Theoretical Analyses of Photoinduced Electron Transfer from Aromatic Amino Acids to the Excited Flavins in Some Flavoproteins

    (2) Here, Iso a and q a are the radii of Iso and the donor (Trp or Tyr), assuming these reactants spherical, and   is the optical dielectric constant of 2 being used). DA  is static dielectric constant

Flavoproteins contain flavin mononucleotide (FMN), flavin adenine dinucleotide, and riboflavin as a cofactor and are ubiquitously distributed in various microorganisms, in leafy vegetables and specific tissues of other multicellular plants, and in the milk, brain, kidney, liver and heart of mammals, where they play an essential role in many redox reactions (Frago et al., 2008).
The fluorescence of flavins was first reported by Weber (1950), along with the fluorescence quenching of flavins by various substances, including aromatic amino acids. Since then many researchers have studied the photochemistry of flavins and flavoproteins (Silva & Edward, 2006). The quenching of flavin fluorescence by an indole ring was reported with isoalloxazine-(CH 2 ) n -indole dyads by McCormick (1977). Time-resolved fluorescence spectroscopy of flavins and flavoproteins has been reviewed by Berg and Visser (2001). However, a number of flavoproteins are practically non-fluorescent, but rather they emit fluorescence with very short lifetimes (sub-picoseconds) upon excitation with an ultra-short laser pulse (Mataga et al., 1998(Mataga et al., , 2000(Mataga et al., , 2002Tanaka et al. 2007;Chosrowjan et al., 2007Chosrowjan et al., , 2008Chosrowjan et al., , 2010. In these flavoproteins tryptophan (Trp) and/or tyrosine (Tyr) residues always exist near the isoalloxazine ring (Iso). The remarkably fast fluorescence quenching in these flavoproteins was demonstrated to be caused by PET from Trp and/or Tyr to the excited state Iso (Iso*), by means of picosecond (Karen et al., 1983(Karen et al., , 1987 and femtosecond (Zhong & Zewail, 2001) transient absorption spectroscopy. The PET phenomena in these flavoproteins are similar to the flavin photo-receptors (Crosson & Moffat, 2001;Masuda & Bauer, 2002), but had been discovered before the flavin photoreceptors.
Since the seminal works on electron transfer theory by Marcus (1956aMarcus ( , 1956bMarcus ( , 1964, several researchers have further developed the electron transfer theory (Hush, 1961;Sumi & Marcus, 1986;Bixon & Jortner, 1991, 1993Bixon et al., 1994;Kakitani & Mataga, 1985;Kakitani et al., 1991Kakitani et al., , 1992. However, they have been modeled for PET in bulk solution and it is not clear whether these theories can be applicable to PET in proteins. Therefore, it is required to establish a method to quantitatively analyze PET in proteins. In any electron transfer theories there are several parameters that are difficult to determine experimentally. The PET rates in flavoproteins have been analyzed experimentally with ultrafast fluorescence dynamics and theoretically by an electron transfer theory using the atomic coordinates obtained by molecular dynamics (MD) simulation. The procedure to determine the unknown PET parameters is as follows (Nunthaboot et al., 2008a(Nunthaboot et al., , 2009a: (1) the time-dependent atomic coordinates of flavoproteins are obtained by MD simulation, (2) the PET rates are then calculated using a PET theory and the atomic coordinates with a set of trial PET parameters, (3) the parameters are then varied until the best-fit between the calculated and observed fluorescence decays is obtained, according to a non-linear least squares method.
In this review article we describe the results of quantitative analyses of PET in wild type (WT) flavodoxin and FMN binding proteins from Desulfovibrio vulgaris, Miyazaki F, and three relevant flavodoxin amino acid substitution mutants (isoforms) and two relevant FMN binding protein amino acid substitution mutants, respectively, and discuss the characteristics of the PET mechanism in flavoproteins.
Note that for brevity, unless stated otherwise, reference to flavodoxin and FMN binding proteins in this article refers to those from Desulfovibrio vulgaris, Miyazaki F.

Marcus-Hush theory
When the j th flavoprotein contains several PET donors, the PET rate from the k th Trp and/or Tyr near Iso to Iso* by Marcus theory as modified by Hush (1961) (MH theory) is expressed by Eq. (1), and the energy diagram for MH theory is shown in Figure 1.
In Eq. (1), q H is the electronic interaction energy between Iso* and Trp (q = Trp) or Tyr (q = Tyr). jk R is the center to center (Rc) distance between Iso and the ET donor k in the j th flavoprotein.  , B k , T and e are the reduced Planck constant, Boltzmann constant, temperature and electron charge, respectively. DA  is the static dielectric constant of medium between the PET donors and acceptor.
() j ES k is the net ES energy between the k th aromatic ionic species and all other ionic groups in the j th flavoprotein, as described below.
Here, Iso a and q a are the radii of Iso and the donor q (Trp or Tyr), assuming these reactants are spherical, and   is the optical dielectric constant (a value of 2 being used). DA  is the static dielectric constant between Iso and a donor. The radii of Iso, Trp and Tyr were determined according to the following procedure. (1) The three dimensional sizes of lumiflavin for Iso, 3-methylindole for Trp, and p-methylphenol for Tyr were obtained by a semi-empirical molecular orbital method (PM3). (2)

Kakitani-Mataga (KM) theory
The PET rate by Kakitani & Mataga (KM theory) is expressed as Eq. (4), which describes the PET rate for both adiabatic and non-adiabatic processes, whilst the MH and Bixon-Jortner (BJ) theories (see below) describe only adiabatic processes.
Here 0 q  is an adiabatic frequency, q  is the PET process coefficient, and 0 q R is a critical distance between the adiabatic and non-adiabatic PET processes. These quantities depend only on q (Trp or Tyr). When 0 q jk RR  the ET process is adiabatic, whereas when 0 q jk RR  it is non-adiabatic. The other quantities are the same as those in the MH theory (section 2.1.1).

Bixon-Jortner (BJ) theory
The BJ theory describes the PET rates from various vibronic states, as shown in Eq. (5), while the MH and KM theories only describe the PET from the lowest vibrational state.
is the vibronic coupling constant, where V  is the reorganization energy associated with the average frequency   , n is the number of vibrational modes in the donor and q  is the van der Waals contact and is given by Eq. (6).
The meanings of all the other notations are the same as that given in the MH theory (section 2.1.1). www.intechopen.com

Electrostatic (ES) energy between the photoproducts and ionic groups in a protein
Proteins, including flavoproteins, contain many ionic groups, which may influence the PET rate. The cofactor in the relevant flavoproteins is FMN, which has two negative charges at the phosphate. The ES energy between the Iso anion or donor k th cation, and all the other ionic groups in the j th flavoprotein is expressed by Eq. (7) where E n , B n , K n and R n are the numbers of Glu, Asp, Lys and Arg residues, respectively, in the flavoprotein. Here, k = 0 for the Iso anion, and k > 0 for the donor cations. 0 j  is the static dielectric constant inside the entire j th flavoprotein, which should be different from DA  . k C is the charge of the aromatic ionic species k, and is -e for k = 0 (Iso anion), +e for k > 1. Glu C (=e), Asp C (=e), Lys C (= + e), Arg C (= + e) and P C (=e) are the charges of Glu, Asp, Lys, Arg and phosphate anions, respectively. It was assumed that these groups are all in an ionic state in solution. The pK a values of the ionic amino acids in water are 4.3 in Glu, 3.9 in Asp, 10.5 in Lys and 12.5 in Arg. However, as residues within proteins these pK a values may be modified in the range of ± 0.3. His displays a pK a of 6.0 in water. All fluorescence measurements were performed in 0.1 M phosphate buffer at pH 7.0, where His should be neutral. Distances between the aromatic ionic species k and the i th Glu are denoted as ()

Observed ultrafast fluorescence dynamics of flavodoxins and FMN binding proteins
Ultrafast fluorescence dynamics of flavodoxins and FMN binding proteins have been measured by means of a fluorescence up-conversion method (Mataga et al., 2002;Chosrowjan et al., 2007Chosrowjan et al., , 2008Chosrowjan et al., , 2010 In Eq. (9), n = 1 or 2, and in Eq. (10) n = 1 to 3, depending on the protein system, j. The decay parameters are listed in  (10), respectively. b Data were taken from the work by Mataga et al. (2002). c Data were taken from the works by Chosrowjan et al. (2007Chosrowjan et al. ( , 2008Chosrowjan et al. ( , 2010

Determination of the PET parameters
The calculated decay function in the j th protein system is expressed by Eq. (11).
... AV means the averaging procedure of the exponential function in Eq. (11) over t'. In Eq. (11) we assumed that the decay function at every instant of time, t', during the MD simulation time range can always be expressed by an exponential function, and thus the MD www.intechopen.com simulation time range must be much longer than experimental decay time range. In Eq. (11) m is the total number of PET donors in the j th flavoprotein. In MH theory, the unknown PET parameters were q H (q = Trp and Tyr), 0

Iso
Here, F N denotes the number of time intervals in the fluorescence decay, and j N is the total number of flavoproteins for simultaneous analysis.

Homology modeling
Flavodoxins are small flavoproteins with a molecular weight of 15 -23 kDa that have been isolated from a variety of microorganisms. Flavodoxins are considered to function as electron-transport proteins in various metabolic pathways (Sancho, 2006). They contain one molecule of non covalently-bound FMN (see Chart 1) as a cofactor, and exhibit a highly negative reduction potential for the semiquinone / hydroquinone couple of FMN, and accordingly the semiquinone state is stable. The redox properties of FMN in flavodoxins are considerably different from those of the free FMN.
The ultrafast fluorescence dynamics of flavodoxins (Mataga et al., 2002) have been extensively investigated in the WT and the Y97F, W59F and W59F/Y97F (DM) substitution isoforms, as described above.
www.intechopen.com Chart 1. Chemical structure of FMN and its atom notations.

Three-dimensional structures of four flavodoxin isoforms
The protein structures of the WT, single amino acid substitution (Y97F and W59F) and the double amino acid substitution (W59F/Y97F; DM) isoforms have been determined by homology modeling method with the Modeler Module of the Discovery Studio 2.0 software package (http://www.discoverystudios.com) using the flavodoxin Desulfovibrio vulgaris, strain Hildenborough structure (PDB code: 1J8Q) as the template. This protein displays 66% amino acid sequence identity and 79% similarity to the WT flavodoxin of Miyazaki reviewed herein. The validities of the structures were examined with a Verified3D analysis (visit for the method, www.proteinstructures.com by Prof. Salam Al-Karadaghi). Verify3D assigns each residue a structural class based on its location and environment (alpha, beta, loop, polar, apolar etc). Then, a database generated from good structures is used to obtain a score for each of the 20 amino acids in this structural class. Figure 2 shows the Verified3D scores at each amino acid residue, where the quality of the structures is satisfactory.
MD simulations were performed for 10 ns in order to investigate the dynamic properties of the proteins and the important interactions that are involved in the binding of the FMN cofactor to the proteins. Figure 3 shows the three-dimensional structures in water that were obtained by MD simulation.  In the WT isoform, Trp59, Tyr97, Tyr99 and Trp16 are potential PET donors to Iso*, whilst these are Trp59, Tyr99 and Trp16 in the Y97F isomer and Tyr97, Tyr99 and Trp16 in the W59F isomer. In the DM, Tyr99 and Trp16 are the potential PET donors. Data were taken from Lugsanangarm et al. (2011a).

Decomposition free energy analysis of amino acid residues at the FMN binding site
In order to evaluate the important amino acid residues for FMN binding, the decomposition free energy per amino acid residue has been obtained. Figure 4 shows the decomposition energy of FMN from FMN-apoflavodoxin complexes. The amino acids near FMN are categorized into three groups, the 10-loop, 60-loop and 90-loop regions (see Figure 4). The decomposition energy is highest in the amino acids in the 10-loop regions (Ser9, Thr10, Thr11, Gly12 and Asn13 and Thr14) in all isoforms ( Figure 4) . These hydrogen bond interactions are considered to contribute the largest proportion of the decomposition free energy. Among the four flavodoxin isoforms, the decomposition energy is highest in Y97F (-9.30 kcal/mol), followed by W59F (-9.25 kcal/mol), DM (-8.60 kcal/mol) and is lowest in the WT (-8.54 kcal/mol).

Structural dynamics of flavodoxins
Potential PET donors in the WT flavodoxin are Trp59, Tyr97 and Tyr99 and Trp16. The protein dynamics of these flavodoxin isoforms have been examined by viewing the timedependent changes in the Rc distances and the inter-planar angles between Iso and these donors. Figure 5 shows the time-evolutions of Rc in the four different flavodoxin isoforms, where the Rc distances clearly fluctuate rapidly but are mostly within ± 10% of the mean values. In the DM the Rc values of Tyr99 and Trp16 vary with long periods in addition to the rapid fluctuation. Since the bulky Tyr97 and Trp59 residues are both replaced by the smaller Phe residue in the DM then the space around Iso may be increased compared to that in the WT, and so may account for the marked fluctuation in the Rc distances of Tyr99 and Trp16. Figure 6 shows the time-evolutions of the inter-planar angles in the WT flavodoxin, where the variation of the inter-planar angles is about ± 30 deg around the mean. The derived mean Rc and edge-to-edge (Re) distances and inter-planar angles over the MD time range are listed in Table 2. The Rc distance was shortest in Tyr97 and then Trp59 in all four flavodoxin isoforms, whilst Tyr99 and Trp16 are quite far from Iso. The inter-planar angle of Trp 59 in the WT is -43 deg, while it is 73 deg in Y97F.

Analysis of PET with crystal structures of flavoproteins
The PET analysis in flavoproteins first starts with their crystal structures . The logarithms of the averaged PET rate (inverse of the averaged lifetimes) in ten flavoprotein systems are plotted against the Re and Rc distances. The logarithms of the PET rates can be expressed with two straight lines when Rc instead of Re is used (see Figure 7). At longer distances the PET rate rapidly decreases with increasing Rc distances, while at shorter distances it decreases slowly with the same sized increments in the Rc value. When Re is used in place of Rc as the distance measure, no such clear distance-dependence is observed in the flavoprotein systems. According to Moser et al. (1992), the logarithm of the PET rate in photosynthesis systems linearly decreases with increasing Re. However, the time domain of the PET rates in their work is much longer than the one in the flavoprotein systems. It is conceivable then that the logarithm of the PET rate in photosynthesis systems increases more slowly with Rc when the distances become shorter.
The PET in the fast phase with low slope was interpreted to be "Coherent PET", where the PET takes place to the Franck-Condon state of Iso* from Trp or Tyr (Mataga et al., 2002).
Of the ten flavoproteins evaluated, the PET donors with a Rc distance of less than 1 nm were all Trp residues, except for Tyr97 in flavodoxin with an exceptionally low PET rate at an Rc value of 0.57 nm. The low rate in Tyr97 was elucidated by the higher ionization potential of Tyr compared to Trp . Moreover, the agreement between ln obs ET k and ln calc ET k were the highest with KM theory (Figure 7) compared to that MH theory (Sumi & Marcus, 1986) or BJ theories (not shown).

PET analysis with MD snapshots of four flavodoxin isoforms
The PET analysis from ultrafast fluorescence dynamics was first conducted by Nunthaboot et al. (2008aNunthaboot et al. ( , 2009a. Time-dependent PET rates in FMN binding proteins were evaluated from the atomic coordinates of the protein as obtained by MD simulation. All PET theories contain several PET parameters that cannot be experimentally determined. Rather these parameters are numerically determined by a non-linear least-square method, as described in Section 2.4. Fluorescence decay functions of four flavodoxin isoforms (WT, Y97F, W59F and DM) were simultaneously analyzed, with the atomic coordinates of these proteins obtained by MD simulation and KM theory, by Lugsanangarm et al. (2011bLugsanangarm et al. ( , 2011c. The PET parameters common among these flavodoxin systems are listed in Table 3. Ultrafast decay functions of the flavodoxins are expressed by Eq. (9) using the decay parameters listed in Table 1. It is noted that the values of 0  and  are quite different between Trp and Tyr, which is related to the electron coupling terms in the KM theory. The quantum basis for the difference is described by Nunthaboot et al. (2008b). In these works it is assumed that the static dielectric constant varies with the protein systems. Table 4 lists the static dielectric constants inside each protein. The dielectric constant of the WT flavodoxin is greatest among the four systems, and that of the DM is the lowest. This is reasonable because Iso in the WT flavodoxin is sandwiched between the polar Trp59 and Tyr97 residues, while both of them are replaced by the non-polar Phe in the DM. Thus, in the DM isoform Iso should be in a relatively non-polar environment, whilst in the Y97F and W59F isoforms the Iso residue may be in a moderately polar environment.

Dynamics of the PET Rate and related physical quantities in flavodoxins
Time-dependent changes in the PET rates of the four flavodoxin isoforms are shown in Figure 8. n the WT and Y97F isoforms, the PET rates from Trp59 are the fastest even though www.intechopen.com the Rc distance between Iso and Tyr97 in the WT is shorter (see Table 2). The mean PET rates over the MD time range (2 ns with 0.1 ps intervals) are listed in Table 5 along with the other mean physical quantities. The mean PET rate is fastest from Trp59 in WT and then in Y97F, as mentioned above, and is then followed by Tyr97 in the W59F isoform. The PET rates from Trp16 and Tyr99 are always negligibly slow.
The net ES energy,

Protein dynamics of FMN binding proteins
The FMN binding protein from Desulfovibrio vulgaris (Miyazaki F) is considered to play an important role in the electron transport process in the bacterium, but the whole picture of the electron flow and coupling of the redox proteins is not yet clear (Kitamura et al., 1998). Three-dimensional structures of the FMN binding protein from D. Vulgaris (Miyazaki F) were determined by X-ray crystallography (Suto et al., 2000) and NMR spectroscopy (Liepinsh et al., 1997). According to these structures, Trp32 is the closest residue to Iso followed by Tyr35 and then Trp106. To examine the effect of Trp32 on the PET rate in the FMN binding protein, Trp32 was replaced by Tyr (W32Y) or Ala (W32A), and further the single negative charge at residue 13, glutamate 13 (E13) was replaced by either Thr (E13T) or Gln (E13Q). The crystal structures of E13T and E13Q were determined by X-ray crystallography (Chosrowjan et al. 2010). The dynamic behavior of these FMN binding protein isoforms were studied by MD simulation (Nunthaboot et al., 2008a(Nunthaboot et al., , 2009a(Nunthaboot et al., , 2011, and Figure 9 shows snapshots of the WT, E13T, E13Q, W32Y and W32A FMN binding protein isoforms.  Table 6. The Rc distance is shortest in Trp32 among the three different aromatic amino acid residue positions (Trp/tyr32, Tyr 35 and Trp106) with mean distances of Trp32 of 0.70, 0.72 and 0.75 nm in the WT, E13T and E13Q isoforms, respectively. The distance between Iso and Tyr32 in W32Y is shorter than that between Iso and Trp32 in the WT. The inter-planar angle between Iso and Trp32 varies from -52 deg in the WT to -38 deg in the E13Q isoform, while that between Iso and Tyr35 varies from 43 deg in W32A to 93 deg in the WT.

Amino acid at position 13 of the FMN binding proteins
The WT FMN binding protein contains Glu13, with a negative charge at neutral pH, whilst in the E13T and E13Q substitution isoforms the amino acids at this position are Thr13 and Gln13 with neutral charges. The distances between the PET donors or acceptor and amino acid residue 13 of the five FMN binding protein isomers are listed in Table 7. The distances between Iso and side chain of amino acid 13 do not significantly vary between the five FMN binding protein isoforms (range 1.5 -1.6 nm), nor does that between Trp32 (0.9 -1.0 nm), Tyr35 (1.0 -1.2 nm) and Trp106 (1.7 -1.97 nm) excepting that of Trp106 in the W32Y isoform that was further away (2.13 nm).
Protein system Trp32 Tyr32 Tyr35 Trp106 b Center-to-center distance (Rc) and edge-to-edge (Re) distance. c Relative standard deviation (RSD), obtained from SD/mean value. Table 6A. Geometrical factor of Iso and the indicated nearby aromatic amino acids of the FMN binding protein isomers a . Data were taken from Nunthaboot et al. (2011). (Reproduced by permission of the PCCP Owner Societies).

Protein system
Inter-planar angle (deg) Trp32 Tyr32 Distances were obtained taking the average over all distances between atoms in the aromatic ring and the center of the two oxygen atoms in the side chain of Glu13. c Obtained by taking the average over all distances between the atoms in the aromatic ring and the oxygen atom of the Thr13 (E13T) or Gln13 (E13Q) side chain. Table 7. Geometry of the amino acid residue at position 13 in the five FMN binding protein isoforms a . Data were taken from Nunthaboot et al. (2011). (Reproduced by permission of the PCCP Owner Societies).

The PET rates and related physical quantities in FMN binding proteins
The common parameters among the five FMN binding protein isoforms are listed in Table 3 Figure 10. Fluctuations of the PET rate are always marked in Tyr35, but not so much in Trp32. In the WT isoform the PET rates vary with rather long periods in addition to the instantaneous fluctuations, which is in accord with the time-evolution of Rc distances in the WT. The mean PET rate and physical quantities related to the PET rates are listed in Table 8, where the PET rate is observed to always be fastest from Trp32, and then from Trp106 whilst that from Tyr35 is always slow (see also Figure 10). Among the WT, E13T and   Figure 11). In contrast, the net ES energies, () j ES k , varied from -0.069 eV in Trp106 (WT) to 0.449 eV in Tyr35 (E13T). This remarkable variation in () j ES k compared to the other physical quantities is also seen in the flavodoxin isoforms.

Effect of changing the negative charge of amino acid residue 13 on the PET rate
The PET rate of Trp32 was fastest in all five FMN binding protein isoforms. The ES energies between the Iso anion and ionic groups in the proteins, ( ) j Ek, fell from 0.071 eV in the WT (and similar values in W32Y and W32A) to -0.023 eV and 0.021 eV in E13T and E13Q, respectively (Table 8; Figure 11), suggesting a potential affect of the charge neutralization at residue position 13. In addition, the ES energies between the Trp32 cation and the ionic groups in the proteins increased dramatically from 0.005 eV in the WT, to 0.335 eV and 0.269 eV in the E13T and E13Q isoforms, respectively. In the WT the ES energy between the negative charge of Glu and Trp32 cation should be negative, which contributes to reduce the value of ( ) j Ek. In the neutral charged (at residue 13) E13T and E13Q isoforms the stabilizing energy found in the WT disappears, again supporting the potential importance of the negative charge at residue 13. It is noted that the absolute values of the net ES energies are quite low in the WT, while they are much higher in the other isoforms. Net ES energies of Trp32, from which the PET rate is fastest, are always positive, while those for Trp106 are negative in the WT, E13T and E13Q isoforms.

Energy gap law in flavodoxin and FMN binding protein systems
The total free energy gap of the k th donor in the j th flavoprotein is expressed by Eq. (12) Table 5 for flavodoxins and Table 8 for FMN binding proteins. Figure 12 shows the modified energy gap law in flavodoxins and FMN binding proteins, as expressed by Eq. (13). The inserts in Figure 12 represent the approximate parabola functions. In the both systems, the PET takes place in the normal region.

Concluding remarks on the PET mechanism in flavoproteins
Quantitative analyses of the PET in proteins have been difficult, because all of the current PET theories contain several unknown parameters which cannot be determined experimentally. In the earlier works the PET rate was qualitatively analyzed from the following two aspects.
1. The donor-acceptor distance-dependence of the PET rate (Dutton law). Hopfield (1974) described biological electron transfer rate in the ground state of a donor in terms of the electron tunneling model. In this model, the rate drops off exponentially with increasing donor-acceptor distance. Hopfield estimated the slope of the logarithm of the rate against the distance to be 14 nm -1 for biological electron transfer reactions. Indeed, Moser et al. (1992) have experimentally demonstrated that logarithms of PET rates linearly decrease with the Re distance between PET donors and acceptors in photosynthetic proteins. In accord, the slope of the logarithm of the PET rate against the free energy gap was also around 14 nm -1 . Gray & Winkler (1996) have reviewed the experimental works on PET rates in ground state donors from various aspects. 2. The free energy gap dependence of PET rate (Energy gap law).
The characteristics of the Marcus theory (1956aMarcus theory ( , 1956bMarcus theory ( , 1964 is that the logarithm of the PET rate is a parabolic function of the reorganization energy and the free energy gap (see Eq. (1)), which is common with the other theories (see Eqs. (4) and (5)). As a test for the Marcus theory many researchers have examined the dependence of the logarithmic values of the PET rates on the free energy gap. Rehm & Weller (1969;1970) first examined the energy gap law with the donor-acceptor systems in organic solvents, but could not find the predicted parabolic dependence. Later Closs et al. (1986) and Mataga et al. (2003) found evidence of the PET processes in the so-called "Inverted region". Interested readers should consult , who have precisely reviewed the current knowledge of PET in solution.

A B
www.intechopen.com The energy gap law in proteins was first experimentally demonstrated in the reaction center of the purple bacterium, Rhodobacter sphaeroides, by Gunner & Dutton (1989), and in both the plant photosystem I and reaction center of the purple bacterium by Iwaki et al. (1996). In these systems, the PET takes place in the normal regions, as in the flavoproteins described above.
We have been trying to quantitatively analyze PET in flavoproteins (Nunthaboot et al., 2008a(Nunthaboot et al., , 2008b(Nunthaboot et al., , 2009a(Nunthaboot et al., , 2009b(Nunthaboot et al., , 2010(Nunthaboot et al., , 2011Lugsanangarm et al., 2011bLugsanangarm et al., , 2011c, using the experimental and theoretical approaches of evaluating the ultrafast fluorescence dynamics of Iso in the flavoproteins and using MD simulation based approaches, respectively.The following conclusions have been derived on the mechanisms of PET in the flavoproteins. 1. The donor-acceptor distance-dependent PET rates were analyzed with MH, KM and BJ theories, whereupon the KM theory was found to be the best for describing PET in the flavoproteins. 2. The ultrafast fluorescence decays of flavoproteins are mostly non-exponential. The nonexponential decay of the WT FMN binding protein was first reproduced with MD snapshots and PET theories, taking an average of the single-exponential decay function over the MD time domain (Nunthaboot et al., 2009a). This suggests that the nonexponential behavior in the decays is caused by the fluctuations of the protein structures with short and longer fluctuation periods. Again, KM theory could best reproduce the observed non-exponential decay. 3. The ultrafast experimental decays in several flavoprotein isoforms are satisfactorily reproduced with common PET parameters in the present method (Nunthaboot et al., 2008a(Nunthaboot et al., , 2010(Nunthaboot et al., , 2011Lugsanangarm et al., 2011c). 4. The introduction of ES energy into the PET theories greatly improves the agreement between the observed and (KM theory) calculated decays in the three FMN binding protein isoforms (Nunthaboot et al, 2008a(Nunthaboot et al, , 2009a. 5. The introduction of the dielectric constant between the donor and acceptor ( DA  ) improved the agreement between the observed and (KM theory) calculated decays (Nunthaboot et al., 2011). DA  is different from the dielectric constant inside the entire protein ( 0 j  ), and always much lower than 0 j  . This is reasonable because normally no amino acid exists between the donor and acceptor. 6. Changes in the single negative charge at residue 13 of the WT FMN binding protein (Glu13) to amino acids with a neutral charge (E13T and E13Q) substantially changed the ultrafast fluorescence decay, which suggests that the ES energy inside the proteins is very important for the PET rate (Chosrowjan et al., 2010;Nunthaboot et al., 2011).

Perspective of the quantitative PET analyses
Method of homology modeling has been useful for the determinations of protein structures, which have been experimentally unable (www.proteinstructures.com). The present method for the quantitative analysis of the PET mechanism may be also applicable to photosynthetic systems and flavin photoreceptors, such as AppA (Nunthaboot et al., 2009b;. Most of the flavoproteins function in the electron transport and electron transfer from a substrate to Iso without light. A number of researchers have been working on the mechanisms of the dark electron transfer in proteins (Grey & Winkler, 1996;Beratan et al., 1991;. These works, however, have mostly focused on the electron coupling term, and not discussed much on the nuclear term. ES energy which is in the nuclear term, should also play an important role on the dark electron transfer rates, and redox potentials of Iso in flavoproteins. Determination of all physical quantities contained in both electronic and nuclear terms of an electron transfer theory could explore a new aspect of the mechanisms of PET and dark electron transfer phenomena in proteins.