Simulation of convoluted and exact emission anisotropy decay profiles

57

Chemistry17 (1983) 57-65 ElsevierBiomedical Press

Biophysical

SIMULATION PROFILES

OF CONVOLUTED

J. PAPENHUIJZEN Department

AND EXACT

* and A.J.W.G.

of Biochemistry,

Agrictdturaf

VISSER

University,

6703

EMISSION

ANISOTROPY

DECAY

** BC

Wageni~zgen,

The

Netherlands

Received 25th January 1982 Revised manuscript received 20th September 1982 Accepted 23rd September 1982 Key

words:

Anisotropy

decay;

Fluorescence

decay;

Rotationul

correlation

time;

Deconcolution

We have Emulated the convolution of the emission anisotropy decay function with both a &puke excitation function (exact solution) a.td a pulse function of either Gaussian or other functional form. It can be readily shown that convolution with a pulse cf iinite width leads to lower r,, values (anisotropy at time zero). Especially in the case of short-lived fluorescence. it can be demonstrated that the convoluted anisotropy lags behind the exact anisotropy leading to longer apparent rotational correlation times. Contour plots of r corrections as a function of both fluorescence lifetime and rotational correlation time were consxucted for two different pulse profiles. inspection of these contour diagrams can lead to an estimate of the relative error involved. when anisotropy data are not deconvoluted.

1. Introzhxction Rotational diffusion coefficients of macromolecules in solution can be determined from the time dependence of the emission anisotropy of internal or external fluorophores [l-3]. Ideally, this is achieved by exciting a fluorescent reporter group in the macromolecule with a s-shaped (Dirac) or infinitely short pulse of plane polarized light. The time evolution of fluorescence intensities parallel ( I,,( 1)) and perpendicular (Z,(t)) to the electrical vector of the excitation pulse yields the ideal anisotropy decay ftinction r(~): r(t)=--

4,(t)--I,(t) 1,,(t)+ZI,(t)

=-

d(t) s(t)

Fluorescence emission can be considered as a first-order rate process. The total fluorescence de-

l

* Present address: Departmen: of Physical and Colloid Agricultural L’niversity. Wageningen. The Chemistry. Netherlands. * To whom correspondence should be addressed.

03014622/83/oooO-0000/503.00

6 1383 Elsevier Biomedical Press

cay (s(t)) is therefore represented by a single or multiple exponential function, depending on local environmental inhomogeneities or on the presence of different fluorescent species (see ref. I): s(t)=

n Ca.exP(--1/q) i-i

(2)

where II is the number of components, tii the relative intensity contribution and 7i the fluorescence lifetime of component i. In a homogeneous environment with only one fluorescent species present, the anisotropy decay function can be written as a sum of exponentials as shown by Tao [4]. The exact number of exponentials in this case depends on the asymmetry of the macromolecule and amounts to maximally five for an asymmetric rotor [5]. Small and Isenberg [6] demonstrated that in practice three or less exponentials are sufficient: r(t)=

5 b,exp(--r/b,) 1-I

(3)

where b, represents the relative contribution of the anisotropy decay with rotational correlation time @, to the overall decay r(r). From eqs. 1-3 it follows that d(r) can also be written as a sum of exponentials. with time constants (I/? + l/9,)-’ that can be considered as combinations of pairs of 5; and C&J,_ The preceding equations only hold when the excitation pulse is infinitely short. In practice, the excitation source generates pulses of finite width. Using the technique of time-correlated single-photon counting and a mode-locked Ar ion laser [7]. pufse widths of the order of 0.3-O-5 ns are obtained and this shoutd be taken into account in the analysis of subnanosecond motion. It implies that the convclution products of s(t) and d(r), respectively. with the apparatus response function g(t) should be considered [ 11: S(r)=1

.q( I’)s( I -

t')dr'

(4n)

r,,. D(;)=I'

g(r’)d(r-r’)dr’ ‘3.m

The experimental time-dependent follows with eqs. 4a and 4b as: R(r)=-

(4b)

anisotropy

now

D(f) S(r)

which is a quotieilt of two convolution products_ A priori it is not obvious how the finite pulse width of g(f) affects the experimental anisotropy R(t) as compared to the true one. r(t)_ The instrumental response function is definitely not cancelIed in the quotient of eq. 5. 4 numerical deconvotution method to retrieve r( I J from R(r) was described by Wahl [S]. Test functions R(r) were obtained from exact expressions for r(r) and s(r). by convolving them with two different experLnenta1 functions g(r). In this study the differences between r(r) and R(r) are more closely investigated. The functions n(r) and S(r) are computed analytically from exact expressions for the response function. Using this approach it is possible to examine visually the conditions under which significant differences between the theoretical r(r) and the experimental Rf r) occur. In addition. for the case of mono-exponential fluorescence and anisotropy decays the

refative deviation between apparent and true anisotropy decay is presented in *; -7 %rm of contour plots. One can then for;naIIy L ecide whether or not experimental data have to be d-convoluted. In order to achieve this simulatic. r~ two computer programs were developed_

2. Methods Exact expressions for S(r) and D( t ) are derived using two different pulse profiles of identical peak height and p&e width: (1) A Gaussian pulse. which is representative for the shape of a mode-locked laser pulse: .q(r)=OSexp(-AA;‘)

(6)

This function reaches its peak value 0.5 for I = 0. The parameter A (in s-‘) is inversely related to the pulse width w at half height through

A=tn

(7)

w2

(2) A modification of the pulse function as used by McIiinnon et al. [‘G] and by Duddeli [IO], exhibiting a ‘tail’ and represemative for the pulse profile of spark gap discharge lamps:

.(f)=$(r+xp[

-A(+)]

This expression also has its m.: :imum 0.5 for I = 0 and again the parameter A (.low in s- ‘) is inversely related to ~1’. However, no analytica expression can be derived and therefore A is evaluated numerically from a given vafue of SC‘. As indicated in section I, the convolution product of d(r) or s(t) with g(z) can generally be written as: F(r) =Ij:,,pCf’,z

qexp(-/3,(r-A-')}dr' I

= ~~~,s:O~,~~P(-P,(~ I

- +))dr'

Lvith F(r) representing either D(r) fl, parameters of d( I ) or s(t)_ Thus, it is sufficient to derive a sion for the convofution product singfe exponential_ Substitu&tn of

(9)

or S(t)

and p,.

general expresof g(r) with a various sets of

parameter values (ai. &) in eq. 9 then yields exact expressions for D(t) and S(z). The convolution product of a Gaussian excitation puke with a single-exponential function is obtained by substitution of eq. 6 for g(r’) with t,;, = -co:

Substitution of the tail excitation pulse (eq. 8) for I,,,;,,= -2/A into eq. 9 yields the following expression for a single exponential:

x cx e2A’

Z---.-~ 8 -

with

exp( - /3(f - r’))dr’

c

-exp(-(A~+Z)) A - .B

2exp{-(Ar+2)) (A-P? 2exp{-(Ar

t2))

+

(A--P)’

The integral on the right-hand side of eq. 10 is a standard Gaussian integral. which has been tabulated [l I] or can be computed using standard subroutines (e.g._ see ref. 12).

In the limiting found:

case A = fi a simpler expression

is

(11)

Fig. I. Flow char! of computer program for the genrrsltion of piots representing ideal and convoluted anisotropy decay. Examples we shown in figs. 3-7.

Fig. 2. Ffow rhnrt of computerprogram to calcukitepoints of contour plots. giving 3 more general impression of the relative error in convolutrd anisotropy decay as compared to the ideal case. Exampks are shown in fig. 8.

1. Papenhuijx=n.

60

A. J. W G.

Visser/Simulation

of cvzission

t

I

.25

t

I

I

1 .!io

m 4

I

Only rotational correlation times determine the ideal aniso?ropy r(t), when the initial anisotropy ( = r,,) is known. However, the shape of a convoluted function R(t) not only varies with these rotational correlat; _-.. times, but also with pulse shape and fluorescence lifetimes. This is illustrated

!G

-r

I

I

2.75

p-of&s

3. Results

.‘F/‘-------3-J’

decay

lated data contour plots can be constructed with the desired accuracy, as the spacing between successive values of T/W and +/rv can be freely chosen A flow chart of this program is shown in fig. 2.

computer program was written in order to plot the functions g(r), r(t) and R(t), respectively, and also the difference R(f) -r(t). Standard subroutines [12] were used to perform some calculattons (A in eq. 8 and the integral in eq. 10) and the plots were generated using a package of graphic subroutines [ 131. A flow chart of this program is presented in fig. 1. For the generation of contour plots the computer program described in fig. 1 was modified in such a way that relative deviations {r(t) R( t )}/r( I ) are calculated as a function of @I/W and T/W. respectively_ The time 6r after the excitation pulse maximum is preset and can be given any va!ue between zero and infinity. From the calcuA

aF2 _ -1.00

onisotropy

4.00

NSEC

NSEC

E224 a .-

z -1

.oo

.25

1.50

2.75

4.00

Fig.

3. Lower

difference tail

(see

R(I)text).

w = 0.5 ns.

Parameter

Left:

results

values:

4.00

NSEC

par.19:pulseresponsefunction r(r).

2.75

1.50

-2.5

NSEC

obtained

rotationnl

(with

!. ideal Gaussian

correlation

time

(-

-

excitation + = 1.0

-)

and pulse.

ns 2nd

nonideal Right:

(-

) anisotropy

results

fluorescence

obtained

lifetime

with

r = 0.1

decay. excitation ns.

i op

panel:

pulse

pulsr width

~h~olt~fz

containing at half

height

a

J. Papenhvijren. A.J. W-C;

Visser/Simula&m

by R(t) plots generated from single-exponential functions r(t) and s(t). The initial anisotropy r, is normalized to 0.3, lower than the maximum value of 0.4. Each figure compares the function R(t) calculated with the two mentioned pulse response functions, having an arbitrary peak value of 0.5 at zero time. The pulse width of 0.5 ns is representative for the pulses generated and detected by a mode-locked Ar ion laser/single-photon counting system [?I_ Figs. 3-5 present a selection of the results. In figs. 3 and 4 the fluorescence lifetimes -r are varied, whereas figs. 4 and 5 demonstrate how different rotational correlation times + influence the results. If the lifetime is much shorter than the pulse width (fig. 3), the discrepancy between exact and convoluted anisotroxy is larger. A Gaussian pulse profile induces a ‘delay in the anisotropy decay, whereas the results obtained with a tail

61

of emissim onisotropy decuy profiles

pump function are even more out of proportion_ A lcnger fluorescence lifetime (fig. 4) does improve the results. From the absolute differences between R(t) and r(t) it can be concluded that the initial anisotropy R, is lower than the true one r,. For the particular set of parameters in fig. 4 the decrease amounts to about 10% of r,. The difference between calculated and ideal anisotropies becomes vanishingly small for longer correlation times (fig. 5); note the expanded scale for the absolute differences_ It should be emphasized that the simulated data are free of noise. Especially the situation depicted in fig. 5 leads in practice to a decay curve in which noise will increse along the decay. since the fluorescence lifetime is relatively short. Finally, we want to present simulated data related to experiments obtained with fluorescent

_

G

m 4

NSEC

NSEC

m <

Yl

-1.00

I

I .25

f

I 1.50

NSEC

NSEC

1

I 2.75

0

I 4.00

62

J. Pupenhuijren. A.J. W G. Visser/Sinrulation of emission unisorropy decay profiks

ftavins and flavoproteins f14J. Free fkvin adenine dinucleotide (FAD) is characterized by a short rotational correlation time (0.46 ns at 10°C) of duration similar to that of the excitation puke. In this case (fig. 6) deconvolution is required to extract the true value of the correlation time. In the particuIar case of Iipoamide dehydrogenase a rapid subnanosecond motion of bound FAD is superimposed on the protein rotation. The experiments carried out at 4 and 40°C were simulated with Gaussian pulse profiles (fig. 7). From the absolute differences in the experiment at 40°C it can be observed that it is difficult to determine the low ampfitude of the rapid motion within the limited experimental time scaie. One should be careful in interpreting observed subnanosecond relaxation processes, as a 0.5 ns

pulse

width

(II.) is usually

found

laser pulses in combination with singIe-photon counting detection. In order to draw more general conclusions from our simulation method we have generated contour plots for both pulse forms. In these plots we portrayed the values of T/W vs. those of +/iv for certain values of the relative error in the anisotropy, {r(t) - R( T))/r(t). CaIcuIations were performed at two distinct times (6~) after the pulse maximum. Firstly, contours were made at 8~ = 0. This set of curves yields the relative error in the initial anisotropy. The second set of curves was obtained at a time 81 = w. These results are col!ected in the four graphs of fig. 8. The four sets of curves in fig. 8 give a general impression of the errors involved, when experimental anisctropy decay curves are approximated by corresponding ideal functions_

for mode-locked

8.00

i7.00

NSEC

I

I a.00

1

I 17.00

NSEC

1

I

26.00

I

I 35.00

I-

NSEC

26.30

35.00

:

co - I-

-1.00

,

I -.25

I

I -50

t

I 1.25

1

I

2.00

NSEC

NSEC Fig. 6. As fig. 3 except that + = 0.46 ns and T = 3.0 ns (FAD

$I; a

4.00

NSEC

NSEC

6.50

9.00

in Hz0

no 10°C).

:i-lr-,,oo :-1 .oo

1.50

4.oc

NS;EC

NSEC

10.0

0.02

0.05

0/w

0.03

0/w

0.05 0.1 _y

1.0

0.1)

0.2

0.2

0.3

0.3

0.5

0.5 0.1 0.1

I

I I I11111

1.0

I

1 Illllll

, , , , , , ,,

0.1 0.1

10.0

T/W

10.0 \ Q/W

\

\

I%

-0.03 _,-O.Ol _- ..-0.02

. -

1

0/w

-0.05 -0.1

-::::.-0.03 ....-0.05

-0.2 -0.3 -0.5

~.I::“,:: ~-‘-+I3 2.

0.1

10.0

\

_

0.1

l1llll-n

T/W

% \ 1.0

I

i-0

I

I I,,,,,,

I 1.0

Illlllrl

'---OS 0.1 .A”

10.0

T/W

0.1

1.0

10.0

-Cc/W

Fig. 8. Contour diagrams describing the relative deviation of the appwent anisotropy R from the true one r. Contour plots ;ire shown for +/w (ordinate) against T/W (absrissa) at constant relative error. (r(f)- R(r)j/r( I). indicated at each curve. Four sets of curves are shoum. The top two curves represent r,, corrections at the excitation maximum. 6f= 0. The louder two curves are for r corrections at a time Sr = w after the excitation maximum. Left: results with Gaussian excitation puke. Right: results with excitation pulse containing tail.

4. Conclusion

From an appraisal of the presented figures one can arrive at the following conclusions_ The initial anisotropy R, is lower than the ideal one ra, when the pulse response function g(t) is of finite width,

regard!ess of its shape. The pulse evidently influences the measured anisotropy decay. A shorter fluorescence lifetime results in larger differences between ideal and nonideal anisotropy decay; longer correlation times cannot be determined if the fluorescence lifetime is too short. When the

Fig. 7. Simulations of nnrsotropy decay of Iipoamide dehydrogenase [14]. Left: at 4OC with 0, = 0.5 ns. Oz = 57 ns and h,/h, = 0.054: 7, = 0.6 ns. .r2 = 4.1 ns and LI,/az = 0.563. Right: at 40°C with 0, = 0.5 ns. r& = 26 ns and b, /b2 = 0.054; 7, = 0.5 ns. 7z = 2.0 ns and U, /a, = 1.041. Only convolutions with Gnussian excitation pulses ( w = 0.5 i-15)are presented.

correlation time is shorter than the pulse width the difference between ideal and nonideal anisotropy increases and r(r) has to be calculated from R(t). For longer correlation times R(t) is a reasonable approximation of r(t). Fig. 8 gives a more general impression of the relative errors involved in experimental anisotropy decay curves as compared to the ideal case. In cases where the fluorescence lifetime is shorter than the pulse width, Wahl [S] found that the nonideal anisotropy passes through a minimum. In this work this could not be coGfirmed (cf. fig. 3). A possible explanation is the very pronounced tail of the experimental pulse response function as employed by Wahl [S].

Acknowledgements We thank Miss Lyda Verstege for preparation of the typescript. This work was supported in part by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO).

References 1 R. RigIer and M. Ehrenberg, Q. Rev. Biophys. 6 (1973) 134. 2 J. Yguerabide, Methods Enzymol. 26. C (I 972) 48. 3 P. Wahl, in: Concepts in biochemical fluorescence, vol. 1, eds. R.F. Chen and H. Edelhcch (Marcel Dekker. New York. 1975) p_ 1. 4 T. Tao. Biopolymers 8 (1969) 609. 5 G.G. Belford, R.L. Belford and G. Weber. Proc. Narl. Acad. Sci. U.S.A. 69 (1972) 1392. 6 E.W. Small and I. Isenberg. Biopolymers 18 (1977) 1907. 7 A.J.W.G. Visser and A. van Hock. J. Biochem. Biophys. Methods 1 (1979? 195. 8 P. Wahl, Biophys. Chem. 10 (1979) 91. 9 A.E. McKinnon. 4-G. Szabo and D.R. Miller. J. Phys. Chem. 81 (1577) 1564. 10 D.A. Duddell. Photochem. Phorobiol. 31 (1980) 121. M. Abramowitz and LA. Stegun, Handbook of mathematical functions (Dover Publica&ns, New York. 1972). ‘International Mathematical and Statistical Library’. IMSL. Inc., Houston. TX (1980). -Komplot’, Communications No. 7. January 1977, Computer Center. University of Groningen. Groningen. The Netherlands. A.J.W.G. Visser. H.J. Grande ?nd C. Veer-r. Biophys. Chem. 12 (1980) 35.