**2. The method of nonlinear laser fluorescence spectroscopy**

The fluorescence signal from fluorophores of complex organic compound (COC) under powerful laser excitation is represented as the nonlinear function of the number of detected fluorescence photons *NFl* (or fluorescence intensity *IFl*) on the photon fluxes *F* of pumping radiation (Filipova et al., 2001). The dependence *NFl(F)* is called fluorescence saturation curve, its typical view is represented in the Fig. 1(a). There are several reasons for that nonlinear dependence: the non-zero lifetime of organic molecules in excited state; intercombination conversion; intermolecular interactions including singlet–singlet annihilation, etc.

<sup>1</sup> The heterogroup is called a chromophore, even when it produces fluorescence, in other words, it is a fluorophore.

In this chapter, a new approach based on the simultaneous use of nonlinear laser fluorimetry, spectrophotometry and conventional fluorimetry methods is presented. The approach allows us to *in vivo* determine the individual photophysical parameters of fluorophores in multi-fluorophore protein complexes. The approach has been applied for investigation of the photophysical properties of the protein molecules of different complexity. Two classes of proteins have been chosen, namely, serum albumins (by the examples of human and bovine serum albumins) and fluorescent proteins (by the example

i. The photophysical parameters such as (a) true absorption cross section (at 266 nm) of tryptophan and intersystem crossing rate in single-tryptophan-containing protein human serum albumin and (b) true absorption cross section (at 266 nm) of tryptophans and rate of energy transfer between them in two-tryptophan-containing protein bovine

ii. The complete solution of the task of determining the photophysical parameters of all mRFP1 spectral forms is given. The mechanism of photophysical processes in the spectral forms under their excitation by UV radiation (at 266 nm) has been clarified. iii. The study of the influence of a single amino acid substitution in mRFP1 protein on individual photophysical parameters of the chromophore (a heterogroup1 responsible for light absorption and fluorescence in the visible wavelength range) of fluorescent spectral form is performed. The 66th amino acid residue (glutamine 66) has been chosen as a position to be replaced. This residue participates in formation of the chromophore and, as was shown in (Banishev et al., 2009), its substitution by polar serine or cysteine changes the spectral and photophysical properties of the resultant mutant of the mRFP1. In the present work this study has been extended. The optical properties of new variants of mRFP1 with polar (asparagine, histidine) and non-polar (alanine, leucine, phenylalanine) substitution have been investigated. It was found that the individual extinction coefficient of the chromophore and the position of the steady-state spectra of the proteins with polar substitution correlate with the volume of the substituted amino

Except for this key target, the methodological task has been put, namely, to demonstrate the unique capabilities of the nonlinear laser fluorimetry method (which is not, so far, well

The fluorescence signal from fluorophores of complex organic compound (COC) under powerful laser excitation is represented as the nonlinear function of the number of detected fluorescence photons *NFl* (or fluorescence intensity *IFl*) on the photon fluxes *F* of pumping radiation (Filipova et al., 2001). The dependence *NFl(F)* is called fluorescence saturation curve, its typical view is represented in the Fig. 1(a). There are several reasons for that nonlinear dependence: the non-zero lifetime of organic molecules in excited state; intercombination conversion; intermolecular interactions including singlet–singlet

1 The heterogroup is called a chromophore, even when it produces fluorescence, in other words, it is a

of monomeric red FP mRFP1). The following new results are presented.

acid residue at position 66. The explanation of this effect is given.

**2. The method of nonlinear laser fluorescence spectroscopy** 

known in a wide circle of opticians) on the specific object.

annihilation, etc.

fluorophore.

serum albumin have been determined.

Fig. 1. (a) The *NFl(F)* dependencies (see text): (1) in the absence of fluorescence saturation, and (1b) when saturation appears (for most COC at *F*>1023 cm−2s−1 (Fadeev et al., 1999)). (b) Photophysical processes in COC (Lakowicz, 1999), without accounting for intermolecular interactions. Solid and dotted vertical lines are radiation and radiationless transitions respectively, Si are singlet states and Ti are triplet states.

The parameters of saturation curves depend on photophysical characteristics of molecules fluorophores, so that such characteristics can be extracted from these curves after resolving an inverse problem (Fadeev et al., 1999). This is a basement of nonlinear laser fluorimetry as a method for investigation of photophysical properties of COC. To solve the inverse problem, we should first calculate (either analytically or numerically) the theoretical saturation curves by using the fluorescence response formation model of an ensemble of fluorescent molecules under their excitation by laser radiation. In present work two models have been used: the conventional model of fluorescence response formation and the model of localized donor-acceptor (LDA) pairs.

The conventional model (Banishev et al., 2008a, 2009) in describing the fluorescence response is represented as a system of equations that describes the kinetics of concentration of COC molecules at the corresponding energy states (Fig. 1(b)). In the case of monomolecular solutions of non-interactive organic compounds, we must give priority to the following photophysical parameters when defining the saturation curve:


The model of the fluorescence response, which takes into account processes pointed out above, can be described by the following set of kinetic equations (Fadeev et al., 1999):

Laser Fluorescence Spectroscopy:

donor and acceptor respectively.

1 2

*t*

 

*t*

*t*

 

*t n n*

Application in Determining the Individual Photophysical Parameters of Proteins 187

Fig. 2. The collective states of the LDA pair (nonmetering the process of singlet-singlet annihilation and intersystem crossing). The S0D, S1D and S0A, S1A are the energy levels of the

respectively; the concentration of such molecules is denoted as 1 1 *n n tr* (, ) .

the state S0A; the concentration of such molecules is denoted as 2 2 *n n tr* (, ) .

mathematically described by the following system of kinetic equations:

1

(, ) (, )

3 3 4

(, ) (, )

(, ) (, )

4 4 4

2 2 4

 

*nn n Ftr n K n Ftr n*

*<sup>n</sup> n n Ftr n Ftr n K n*

 

*A DA D*

*A D SS*

where *τD*, *τA* and *σD*, *σ*A are the lifetime and absorption cross section of the donor (denoted by D) and acceptor (denoted by A) as it defined above; *KDA* is the rate of the energy transfer from the excited donor to the unexcited acceptor; *KSS* is the rate of energy transfer from the excited donor to the excited acceptor (singlet–singlet annihilation (Fadeev et al., 1999); *Ftr* (, ) is the photon flux density (see Eqs. (1a)); and *n0* is the total concentration of molecules containing a LDA pair. In this model, the following photophysical parameters are

*n n <sup>n</sup> Ftr n Ftr n K n K n*

22 1

 

*D A*

23 4

 

3 12 4

   

> 

> > (1b)

*D A*

*D A*

*D A DA SS A D*

1 2 3

*n n <sup>n</sup> Ftr <sup>n</sup>*

*D A*

(, )( )

 

<sup>340</sup> *nnn* ,




As a result, there is no need to describe the fluorescence response from a sub-ensemble of donor and acceptor molecules separately, i.e. it is unnecessary to create two systems of equations (one for a donor sub-ensemble and one for an acceptor sub-ensemble) similar to (1a), as it takes place in the conventional approach. Instead of this, the system of equations that describes the populations of the collective states can be written. The dynamics of variation in the concentrations of these four collective states of the LDA pair (nonmetering the process of intersystem crossing in molecules of the donor and acceptor) is

$$\begin{aligned} \frac{\partial n\_1(t, \vec{r})}{\partial t} &= -F(t, \vec{r}) \cdot \sigma \cdot [n\_0(t, \vec{r}) - n\_3(t, \vec{r}) - n\_2(t, \vec{r})] + (K\_3 - K\_{32}^\circ) \cdot n\_3(t, \vec{r}) \\ \frac{\partial n\_3(t, \vec{r})}{\partial t} &= F(t, \vec{r}) \cdot \sigma \cdot [n\_0(t, \vec{r}) - n\_3(t, \vec{r}) - n\_2(t, \vec{r})] - K\_3 \cdot n\_3(t, \vec{r}) \\ \frac{\partial n\_2(t, \vec{r})}{\partial t} &= K\_{32}^\circ \cdot n\_3(t, \vec{r}) \\ n\_0 &= n\_1 + n\_2 + n\_3 \end{aligned} \tag{1a}$$

where *n0* is the total concentration of molecules; *n3*, *n2*, and *n1* are concentrations of molecules in the S1, T1 and S0 states, respectively; *Ftr* (, ) is the photon flux density of exciting radiation at the coordinate point *r* at instant of time *t*. The rest of parameters are defined above. In model (1a), the transition from T1 to S0 is neglected. This assumption is valid if the light pulse duration (*tp*) is much less than the lifetime in the T1 state, i.e. *tp* is much less then *(K21+K'21)-1*. For pulse lasers often used in laser fluorescence spectroscopy, the *tp* is ~10 ns, and this condition is fulfilled.

The conventional model (1a) describes the processes in a system in the absence of interaction between molecules. If there is the interaction and an ensemble of fluorophores generating the fluorescence response consists of subensembles of the donor and acceptor molecules, then the conventional approach is reduced to two systems of kinetic equations, i.e. separately for each subensemble. The term describing the energy transfer is in this case a "cross term" that connects these two systems of equations. Such model, based on separate mathematical description of two subensembles, is able to describe the fluorescence response when each molecule of the donor is surrounded by a large number of the acceptor molecules onto which the energy transfer can occur (Agranovich & Galanin, 1982) (i.e. the possibility that the donor molecule and all the locally surrounding it acceptor molecules simultaneously stay in the excitation state is excludes). The situation like this is typical for a concentrated binary solution of single-fluorophore molecules (for example, dye solutions with the concentration higher than 10-4 M) or for complexes with high local concentration (Fadeev et al., 1999), such as phytoplankton. The energy transfer process in that case is called the intermolecular one.

If there is a donor–acceptor pair within a single molecule (i.e. we have a molecule with a LDA pair), the situation is possible when the donor and the acceptor are simultaneously in an excited state. Therefore, the description of the energy transfer in the framework of a conventional scheme is impossible and the model (1a) should be modified. Let us note that the molecular objects with LDA pair are finding more and more wide applications at present time. Commonly, systems of this kind are constructed artificially from pairs of organic compounds, for example, from dye molecules (Srinivas et al., 2001) or FP macromolecules (Truong & Ikura , 2001). In (Banishev et al., 2008b) a fluorescence response formation model of an ensemble of LDA pairs has been suggested by the author. The model makes it possible to describe the energy transfer inside a LDA pair, disregarding the energy transfer between the pairs. The main idea of this approach consists in the following. Let us introduce a notion of the *collective states* of a LDA pair (Fig. 2); each of these states simultaneously describes both the donor state and the acceptor state:

(, ) ( , ) [ ( , ) ( , ) ( , )] ( ) ( , )

*n tr Ftr n tr n tr n tr K K n tr*

where *n0* is the total concentration of molecules; *n3*, *n2*, and *n1* are concentrations of molecules in the S1, T1 and S0 states, respectively; *Ftr* (, ) is the photon flux density of

defined above. In model (1a), the transition from T1 to S0 is neglected. This assumption is valid if the light pulse duration (*tp*) is much less than the lifetime in the T1 state, i.e. *tp* is much less then *(K21+K'21)-1*. For pulse lasers often used in laser fluorescence spectroscopy,

The conventional model (1a) describes the processes in a system in the absence of interaction between molecules. If there is the interaction and an ensemble of fluorophores generating the fluorescence response consists of subensembles of the donor and acceptor molecules, then the conventional approach is reduced to two systems of kinetic equations, i.e. separately for each subensemble. The term describing the energy transfer is in this case a "cross term" that connects these two systems of equations. Such model, based on separate mathematical description of two subensembles, is able to describe the fluorescence response when each molecule of the donor is surrounded by a large number of the acceptor molecules onto which the energy transfer can occur (Agranovich & Galanin, 1982) (i.e. the possibility that the donor molecule and all the locally surrounding it acceptor molecules simultaneously stay in the excitation state is excludes). The situation like this is typical for a concentrated binary solution of single-fluorophore molecules (for example, dye solutions with the concentration higher than 10-4 M) or for complexes with high local concentration (Fadeev et al., 1999), such as phytoplankton. The energy transfer process in that case is

If there is a donor–acceptor pair within a single molecule (i.e. we have a molecule with a LDA pair), the situation is possible when the donor and the acceptor are simultaneously in an excited state. Therefore, the description of the energy transfer in the framework of a conventional scheme is impossible and the model (1a) should be modified. Let us note that the molecular objects with LDA pair are finding more and more wide applications at present time. Commonly, systems of this kind are constructed artificially from pairs of organic compounds, for example, from dye molecules (Srinivas et al., 2001) or FP macromolecules (Truong & Ikura , 2001). In (Banishev et al., 2008b) a fluorescence response formation model of an ensemble of LDA pairs has been suggested by the author. The model makes it possible to describe the energy transfer inside a LDA pair, disregarding the energy transfer between the pairs. The main idea of this approach consists in the following. Let us introduce a notion of the *collective states* of a LDA pair (Fig. 2); each of these states

simultaneously describes both the donor state and the acceptor state:

032 3 32 3

at instant of time *t*. The rest of parameters are

(1a)

0 3 2 33

1 '

*n tr Ftr n tr n tr n tr K n tr*

(, ) ( , ) [ ( , ) ( , ) ( , )] ( , )

3

*t*

*t*

*t nnnn*

2 '

 

0123

exciting radiation at the coordinate point *r*

the *tp* is ~10 ns, and this condition is fulfilled.

called the intermolecular one.

32 3

,

(, ) (, )

*n tr K n tr*

Fig. 2. The collective states of the LDA pair (nonmetering the process of singlet-singlet annihilation and intersystem crossing). The S0D, S1D and S0A, S1A are the energy levels of the donor and acceptor respectively.


As a result, there is no need to describe the fluorescence response from a sub-ensemble of donor and acceptor molecules separately, i.e. it is unnecessary to create two systems of equations (one for a donor sub-ensemble and one for an acceptor sub-ensemble) similar to (1a), as it takes place in the conventional approach. Instead of this, the system of equations that describes the populations of the collective states can be written. The dynamics of variation in the concentrations of these four collective states of the LDA pair (nonmetering the process of intersystem crossing in molecules of the donor and acceptor) is mathematically described by the following system of kinetic equations:

$$\begin{aligned} \frac{\partial n\_1}{\partial t} &= -F(t, \vec{r}) \cdot (\boldsymbol{\sigma}\_D + \boldsymbol{\sigma}\_A) \cdot n\_1 + \frac{n\_2}{\tau\_D} + \frac{n\_3}{\tau\_A} \\ \frac{\partial n\_2}{\partial t} &= -F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_A \cdot n\_2 - \frac{n\_2}{\tau\_D} - K\_{DA} \cdot n\_2 + F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_D \cdot n\_1 + \frac{n\_4}{\tau\_A} \\ \frac{\partial n\_3}{\partial t} &= -F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_D \cdot n\_3 - \frac{n\_3}{\tau\_A} + F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_A \cdot n\_1 + \frac{n\_4}{\tau\_D} + K\_{DA} \cdot n\_2 + K\_{SS} \cdot n\_4 \\ \frac{\partial n\_4}{\partial t} &= F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_A \cdot n\_2 + F(t, \vec{r}) \cdot \boldsymbol{\sigma}\_D \cdot n\_3 - \frac{n\_4}{\tau\_D} - \frac{n\_4}{\tau\_A} - K\_{SS} \cdot n\_4 \\ \boldsymbol{n}\_1 + \boldsymbol{n}\_2 + n\_3 + n\_4 &= n\_{0, \prime} \end{aligned} \tag{1b}$$

where *τD*, *τA* and *σD*, *σ*A are the lifetime and absorption cross section of the donor (denoted by D) and acceptor (denoted by A) as it defined above; *KDA* is the rate of the energy transfer from the excited donor to the unexcited acceptor; *KSS* is the rate of energy transfer from the excited donor to the excited acceptor (singlet–singlet annihilation (Fadeev et al., 1999); *Ftr* (, ) is the photon flux density (see Eqs. (1a)); and *n0* is the total concentration of molecules containing a LDA pair. In this model, the following photophysical parameters are

Laser Fluorescence Spectroscopy:

(Boychuk at al., 2000).

*(F)*]). If in measurements of

*Fl del*

The *tdel* changes discretely and proportionally to the detector gate step: *tdel=i*

shorter than the lifetime of a fluorophore in the excitation state (picoseconds).

complex organic compound (Filipova et al., 2001).

(*0lim|F0*[

Application in Determining the Individual Photophysical Parameters of Proteins 189

as a reference signal is used, then it is possible to find the fluorescence quantum yield of a

For the reasons pointed out in (Banishev et al., 2008a), the same laser fluorimeter has been optimized for measuring the nanosecond fluorescence decay (the kinetic mode of the fluorimeter operation). The curve represents the dependence of the number *NFl(tdel)* of fluorescence photons in the detector gate (with wide *tg*) on the gate delay time *tdel* with respect to a laser pulse. For the model (1a) an expression for kinetic curve can be written as:

> /2 31 3 /2

*t t*

*g del*

*g del*

( ) (, )

number of the detector gate step. Similar expressions can be written for (2b, c). Gate position at which its centre coincided with the laser pulse maximum was taken as the zero delay (*tdel*=0). This was detected by the maximum of water Raman line (Banishev et al., 2006).

By solving the inverse problem, the fluorescence lifetime *τ* of a fluorophore can be determined independently from the dependence *NFl(tdel)*. In the experiment the fluorescence signal is measured in relative units. For comparison of the experimental data with the theoretical ones it is necessary to normalize the obtained experimental curve to the fluorescence intensity at some fixed time delay. This procedure, the fluorimeter capabilities in the kinetic mode and the corresponding theory can be found elsewhere (Banishev et al., 2006). The difference of such variant of kinetic fluorimetry from the conventional timeresolved fluorimetry is that the fluorescence is excited by a pulse with rather long duration (~10 ns), and for fluorescence registration an optical gated multichannel analyser is used. Whereas, the conventional time-resolved fluorimetry (Lakowicz, 1999) is based on the analysis of fluorescence decay curves after the excitation pulse, whose duration is much

For determination of the photophysical parameters from experimental curves (solution of the inverse problem) the variation algorithm (Banishev et al., 2008a) was used. It is based on the procedure of minimizing the functional of the residue between the experimental curves and curves calculated from models (1a) or (1b) by varying the photophysical parameters

It is necessary to point out two distinctive features of nonlinear laser fluorimetry: (i) as the method implies detection of fluorescence photons (see Eqs. (2)), the photophysical parameters derived from the saturation curve relate only to a fluorescent molecule of COC; (ii) information on the concentration of fluorescent molecules is not used in deriving the photophysical parameters from the saturation curve (Banishev et al., 2009; Filipova et al., 2001). Thus, the method allows one to determine individual photophysical parameters of a molecule in the case when the following complex situation takes place: (i) the sample under study is a multicomponent ensemble of molecules, the absorption bands of its subensembles overlapping (i.e. when the sample is excited, all the subensembles absorb light); (ii) the concentrations of molecules from the subensembles in the mixture are *a priori* unknown.

*Vtt N t K dr n t r dt* *0*

*<sup>0</sup>* the Raman scattering band of water molecules

(3)

*tstep*, where *i* is a

being dependent on photon flux density *F* and tends to a constant which is denoted by

presented: the absorption cross section and the excited state lifetime of the donor and acceptor, the energy transfer rates *KDA* and *KSS*.

By solving systems (1a) and (1b) numerically, one can find the concentration of the fluorescent molecules in the excited state and calculate the number of fluorescence photons *NFl*, emitted from the volume *V* after the action of the laser pulse (Filipova et al., 2001; Fadeev et al., 1999). The theoretical saturation curve for the model (1a) can be calculated from following equation:

$$N\_{Fl}(\mathcal{X}) = K\_{31} \cdot \int\_{V} d\vec{r} \int\_{-\infty}^{+\infty} n\_3(t, \vec{r}) dt \tag{2a}$$

For the model (1b), for the donor (2b) and the acceptor (2c) curves, respectively:

$$N\_{\rm FI}^{D}(\mathcal{A}) = \tau\_{D}^{-1} \cdot \eta\_{D} \cdot \int\_{V} \vec{dr} \int\_{-\infty}^{\omega\_{0}} (n\_{2}(t, \vec{r}) + n\_{4}(t, \vec{r}) \text{ )}dt\tag{2b}$$

$$N\_{\rm FI}^{\Lambda}(\mathcal{L}) = \tau\_A^{-1} \cdot \eta\_A \cdot \int\_V \vec{dr} \int\_{-\infty}^{+\infty} (n\_3(t, \vec{r}) + n\_4(t, \vec{r})) \, dt \tag{2c}$$

where *ηD* and *ηA* are the fluorescence quantum yield, which is defined as the ratio of the radiation decay rate of S1 state to the sum of all rates of S1 state decay (i.e.  *K31/K3)*, of the donor and acceptor, correspondingly; is the fluorescence registration wavelength. Other symbols are defined in Eqs. (1a).

In considered model (1a), the fluorescence saturation is caused by a finite lifetime *τ* and by intercombination conversion. In model (1b), due to the finite fluorescence lifetime and due to the saturation of the energy transfer channels. Let us note that the model (1b) could be also supplemented with the intersystem crossing mechanisms, but preliminary experiments showed that at the given parameters of the laser radiation the process for albumins and mRFP1 is small compared to the mechanisms under study and contributes little to fluorescence saturation. Therefore, this mechanism has been excluded to increase the stability of the inverse problem solution (details and mathematical basement of inverse problem solution of nonlinear laser fluorimetry can be found elsewhere (Boychuk at al., 2000). For the same reason the induced processes from the excited states (two-photon absorption or photoizomerization, etc) have been excluded.

As was mention above, the photophysical parameters of fluorophores (, *K'32* and in the model (1a) and *τD*, *τA*, *σD, σA*, *KDA* and *KSS* in the model (1b)) can be determined from the dependence *NFl(F)*, by solving the inverse problem. However, in experiments, it is convenient to normalize the number of detected fluorescence photons *NFl* to the reference signal (will denote as *NRef)*, which can represent a part of exciting radiation directed to the reference channel of the detection system by a beamsplitter or a Raman scattering signal from water molecules (Fadeev et al., 1999). In this case, one has to deal with the dependence *[Ф(F)]-1=NRef/NFl* (which is also called a saturation curve, *Ф(F)* is the fluorescence parameter) rather than *NFl(F)*. According to the practical experience such normalization also helps to increase the stability of the inverse problem solution. In the absence of saturation, stops

presented: the absorption cross section and the excited state lifetime of the donor and

By solving systems (1a) and (1b) numerically, one can find the concentration of the fluorescent molecules in the excited state and calculate the number of fluorescence photons *NFl*, emitted from the volume *V* after the action of the laser pulse (Filipova et al., 2001; Fadeev et al., 1999). The theoretical saturation curve for the model (1a) can be calculated

> <sup>31</sup> <sup>3</sup> ( ) (, ) *Fl V N K dr n t r dt*

2 4 ( ) ( ( , ) ( , ) ) *<sup>D</sup>*

 

3 4 ( ) ( ( , ) ( , )) *<sup>A</sup>*

 

where *ηD* and *ηA* are the fluorescence quantum yield, which is defined as the ratio of the

In considered model (1a), the fluorescence saturation is caused by a finite lifetime *τ* and by intercombination conversion. In model (1b), due to the finite fluorescence lifetime and due to the saturation of the energy transfer channels. Let us note that the model (1b) could be also supplemented with the intersystem crossing mechanisms, but preliminary experiments showed that at the given parameters of the laser radiation the process for albumins and mRFP1 is small compared to the mechanisms under study and contributes little to fluorescence saturation. Therefore, this mechanism has been excluded to increase the stability of the inverse problem solution (details and mathematical basement of inverse problem solution of nonlinear laser fluorimetry can be found elsewhere (Boychuk at al., 2000). For the same reason the induced processes from the excited states (two-photon

model (1a) and *τD*, *τA*, *σD, σA*, *KDA* and *KSS* in the model (1b)) can be determined from the dependence *NFl(F)*, by solving the inverse problem. However, in experiments, it is convenient to normalize the number of detected fluorescence photons *NFl* to the reference signal (will denote as *NRef)*, which can represent a part of exciting radiation directed to the reference channel of the detection system by a beamsplitter or a Raman scattering signal from water molecules (Fadeev et al., 1999). In this case, one has to deal with the dependence *[Ф(F)]-1=NRef/NFl* (which is also called a saturation curve, *Ф(F)* is the fluorescence parameter) rather than *NFl(F)*. According to the practical experience such normalization also helps to

increase the stability of the inverse problem solution. In the absence of saturation,

*V*

*V*

(2a)

(2b)

(2c)

is the fluorescence registration wavelength. Other

, *K'32* and

in the

stops

 *K31/K3)*, of the

*r n t r n t r dt*

*r n t r n t r dt*

1

*Fl D D*

*Fl A A*

absorption or photoizomerization, etc) have been excluded.

As was mention above, the photophysical parameters of fluorophores (

*N d* 

radiation decay rate of S1 state to the sum of all rates of S1 state decay (i.e.

*N d* 

For the model (1b), for the donor (2b) and the acceptor (2c) curves, respectively:

acceptor, the energy transfer rates *KDA* and *KSS*.

from following equation:

<sup>1</sup>

donor and acceptor, correspondingly;

symbols are defined in Eqs. (1a).

being dependent on photon flux density *F* and tends to a constant which is denoted by *0* (*0lim|F0*[*(F)*]). If in measurements of *<sup>0</sup>* the Raman scattering band of water molecules as a reference signal is used, then it is possible to find the fluorescence quantum yield of a complex organic compound (Filipova et al., 2001).

For the reasons pointed out in (Banishev et al., 2008a), the same laser fluorimeter has been optimized for measuring the nanosecond fluorescence decay (the kinetic mode of the fluorimeter operation). The curve represents the dependence of the number *NFl(tdel)* of fluorescence photons in the detector gate (with wide *tg*) on the gate delay time *tdel* with respect to a laser pulse. For the model (1a) an expression for kinetic curve can be written as:

$$N\_{Fl}(t\_{del}) = K\_{31} \cdot \int\_{V} d\vec{r'} \int\_{-t\_g/2 + t\_{dd}}^{t\_g/2 + t\_{dd}} n\_3(t, \vec{r'}) dt\tag{3}$$

The *tdel* changes discretely and proportionally to the detector gate step: *tdel=itstep*, where *i* is a number of the detector gate step. Similar expressions can be written for (2b, c). Gate position at which its centre coincided with the laser pulse maximum was taken as the zero delay (*tdel*=0). This was detected by the maximum of water Raman line (Banishev et al., 2006).

By solving the inverse problem, the fluorescence lifetime *τ* of a fluorophore can be determined independently from the dependence *NFl(tdel)*. In the experiment the fluorescence signal is measured in relative units. For comparison of the experimental data with the theoretical ones it is necessary to normalize the obtained experimental curve to the fluorescence intensity at some fixed time delay. This procedure, the fluorimeter capabilities in the kinetic mode and the corresponding theory can be found elsewhere (Banishev et al., 2006). The difference of such variant of kinetic fluorimetry from the conventional timeresolved fluorimetry is that the fluorescence is excited by a pulse with rather long duration (~10 ns), and for fluorescence registration an optical gated multichannel analyser is used. Whereas, the conventional time-resolved fluorimetry (Lakowicz, 1999) is based on the analysis of fluorescence decay curves after the excitation pulse, whose duration is much shorter than the lifetime of a fluorophore in the excitation state (picoseconds).

For determination of the photophysical parameters from experimental curves (solution of the inverse problem) the variation algorithm (Banishev et al., 2008a) was used. It is based on the procedure of minimizing the functional of the residue between the experimental curves and curves calculated from models (1a) or (1b) by varying the photophysical parameters (Boychuk at al., 2000).

It is necessary to point out two distinctive features of nonlinear laser fluorimetry: (i) as the method implies detection of fluorescence photons (see Eqs. (2)), the photophysical parameters derived from the saturation curve relate only to a fluorescent molecule of COC; (ii) information on the concentration of fluorescent molecules is not used in deriving the photophysical parameters from the saturation curve (Banishev et al., 2009; Filipova et al., 2001). Thus, the method allows one to determine individual photophysical parameters of a molecule in the case when the following complex situation takes place: (i) the sample under study is a multicomponent ensemble of molecules, the absorption bands of its subensembles overlapping (i.e. when the sample is excited, all the subensembles absorb light); (ii) the concentrations of molecules from the subensembles in the mixture are *a priori* unknown.

Laser Fluorescence Spectroscopy:

**3.1.2 Registration system** 

**3.1.3 Laser radiation parameters** 

according to the equation *F-1=E-1*

radiation output intensity at the outlet from the Pockels cell.

could be varied from 10 to 1200 ns; in our experiments it was set to 10 ns.

*ћS*

The laser pulse duration at the wavelengths of 532 and 266 nm were fitted well by Gaussian function with the full width at half maximum (*tp*) of 12 and 10 ns, respectively. When measuring (i) the kinetic curves, the laser beam diameter was 3 mm and (ii) the saturation curves, the beam was focused to a spot with a diameter from 600 to 800 µm, depending on the protein under investigation. The values of the photon flux density *F* were determined

*tp*, where *ћ*

is a photon energy, *E* is average pulse

Application in Determining the Individual Photophysical Parameters of Proteins 191

mirror. Then the beam arrived at the polarizer, which reflects only vertical polarization. The reflected beam was directed through the quarter-wave plate for changing the vertical polarization to the circular one. That improves the efficiency of frequency-doubling in a KTP crystal. The KDP and BBO nonlinear crystals were used for generating the 3d and 4th harmonics. After the frequency conversion, the radiation of the 4th (or the 3d), 2nd and the fundamental harmonics was transmitted through the quartz prism for their spatial divergence in the horizontal plane. The continuous adjustment of the laser intensity during the saturation curves measurement was carried out by the Pockels cell, which was placed in the beam way right after the KTP crystal. The cell consists of the electro-optical component (DKDP crystal) and two Glan prisms. Changing the voltage on the DKDP, one can adjust the

The laser radiation was focused on the cuvette with the sample solution by a long-focallength lens (the focal length is 20 cm). For collecting the fluorescence photons, the lightguide cable (the length is 5 m), consisting of seven quartz fibers (the diameter of each fiber is 600 micron), which were laid out in a row (like a slit) at both ends, was used. The cable inlet was fixed at the cuvette side, and the outlet was clasped to the polychromator entrance slit; the sample solution was excited by transmitted laser radiation. As a detector of radiation, the optical multichannel analyser (OMA) was used. The optical chamber (DeltaTech, Scientific Park of MSU) of the analyser consists of the electro-optical converter based on a gated microchannel plate (MCP), CCD matrix, and optical device for transferring an image from the MCP to the CCD matrix (a pixel size is 11×11µm2). The chamber was fixed to the polychromator (MUM without the output slit, reciprocal linear dispersion is 0.15 nm per channel) optical output. The multichannel analyser was connected to the PC. As a result, optical image in the polychromator output slit plane could be obtained as a 2D picture on the PC monitor. The software installed on the PC allowed the OMA to operate both in the continuous and gated modes. When working in gated mode, a part of light was sent to the silica photodiode PD-265 (the building-up time of the leading edge is less than 2 ns). The photodiode was connected to the nanogenerator triggering inlet (the trigger level is 0.6 V). The gating of the MCP was implemented by high-voltage pulses from the nanogenerator (the amplitude is 800 V). The detector gate delay time could be adjusted through the nanogenerator over the range of 50 ns (the dead time of the detector) to 1200 ns with the step *tstep*=2.5 ns. Exactly because of the dead time of the registration system, when it operates in the gated mode (i.e. in the case of kinetic curves measurements), the light-guide cable as the dead time compensator (the optical delay line) was used. The detector gate width *tg*
