# Simulation of Decay-Associated Spectra

According to the formalism presented in chapters 1.3, 1.4, and 2.1, the decay-associated spectra can be simulated for each pigment or compartment in a coupled system. First we start with a simulation of the DAS of the coupled chain shown in Figure 11 but without the quencher molecule bound to the state **N**_{4}. The DAS of the individual pigments exhibit multiphasic relaxation dynamics at each molecule of the coupled chain:

Figure 12. Simulated decay-associated spectra for the system shown in Figure 11 (without the quencher at * N_{4})* with a Gaussian lineshape function in the wavelength domain (upper panel) and in the “pigment” domain (lower panel). The black curve in the upper panel denotes a single exponential decay component with a time constant of 4 ns. In the lower panel we see a complex DAS pattern distributed along all coupled states. The time constants are 2.7 ps (cyan), 3.3 ps (red), 5 ps (green), 37 ps (magenta) and 4 ns (black).

In the simulation shown in Figure 12, it is assumed that * k_{F}* = 4 ns and

*= 10 ps. The simulation is performed without the quencher located at the molecule*

**k**

**N**_{4}. The initial excitation is set to pigment

**N**_{1}as indicated in Figure 11. In the spectral domain there is only one monoexponential decay with a typical 4 ns decay time because all molecules are isoenergetic (Figure 12, upper panel). In the spatial domain the DAS is rather complex and each pigment exhibits an individual time dependent population and therefore fluorescence emission. The structure of the spatially resolved DAS pattern, especially the symmetry of the DAS shown in Figure 12, lower panel, is briefly discussed in the following.

The interpretation of the DAS in Figure 12 is in full accordance with the findings that the dominant decay of each pigment exhibits a time constant of 4 ns (decay) and that all other time constants describe the EET inside of the coupled structure. The excited pigment *N _{1}* transfers the excitation energy fast to the neighbouring pigments N

_{0}and N

_{2}(red curve, 3.3 ps). These exhibit the characteristic rise kinetics with 3.3 ps time constant (red curve). The red curve shows a mirror symmetry along the axis between

*N*

_{2}and

*N*

_{3}(which is not clearly visible as the components with other colors overlay the negative amplitude at pigment

*N*

_{0 }and N

_{3}). Therefore the rise kinetics (negative amplitude) is also found at N

_{3}and N

_{5}whereas N

_{4}contains a high amplitude of the 3.3 ps decay kinetics (positive amplitude) similar to the initially excited pigment

*N*

_{1}.

From the DAS shown on the right side of Figure 12, it can be seen that the DAS components with time constants of 4 ns (black) and 3.3 ps (red) appear mirror symmetric to the center of the chain while the time constants with values of 2.7 ps (cyan), 5 ps (green) and 37 ps (magenta) are anti-symmetric according to the mirror plane between *N*_{2} and *N*_{3} . In addition, the evaluation delivers a 10 ps component which has amplitude zero in the whole range of states and which therefore is not visible in the fluorescence.

It has to be strictly pointed out that the complexity that is found in the DAS of Figure 12 cannot yet be resolved experimentally. Even if it would be possible to resolve single molecules in the spatial domain (e.g. the states denoted as N in Figure 11; this might be possible with time resolved STED microscopy or with tip enhanced single molecule fluorescence spectroscopy) one cannot expect to find a DAS as depicted in Figure 12, lower panel, because the time constants of 2.7 ps, 3.3 ps and 5 ps are hardly distinguishable with the technique of TSCSPC employing a laser system with 70 ps FWHM of the IRF. Here fluorescence up conversion would be a technique that delivers the time resolution, but applied high resolution techniques like STED or tip enhanced AFM would influence the excited state lifetime of the single molecules and therefore also do not represent the natural state of the system. However the expec?tation for any given resolution can be calculated from the DAS shown in Figure 12 by mathematic convolution of the temporal and spatial sensor elements with the DAS shown in Figure 12, lower panel.

Most probably the experimenter will find a clearly resolved 4 ns decay kinetic (black curve) and the pronounced 37 ps component (magneta curve) with high positive amplitude at the states *N _{0}* and

*N*and negative amplitude at states N

_{1}_{4}and N

_{5}. All other (fast) rise and decay components will appear as one additional very fast component. The amplitude of this fast component is comparable to the sum of the amplitudes of all fast components (2.7 ps (cyan), 3.3 ps (red) and 5 ps (green)) shown in Figure 12, lower panel.

This situation of a DAS that could be resolved experimentally (still assuming single molecule resolution in the spatial domain) is presented in Figure 13.

Figure 13. DAS for the system shown in Figure 11 (without the quencher position at N_{4}) with a Gaussian lineshape function in the “pigment” domain. The fast components shown in Figure 12 (cyan, red, green curve) are summed to one component here (shown in green). In addition the 37 ps (magenta) and 4 ns (black) component are shown.

One expects that the positive decay kinetics of all pigments that are not directly excited is mainly dominated by the 4 ns fluorescence decay (black curve in Figure 13). The initially excited pigment *N*_{1} decays with very fast (3-5 ps, green curve) decay kinetics which appears as fluorescence rise at the neighbouring molecules (negative amplitude). The 37 ps component represents the overall energy transfer along the coupled chain of molecules with positive amplitude at pigment *N*_{0} and *N*_{1} which exhibit mainly donor character, and negative amplitude at pigment *N*_{4}

and * N_{5}* which exhibit an acceptor character for the overall energy equilibration process. This overall time spread in the ensemble measurement is restricted by the geometry and symmetry of the pigment chain as shown in Figure 11 rather than by certain transfer “steps” inside the system. The interaction time between the single subunits is much faster than this averaged overall energy transfer (magneta curve) and the amplitude distribution reflects the average pathway of the energy caused by the geometrical constitution of the sample.

It is interesting to see how positive and negative amplitudes of the corresponding fluorescence components are distributed along the pigment chain if the initial excitation changes. One might investigate a situation where * N_{1}* and N

_{4}are initially excited instead of only

*as presented in Figure 14, upper panel.*

**N**_{1}**Figure 14. **Simulated decay-associated spectra for the system shown in Figure 11 (without the quencher position at N_{4}) with a Gaussian lineshape function according to Figure 12 but with initial population * N_{0}(t* = 0) = N

_{2}(0) = N

_{3}(0) = N

_{5}(0) = 0, N1(0) = 1, N4(0) = 1 (upper panel, full symmetric case) and N0(1 = 0) = N1(0) = N2(0) = 1, N3(0) = N4(0) = N5(0) = 0 (lower panel, full antisymmetric case).

In the upper panel of Figure 14, the DAS pattern of a full mirror symmetric initial condition is simulated with N0(1 = 0) = N2(0) = N3(0) = N5(0) = 0, N1(0) = 1, N4(0) = 1. In the lower panel, the situation of a primary excitation of N_{0}, N1 and N_{2} is shown, i.e. a fully antisymmetric condition N0(1 = 0) = N1(0) = N2(0) = 1, N3(0) = N4(0) = N5(0) = 0.

All fast decay components, except the dissipative decay component for the relaxing system after equilibration (which is independent from the initial condition, 4 ns component, black curve), represent the symmetry of the initial condition according to a suggested mirror plane in the middle of the structure.

Functional PS II preparations exhibit a markedly reduced excited state lifetime due to the presence of photochemical quenchers, i.e. the reaction center of PS II where charge separation occurs.

This situation is simulated in Figure 15 for the coupled pigment chain shown in Figure 11 assuming the initial excitation of pigment N_{t} and the quencher located at pigment N_{4}.

The simulation was performed with the rate constants *k _{F}* = (4 ns)

^{-1},

*kQ*= (2 ps)

^{1}and

*k*= (10 ps)

^{-1}.

As seen in the upper panel of Figure 15, the decay is no longer monoexponential in the wavelength domain after introduction of the quencher. The main decay component exhibits a lifetime of 88 ps which is composed by the quencher efficiency kQ, the EET rate constant *k* and the distance between the initially excited state and the quencher. The model simulates a “random walk” of the initially excited electron until it is quenched to the ground state (for a description of such random walk models in photosynthesis research see Bergmann, 1999).

On account of the coupling of several pigments, the amplitude distribution of the decay components that are found in the fluorescence kinetics becomes more complex when the quencher is present (see Figure 15, lower panel).

As described, the solution of the rate equation systems delivers the time-resolved fluorescence spectra and DAS. Therefore one can determine *k _{FRET}* from the experimental data. In chapter 2.2.3 we will see how the Forster Energy Transfer rate constant can be calculated independently from fluorescence spectra. It can also be found with the help of rate eq. 38 and the reverse transformation denoted in eq. 41. The Forster Resonance Energy Transfer can be evaluated independently using the time-integrated spectra of the donor and the acceptor, or it also can be fitted onto the time-resolved spectra comparing the solution of the rate equations with the measurement results.

Figure 15. Simulated decay-associated spectra for the system shown in Figure 11 (including the quencher position at * N_{4})* with a Gaussian lineshape function in the wavelength domain (upper panel) and in the “pigment” domain (lower panel). The black curve denotes the main relaxation of the coupled system, occurring as a strictly positive decay component with a time constant of 88 ps. In addition we find fast time constants with 3 ps (red), 4 ps (green) and 13 ps (blue).

In many cases there is a lack of information for the direct calculation of the distance (*R*_{12}) dependent rate constant *k _{FRET}* (R

_{12}) due to the fact that there are missing parameters like the pigment dipole orientation factor

*к*(see chapter 2.2.3). For a full understanding of the advantages of the Forster Resonance Energy Transfer and its limitations we will shortly outline the basic concept of FRET.