equations and Excited State Dynamics in Coupled Systems

If there is any physical potential between two states, then these states are coupled. For our present interest, only the general electromagnetic cou?pling is relevant. This means that two states are coupled if the force field of one of the two states is seen by the other state. In such cases, the dynamics of one state modulate the dynamics of the other.

Coupled 2-level states can undergo several transitions, e.g. by spontaneous emission of light or phonons or by Inter-System Crossing (ISC). These transitions are discussed later and evaluated with regards to the experimental data. Schematic illustrations of excitation energy transfer transitions, including electron delocalisation and electron transfer, are shown in Figure 10, 4^{th} row.

Figure 10. Interactions and EET/ET processes of coupled 2-Niveau systems bound to a protein matrix.

The electron-phonon interaction shown in Figure 10 shifts lifetimes (i.e. the transition probabilities) and the energy of the states. In photosynthetic systems, the formalism of rate equations can be incorporated by treating different molecules as localised states. Generally this will lead to rather complex equation systems of up to a thousand coupled differential equations. A simplified model respects the given resolution of the measurement setup by employing the concept of “compartments” as suggested e.g. by (Hader, 1999).

According to this compartmentation model (see chapter 1.4, Figure 5) a complex system can be treated as a form of “compartment” if the ener?getic equilibration between the coupled molecules that are treated as a compartment is fast in comparison to the achievable resolution of the measurement setup.

The advantage of the compartmentation model is a reduction of the number of states. For example one can treat the LHC, the core antenna complexes and the reaction center (see Figure 18 and Figure 18) as compartments of thylakoid membrane fragments and therefore as single states instead of treating each single molecule separately. The whole lightharvesting complex, including the core antenna and the Chl molecules in the reaction center, might even be treated as one single compartment.

The states and their couplings are visualized in the scheme as shown in Figure 11. Here five arbitrary states are coupled via EET exhibiting one and the same rate constant *k* for all transitions to neighbouring states. Additionally, there is a relaxation to the ground state; this is (*k _{Q}+k_{F}*) when the excitation energy interacts with a quenched state as shown in Figure 11. The probability for relaxation to the ground state of all states without quencher is

*k*.

_{F}In the scheme shown in Figure 11 the state *N _{4}* is bound to a quencher (e.g. a reaction center) and therefore exhibits reduced lifetime. The asterix (*) indicates that initially (time

*t*= 0) the state N is excited while all other molecules are found in the ground state.

**Figure 11. **Scheme of five coupled states that can transfer energy to neighbouring molecules. States 1, 2, 3, and 5 have equal relaxation probabilities from the excited to the ground state (rate constant k_{F}), while state 4 is connected with a quencher and therefore exhibits a higher relaxation probability * kF* +

*.*

**kQ**From the schematic model presented in Figure 11 one can formulate the rate equations for the time derivative of each state. For molecule N_{2 }this equation exemplarily denotes to

The final set of five coupled, linear, differential equations of the first order can be solved to obtain the excited state dynamics of all five mole- cules/states.

The fluorescence intensity is proportional to the time-dependent population density of the fluorescing states. This is calculated from the solution of such a set of coupled linear differential equations of the first order (for mathematical details, see Schmitt, 2011).

This set of equations is solved by a sum of exponential functions with different time constants. In a nondegenerated set with *n* pairwise different eigenvalues of the transfer (imaging) matrix *T* describing the change *N* of all *n* state population densities, the set of differential equations can be written according to eq. 38:

If the entries of *T* are not known, we at least have some information about the symmetry of *T* according to eq. 8. It is a necessary condition

*T *(^{E}‘~^{E}k)

for the equilibrium case that *к~ ^{= е k}‘^{T}* according to eq. 9. The deriva-

^{T}ik

tion of eq. 9 is shown in chapter 1.3.

For the sake of simplicity, the relaxing system might be treated as locally equilibrated.

In that case eq. 8 helps us reduce the nondiagonal elements of *T* to * ^{x}h *of their quantity.

The solution for the *i*^{th} state population density is given by

where *y.* is denoting the /^{h} eigenvalue and *U.* the 1^{th} component of

the /^{h} eigenvector of *T*. For a nondegenerated system eq. 39 is proportional to the assumed fluorescence decay:

with *у.* = *—^Tj*) according to the proportionality between fluorescence intensity and excited state population density. If one or more *y.* = 0, one can reduce the dimensionality of *T* to a smaller size with all eigenvalues being nonzero and nondegenerated. In such cases, eq. 40 is suitable to calculate the expected fluorescence emission in the time domain from the

transfer matrix *T*. Vice versa *T* can be reconstructed if the fluorescence kinetics of all molecules is measured.

For that purpose the eigenvector matrix *U* and the eigenvalues in the diagonal matrix Г have to be used to calculate the transfer matrix *T*:

Eq. 41 is unique if one knows the amplitudes and time constants of all exponential decay components for each compartment in the sample.

The time-dependent fluorescence of emitters described by a coupled system of differential equations as given by eq. 40 occurs multiexponentially with up to *n* decay components (i.e. exponential decay time constants) for *n* coupled subsystems (i.e. *n* coupled excited states that define the dimensionality of *N* with respect to eq. 38).

Therefore the average decay time shown in eq. 35 cannot be used. In the case of coupled states the average decay time can be defined as a time constant proportional to the overall fluorescence intensity (for further details see Lakowicz, 2006):

If one evaluates eq. 42 with the multiexponential decay curves as given by eq. 40 it follows that

at a certain wavelength position.

Usually one does not find all the eigenvector components (i.e. when some state populations do not show fluorescence or when not all fluorescence decay components can be resolved). In that case eq. 41 cannot be

used to find a unique transfer matrix *T*. But a reasonable assumption of *T* with iterative comparison of the eigensystem (i.e. eigenvectors and eigenvalues) *(U* , *T*) with the fluorescence decay helps find a solution for *T* which is consistent with the globally observed fluorescence dynamics in all wavelength sections according to eq. 40. Such an iterative comparison of the measurement data (eigenvectors and eigenvalues) and

a reasonable transfer matrix *T* can be used to calculate a suggestion for the transfer rates of a system of arbitrary complexity which leads to observable fluorescence decay.

Our computation was highly efficient when employing an algorithm that starts with an arbitrary reasonable transfer matrix *T* start. First the eigenvectors and eigenvalues of *T* Start are calculated. In the next step, one will use the observed decay times for all eigenvalues *y. =* -(г. ) instead

of the eigenvalues of *T* Start. Therefore the eigenvalues are changed but the eigenvectors are not. Doing the transformation according to eq. 41 with the changed (experimentally observed) eigenvalues one will find an improved suggestion *T* for the transfer matrix *T* Start.

There are some symmetry constraints for the transfer matrix *T*. For example, the rate constant for the energy transfer from pigment one to pigment two has to be the same as the energy reception at pigment two according to the energy transfer. Additionally, we have already mentioned the Boltzmann equilibrium which gives rise to eq. 8 as a symmetry argument for the transposed entries. Therefore, this symmetry condition might also be used as constraint.

According to the suggested algorithm, all violated constraints are corrected in the transfer matrix *T*. After that one can again calculate the eigenvectors and eigenvalues of *T*. The observed decay times for all eigenvalues are again set to *у.* = *-(г*.) . Then the algorithm starts from

the beginning. Doing the transformation according to eq. 41 with the changed (experimentally observed) eigenvalues leads to a further improvement of *T*.

The described algorithm for calculating the transfer rates in *A.marina *converged after five iterations and therefore was faster than other algorithms found in the literature. There are several other correlations between experiment and calculation which could be implemented into the algorithm. For example, one could also use the observed amplitudes of the different fluorescence components to correct the eigenvectors

of *T* _{Star}t instead of the eigenvalues.