Decomposition of Protein Tryptophan Fluorescence Spectra into [608816]

Decomposition of Protein Tryptophan Fluorescence Spectra into
Log-Normal Components. I. Decomposition Algorithms
Edward A. Burstein, Sergei M. Abornev, and Yana K. Reshetnyak
Institute of Theoretical and Experimental Biophysics, Russia Academy of Sciences, Pushchino, Moscow region, Russia 142290
ABSTRACT Two algorithms of decomposition of composite protein tryptophan fluorescence spectra were developed
based on the possibility that the shape of elementary spectral component could be accurately described by auniparametric log-normal function. The need for several mathematically different algorithms is dictated by the fact thatdecomposition of spectra into widely overlapping smooth components is a typical incorrect problem. Only thecoincidence of components obtained with various algorithms can guarantee correctness and reliability of results. In thispaper we propose the following algorithms of decomposition: (1) the SImple fitting procedure using the root- Mean-
Square criterion (SIMS) operating with either individual emission spectra or sets of spectra measured with various
quencher concentrations; and (2) the pseudo-graphic analytical procedure using a PHase plane in coordinates of
normalized emission intensities at various wavelengths (wavenumbers) and REsolving sets of spectra measured with
various Quencher concentrations (PHREQ). The actual experimental noise precludes decomposition of protein spectra
into more than three components.
INTRODUCTION
The fluorescence parameters of tryptophan residues are
sensitive to the microenvironment of fluorophore in proteinstructure. For this reason, fluorescence characteristics arewidely used to study physico-chemical and dynamic prop-erties of tryptophan microenvironment in proteins and thestructural transitions and behavior of protein molecules as awhole (Lakowicz, 1983, 1999; Demchenko, 1986). Theoverwhelming majority of proteins exhibit smooth, non-structured spectra of tryptophan fluorescence, which oftencontain more than one component. The multicomponentnature of protein spectra makes the unequivocal interpreta-tion of them difficult and poses a task of development ofmethods for the decomposition of tryptophan fluorescencespectra into elementary components (Burstein et al., 1973;Burstein, 1977).
The problem of decomposition of multicomponent spec-
tra belongs to the class of typical, so-called reverse prob-lems,becauseonemustdeterminetheparametersofspectralcomponents from the overall experimental spectrum, wherethe components are indirectly manifested. Solutions of suchproblems are, as a rule, unstable against slight variations intheinputdata(noise).Becausetherealinputdataareknownapproximately (i.e., with some experimental error), thisinstability results in an inevitable ambiguity of the solutionwithin the given accuracy. In this respect Tikhonov andArsenin (1986) classified such a problem as an incorrectone. To obtain a sufficiently stable solution, it is necessaryto formulate a principle of choosing among the possiblesolutions, based on an additional information about the
systemunderstudyandthesolutionquality.Theapplicationof additional information forms a basis for regularizing thesolving (Tikhonov and Arsenin, 1986). The regularizingfactors (functions, algorithms, or logical premises) allowonetodeveloppracticalwaysofsolvingincorrectproblems.Todecomposethemulticomponentproteintryptophanspec-tra,weusedthefollowingregularizingfactors(AbornevandBurstein, 1992; Abornev, 1993):
1.The spectrum of an elementary component on the
frequency (wave number) scale is described by a bipa-rametric (maximal amplitude and maximum position)log-normal function (Burstein and Emelyanenko, 1996).
The quadriparametric (maximal amplitude, I
m, spectral
maximum position, nm, and positions of half-maximal
amplitudes, n2and n1; see Fig. 1) log-normal function
has been proposed by Siano and Metzler (1969) fordescribing the absorption spectra of complex moleculesand was later successfully used to resolve multicompo-nent absorption spectra, including those of biologicalsystems (Metzler et al., 1972, 1985, 1991; Morozov andBazhulina, 1989). The log-normal function used in itsmirror-symmetric form has been shown to accuratelydescribe fluorescence spectra as well (Burstein, 1976;Burstein and Emelyanenko, 1996). The straight linearrelationships between positions of maximal (
nm) and two
half-maximal amplitudes ( n2andn1) have been revealed
for a large series of monocomponent spectra of smalltryptophan derivatives in various solvents and allowed toreduce the number of unknown parameters from four totwo (Burstein and Emelyanenko, 1996). Such a reductionof number of parameters sought is known to make adecomposition much more unambiguous (Antipova-Ko-rotaeva and Kazanova, 1971). As a result, the biparamet-ric log-normal function (uniparametric one for the spec-
Received for publication 12 January 2001 and in final form 7 June 2001.
AddressreprintrequeststoDr.EdwardA.Buzstein,InstituteofTheoretical
and Experimental Biophysics, Russia Academy of Sciences, Pushchino,Moscow region, Russia 142290. Tel.: 0967-733319; Fax: 0967-790553;E-mail: burstein@fluoz.iteb.sezpukhov.su.
© 2001 by the Biophysical Society0006-3495/01/09/1699/11 $2.001699 Biophysical Journal Volume 81 September 2001 1699–1709

tral shape) for fluorescence spectra of tryptophan and its
residues in proteins appears as follows:
HI~n!5ImzexpH2ln2
ln2rzln2Sa2n
a2nmDJ~atn,a!
I~n!50 ~atn$a!
(1)
whereImis the maximal intensity; nis the current wave –
number; ris the band asymmetry parameter r5(nm2
n2)/(n12nm);aisthefunction-limitingpointposition a5
nm1(rz(n12n2))/(r221). Therewith, nm,n1, and n2
are related as (Burstein and Emelyanenko, 1996):
n150.831 znm17070 ~cm21!
n251.177 znm27780 ~cm21! (2)
2.The shape and position of tryptophan emission spectra
remain unchanged at fluorescence quenching with water-soluble small quenchers (Burstein, 1968, 1977; Lehrer,
1971; Lehrer and Leavis, 1978). A series of spectra mea-sured at various quencher concentrations represents a set ofdata wherein the position and shape parameters of compo-nents are constant and only their relative contributions arevariable. Such an expansion of the statistical mass of data,compared with that of an individual spectrum, also aids therise of solution reliability.
3.The change of amplitudes of individual components
with quenching obeys the Stern-Volmer law (Burstein,
1977; Lehrer, 1971; Lehrer and Leavis, 1978).
4. An additional important factor, which results in rise
of reliability of the decomposition procedure, is the useofstatistically redundant information : practically all ex-
perimentally determined points in the input spectra areused for the decomposition, i.e., the number of experi-mental points under analysis far exceeds the number ofparameters sought. This approach attenuates the effect ofoccasional noise and improves the accuracy of results(Akseenko et al., 1989).
The above-mentioned factors allowed us to develop sev-
eral methods for sufficiently stable decomposition of com-posite tryptophan fluorescence spectra of proteins withoutexceeding the experimental error. The practical need inmore than one mathematically different algorithm is a con-sequence of the fact that the individual component spectraare wide (25–61 nm) compared with the spectral intervalwithin which their maxima may be positioned (307–355nm). Thus, the coincidence of parameters of componentsrevealed with diverse methods can guarantee the reliabilityof decomposition results for a given protein. Here we de-scribe two methods, first of which is based on the fitting ofexperimental spectra by a sum of log-normal componentsusing the root-mean-square criterion of fitting quality, andthe second algorithm uses an analytical pseudo-graphicsolving the task.
MATERIALS AND METHODS
Materials
The annexin VI was supplied by Dr. Andrzej Sobota, N. Nencki Institute
of Experimental Biology, Warsaw, Poland (Bandorowicz et al., 1992;Sobota et al., 1993). KI, KC, and Na
2S2O3were commercial preparations
of ultra-high purity of Russian industry production.
Fluorescence spectra
The fluorescence spectra were measured using the lab-made spec-
trofluorimeter with registration from the front surface of the cell(Bukolova-Orlova et al., 1974). The 296.7-nm mercury line from theultra-high-pressure mercury lamp SVD-120A (The Moscow Electro-lamp Factory, Moscow) was used for excitation. The slit widths of theexcitation and output monochromators did not exceed 2 nm. Aftercorrection for instrument spectral sensitivity, the intensities were pro-portional to the number of photons emitted in the unit wavelengthinterval. Under the measurements of fluorescence spectra with differentconcentrations of KI the total ionic strength was kept constant (0.4 M)by addition of KCI. The stock solution of KI contained Na
2S2O3to
prevent oxidation of I2. The decomposition algorithms were used as
Visual Basic programs in personal computers.
THE FITTING ALGORITHM WITH MINIMIZATION
OF ROOT-MEAN-SQUARE RESIDUES (SIMS)
Algorithm descriptionThe below-described algorithm we called SIMS ( SImple
fitting procedure using the root- Mean-Square criterion)
(Abornev and Burstein, 1992; Abornev, 1993).
FIGURE 1 The quadriparametric (maximal amplitude, Im, spectral max –
imum position, nm, and positions of half-maximal amplitudes, n2andn1)
log-normal function.1700 Burstein et al.
Biophysical Journal 81(3) 1699–1709

Because, under the different concentrations of external
fluorescence quenchers, the position and shape of spectralcomponents remain unchanged while the relative contribu-tions of components to the overall spectrum are changed,then the experimental spectra can generally be described asfollows:
F~i,j!5O
k51L
I~k,i!zw~k,j! (3)
wherei51 ,…,Nis the number of spectra correspond-
ing to the ith quencher concentration, c(i);j51 ,…,M
is the number of current frequency (wave number), n(j);
k51 ,…,Listhenumberofcomponentsdeterminedby
the position of its spectral maximum, nm(k);F(i,j)i st h e
experimental intensity of fluorescence on the wave num-
ber scale in the ith spectrum at the jth current frequency
n(j);w(k,j) is the value of the log-normal function with
a maximum at nm(k) at current frequency n(j) with unite
maximal amplitude (at given kandj, this value is the
same for any of the Nspectra); and I(k,i) is the maximal
amplitude of the kth component in the ith spectrum.
The problem is to find the positions nm(k) and the
maximal intensities I(k,i) of log-normal spectral compo-
nents from the set of experimental spectra F(i,j); Eq. 3,
however, cannot be solved analytically because the log-normal function
w(k,j) is essentially transcendental with
respect to the unknown nm(k) (see Eq. 1). Hence, the
solution can be sought for by approximation, e.g., using
the fitting of nm(k) values by minimization of residuals.
At each step of this process, the transcendental w(k,j)
terms are computed for a given nm(k) value, and then the
corresponding I(k,i) values are easily determined analyt-
ically, solving the set of linear equations.
The fact that individual components in protein fluores-
cence spectra are very broad and mutually overlappedposesseverelimitationsontheprocedureofsearchingfora functional minimum. Attempts to use modern fast fit-ting methods revealed a strong dependence of solutionson the initial conditions. Only the exhaustive enumera-tion of
nmvalues (with successively diminishing steps
from ;8 nm down to 0.1 nm) avoided “trapping” in the
local minima of the functional (rms residuals) and, thus,to find its global minimum. It is essentially important inthe presence of experimental noise. Moreover, it obviatesthe need to set any arbitrary initial conditions, whichoften leads to the erroneous result of solving an incorrectreverse problem (Tikhonov and Arsenin, 1986). Thisnotwithstanding, the results of decomposition of experi-mental and simulated multicomponent spectra showedthat the typical experimental noise of ;0.5–1.5% does
not permit a sufficiently reliable decomposition for morethan three spectral components. Therefore, we shall con-sider this limiting case with L#3 in describing the
algorithm. Uni-, bi-, and tri-component solutions aresearched independently by turn for the set of experimen-
tal spectra. However, we shall consider below the tri-component solution as a more general case.
Withfixed
nm(1),nm(2),and nm(3)valuesatcurrentwave
numberjat each fitting step, the solution can be found on
the basis of the minimal least-square formalism, i.e., whentheSwould be minimal:
S5O
i51NO
j51MFO
k513
I~k,i!zw~k,j!2F~i,j!G2
The unknowns are I(k,i). The w(k,j) values are calculated
from Eq. 1 at given nm(k) and n(j) values. The criterion Sis
minimal when dS/dI(k,i)50. These conditions allow
construction of Nsets of three equations in each:
dS
dI~k,i!5O
j51MFO
k513
I~k,i!zw~k,j!2F~i,j!Gzw~k,j!50
(4)
or, after opening the brackets,
O
j51M
w~k,j!zO
k513
I~k,i!zw~k,j!5O
j51M
F~i,j!zw~k,j!
Transposing the summation over kandjin the left part, we
can write down the whole set of Eq. 4:
5O
k513FI~k,i!zO
j51M
w~k,j!zw~1,j!G5O
j51M
F~i,j!zw~1,j!
O
k513FI~k,i!zO
j51M
w~k,j!zw~2,j!G5O
j51M
F~i,j!zw~2,j!
O
k513FI~k,i!zO
j51M
w~k,j!zw~3,j!G5O
j51M
F~i,j!zw~3,j!
(4a)
Because all the sums over jare known, we obtain N
nonuniform sets ( i51 ,…,N) of linear equations,
where each set contains three equations and can be solved
independently, the main determinant being the same for all N
sets.These canonicalsetsoflinearequationsarethensolved
(i.e., theI(k,i) amplitudes are evaluated) using the routine
Gaussmethod.Ananalogousalgorithmwasdevelopedforthe
decomposition of an individual emission spectrum (the pro-gram SIMS-MONO). In this case, the set of equations isconstructed using an expanded set of points of a single spec-trum.
Then, at each step we determine the sum of absolute
values (modules) of residuals S(differences between calcu-
lated and experimental intensities):
S5O
i51NO
j51MO
k51L
uI~k,i!zw~k,j!2F~i,j!u(5)Decomposition of Protein Fluorescence Spectra 1701
Biophysical Journal 81(3) 1699–1709

and the parameter D, which characterizes the quality of
accordance of spectral components quenching with theStern-Volmer law, i.e.,
D51
3zO
k51L
Rsd~k! (6)
Rsd51
Y~k,N!zFO
i51N~Y~k,i!2X~k,i!!2
NG1/2
(6a)
HereRsdis the relative root-mean-square residual between
the values X(k,i)5I(k, 1)/I(k,i) determined by solving Eq.
4a and the Y(k,i) values calculated from the linear equation
of the Stern-Volmer law:
Y~k,i!5Ksn~k!za~i!1B~k! (7)
wherea(i)isanactivityofionicquenchersthatiscalculated
using values of concentrations c(i) (Hodgman et al., 1955).
In the programs, the relations between c(i) anda(i) values
were included analytically in the polynom forms.
The resulting combined minimization criterion (function-
al) is used in the form:
S15Sz~11D!
A set of components, i.e., the values of spectral maxima
positions nmand maximal amplitudes Im, corresponding to
the global minimum of S1is considered as the solution of
Eq. 4a. The above-described algorithm of three-component
decomposition can be, in principle, expanded over an arbi-trary number of components.
The procedure of searching for a sufficient number of
components describing a series of experimental spectraof a protein is carried out as follows. The experimentalseries of spectra is consecutively decomposed into one,two, and three components. Because the experimentalspectra are measured with a constant wavelength incre-ment, each value set (i.e.,
nm(1),nm(2), and nm(3)) on the
frequency scale ( nm(cm21)5107/lm(nm)) is determined
by exhaustion in the wavelength range from 300 to 370
nm with consecutive three-times shortening steps from8.1 to 0.1 nm. The intensities on the frequency ( F
n) and
the wavelength ( Fl) scales are related as Fn5Flzl2;
hence the spectra of individual components are converted
onto the wavelength scale and the maximum position ofcomponents (
nm(k)) are presented on the wavelength
scale ( lm(k)).
To estimate the quality of decomposition, the relative
rms residual of theoretical and experimental spectra isexpressed as a percentage of maximal amplitude F
mof
the spectrum, measured in the absence of quencher:
Ts5Tz~11D! (8)WhereDis determined with Eq. 6 and
T51
MzO
j51M
S2~j! (9)
S~j!51
NzO
i51N
s~i,j! (10)
s~i,j!5O
k51L
I~k,i!zw~k,j!2F~i,j!
Fm~c50!(11)
To choose among the one-, two-, or three-component solu-
tions as being more reliable, we used the discriminant Ds
values equal to the product of functional Tsby the number
of components searched for, i.e., the number of parameters
(L) varied under fitting:
Ds5LzTs (12)
Before determining the final results, the same procedure is
used for the smoothing experimental spectra. As a rule, thespectra contain some points distorted by Raman line, scat-tered mercury lines from the light source, and/or by arandomnoise.Aftereachdecompositioncycle,theintensityvalues differing by .2% from the theoretical ones were
changed to be equal to the latter. The smoothing cycles arerepeated while such differences disappear, but their numberis not to exceed 10 to avoid an eventual distortion of thespectral shape.
Properties of the algorithm
To test how various factors affect the accuracy of solu-
tion, a series of decompositions were carried out forsimulated spectra. The latter were sums of two or threelog-normal curves varying in the positions of maximaand relative amplitudes. Because, above all, we wereinterested in the quality of spectral resolution of compo-nents, the precision was evaluated as a value of D
l,
which is a mean absolute difference between the posi-tions of component maxima (in nanometers) that werepreset in simulation and those obtained after the decom-position. Fig. 2 shows the D
lvalue dependencies for
two- and three-component decompositions, crosses andsquares, respectively, of the amplitude of randomly in-troduced noise, S% (panel A); the number of spectra ( N)
with various “quencher concentrations” used (panel B);
the distance between the component spectral maxima,D
lmax, nm (panel C); the ratio of Stern-Volmer constants
of quenching for the components, K2/K1 (panel D); the
number of registered points in each spectrum, M(panel
E); and the contribution of one component in the total
spectrum, as an intensity ratio I1/(I11I2) (panel F). The
noise was introduced as random equally probable posi-tive and negative deviations with amplitude from 0 to S%1702 Burstein et al.
Biophysical Journal 81(3) 1699–1709

of the theoretical amplitude. The maximal “quencher
concentrations” were such that would reduce the ampli-tude of a total spectrum approximately by half. Varyingone parameter, the others were held constant at the fol-lowing standard values: S%50.6%;N55;M515;
D
lmax510 nm;K150.1 M21;K253.0 M21;I15I2.
For three-component decomposition the standard values
ofS%,N,M, and Dlmaxare the same as for two-compo –
nent ones, but K151M21,K255M21,K350M21,
andI15I25I3.OnlyK1andK2changeinpanel D,and
I1 andI2 in panel F. As can be seen, the method provides
an acceptable accuracy ( Dl,1 nm in the two-compo-
nent and ,1.5 nm in the three-component decomposi-
tion) in determining the true positions of componentmaxima at values of initial parameters usually existing inpractice: S%50.5–1.5;N53–10;M510–20; D
lmax.
7 nm; contribution of an individual component of 10–
90%. The accuracy of decomposition into three compo-nents is somewhat worse than that into two components;however, it is also quite satisfactory, taking into accountthe large overlapping of the component spectra.THE ANALYTICAL ALGORITHM BASED ON THE
PSEUDO-PHASE REPRESENTATION (PHREQ)
The algorithm PHREQ (the PHase-plot-based REsolution
usingQuenchers) is an analytical realization of a graphical
way of two-component decomposition of a set of protein-tryptophan fluorescence spectra measured at various con-centrations of quencher (Abornev, 1993). The method usesquasi-phase representation of parameters characterizing theshape of fluorescence spectra (Burstein, 1976), which wasalready successfully applied for analysis of protein structuraltransitions (Kaplanas et al., 1975; Permyakov et al., 1980a,b).
In the most general case, to use any physical parameter P
eithertocharacterizetheshapeoftransitioncurveA 3Bor
to estimate the proportion of components in the mixture ofA and B components, this parameter ( P) should be linearly
related to either extent of transition completing (
a) or con-
tribution of a mixture component into total concentration.
a5cB
cA1cB
FIGURE 2 The properties of the SIMS algorithm. The dependencies for two- and three- component decompositions, crosses and squares, respectively,
between Dlvalues (mean absolute difference between positions of component maxima (in nanometers) that were preset in simulation and those obtained
after the decomposition) and the amplitude of randomly introduced noise, S%(A); the number of spectra ( N) with various “quencher concentrations” used
(B); the distance between the component spectral maxima, Dlmax,n m(C); the ratio of Stern-Volmer constants of quenching for the components, K2/K1
(D); the number of registered points in each spectrum, M(E); and the contribution of one component in the total spectrum, as an intensity ratio I1/(I11
I2) (F).Decomposition of Protein Fluorescence Spectra 1703
Biophysical Journal 81(3) 1699–1709

wherecAandcBare concentrations of A and B. Then, the
composite parameter Pfor the system can be expressed as:
P5bzPA1azPB
5~12a!zPA1azPB
5PA1az~PB2PA! (13)
Therefore, Pistheweightedmeanofvalues PAandPB,which
characterize the pure A and B components, respectively. The
weight factors are b512aanda, respectively.
A two-state transition or a mixture can be characterized
bytwomutuallyindependentphysicalparameters P1andP2
linearly related to a:
P15bzPA11azPB1
P25bzPA21azPB2
It can be simply shown that parameters P1andP2are linear
functions of one another at the same avalue, i.e.,
P15kzP21m (14)
Wherekandmare factors expressed through the values of
PA1,PA2,PB1, andPB2, and the plane with coordinates ( P1,
P2) possesses a property of phase-plane. In such a plane the
states A and B are represented by points ( PA1,PA2) and
(PB1,PB2), respectively. The transition between A and B is
reflected by the totality of phase-points, which should be
onto the straight line connecting points A and B. At anyintermediate point Z with coordinates ( P
1,P2) located on
thisline(0 ,a,1),aisproportionaltotheratiooflengths
of segment between the points Z and A and segment be-tween points A and B, i.e.,
a5AZ/AB. These simple
relationships allow graphic determination of contributionsof components A and B at any step of transition or intwo-component mixtures.
By the way, in the case of transitions measured by kinet-
ics or equilibrium shift, the deviation from linearity oftrajectory AB suggests the existence of one or more inter-mediate states in the process. In the simplest cases it couldbe evaluated P
1andP2for the intermediate state by extrap –
olating the initial and last linear parts of the trajectory topoint of their intersection, assumedly representing the in-termediate in the plane.
In case of decomposition of fluorescence spectra, the
“physical” state is a position and shape of a spectral com-ponent, which remains unchanged under different concen-trations of fluorescence quenchers. However, addedquenchers perturb the “spectral” state, i.e., change the ratiosof component contributions. The role of parameters P
1and
P2for two-component fluorescence spectra plays the emis –
sion intensities F(n1) andF(n2) measured at different wave –
numbers n1andn2. Such an approach demands the equality
of the number of photons absorbed by both components in
the unit time interval. The parameters F(n1) andF(n2)
reflect the contributions of aandbof two components intothe total emission spectrum. To obtain the linear trajectory
on the quasi-phase plane [ F(n1),F(n2)] it can change the a
value by measuring spectra at various concentrations of
fluorescence quenchers (Cs1,I2, acrylamide, etc.), which
change intensities of two components in different degrees
depending on solvent accessibility of fluorophore(s) fromwhich a component originates, but do not affect the shapeand maximum positions of spectral components (Burstein,1968, 1976, 1977, Lehrer, 1971; Lehrer and Leavis, 1978).
In the ideal imaginary case, when the quenching does not
change summary emission quantum yield, on the quasi-phase plane [ F(
n1),F(n2)] the points obtained at several
quencherconcentrationslieonastraightlineconnectingthe
points of “pure” spectral forms (“ nm1” and “ nm2” in Fig. 3),
of which the total spectra consist ( a50 and a51). In
reality,toexcludethepervertedeffectsofquenchersontotalquantum yield it is necessary to normalize F(
n1) andF(n2)
values by either total surface area under spectrum or by
emission intensity at any third, constant wave number nn,
F(nn). Because the precise measurement of the area under
an experimental spectrum is almost impossible, we used
normalizing by F(nn):
Pn15F~n1!
F~nn!
Pn25F~n2!
F~nn!(15)
TheF(n1),F(n2), andF(nn) can be represented as the
combinations of normalized log-normal functions ( w(nmi,
nj)) at wavenumbers n1,n2, and nnwith maxima at nm1and
nm2according to Eq. 3, i.e.,
Pn15I1zw~nm1,n1!1I2zw~nm2,n1!
I1zw~nm1,nn!1I2zw~nm2,nn!
Pn25I1zw~nm1,n2!1I2zw~nm2,n2!
I1zw~nm1,nn!1I2zw~nm2,nn!
Thus,Pn1andPn2could be presented as:
Pn15wr~nm1,n1!1az@wr~nm2,n1!2wr~nm1,n1!#
Pn25wr~nm1,n2!1az@wr~nm2,n2!2wr~nm1,n2!#(16)
where wr(nmi,nj)5w(nmi,nj)/w(nmi,nn) and w(nmi,nj) are
values of log-normal functions with maximal amplitudes
equal 1 (see Eqs. 1 and 2) and maximum positions nmiat
current wavenumber nj. In such a representation, ameans
the contribution of the component with the maximum posi-
tion at nm2(f(2)) in the normalizing fluorescence intensity at
nn,F(nn).
a;f~2!5I2zw~nm2,nn!
I1zw~nm1,nn!1I2zw~nm2,nn!(17)1704 Burstein et al.
Biophysical Journal 81(3) 1699–1709

Aswellasinthecaseof P15F(n1)andP25F(n2),Pn1and
Pn2are mutually linearly related (the equation that is anal –
ogous to Eq. 14):
Pn15kzPn21m
where:
k5wr~nm2,n1!2wr~nm1,n1!
wr~nm2,n2!2wr~nm1,n2!
m5wr~nm1,n1!zwr~nm2,n2!2wr~nm2,n1!zwr~nm1,n2!
wr~nm2,n2!2wr~nm1,n2!
Therefore, the points obtained at various quencher concen-
trations form the linear track on the phase-plane ( Pn1,Pn2)
(see Fig. 3). Thus, the phase-plot in coordinates ( Pn1,Pn2)
(Fig. 3) can be used for estimating the main parameters of
the two-component spectrum, i.e., nm1and nm2and their
relative contributions aand (1 2a). To estimate the com-
ponents’ maximal positions nm1andnm2we used the ex –
trapolation of the linear track through the experimentalpoints,obtainedwithvariousquencherconcentrations,upto
FIGURE 3 The representation of fluorescence spectra measured at dif-
ferent concentration of quenchers as points on the quasi-phase plane. ThecurveScorresponds to the totality of all possible elementary log-normal
functions.
FIGURE 4 ThepropertiesofthePHREQalgorithm.Thedependenciesfortwo-componentdecompositionsbetween Dlvalues(meanabsolutedifference
between positions of component maxima (in nanometers) that were preset in simulation and those obtained after the decomposition) and the amplitude o f
randomly introduced noise, S%(A); the number of spectra ( N) with various “quencher concentrations” used ( B); the distance between the component
spectral maxima, Dlmax,n m(C); the ratio of Stern-Volmer constants of quenching for the components, K2/K1(D); the number of registered points in each
spectrum, M(E); and the contribution of one component in the total spectrum, as an intensity ratio I1/(I11I2) (F).Decomposition of Protein Fluorescence Spectra 1705
Biophysical Journal 81(3) 1699–1709

its intersection with the curve S, which corresponds to the
totality of all possible elementary log-normal functions nor-malized by the same F(
nn) value as the coordinate values
Pn1andPn2. From the distances between the experimental
pointandthepoints nm1andnm2,d2andd1,respectively,the
contributions of components can be calculated:
a;f~2!5d2
d11d2
and
12a;f~1!5d1
d11d2Using the PHREQ algorithm all points of experimental
spectra are analyzed and the solutions obtained at anydifferent
n1,n2, and nnare averaged.
Properties of the algorithm
Analogously to the testing of the various factors affecting
the accuracy of solution obtained by the SIMS algorithm,we carried out a series of decompositions for simulatedspectra (the sums of two log-normal curves with variouspositions of maxima and relative amplitudes) by thePHREQ method. Fig. 4 shows the dependencies for two-component decomposition D
lvalues (mean absolute differ-TABLE 1 The results of decomposition of tryptophan fluorescence spectra of annexin VI (50 mM cacodylate, pH 7.0) measured
at three different concentrations of KI (0.0, 0.2, and 0.4 M) into log-normal components by the SIMS and PHREQ methods
N lm(nm) nm(cm21) Ii AiSi(%) Ksn(M21) Ksnrel(%) RB Rsd
SIMS, 1-component solution ( Ds53.222)
1 331.3 60.5 29949 645 10225 494795 100 1.05 60.17 7.2 61.2 0.986 1.01 0.015
8571 414728 1007939 384178 100
SIMS, 2-component solution ( D
s50.757)
1 326.0 60.5 30460 646 8200 441495 74.0 0.82 60.08 5.6 60.5 0.995 1.01 0.008
7203 387793 77.66692 360302 77.9
2 347.7 60.5 28458 640 2875 154778 26.0 1.88 60.48 12.9 63.3 0.969 1.04 0.035
2075 111720 22.41897 102159 22.1
SIMS, 3-component solution ( D
s51.045)
1 317.2 60.5 313.48 649 1144 64681 10.3 0.06 60.39 0.4 62.7 0.159 1.31 0.042
1040 58768 11.21128 63779 13.9
2 329.4 60.5 30139 645 8422 478038 75.6 1.05 60.07 7.2 60.5 0.998 1.01 0.006
7210 407500 77.56528 369004 75.4
3 356.3 60.5 27724 638 1575 88995 14.1 2.09 60.79 14.3 65.4 0.936 1.06 0.054
1057 59734 11.41001 56595 11.6
PHREQ ( D
s50.527)
1 325.5 62.9 30514 6270 7660 363370 67.7 0.75 60.02 5.1 60.1 1.0 1.00
6875 326121 72.86346 301067 72.7
2 346.5 63.1 28564 6253 3586 173357 32.3 1.94 60.60 13.3 64.1 0.956 1.04
2522 121928 27.22343 113292 27.3
N5the number of log-normal components;
lm5the maximum position of log-normal components in wavelength scale;
nm5the maximum position of log-normal components in frequency scale;
Ii5the maximal intensity of log-normal components at ith concentration of KI, i.e., at 0, 0.2, and 0.4 M;
Ai5the area of log-normal component under the total spectrum at ith concentration of KI;
Si5the contribution of log-normal component (in percent) into the area under the total spectrum at ith concentration of KI;
Ksn5the Stern-Volmer quenching constant for each log-normal component;
Ksnrel5the relative Stern-Volmer quenching constant, i.e., the ratio of the Ksnfor each log-normal component to the Ksnfor free aqueous tryptophan
emission quenching with KI, which was taken as 14.6 M21(Burstein, 1977);
R5the coefficients of linear correlation of the parameters X(k,i)5I(k, 1)/I(k,i) andc(i) on the plots in Stern-Volmer coordinates;
B5the free parameter in Eq. 7;
Rsd5the relative root-mean-square residuals, see Eq. 6a.1706 Burstein et al.
Biophysical Journal 81(3) 1699–1709

FIGURE 5 Decomposition of tryptophan fluorescence spectra of annexin VI measured at different concentrations of KI (0, 0.2, and 0.4 M) by one-, two-
and three-components by the SIMS algorithm. Panels Arepresent the experimental spectra measured without fluorescence quencher ( points) and calculated
theoretical spectra, which are the sum of log-normal components ( curves), and root-mean-square deviations between two kinds of spectra. Panels B
represent the experimental spectra at different concentrations of KI ( points) and calculated theoretical spectra ( curves), and root-mean-square deviations
between two kinds of spectra. Panels Crepresent the Stern-Volmer plots for each calculated component.Decomposition of Protein Fluorescence Spectra 1707
Biophysical Journal 81(3) 1699–1709

ence between positions of component maxima (in nanome-
ters) that were preset in simulation and those obtained afterthe decomposition) and the amplitude of randomly intro-duced noise, S% (panel A); the number of spectra ( N) with
various “quencher concentrations” used (panel B); the dis-
tance between the component spectral maxima, D
lmax,n m
(panelC); the ratio of Stern-Volmer constants of quenching
for the components, K2/K1 (panelD); the number of regis-
tered points in each spectrum, M(panelE); and the relative
contribution of one component in the total spectrum, as anmaximal intensity ratio I1/(I11I2) (panel F). The noise
was introduced as random equally probable positive andnegative deviations with amplitude from 0 to preset S% (in
percent to the maximal spectrum amplitude). The maximal“quencher concentration” was chosen so that it would re-duce the amplitude of a total spectrum approximately byhalf. Varying one parameter, the others were held constantat the following standard values: S%50.6%;N55;M5
15;D
lmax510 nm;K150.1 M21;K253.0 M21;I15
I2. As well as the SIMS algorithm, the PHREQ provides an
acceptable accuracy ( Dl,1 nm) of decomposition at real
conditions.AN EXAMPLE OF DECOMPOSITION OF
EXPERIMENTAL SPECTRA BY SIMS ANDPHREQ ALGORITHMS
As an example, we present here results of decomposition of
tryptophan fluorescence spectra of annexin VI measured atthree different concentrations of quencher KI (0.0, 0.2, and0.4 M) by SIMS and PHREQ algorithms (Table 1 and Figs.5 and 6). From the mono-, bi-, and tri-component decom-positions obtained by SIMS, the bi-component solution waschosen as the most reliable because it has the least value ofdiscriminant D
s(0.757), i.e., corresponding to the best fit of
experimental spectra by theoretical curves. Table 2 summa-
rizestheresultsoftwo-componentdecompositionsobtainedusing both SIMS and PHREQ methods. The fitting algo-rithm SIMS and the analytical one PHREQ gave very sim-ilar results: obtained maximum positions
lmdiffer within
1.2 nm, the contributions Sof components differ within
6.3%, and the relative Stern-Volmer quenching constantsK
snrelwithin 0.5%. In the next papers of this series it will be
demonstratedthatthesamevaluesofmaximumpositionsof
two spectral components ( ;325 and 347 nm) will be pre-
FIGURE 6 Decomposition of tryptophan fluorescence spectra of annexin VI measured at different concentrations of KI (0, 0.2, and 0.4 M) by two
components by the PHREQ algorithm. Panel Arepresents the experimental spectra without quencher ( points) and calculated theoretical spectra, which are
thesumoftwolog-normalcomponents( curves),androot-mean-squaredeviationsbetweentwokindsofspectra.Panel Brepresentstheexperimentalspectra
at different concentrations of KI (points) and calculated theoretical spectra ( curves), and root-mean-square deviations between two kinds of spectra. Panel
Crepresents the Stern-Volmer plots for two calculated components.
TABLE 2 The summary of the decomposition of tryptophan fluorescence spectra of annexin VI into two log-normal
components by the SIMS and PHREQ methods
Comp.SIMS PHREQ
lm(nm) S(%) Ksnrellm(nm) S(%) Ksnrel
1 326.0 60.5 74.0 61.0 5.6 60.5 325.5 62.9 67.7 60.5 5.1 60.1
2 347.7 60.5 26.0 62.0 12.9 63.3 346.5 63.1 32.3 61.0 13.3 64.11708 Burstein et al.
Biophysical Journal 81(3) 1699–1709

dicted based on the analyses of the physical and structural
parameters of microenvironments of two tryptophan resi-dues (W192 and W343) in the crystal structure of annexinVI.
The authors are thankful to Dr. Andrzej Sobota for preparation of annexin
VI and to Drs. V. I. Emelyanenko and O. A. Andreev for fruitful discus-sion. Moreover, we are indebted to Drs. D. B. Veprintsev and D. S.Rykunov for their help with computer maintenance.
ThisworkwassupportedinpartbyGrants95-04-12935,97-04-49449,and
00-04-48127 from the Russia Foundation of Basic Research.
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