# Rate Equations in Photosynthesis

The mathematical concept of rate equations is suitable to analyze the synergetic behaviour of a complex coupled network if the coupling strength and the transition probabilities between single states are known, or if measurement results for the fluorescence dynamics or other experimental results that deliver information on the transfer probabilities exist. We will now have a look at the underlying concept of the theory of coupled pigments and the derivation of the rate constants for excitation energy (EET) and electron transfer (ET) and proton transfer (PT) processes in photosynthesis from experimental results. As one example the theory of Forster Resonance Energy Transfer is elaborated (see chap. 2.2.3) while other concepts like the famous Marcus theory are omitted for the sake of a compact description.

In earlier works we performed a theoretical description of the EET dynamics and ET in systems with pigment-pigment and pigment-protein interaction using rate equations that were applied to structures with increasing hierarchical complexity (Schmitt, 2011). The study started with a system consisting of two excitonically coupled Chl molecules in a tetrameric protein environment represented by the recombinant water soluble Chl binding protein (WSCP) of type Ila (Theiss et al., 2007b; Schmitt et al., 2008; Renger et al., 2009, 2011). It was completed with a study of the photosystem II (PSII) dynamics in whole cells of *Chlorella pyrenoidosa* Chick and whole leaves of the higher plant *Arabidopsis thaliana* (Belyaeva et al., 2008, 2011, 2014, 2015). In this way a quantification of dissipative excited state relaxation processes as a function of increasing excitation light intensity was achieved. The approach permits the determination of selected parameter values, their probability and stability in dynamical systems. A way to calculate thermodynamic quantities (e.g. entropy) under nonequilibrium conditions from rate equations was proposed (Schmitt, 2011) and is presented in chapter 4.4.

Photosynthetic EET and ET steps are predestined for a description by classical rate equations as it is assumed that (incoherent) excitons or electrons are transferred from one pigment protein complex (PC) to another one with given rate constant (probability per time unit). Coherence effects as studied in quantum mechanical approaches are omitted here since the description focuses on the approach to describe the behaviour of individual elements on timescales where coherence effects seem to be of minor relevance. Polarizations and coherence can be added into the formalism and extend it to optical Bloch equations and quantum mechanical approaches. Since the typical coherence times for the quantum mechanical states that are excited by light or due to energy or particle transfer are short in comparison to ET and most EET steps, the rate equations are treated classically and the formalism as presented by eq. 1 and eq. 7 represents an useful approach. This is especially assumed for ROS signaling in a thermalized environment. For fast EET it is known that coherence effects might play a role for efficient transfer (Calhoun et al., 2009). The photochemically active chlorophyll containing photosynthetic pigment protein complexes (PPCs) are classified into the photosystems (PS) PS I and PS II (see Figure 4). Electrons flow from PS II to PS I. Light absorption changes the redox properties of a single electron in each photosystem. The nomenclature has developed historically denoting the “first” PS as PS II which delivers electrons to the PS I (Byrdin, 1999).

It should be mentioned that in contrast to the simplified schemes shown in Figure 4, the distribution of different pigment-protein complexes with different optical properties is not homogeneous along the thylakoid membrane (Steffen, 2001) and that there exist PS II complexes with different antenna sizes, so called alpha centers and beta centers (Albertsson et al., 1990) which are inhomogeneously distributed along the thylakoid membrane.

The light energy is absorbed by chromophores bound to the photosynthetic complexes of PS I and PS II. These chromophores are mainly chlorophyll and carotenoid molecules. For a description of PS I see (Byrdin, 1999) and references therein.

The electronically excited singlet states formed by light absorption of the Chl molecules are not completely transformed into Gibbs free energy. A fraction is emitted as red fluorescence and the dynamics of the fluorescence emission of all samples containing PS I and PS II is mainly determined by the properties (organization and coupling) of the photosynthetic pigment-protein complexes of PS II (see Schatz et al., 1988). Therefore, fluorescence emission delivers a strong tool to monitor the excited state dynamics in photosynthetic complexes, especially PS II.

**Figure 4. **Membrane proteins inside the lipid double membrane of the thylakoids and light reaction in PS II, PS I and ATP-synthase.

The “light” reaction performs the exploitation of solar energy by highly functionalized PPCs. Solar energy represents the unique Gibbs free energy source of the earth’s biosphere. The Gibbs free energy is converted into high energy chemical compounds via the process of photosynthesis. This is achieved perfectly by incorporation of suitable chromophores into protein matrices. The PPCs are optimized to energy absorption and transfer, thereby producing the high energetic compounds ATP and NADP H2 (see Figure 4).

PS II and PS I are functionally connected by the Cytbef (Cyt) complex where plastoquinol PQH2 formed at PS II is oxidized and the electrons are transferred to PS I via plastocyanin (PC) as mobile carrier (see Figure 4). This process is coupled with proton transport from the stroma to the lumen, thus increasing the proton concentration in the lumen. At PS I the light driven reaction leads to the reduction of NADP+ to NADP H_{2}. The proton gradient provides the driving force for the ATP synthase where ATP is formed from ADP + P. NADP H_{2} is used in the dark reactions for CO2 reduction to produce glucose inside the chloroplast stroma.

The spatial separation between lumen and stroma, and the oriented arrangement of PS I, PS II and Cytbef enables a directional electron flow coupled with the formation of a transmembrane electrochemical potential difference. This way the absorbed Gibbs free energy of the photons is partially and transiently “stored”. For further thermodynamic considerations of the initial processes of photosynthesis, see chapter 2 and chapter 4.4.

A simplified model for EET and ET processes employs the concept of “compartments” as suggested by (Hader, 1999). According to this com- partmentation model, a rather complex system can be treated in form of “compartments” if the energetic equilibration between the coupled molecules which are treated as a compartment is fast in comparison to the achievable resolution of the measurement setup. Then there generally occurs an incoherent EET between different compartments which can be understood as a single “transfer step”.

The advantage of the compartment model is a reduction of the number of states that needs to be analyzed to characterize the system. For example, one can treat the light-harvesting antenna as a composition of compartments which are represented as intrinsic antenna systems or extrinsic antenna systems (see Figure 5). The compartments of the thyla- koid membrane as shown in Figure 4 typically divide into the mentioned photosystems PS I and PS II or Cytbef or the ATPase. Also lumen or stroma could be understood as compartments if one speaks about the transfer of single protons modeled during equilibration across the membrane. In this way single states do not necessarily demand a treatment of

**Figure 5. **Compartment model according to (Hader, 1999).

each single molecule. The whole light-harvesting complex, including the core antenna and the Chl molecules in the reaction center (RC), might even be treated as one single compartment. In a more complex approach one can treat a whole cell as a single compartment. The question is what kind of dynamics we intend to describe and if there is need for the description of the intrinsic dynamics of the compartments. Such simplifications are absolutely necessary to achieve an unambiguous description of the biological structures and to have a chance to derive clear results from observed trajectories, such as time resolved fluorescence signals. As can be seen in the discussion on the description of the overall PS II dynamics from a set of trajectories of time resolved fluorescence emission presented in chapter 2.7, such schemes are calculated on a huge parameter space and a reduction of this parameter space is inevitably necessary to avoid overparametrization and concomitant arbitrariness in the gathered results. If there are too many parameters imbedded into a system one can use it to “fit an elephant” or even “let him wiggle his trunk” as stated in (Dyson, 2004). The phrase is attributed to John von Neumann who already claimed the possibility to fit an elephant from only four parameters, while five parameters are enough to let the elephant dance. Compartment models help to control the elephant’s trunk and to draw clear conclusions from measurement results in the framework of the simplified system provided that the system is a suitable framework for the description of the results.

Figure 6 shows how EET conducts excitation energy between different compartments of cyanobacterial phycobiliprotein (PBP) antenna complexes. On the left panel EET inside the PBP antenna rod and between the PBP antenna and Chl *d* in *A.marina* is shown. At the top, the model scheme of the phycocyanin (PC) trimer with its bilin chro- mophores is shown. The right panel represents a model scheme and time constants for the EET inside the phycobilisomes of *Synechococcus *6301, and from there to the RC. This gives a resume of the extant literature on EET processes inside the PBP antenna complexes of cyanobacteria (Gillbro et al., 1985; Suter and Holzwarth, 1987; Holzwarth, 1991; Mullineaux and Holzwarth, 1991; Debreczeny at al., 1995a, 1995b; Sharkov et al., 1996). The calculations that lead to the results shown on the left side of Figure 6, are presented in (Schmitt, 2011). To avoid redundant repetition, we refrain from presenting the exact calculated equations here. Applications, the calculation, and the comparison between experimental results and the theoretical model are shown in chapters 2.6 and 2.7.

**Figure 6. **Compartment models describing the kinetics of excitation energy transfer (EET) processes in *A.marina* (left side) and in the cyanobacterium *Synechococcus* 6301 (right side) as published in (Theiss et al., 2011). Image reproduced with permission.