- Email: [email protected]
Bacteriorhodopsin photocycle kinetics analyzed by the maximum entropy method
Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16 www.elsevier.com/locate/jphotobiol
Bacteriorhodopsin photocycle kinetics analyzed by the maximum entropy method A. Luka´cs, E. Papp
*
Department of Biological Physics, Eo¨tvo¨s University, Pa´zma´ny P. Se´ta´ny 1/A, Budapest H-1117, Hungary Received 15 June 2004; received in revised form 27 July 2004; accepted 9 August 2004 Available online 7 October 2004
Abstract A maximum entropy method (MEM) was developed for the study of the bacteriorhodopsin photocycle kinetics. The method can be applied directly to experimental kinetic absorption data without any assumption for the number of the intermediate states taking part in the photocycle. Though this method does not give a specific kinetics, its result is very useful for selection between possible photocycle kinetics. Using simulated data, it is shown that MEM gives correct results for the number of the intermediate states and the amplitude distributions around the characteristic lifetimes. Analyzing experimental absorption data at five different wavelengths, MEM gives seven or eight characteristic lifetimes, which means that at least so many distinct intermediate states exist during the photocycle. Many possible photocycle kinetic models were studied and compared with the MEM result. The best agreement was found with a branching photocycle model of eight intermediate states (K, L, M1, M2, M3, M4, N, O). The branching occurs at the L intermediate state (M1 and M2 being in one branch and M3 and M4 in the other branch), but at high pH it occurs already at the K state. 2004 Elsevier B.V. All rights reserved. Keywords: Maximum entropy method; Bacteriorhodopsin; Photocycle kinetics
1. Introduction Bacteriorhodopsin (BR) is a light driven proton pump built in the cell membrane (purple membrane) of Halobacterium salinarum [1–3]. After excitation it undergoes a series of molecular changes, the photocycle, which results in the translocation of a proton through the membrane. During the photocycle (10– 100 ms at room temperature) several spectrally different intermediate states (J, K, L, M, N and O) can be distinguished. These intermediates have broad and
*
Corresponding author. Tel.: +36 1 372 2785; fax: +36 1 372 2757. E-mail address: [email protected] (E. Papp).
1011-1344/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jphotobiol.2004.08.004
strongly overlapping absorption bands [4] except the M state which is shifted to the blue. The process is thermally activated from the J state. To understand better the mechanism of the proton pump in BR a lot of work was devoted to the study of the photocycle [5]. Measuring and comparing the transient absorption changes at different wavelengths during the photocycle, different kinetic models can be tested. The generally accepted procedure is the following. Assuming a given number of intermediate states and first order reaction steps between them (kinetic model) the time dependence of concentrations of the intermediate states can be calculated. The input parameters are the rate constants of the given kinetic model and these parameters are varied during the fitting procedure. From the concentrations the transient
2
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
absorption change can be calculated for any wavelength knowing the absorption spectra of the intermediate states. This calculated absorption change is compared with the measured changes and the rate constants of the assumed kinetic model are determined by minimizing the deviation between the measured and calculated absorption changes. But, mainly because of the high number of the intermediate states and their broad, strongly overlapping absorption bands, there are several quite different kinetic models which give almost the same goodness of the fit. This analysis showed that the BR photocycle kinetics is a complex problem and may have several solutions [6]. To resolve this problem, a thorough analysis was carried out in [7] introducing more control parameters (pH and temperature) beside the wavelength and time. Three kinetic models were compared with the conclusion that the sequential model with reversible reactions between the intermediates (K, L, M1, M2, N, O) gives the most acceptable fit with a linear Arrhenius plot for the rate constants. In the above ‘‘kinetic model analysis’’ the time dependence is given by discrete exponentials. The number of exponentials and the values of the characteristic lifetimes in the exponentials are uniquely determined by the kinetic model and the values of the rate constants. The number of exponentials equals with the number of intermediate states assumed in the kinetic model. So a kinetic model can be represented by discrete lines in the lifetime ‘‘space’’. We will call this a discrete line distribution. There exists a different approach to data analysis of noisy experimental data, the maximum entropy method (MEM) [8]. In a previous paper [9], the first result of the possible application of MEM to BR absorption kinetic rate processes was reported. Instead of discrete lines the MEM gives a distribution with characteristic peaks in the lifetime space. This distribution can be called as the ‘‘image’’ [8] of the experimental data. The discrete lines resulting from kinetic model analysis represent a special image. There are many possible images (distributions) that can more or less correctly (minimum misfit) describe the given experimental data. The MEM selects one image from the many possible ones. Comparing the kinetic model analysis and the MEM, there are some very important differences. In the kinetic model analysis the number of the intermediate states is fixed at the start and their absorption spectra have to be known. In MEM the process itself selects the peaks (image) without any constraint for the number and spectral properties of the intermediate states. On the other hand the MEM does not give a complete information about the kinetics, it gives only the image. This problem will be analyzed in more details later in this paper. It means that the rate con-
stants (kinetic model) uniquely determine the image but the opposite is not true. Therefore the results from kinetic model analysis and from the MEM can be compared only at the level of images. In a previous paper [9], the MEM was applied for the time dependence of intermediate state concentration (M state). This analysis was restricted in several relations. In this paper, the MEM analysis is applied directly to the measured absorption kinetic experimental data of the BR photocycle. This analysis shows that there are at least 7 or 8 intermediate states taking part in the photocycle. A search for a corresponding kinetic model, resulting in a similar image as the MEM image, was also made. A branching photocycle kinetic model, with four M intermediate states, gives similar image, but there is no complete agreement between the two images.
2. Materials and methods The purple membrane suspensions isolated from H. salinarium strain S9 were gifts from the Biophysics Institute, Biological Research Center, Szeged, Hungary. For excitation of the photocycle a Nd-YAG laser, frequency doubled to 530 nm, was applied and the data were averaged. For MEM analysis, we used also the absorption data recorded in the Biophysics Institute, Szeged [7]. Spectral data for the extinction coefficients of the intermediate states were taken from the literature [4]. For the kinetic model analysis the RATE program [10–12] was used. For the MEM analysis, a program described elsewhere [9] was used with some modifications.
3. Maximum entropy method and absorption kinetic analysis Any photocycle model can be described and solved by the methods of chemical kinetics. For example, the reversible photocycle model with n = 6 intermediate states (K, L, M1, M2, N, O): k1
K
k2
O
k 11
!
L
k3
k4
BR
k5
M1 and
k6
N
k 12
!
M2 BR
k7
k8
N
k9
k 10
ð1Þ
can be formulated by the following system of linear ordinary differential equations (in matrix representation and assuming first-order reactions between the intermediate states):
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
0 B B B B dB B B dt B B B B @
C1
1
0
C B C B C B C B C B C B C4 C ¼ B C B B C5 C C B C B C6 A @ BR C2 C3
k 1 k1 0 0 0
k2
0
ðk 2 þ k 3 Þ k4 k3 ðk 4 þ k 5 Þ 0 0
k5 0
0
0
0
0 k6
0 0
0 0
ðk 6 þ k 7 Þ k8 k7 ðk 8 þ k 9 þ k 12 Þ
3
0
0 k 10
0
0
0
0
k9
ðk 10 þ k 11 Þ
0
0
0
0
k 12
k 11
10
C1
1
CB C 0 CB C 2 C CB C C B 0C CB C 3 C CB C 0 CB C 4 C: CB C C B 0C CB C 5 C CB C 0 A@ C 6 A 0 BR
ð2Þ
Here C1, . . ., C6 are the time dependent concentrations of the intermediates and BR denotes the same for molecules in the ground state. As the matrix is redundant in Eq. (2) (all elements in the last column are zero, this is because there is no backreaction from the ground state at the end of the photocycle), it is better to write Eq. (2) in the following form:
1 0 k 1 C1 BC C B k B 2C B 1 B C B C C B 0 dB B 3C¼ B B B dt B C 4 C C B 0 B C B @ C5 A @ 0 0
0
C6
k2 ðk 2 þ k 3 Þ
0 k4
0 0
0 0
0 0
k3
ðk 4 þ k 5 Þ
k6
0
0
0 0
k5 0
ðk 6 þ k 7 Þ k7
k8 ðk 8 þ k 9 þ k 12 Þ
0 k 10
1 C1 CB C C CB 2 C CB C CB C 3 C CB C: CB C C CB 4 C CB C A@ C 5 A
0
0
0
k9
ðk 10 þ k 11 Þ
C6
In that case BR(t) is determined by the mass conservation: n X BRðtÞ ¼ C 0 C i ðtÞ; ð4Þ i¼1
where C0 is the total concentration of the BR molecules (in the ground and intermediate states). Eq. (3) can be written in a compact form: dCðtÞ ¼ K CðtÞ; dt
ð5Þ
where C is a vector with (C1, . . ., Cn) components for the concentrations of the intermediate states and K is the rate matrix with n by n elements determined by the rate constants (ki) of the given photocycle kinetic model. The linear differential equation system, Eq. (5), can be solved in terms of the eigenvectors and eigenvalues of the K matrix [13]. In general, an n by n matrix can have n linearly independent eigenvalues (k j , apparent rate constants) and eigenvectors (nj): K nj ¼ k j nj ; with j = 1, . . ., n and:
ð6Þ
CðtÞ ¼ expðk j tÞ nj
10
ð3Þ
ð7Þ
is a solution of Eq. (5). The general solution of Eq. (5) is a linear combination of Eqs. (7): n X C i ðtÞ ¼ aj expðk j tÞ nji j¼1
¼
n X
cij expðk j tÞ;
ð8Þ
j¼1
where cij = aj Æ nji. The constants aj are determined by the initial condition. (For BR photocycle with short laser pulse excitation the initial condition at t = 0 is the following: C16¼0 and C2 = = Cn = 0.) The absorption change during the photocycle can be calculated using Eq. (8) and the extinction coefficients, ei(k), of the intermediate states and eBR(k) of the ground state. Then the absorption, A0(k), at wavelength k before excitation is: A0 ðkÞ ¼ C 0 eBR ðkÞ
ð9Þ
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
and after excitation: n X C i ðtÞ ei ðkÞ þ BRðtÞ eBR ðkÞ: Aðt; kÞ ¼
ð10Þ
i¼1
The absorption change during the photocycle is given by the difference of Eqs. (10) and (9): DAðk; tÞ ¼ Aðk; tÞ A0 ðkÞ ¼
n X
Dei ðkÞ C i ðtÞ
ð11Þ
i¼1
(Eq. (4) was used here and Dei(k) = (ei(k) eBR(k))). The absorption change, DA(k,t), can be measured during the photocycle at different k wavelengths. In a kinetic model analysis, one starts with an assumed K matrix (Eq. (2) or (3)), numerically calculates the time dependence of the concentrations C(t), and the expected absorption changes (Eq. (11)). Then this is compared with the measured absorption changes and the initial rate constants are varied until a minimum is reached in the square deviation. If Eq. (8) is put into Eq. (11) then we get for the absorption change: DAðk; tÞ ¼
n X n X i¼1
¼
n X
Dei ðkÞ cij expðk j tÞ
j¼1
bj ðkÞ expðk j tÞ;
ð12Þ
j¼1
Pn where bj ðkÞ ¼ i¼1 Dei ðkÞ cij . Eqs. (8) and (12) have a very similar form and this form is a suitable one for the application of MEM. The MEM analysis gives a distribution of the amplitudes on the quasi continuous lifetime scale with characteristic peaks around the sj ¼ 1=k j lifetimes. MEM is a conditional extremum procedure [8,9]. An entropy function, S, is defined on the amplitude (cij or bj(k)) distribution. Starting with a flat (structureless) distribution the procedure selects such a distribution, which minimizes the square deviation between the measured C mi ðtÞ (or DAm(k,t)) and the calculated (Eq. (8) or (12)) values, while keeping the entropy at a maximum level. In this analysis, there is no starting assumption about the number of the eigenvalues (intermediate states) or the kinetics. In principle, from a MEM analysis based on Eq. (12), the bj(k) amplitude distributions, the k j eigenvalues and the number (n) of the intermediate states can be determined. But the knowledge of these quantities is not necessarily enough to determine the true rate constants (K matrix) [14]. One problem is that Eq. (12) cannot be reversed without assumptions for the n intermediate states. The other problem is that some intermediate states have identical spectral properties. For these intermediate states, the differential extinction coefficients are the same and Eq. (12) becomes underdetermined. For
these reasons (and that the MEM results have some uncertainties), a MEM analysis cannot give specific kinetics. It yields results which can be compared with a given kinetics in the following sense. A kinetics (K matrix) is assumed, from that the eigenvalue problem is solved and the expected absorption change, Eq. (12), is determined. Only this result can be compared with the MEM result. In a previous paper [9], the result of a MEM analysis was reported in a special case. It was assumed that the absorption change at k = 410 nm is approximately proportional to the M state concentration (Eq. (11)) and that the negative and positive amplitudes (rise and decay) in Eq. (8) are well separated on the lifetime scale. To apply this method for the other intermediate states, the knowledge of their concentrations would be necessary. In principle it is possible to solve Eq. (11) for the concentrations. For example, if n = 5 intermediate states are assumed and measuring DA(k,t) at 5 different wavelengths then Eq. (11) gives a linear equation system for every time and this equation system can be solved for Ci(t). The result of this calculation is shown in Fig. 1. As can be seen we get negative values for some concentrations and this is an unacceptable and unphysical result. The origin of that false result is not quite clear. It may be caused by the similarity and strong overlap of the absorption bands (mainly the L and N intermediate states) and perhaps by that a small (and systematic) error in the absorption coefficients is strongly amplified in that kind of solution. (This view is supported by the result for the M state – where the absorption band is well separated – and we get an acceptable result, except for the shortest times.) To try to overcome these difficulties in the direct determination of the concentration the following approximation was made. We assumed that in the early part of the photocycle only the K, L, M, while in the later part only the M, N, O intermediate states are present. In that case Eq. (11) is overdetermined but it can be K L M N O BR
300 250
Concentration [a.u.]
4
200 150 100 50 0 -50 -100 -7
-6
-5
-4 -3 log time [s]
-2
-1
0
Fig. 1. Direct calculation of the concentrations of the intermediate states. (For BR the sign is changed.)
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16 350
4.5 pH 5C
Concentration [a.u.]
300 250
K L M N O BR
4.5 pH 30C
5
K L M N O BR
200 150 100 50 0
300
Concentration [a.u.]
7.0 pH 20C 250 200
K L M N O BR
7.0 pH 30C
K L M N O BR
150 100 50 0
300
K L M N O BR
Concentration [a.u.]
9.0 pH 5C 250
9.0 pH 30C
200
K L M N O BR
150 100 50 0 -7
-6
-5
-4
-3
-2
-1
0
-7
-6
log time [s]
-5
-4
-3
-2
-1
0
log time [s]
Fig. 2. Direct concentration calculations at different pH and temperatures excluding some intermediate states in the early and later times of the photocycle. (For BR the sign is changed.)
solved for Ci(t) by a least square fit. The solution is presented in Fig. 2 and it has the following characteristic features: There is a window in the middle time range where the solution is quite unreliable (except for the M state) for reasons mentioned above. The M state concentrations are still negative in some cases at the short times. The behavior of the N and O intermediate states is interesting. At low pH the N state concentration decreases with increasing temperature (and it is very low at 30 C). At high pH the O state concentration decreases with decreasing temperature (and it is very low below 15 C). But the maximum values in the N and O states concentrations appear nearly at the same time during the photocycle. After excitation the BR concentration starts to increase very early with the decay of the L or M1 states. This can be a side effect caused by the measuring light
leading to a photoinduced transition from L or M1 to the BR ground state. We include the possibility of a photoinduced transition from M1 to BR here and later in the photocycle kinetics. This improves the fit to the experimental data but it does not have an essential effect on the photocycle kinetics. These features will be taken into account later in the photocycle analysis.
4. Results As the attempt for a direct determination of the concentration failed the previously reported MEM analysis [9] cannot be applied for the whole photocycle. Instead the MEM has to be applied directly to the measured DA(k,t) absorption changes using Eq. (12).
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
6
According to Eq. (12) the image of the absorption change DA(t) at a given wavelength, k, is: DAðtÞ ¼
N X
bi expðt=si Þ;
ð13Þ
i¼1
where s is on the quasi continuous lifetime scale and bi are the amplitudes. (In the following analysis the s scale is usually restricted to 107–101 s and the log scale is divided to N = 200 equal intervals.) This form will be used for the MEM analysis. But this differs in two essential points from the form used in [9], therefore the earlier MEM procedure has to be modified. The first modification is connected with the sign of the amplitudes, bi. In the previous report, the amplitudes were restricted to positive values so a simple entropy definition was possible. In Eq. (13) there is no preliminary information about the sign of bi, but the entropy can be defined only on positive numbers. So we are forced to define the entropy (S) by the absolute values of bi: N X bi ð14Þ S¼ jbi j ln 1 ; Ai i¼1 where Ai are the initial values for bi which maximizes the entropy at the start. But using the absolute value in Eq. (14) introduces some difficulties in the MEM procedure. The problem is with the change of the sign of the bi amplitudes. This problem arises because of the use of the entropy metric [8,9]. The entropy metric is an essential point in the MEM algorithm. It discriminates in favor of allowing high values of bi in Eq. (13) to change more rapidly than low ones between the iterations. There is no problem if the bi amplitudes take positive values by definitions. Fortunately in many applications this is the case. But in absorption kinetics, the bi amplitudes can take both positive and negative values. In that case starting with a flat (constant) bi amplitudes with a given (e.g., positive) sign, during the iterations some bi will increase to give the positive peaks, other bi amplitudes will decrease. To reproduce the negative peaks, the sign of bi at some places on the lifetime scale (see, e.g., Fig. 4) has to be changed. But the entropy metric resists to change the sign of bi. This may result some distortion of the image, i.e., the amplitude distribution. To circumvent this problem, it is advisable to divide the MEM analysis into two steps. In the first step such a division of the lifetime scale is determined where the sign of the bi is positive or negative. To get this information, the following procedure was made. In Eq. (13), a constant level was added to the bi amplitudes shifting up the amplitude spectra and a corresponding contribution was added to the absorption change. From the MEM fitting to this modified absorption change the zero points of bi could be determined approximately. In the second step – the real MEM anal-
ysis – the sign information about the bi amplitudes was used from the start, i.e., the initial Ai amplitudes in Eq. (14) have a flat (Ai = const) distribution but signed according to the previous result. The second modification is connected with the mass conservation, Eq. (4). It was shown earlier [9] that, if the MEM is applied to the concentrations, Eq. (8), then the mass conservation can be taken into account by reducing the three dimensional search subspace to a two dimensional subspace in a proper way. This restriction cannot be applied in the case of Eq. (12). So in the following the MEM search will be made in the full three dimensional subspace as described in [8,9]. 4.1. Simulated data analysis Before applying the MEM to experimental absorption data a reliability test was made on generated data in the following way. Starting from a known kinetics – in that case a reversible photocycle, Eq. (1) – and assuming numerical values for the rate constants given in Table 1, Eq. (6) is solved for the eigenvalues and eigenvectors. (These numerical values correspond approximately to that reported in [7] for pH 7 and 20 C. For pH 9; 20 C simulated data the rate constants are not shown but they correspond to the reversible model, too.) Fig. 3 shows the characteristic lifetimes sj corresponding to the k j eigenvalues ðsj ¼ 1=k j Þ and the amplitudes cij in Eq. (8) for the intermediate states concentrations. As expected there are n=6 eigenvalues coinciding with the number of the intermediate states, but the amplitudes differ widely, in some cases they are very small. (For example, for the O state there are only two significant amplitudes, the other four are very small.) Knowing the cij amplitudes the expected absorption change during this simulated photocycle can be calculated for any wavelength using Eq. (12). This calculation was done for five different wavelengths (410, 500, 570, 600 and 650 nm). The result for different wavelengths is similar to that shown in Fig. 3, but the amplitudes are different (see Fig. 4, dotted lines with an x at the end).
Table 1 Rate constants for simulated data generation ki
½1s
ki
½1s
k1 k2 k3 k4 k5 k6
4.055 · 105 9.45 · 104 1.135 · 105 8.072 · 105 6.56 · 104 0
k7 k8 k9 k10 k11 k12
3.5 · 103 1.05 · 104 0.17 · 103 0.9 · 102 0.8 · 102 4.9 · 102
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16 4
K
L
M1
M2
N
O
7
amplitude [a.u.]
3 2 1 0 -1 -2 -3 -4 4 3
amplitude [a.u.]
2 1 0 -1 -2 -3 -4 4
amplitude [a.u.]
3 2 1 0 -1 -2 -3 -4 -7
-6
-5
-4
-3
-2
log time [s]
-1 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 3. The characteristic lifetimes and the discrete amplitudes for the intermediate states.
A Gaussian random noise was added to the simulated absorption changes and these simulated data were analyzed by the MEM. The images (discrete amplitudes and the amplitude distributions on the lifetime scale) are shown in Figs. 4 and 5 for the two simulated data sets (pH 7; 20 C and pH 9; 20 C, respectively). These figures show the discrete amplitudes at the characteristic lifetimes (calculated by Eqs. (6), (8) and (12) using the assumed rate constants) and the distribution of amplitudes (peaks) from the MEM analysis (Eq. (13)) at five different wavelengths. In Figs. 4f and 5f the MEM peaks for different wavelengths are presented together showing that the peaks appear approximately around the same characteristic lifetimes. To compare the discrete amplitudes with the peaks of the MEM analysis Gaussian curves were fitted to these peaks and the area under the peaks were determined. This result is shown in Fig. 5, too.
This analysis is concluded with the following remarks: Instead of discrete amplitudes we get distribution (approximately a Gaussian peak) around a given characteristic lifetime. The characteristic lifetimes (si), or eigenvalues of the kinetic model can be reproduced by MEM analysis: the peaks appear nearly at the same lifetimes for the different wavelengths. But to recover all eigenvalues absorption data are needed for more wavelengths because if an amplitude at a given wavelength is too small it may not be recovered. There may be such a case that the amplitudes for a given characteristic lifetime are very small at all measuring wavelengths. For example, in the case shown in Fig. 5 the amplitudes for the fourth characteristic lifetime (log s 3.5) are very small and the MEM hardly detects it. In those cases the MEM underestimates the number of the intermediate states. If two characteristic
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
8
10
10
410 nm
amplitude [a.u.]
(a)
(b)
500 nm
15 (d)
610 nm
5
5
0
0
-5
-5
-10
-10
30
570 nm
(c)
amplitude [a.u.]
20
10
10
5
0
0 -5
-10
-10
-20 -15
-30 20
650 nm
10 (e)
410 nm 500 nm 570 nm 610 nm 650 nm
(f)
amplitude [a.u.]
10
5 0
0 -10
-5 -20
-10 -7
-6
-5
-4
-3
-2
-1
log time [s]
0
-30
-7
-6
-5
-4
-3
-2
-1
0
log time [s]
Fig. 4. Discrete amplitudes of the kinetic analysis (dotted lines with an x at the end) and the amplitude distributions from the MEM analysis (solid lines) at different wavelengths for simulated absorption data. (For Fig. 4f see the text.)
lifetimes (eigenvalues) are too close, the MEM may not resolve them into two peaks. The MEM analysis is able to recover the main features of the kinetics behind the (simulated) absorption change. 4.2. Experimental data analysis A MEM analysis was done on experimental absorption data taken at different temperatures and pH. Data measured in our laboratory and data reported in [7] were used. Similarly to the simulated data at every pH and temperature five absorption data sets, measured at different wavelengths (410, 500, 570, 610 and 650 nm), were analyzed. Fig. 6 shows the amplitude distributions at low pH. For a better comparison in Fig. 6(f), the sum of the absolute values of the five different amplitude distributions is presented. Seven different characteristic life-
times with differing amplitudes can be distinguished and not all peaks are seen at every wavelengths. The peak at the shortest lifetime is very weak and it appears only at two wavelengths (610 and 650 nm) and its position on the lifetime scale is rather uncertain but the noise of the experimental data at these early times are rather high. All these peaks can be detected at different temperatures with shifting characteristic lifetimes. Fig. 7 shows the result at pH 7 in a similar representation. At this pH eight characteristic lifetimes are detected (Fig. 7(f)) with six stronger and two weak amplitudes. The behavior of the amplitude distributions at pH 7 is similar to that at pH 4.5 except that at pH 7 a new and weak peak (peak 3 in Fig. 7(f)) seems to appear, but only at one wavelength (570 nm). It was found that at pH 7 and at lower pH there is a very characteristic triple peak appearing at every measured temperature at the longest lifetime range and most significantly at 650 nm (Figs. 6(e) and 7(e)). Later in seeking a possible kinetics
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
410 nm
(b)
500 nm
(c)
570 nm
(d)
610 nm
20 (e)
650 nm
(f)
amplitude [a.u.]
20 (a)
9
10 0 -10 -20
20
amplitude [a.u.]
10 0
-10
amplitude [a.u.]
-20
410 nm 500 nm 570 nm 610 nm 650 nm
10 0 -10 -20 -7
-6
-5
-4
-3
-2
-1
log time [s]
0
-7
-6
-5
-4
-3
-2
-1
0
log time [s]
Fig. 5. Discrete amplitudes of the kinetic analysis (dotted lines with an x at the end), the amplitude distributions from the MEM analysis (solid lines) and the amplitudes (solid discrete lines) from the Gauss fit to the peaks at different wavelengths for simulated absorption data. (For Fig. 5f see the text.)
the presence of this triple peak will be a selecting condition. At high pH (pH 9, Fig. 8) at least eight peaks can be detected in the amplitude distribution. This is summarized in Fig. 8(f). Here the peaks are broader than at lower pH. The behavior of the characteristic lifetimes in dependence of the temperature is very similar to the Arrhenius temperature dependence of the rate constants. The MEM fit to the experimental absorption data is very good, the root mean square deviation is usually and approximately an order of magnitude less than the deviation found at kinetic model analysis (Fig. 9). Summarizing the result of the MEM analysis, we can say that there are at least eight (or seven at low pH) peaks in the amplitude distributions on the lifetime scale. These peaks appear distinctly at approximately the same characteristic lifetimes for a given pH and temperature (Figs. 6(f), 7(f) and 8(f)). This is a very important requirement for a characteristic lifetime to correspond to an eigenvalue. The consequence of that
result is that at least eight (or seven at lower pH) different intermediate states have to be assumed in the BR photocycle kinetics. The MEM result can be compared with that of the multiexponential global analysis. In global analysis, the number of exponentials is decided by the goodness of the fit. In Fig. 10, the characteristic lifetimes from MEM analysis (taken as the average of the distributions), the result of the global fits of [15, Fig. 6(b)] and [16] are compared as an Arrhenius plot (instead of lifetimes the apparent rate constants k = 1/s are shown at neutral pH). In [15], seven characteristic lifetimes were found. Six rate constants agree quite well in the two (MEM and [15]) data sets, but there are two other rapid rate constants in the MEM and the slowest one is missing. (Later in [16] it was proposed that this slow rate constant is connected with the BR 13-cis photocycle.) In [16], eight characteristic lifetimes were found. There is an approximate correspondence between seven rate constants. As mentioned above the slowest rate constant in [16] was identified as one which belongs to the BR
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
10
10
amplitude [a.u.]
10 410 nm
500 nm
(a)
(b)
5 5
0
0
-5 -10
-5
-15 20
570 nm
(c)
amplitude [a.u.]
10 0
610 nm
15
(d)
10
-10
5 -20
0
-30 -40
-5
-50
15 650 nm
(e)
7. 1.2e-2
100
(f)
amplitude [a.u.]
10
80
5 0
60
-5
-20 -7
-6
-5
-4
-3
-2
log time [s]
-1
4. 1.15e-4 3. 6.2e-5
20
-15
6. 4.8e-3
2. 1.6e-6
40
-10 1. 3.0e-7
5. 1.5e-3
0 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 6. MEM result for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 4.5, 20 C). In Fig. 6f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
13-cis photocycle, this is missing in the MEM. In summary these two results from global analysis recover at least six or seven characteristic lifetimes and the above mentioned triplet is present in both around 650 nm. 4.3. Possible photocycle kinetics With the photocycle kinetics a problem arises: both the multiexponential analysis and the MEM give more (6, 7, 8) characteristic lifetimes (eigenvalues) than the number (5) of the known, spectrally distinct intermediate states. This problem may be resolved by several ways. One can assume that there are more than five intermediate states but some of them are spectrally very similar. The reversible sequential model [7] assumes six intermediate states with two M forms. A parallel, irreversible model was proposed in [17] with Lf, Mf and Ls, Ms in the two branches. In [16] this problem is solved quite differently. Only five intermediate states (K, L, M, N, O) are assumed in a linear photocycle but fast quasiequilibration between L M M, M M N, N M O produce
more (7) characteristic lifetimes in the multiexponential global fitting. It was shown in Section 3 that there is no direct way to determine the photocycle kinetics (e.g., the K matrix) from the result of the MEM analysis. To find a kinetics which would agree with the MEM, we have to carry out a time consuming procedure similar to that what was made in the case of the simulated data. A K matrix is assumed and the rate constants are determined by fitting to the measured absorption data, then solving the eigenvalue problem the discrete amplitudes on the lifetime scale can be calculated. This is the result what can be compared with the MEM result. The MEM serves partly as a filter, it can accept or reject an assumed kinetics. From MEM it is known that there are at least seven or eight intermediate states appearing during the photocycle. One can select only from the five known, spectrally distinct intermediate states: K, L, M, N, O, and assume that some of them appear more than once, with the same spectral property. (As e.g., in the reversible model, Eq. (1), the M state is doubled M1 and M2.) Tak-
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
amplitude [a.u.]
10 500 nm
(a)
10 410 nm
11
(b)
5
5 0
0
-5
-5
-10
-10
15
15 10
610 nm
(c)
570 nm
(d)
10
amplitude [a.u.]
5 0
5
-5 -10
0
-15
-5
-20 -25
-10
amplitude [a.u.]
15
80
650 nm
(e)
8. 8.1e-3
70
10
60
5
50
0
40
-5
30
5. 1.7e-4
2. 1.45e-6
4. 7.5e-5
20
-10
1. 6.8e-7
10 -15 -7
(f)
-6
-5
-4
-3
-2
log time [s]
-1
3. 7.5e-6
7. 3.1e-3
6. 8.6e-4
0 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 7. MEM result (solid lines) for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 7, 20 C), the characteristic lifetimes and discrete amplitudes (dotted lines with an x at the end) of the reversible kinetic model (Eq. (1)) and the characteristic lifetimes and discrete amplitudes (solid discrete lines) of 4M kinetic model (Eq. (15)). In Fig. 7f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
ing into account this selection and the possible different kinetic connections between the intermediate states we get a huge number of possibilities. Around a hundred different kinetic models were checked. In this procedure, the following selection rules were applied: A good fit to the measured absorption change. As a standard the rms deviations were compared to that of the reversible model (Eq. (1)). The kinetic model should be applicable in a wide range of pH and temperature and the rate constants should show an Arrhenius type behavior. An acceptable similarity to the MEM result for the eigenvalues and the amplitude distribution. We followed especially the characteristic triplet eigenvalues observed at the longer lifetimes (see, e.g. Fig. 7e). Similarity of concentrations with the result of the direct concentration determinations. Fig. 7 shows partly the MEM result with the discrete amplitudes of the reversible kinetic model (Eq. (1)) with
two M states. There is a rather good similarity between them, but at least one line is missing in the kinetic model (from the triplet, see Fig. 7 at 650 nm). Though many different kinetic models were tried (with 2O, 2N, 2L, etc. intermediate states), the result shown in Fig. 7 suggests that a generalized kinetic model based on the reversible model, Eq. (1), would give better results. Finally, from all the analyzed kinetic models, a branching model, Eq. (15), with four M states gave the most satisfactory result. In this model there is a branching at the L state. K $ L $ M 1 $ M 2 $ N $ O ! BR and L $ M 3 ! M 4 ! N
ð15Þ
Fig. 7 shows the discrete amplitudes of the 4M kinetic model, Eq. (15), too, at pH 7. This model reproduces the characteristic triplet (Fig. 7, 650 nm) but the agreement is not so good as with the simulated data. For example, one discrete amplitude of the kinetic model at 500 nm (Fig. 7) is not seen by the MEM.
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
12
10 410 nm
6
(a)
500 nm
(b)
610 nm
(d)
amplitude [a.u.]
4
5
2 0
0
-2
-5
-4
-10
-6
570 nm
8
(c)
amplitude [a.u.]
6
10
4 2
5
0 -2 0
-4 -6
-5
-8
5
amplitude [a.u.]
4
650 nm
(e)
2. 1.8e-6
30
5. 1.9e-4
(f)
25
3
6. 3.2e-3
20
2
8. 5.9e-2 7. 1.4e-2
15
1
10
0
4. 3.05e-5 1. 7.5e-7
5
3. 7.16e-6
-1 0
-2 -7
-6
-5
-4
-3
-2
-1
0
-7
-6
log time [s]
-5
-4
-3
-2
-1
0
log time [s]
Fig. 8. MEM result for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 9, 20 C). The characteristic lifetimes and discrete amplitudes (dotted lines with an x at the end) of the reversible kinetic model (Eq. (1)). In Fig. 8f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
410 nm
100
500 nm
absorption change [a.u.]
570 nm 610 nm
50
650 nm
0 -50 -100 -150 -200
-6
-5
-4 -3 log time [s]
-2
-1
Fig. 9. Experimental absorption change data measured at five different wavelength (symbols, pH 7, 20 C) and the MEM fit to the data (continuous line).
Apparent rate constants log k [1/s]
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
13
6 5 4 3 2 1 3.20
3.30 3.40 3.50 1/Temperature [1000/K]
3.60
Fig. 10. Comparison of the seven apparent rate constants (up triangles) taken from [15] (pH 7), the eight rate constants (circles) from [16] (pH 7.2) and the result of MEM (eight rate constants, filled squares, pH 7).
100
80
410 nm
500 nm
60
amplitude [a.u.]
40
50
20 0
0
-20 -40
-50
-60 -80
-100
100
570 nm
610 nm
100
amplitude [a.u.]
50 50
0 0
-50 -50
-100 -100
-7
-6
-5
-4
-3
-2
-1
0
-7
-6
amplitude [a.u.]
70 60 50 40 30 20 10 0 -10 -20 -7
-5
-4
-3
-2
-1
0
log time [s]
log time [s]
650 nm
-6
-5
-4
-3
-2
-1
0
log time [s] Fig. 11. The characteristic lifetimes and the discrete amplitudes for the Gauss fitted MEM result (solid lines) and for the 4M kinetic model (Eq. (16), dotted lines with an x at the end).
14
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
Concentration [a.u.]
250 200
200
150
150
100
100
50
50
0
0
250
Concentration [a.u.]
250
K
M
N
200
200
150 150
100 100
50
50 0
0
5
300
O Concentration [a.u.]
L
BR
4
250
3
200
2
150 100
1
50
0
0
-1
-7
-6
-5
-4
-3
-2
-1
-7
0
-6
-5
-4
-3
-2
-1
0
log time [s]
log time [s]
Fig. 12. Comparison of the concentrations for the intermediate states. Direct concentration calculations (x), concentration for the 4M model (solid line), concentration for the reversible kinetic model (square).
It was found that at pH 4.5 the L M M3 is a stand alone transition and the M4 intermediate state does not appear at all. This may be the reason that at low pH the MEM detects only seven peaks (Fig. 6). At high pH a 4M kinetic model with branching gave the best result, but the branching is at the K state (Eq. (16)). This branching is markedly preferred at high pH against the branching at the L state.
7
KL LK
ð16Þ
Fig. 11 shows the discrete amplitudes after the Gauss fits to the MEM result (Fig. 8) together with the characteristic lifetimes and discrete amplitudes of the 4M kinetic model, Eq. (16). As can be seen there is a high similarity between the kinetic model and the MEM result. The kinetic model of Eq. (16) reproduces all eigenvalues and amplitudes of the MEM analysis, though there are some shifts in the eigenvalues. For comparison Fig. 8 shows the eigenvalues and amplitudes of the reversible kinetic model (Eq. (1)) together with the MEM result. As can be seen some of the eigenvalues are missing in this kinetic model. There is some difference between the 2M and 4M kinetic models in the
Rate constants log k [1/s]
K $ L $ M 1 ! M 2 $ N ! O ! BR and K $ M 3 ! M 4 ! N
goodness of the fit to the experimental absorption data, too. The 4M model gives at least the same (or better) goodness of the fit as the reversible 2M model. A comparison between the concentrations determined from the different kinetic models and from the direct concentration calculation (Section 3) is shown in Fig. 12.
6
LM1 M1L M1M2
5
M2N NM2
4
NO OBr KM3
3
M3K M3M4 M4N
2
M4O
1 0
3.30
3.40 3.50 3.60 1/Temperature [1000/K]
3.70
Fig. 13. Arrhenius plot of the rate constants of the 4M kinetic model at pH 9.
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
As it was emphasized, from the possible kinetic models we accepted only that which gave approximately linear reciprocal temperature dependence in the Arrhenius plot of the rate constants. Such a plot is given in Fig. 13 for pH 9. It is interesting to note, that the branching at the N state (N ! BR transition) in the reversible model (Eq. (1)) was not supported by the rate constants fitting on the basis of the 4M kinetic models.
5. Conclusion In this and a previous paper [9] a MEM procedure was applied to analyze absorption kinetic data for the BR photocycle. Though this method, the MEM, is not a kinetic analysis in the sense that it does not give a specific kinetic model, but the information resulting from MEM is very useful for selecting a possible kinetic model and it is free from any assumption. The most important result following from the MEM analysis is that at least eight intermediate states are present in the BR photocycle. Many kinetic models with eight intermediate states were analyzed and compared with the MEM result. A branching kinetic model with four different M states gave the best agreement. (Perhaps, it is interesting to remark that in an earlier paper [18] it was reported that within a simplified and local analysis taking into account only two M states a branching model gave better result than a sequential model.) The proposed 4M kinetic model (Eqs. (15) and (16)) includes most of the properties of the reversible 2M model given by Eq. (1) [7] as one part of the branching. The branching is most pronounced at high pH. This may be the reason that the cooperativity in the BR photocycle is most pronounced at high pH, too [19]. A MEM analysis results in an amplitude distribution (peaks) on the lifetime scale instead of discrete lines as the case in a kinetic model analysis. The entropy, Eq. (14), is determined by the number of the peaks and by the width of the peaks. The broadening of the peaks will increase the entropy. There may be two sources of the broadening of the peaks and the increase of the entropy [9]: a noise generated entropy and what was called in [9], the inherent entropy. The inherent entropy may be connected with the protein dynamics and heterogeneity. In that case the kinetic processes are described by rate distributions instead of rate constants [20]. With simulated absorption kinetic data there is only noise generated broadening and the peaks are rather narrow (Figs. 4 and 5). For real absorption data the peaks are broader and there is an increase of entropy with increasing pH (Figs. 6–8), though the noise level is approximately the same. It was remarked earlier that the MEM fit to the experimental absorption change is much better than the discrete exponential fit (kinetic model analysis). This
15
is probably because the MEM takes into account the rate distribution. The molecular background for the 4M kinetic model is not clear. The branching occurs probably at the proton release side of the Schiff-base. The M3 ! M4 transition may be similarly connected to the proton switch as in the case of the M1 ! M2 transition, but it is pH dependent. At low pH there is no M3 ! M4 transition and the M4 intermediate state does not appear. To find a kinetics which would agree with the MEM result, it was assumed that eight intermediate states exist. But the MEM result does not exclude the existence of more intermediate states functionally important in the photocycle. If there are more such intermediate states the MEM may not detect them either because the corresponding peaks are too small or the corresponding eigenvalues are degenerate. In this work there was no attempt to analyze photocycles with more than eight intermediate states.
Acknowledgements We are grateful to Dr. G. Va´ro´ (Biophysics Institute of the Hungarian Academy of Sciences) to provide the experimental absorption data set and to allow to use the RATE program for kinetic analysis. We thank Dr. Z. Dancsha´zy (Biophysics Institute of the Hungarian Academy of Sciences) for providing us with purple membrane preparations and for valuable discussions.
References [1] W. Stoeckenius, R.H. Lozier, R.A. Bogomolni, Biochem. Biophys. Acta 505 (1979) 215–278. [2] W. Ku¨hlbrandt, Nature 406 (2000) 569–570. [3] S. Subramaniam, R. Henderson, Nature 406 (2000) 653–657. [4] C. Gergely, L. Zima´nyi, G. Va´ro´, J. Phys. Chem. B 101 (1997) 9390–9395. [5] J.K. Lanyi, G. Va´ro´, Isr. J. Chem. 35 (1995) 365–385. [6] J.F. Nagle, Biophys. J. 59 (1991) 476–487. [7] K. Ludmann, C. Gergely, G. Va´ro´, Biophys. J. 75 (1998) 3110– 3119. [8] J. Skilling, R.K. Bryan, Mon. Not. R. Astron. Soc. 211 (1984) 111–124. [9] Zs. Ablonczy, A. Luka´cs, E. Papp, Biophys. Chem. 104 (2003) 249–258. [10] C. Gergely, C. Ganea, G.I. Groma, G. Va´ro´, Biophys. J. 65 (1993) 2478–2483. [11] C. Gergely, C. Ganea, G. Va´ro´, Biophys. J. 67 (1994) 855–861. [12] G. Va´ro´, J.K. Lanyi, Biochemistry 34 (1995) 12161–12169. [13] Tyson J.J., The Belousov–Zhabotinskii reaction, in: Levin S. (Ed.), Lecture Notes in Biomathematics, vol. 10, Springer, Berlin , Heidelberg, New York, MA, 1976. [14] I. Szundi, J.W. Lewis, D.S. Kliger, Biophys. J. 73 (1997) 688– 702. [15] A.H. Xie, J.F. Nagle, R.H. Lozier, Biophys. J. 51 (1987) 627–635.
16
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
[16] I. Chizov, D.S. Chernavskii, M. Engelhard, K.-H. Mueller, B.V. Zubov, B. Hess, Biophys. J. 71 (1996) 2329–2345. [17] R.W. Hendler, R.I. Shrager, S. Bose, J. Phys. Chem. B 105 (2001) 3319–3328. [18] E. Papp, V.H. Ha, Biophys. Chem. 57 (1996) 155–161.
[19] Zs. Ablonczy, V.H. Ha, E. Papp, Biophys. Chem. 71 (1998) 235– 243. [20] P.J. Steinbach, K. Chu, H. Frauenfelder, J.B. Johnson, D.C. Lamb, G.U. Nienhaus, T.B. Sauke, R.D. Young, Biophys. J. 61 (1992) 235–245.
Bacteriorhodopsin photocycle kinetics analyzed by the maximum entropy method A. Luka´cs, E. Papp
*
Department of Biological Physics, Eo¨tvo¨s University, Pa´zma´ny P. Se´ta´ny 1/A, Budapest H-1117, Hungary Received 15 June 2004; received in revised form 27 July 2004; accepted 9 August 2004 Available online 7 October 2004
Abstract A maximum entropy method (MEM) was developed for the study of the bacteriorhodopsin photocycle kinetics. The method can be applied directly to experimental kinetic absorption data without any assumption for the number of the intermediate states taking part in the photocycle. Though this method does not give a specific kinetics, its result is very useful for selection between possible photocycle kinetics. Using simulated data, it is shown that MEM gives correct results for the number of the intermediate states and the amplitude distributions around the characteristic lifetimes. Analyzing experimental absorption data at five different wavelengths, MEM gives seven or eight characteristic lifetimes, which means that at least so many distinct intermediate states exist during the photocycle. Many possible photocycle kinetic models were studied and compared with the MEM result. The best agreement was found with a branching photocycle model of eight intermediate states (K, L, M1, M2, M3, M4, N, O). The branching occurs at the L intermediate state (M1 and M2 being in one branch and M3 and M4 in the other branch), but at high pH it occurs already at the K state. 2004 Elsevier B.V. All rights reserved. Keywords: Maximum entropy method; Bacteriorhodopsin; Photocycle kinetics
1. Introduction Bacteriorhodopsin (BR) is a light driven proton pump built in the cell membrane (purple membrane) of Halobacterium salinarum [1–3]. After excitation it undergoes a series of molecular changes, the photocycle, which results in the translocation of a proton through the membrane. During the photocycle (10– 100 ms at room temperature) several spectrally different intermediate states (J, K, L, M, N and O) can be distinguished. These intermediates have broad and
*
Corresponding author. Tel.: +36 1 372 2785; fax: +36 1 372 2757. E-mail address: [email protected] (E. Papp).
1011-1344/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jphotobiol.2004.08.004
strongly overlapping absorption bands [4] except the M state which is shifted to the blue. The process is thermally activated from the J state. To understand better the mechanism of the proton pump in BR a lot of work was devoted to the study of the photocycle [5]. Measuring and comparing the transient absorption changes at different wavelengths during the photocycle, different kinetic models can be tested. The generally accepted procedure is the following. Assuming a given number of intermediate states and first order reaction steps between them (kinetic model) the time dependence of concentrations of the intermediate states can be calculated. The input parameters are the rate constants of the given kinetic model and these parameters are varied during the fitting procedure. From the concentrations the transient
2
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
absorption change can be calculated for any wavelength knowing the absorption spectra of the intermediate states. This calculated absorption change is compared with the measured changes and the rate constants of the assumed kinetic model are determined by minimizing the deviation between the measured and calculated absorption changes. But, mainly because of the high number of the intermediate states and their broad, strongly overlapping absorption bands, there are several quite different kinetic models which give almost the same goodness of the fit. This analysis showed that the BR photocycle kinetics is a complex problem and may have several solutions [6]. To resolve this problem, a thorough analysis was carried out in [7] introducing more control parameters (pH and temperature) beside the wavelength and time. Three kinetic models were compared with the conclusion that the sequential model with reversible reactions between the intermediates (K, L, M1, M2, N, O) gives the most acceptable fit with a linear Arrhenius plot for the rate constants. In the above ‘‘kinetic model analysis’’ the time dependence is given by discrete exponentials. The number of exponentials and the values of the characteristic lifetimes in the exponentials are uniquely determined by the kinetic model and the values of the rate constants. The number of exponentials equals with the number of intermediate states assumed in the kinetic model. So a kinetic model can be represented by discrete lines in the lifetime ‘‘space’’. We will call this a discrete line distribution. There exists a different approach to data analysis of noisy experimental data, the maximum entropy method (MEM) [8]. In a previous paper [9], the first result of the possible application of MEM to BR absorption kinetic rate processes was reported. Instead of discrete lines the MEM gives a distribution with characteristic peaks in the lifetime space. This distribution can be called as the ‘‘image’’ [8] of the experimental data. The discrete lines resulting from kinetic model analysis represent a special image. There are many possible images (distributions) that can more or less correctly (minimum misfit) describe the given experimental data. The MEM selects one image from the many possible ones. Comparing the kinetic model analysis and the MEM, there are some very important differences. In the kinetic model analysis the number of the intermediate states is fixed at the start and their absorption spectra have to be known. In MEM the process itself selects the peaks (image) without any constraint for the number and spectral properties of the intermediate states. On the other hand the MEM does not give a complete information about the kinetics, it gives only the image. This problem will be analyzed in more details later in this paper. It means that the rate con-
stants (kinetic model) uniquely determine the image but the opposite is not true. Therefore the results from kinetic model analysis and from the MEM can be compared only at the level of images. In a previous paper [9], the MEM was applied for the time dependence of intermediate state concentration (M state). This analysis was restricted in several relations. In this paper, the MEM analysis is applied directly to the measured absorption kinetic experimental data of the BR photocycle. This analysis shows that there are at least 7 or 8 intermediate states taking part in the photocycle. A search for a corresponding kinetic model, resulting in a similar image as the MEM image, was also made. A branching photocycle kinetic model, with four M intermediate states, gives similar image, but there is no complete agreement between the two images.
2. Materials and methods The purple membrane suspensions isolated from H. salinarium strain S9 were gifts from the Biophysics Institute, Biological Research Center, Szeged, Hungary. For excitation of the photocycle a Nd-YAG laser, frequency doubled to 530 nm, was applied and the data were averaged. For MEM analysis, we used also the absorption data recorded in the Biophysics Institute, Szeged [7]. Spectral data for the extinction coefficients of the intermediate states were taken from the literature [4]. For the kinetic model analysis the RATE program [10–12] was used. For the MEM analysis, a program described elsewhere [9] was used with some modifications.
3. Maximum entropy method and absorption kinetic analysis Any photocycle model can be described and solved by the methods of chemical kinetics. For example, the reversible photocycle model with n = 6 intermediate states (K, L, M1, M2, N, O): k1
K
k2
O
k 11
!
L
k3
k4
BR
k5
M1 and
k6
N
k 12
!
M2 BR
k7
k8
N
k9
k 10
ð1Þ
can be formulated by the following system of linear ordinary differential equations (in matrix representation and assuming first-order reactions between the intermediate states):
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
0 B B B B dB B B dt B B B B @
C1
1
0
C B C B C B C B C B C B C4 C ¼ B C B B C5 C C B C B C6 A @ BR C2 C3
k 1 k1 0 0 0
k2
0
ðk 2 þ k 3 Þ k4 k3 ðk 4 þ k 5 Þ 0 0
k5 0
0
0
0
0 k6
0 0
0 0
ðk 6 þ k 7 Þ k8 k7 ðk 8 þ k 9 þ k 12 Þ
3
0
0 k 10
0
0
0
0
k9
ðk 10 þ k 11 Þ
0
0
0
0
k 12
k 11
10
C1
1
CB C 0 CB C 2 C CB C C B 0C CB C 3 C CB C 0 CB C 4 C: CB C C B 0C CB C 5 C CB C 0 A@ C 6 A 0 BR
ð2Þ
Here C1, . . ., C6 are the time dependent concentrations of the intermediates and BR denotes the same for molecules in the ground state. As the matrix is redundant in Eq. (2) (all elements in the last column are zero, this is because there is no backreaction from the ground state at the end of the photocycle), it is better to write Eq. (2) in the following form:
1 0 k 1 C1 BC C B k B 2C B 1 B C B C C B 0 dB B 3C¼ B B B dt B C 4 C C B 0 B C B @ C5 A @ 0 0
0
C6
k2 ðk 2 þ k 3 Þ
0 k4
0 0
0 0
0 0
k3
ðk 4 þ k 5 Þ
k6
0
0
0 0
k5 0
ðk 6 þ k 7 Þ k7
k8 ðk 8 þ k 9 þ k 12 Þ
0 k 10
1 C1 CB C C CB 2 C CB C CB C 3 C CB C: CB C C CB 4 C CB C A@ C 5 A
0
0
0
k9
ðk 10 þ k 11 Þ
C6
In that case BR(t) is determined by the mass conservation: n X BRðtÞ ¼ C 0 C i ðtÞ; ð4Þ i¼1
where C0 is the total concentration of the BR molecules (in the ground and intermediate states). Eq. (3) can be written in a compact form: dCðtÞ ¼ K CðtÞ; dt
ð5Þ
where C is a vector with (C1, . . ., Cn) components for the concentrations of the intermediate states and K is the rate matrix with n by n elements determined by the rate constants (ki) of the given photocycle kinetic model. The linear differential equation system, Eq. (5), can be solved in terms of the eigenvectors and eigenvalues of the K matrix [13]. In general, an n by n matrix can have n linearly independent eigenvalues (k j , apparent rate constants) and eigenvectors (nj): K nj ¼ k j nj ; with j = 1, . . ., n and:
ð6Þ
CðtÞ ¼ expðk j tÞ nj
10
ð3Þ
ð7Þ
is a solution of Eq. (5). The general solution of Eq. (5) is a linear combination of Eqs. (7): n X C i ðtÞ ¼ aj expðk j tÞ nji j¼1
¼
n X
cij expðk j tÞ;
ð8Þ
j¼1
where cij = aj Æ nji. The constants aj are determined by the initial condition. (For BR photocycle with short laser pulse excitation the initial condition at t = 0 is the following: C16¼0 and C2 = = Cn = 0.) The absorption change during the photocycle can be calculated using Eq. (8) and the extinction coefficients, ei(k), of the intermediate states and eBR(k) of the ground state. Then the absorption, A0(k), at wavelength k before excitation is: A0 ðkÞ ¼ C 0 eBR ðkÞ
ð9Þ
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
and after excitation: n X C i ðtÞ ei ðkÞ þ BRðtÞ eBR ðkÞ: Aðt; kÞ ¼
ð10Þ
i¼1
The absorption change during the photocycle is given by the difference of Eqs. (10) and (9): DAðk; tÞ ¼ Aðk; tÞ A0 ðkÞ ¼
n X
Dei ðkÞ C i ðtÞ
ð11Þ
i¼1
(Eq. (4) was used here and Dei(k) = (ei(k) eBR(k))). The absorption change, DA(k,t), can be measured during the photocycle at different k wavelengths. In a kinetic model analysis, one starts with an assumed K matrix (Eq. (2) or (3)), numerically calculates the time dependence of the concentrations C(t), and the expected absorption changes (Eq. (11)). Then this is compared with the measured absorption changes and the initial rate constants are varied until a minimum is reached in the square deviation. If Eq. (8) is put into Eq. (11) then we get for the absorption change: DAðk; tÞ ¼
n X n X i¼1
¼
n X
Dei ðkÞ cij expðk j tÞ
j¼1
bj ðkÞ expðk j tÞ;
ð12Þ
j¼1
Pn where bj ðkÞ ¼ i¼1 Dei ðkÞ cij . Eqs. (8) and (12) have a very similar form and this form is a suitable one for the application of MEM. The MEM analysis gives a distribution of the amplitudes on the quasi continuous lifetime scale with characteristic peaks around the sj ¼ 1=k j lifetimes. MEM is a conditional extremum procedure [8,9]. An entropy function, S, is defined on the amplitude (cij or bj(k)) distribution. Starting with a flat (structureless) distribution the procedure selects such a distribution, which minimizes the square deviation between the measured C mi ðtÞ (or DAm(k,t)) and the calculated (Eq. (8) or (12)) values, while keeping the entropy at a maximum level. In this analysis, there is no starting assumption about the number of the eigenvalues (intermediate states) or the kinetics. In principle, from a MEM analysis based on Eq. (12), the bj(k) amplitude distributions, the k j eigenvalues and the number (n) of the intermediate states can be determined. But the knowledge of these quantities is not necessarily enough to determine the true rate constants (K matrix) [14]. One problem is that Eq. (12) cannot be reversed without assumptions for the n intermediate states. The other problem is that some intermediate states have identical spectral properties. For these intermediate states, the differential extinction coefficients are the same and Eq. (12) becomes underdetermined. For
these reasons (and that the MEM results have some uncertainties), a MEM analysis cannot give specific kinetics. It yields results which can be compared with a given kinetics in the following sense. A kinetics (K matrix) is assumed, from that the eigenvalue problem is solved and the expected absorption change, Eq. (12), is determined. Only this result can be compared with the MEM result. In a previous paper [9], the result of a MEM analysis was reported in a special case. It was assumed that the absorption change at k = 410 nm is approximately proportional to the M state concentration (Eq. (11)) and that the negative and positive amplitudes (rise and decay) in Eq. (8) are well separated on the lifetime scale. To apply this method for the other intermediate states, the knowledge of their concentrations would be necessary. In principle it is possible to solve Eq. (11) for the concentrations. For example, if n = 5 intermediate states are assumed and measuring DA(k,t) at 5 different wavelengths then Eq. (11) gives a linear equation system for every time and this equation system can be solved for Ci(t). The result of this calculation is shown in Fig. 1. As can be seen we get negative values for some concentrations and this is an unacceptable and unphysical result. The origin of that false result is not quite clear. It may be caused by the similarity and strong overlap of the absorption bands (mainly the L and N intermediate states) and perhaps by that a small (and systematic) error in the absorption coefficients is strongly amplified in that kind of solution. (This view is supported by the result for the M state – where the absorption band is well separated – and we get an acceptable result, except for the shortest times.) To try to overcome these difficulties in the direct determination of the concentration the following approximation was made. We assumed that in the early part of the photocycle only the K, L, M, while in the later part only the M, N, O intermediate states are present. In that case Eq. (11) is overdetermined but it can be K L M N O BR
300 250
Concentration [a.u.]
4
200 150 100 50 0 -50 -100 -7
-6
-5
-4 -3 log time [s]
-2
-1
0
Fig. 1. Direct calculation of the concentrations of the intermediate states. (For BR the sign is changed.)
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16 350
4.5 pH 5C
Concentration [a.u.]
300 250
K L M N O BR
4.5 pH 30C
5
K L M N O BR
200 150 100 50 0
300
Concentration [a.u.]
7.0 pH 20C 250 200
K L M N O BR
7.0 pH 30C
K L M N O BR
150 100 50 0
300
K L M N O BR
Concentration [a.u.]
9.0 pH 5C 250
9.0 pH 30C
200
K L M N O BR
150 100 50 0 -7
-6
-5
-4
-3
-2
-1
0
-7
-6
log time [s]
-5
-4
-3
-2
-1
0
log time [s]
Fig. 2. Direct concentration calculations at different pH and temperatures excluding some intermediate states in the early and later times of the photocycle. (For BR the sign is changed.)
solved for Ci(t) by a least square fit. The solution is presented in Fig. 2 and it has the following characteristic features: There is a window in the middle time range where the solution is quite unreliable (except for the M state) for reasons mentioned above. The M state concentrations are still negative in some cases at the short times. The behavior of the N and O intermediate states is interesting. At low pH the N state concentration decreases with increasing temperature (and it is very low at 30 C). At high pH the O state concentration decreases with decreasing temperature (and it is very low below 15 C). But the maximum values in the N and O states concentrations appear nearly at the same time during the photocycle. After excitation the BR concentration starts to increase very early with the decay of the L or M1 states. This can be a side effect caused by the measuring light
leading to a photoinduced transition from L or M1 to the BR ground state. We include the possibility of a photoinduced transition from M1 to BR here and later in the photocycle kinetics. This improves the fit to the experimental data but it does not have an essential effect on the photocycle kinetics. These features will be taken into account later in the photocycle analysis.
4. Results As the attempt for a direct determination of the concentration failed the previously reported MEM analysis [9] cannot be applied for the whole photocycle. Instead the MEM has to be applied directly to the measured DA(k,t) absorption changes using Eq. (12).
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
6
According to Eq. (12) the image of the absorption change DA(t) at a given wavelength, k, is: DAðtÞ ¼
N X
bi expðt=si Þ;
ð13Þ
i¼1
where s is on the quasi continuous lifetime scale and bi are the amplitudes. (In the following analysis the s scale is usually restricted to 107–101 s and the log scale is divided to N = 200 equal intervals.) This form will be used for the MEM analysis. But this differs in two essential points from the form used in [9], therefore the earlier MEM procedure has to be modified. The first modification is connected with the sign of the amplitudes, bi. In the previous report, the amplitudes were restricted to positive values so a simple entropy definition was possible. In Eq. (13) there is no preliminary information about the sign of bi, but the entropy can be defined only on positive numbers. So we are forced to define the entropy (S) by the absolute values of bi: N X bi ð14Þ S¼ jbi j ln 1 ; Ai i¼1 where Ai are the initial values for bi which maximizes the entropy at the start. But using the absolute value in Eq. (14) introduces some difficulties in the MEM procedure. The problem is with the change of the sign of the bi amplitudes. This problem arises because of the use of the entropy metric [8,9]. The entropy metric is an essential point in the MEM algorithm. It discriminates in favor of allowing high values of bi in Eq. (13) to change more rapidly than low ones between the iterations. There is no problem if the bi amplitudes take positive values by definitions. Fortunately in many applications this is the case. But in absorption kinetics, the bi amplitudes can take both positive and negative values. In that case starting with a flat (constant) bi amplitudes with a given (e.g., positive) sign, during the iterations some bi will increase to give the positive peaks, other bi amplitudes will decrease. To reproduce the negative peaks, the sign of bi at some places on the lifetime scale (see, e.g., Fig. 4) has to be changed. But the entropy metric resists to change the sign of bi. This may result some distortion of the image, i.e., the amplitude distribution. To circumvent this problem, it is advisable to divide the MEM analysis into two steps. In the first step such a division of the lifetime scale is determined where the sign of the bi is positive or negative. To get this information, the following procedure was made. In Eq. (13), a constant level was added to the bi amplitudes shifting up the amplitude spectra and a corresponding contribution was added to the absorption change. From the MEM fitting to this modified absorption change the zero points of bi could be determined approximately. In the second step – the real MEM anal-
ysis – the sign information about the bi amplitudes was used from the start, i.e., the initial Ai amplitudes in Eq. (14) have a flat (Ai = const) distribution but signed according to the previous result. The second modification is connected with the mass conservation, Eq. (4). It was shown earlier [9] that, if the MEM is applied to the concentrations, Eq. (8), then the mass conservation can be taken into account by reducing the three dimensional search subspace to a two dimensional subspace in a proper way. This restriction cannot be applied in the case of Eq. (12). So in the following the MEM search will be made in the full three dimensional subspace as described in [8,9]. 4.1. Simulated data analysis Before applying the MEM to experimental absorption data a reliability test was made on generated data in the following way. Starting from a known kinetics – in that case a reversible photocycle, Eq. (1) – and assuming numerical values for the rate constants given in Table 1, Eq. (6) is solved for the eigenvalues and eigenvectors. (These numerical values correspond approximately to that reported in [7] for pH 7 and 20 C. For pH 9; 20 C simulated data the rate constants are not shown but they correspond to the reversible model, too.) Fig. 3 shows the characteristic lifetimes sj corresponding to the k j eigenvalues ðsj ¼ 1=k j Þ and the amplitudes cij in Eq. (8) for the intermediate states concentrations. As expected there are n=6 eigenvalues coinciding with the number of the intermediate states, but the amplitudes differ widely, in some cases they are very small. (For example, for the O state there are only two significant amplitudes, the other four are very small.) Knowing the cij amplitudes the expected absorption change during this simulated photocycle can be calculated for any wavelength using Eq. (12). This calculation was done for five different wavelengths (410, 500, 570, 600 and 650 nm). The result for different wavelengths is similar to that shown in Fig. 3, but the amplitudes are different (see Fig. 4, dotted lines with an x at the end).
Table 1 Rate constants for simulated data generation ki
½1s
ki
½1s
k1 k2 k3 k4 k5 k6
4.055 · 105 9.45 · 104 1.135 · 105 8.072 · 105 6.56 · 104 0
k7 k8 k9 k10 k11 k12
3.5 · 103 1.05 · 104 0.17 · 103 0.9 · 102 0.8 · 102 4.9 · 102
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16 4
K
L
M1
M2
N
O
7
amplitude [a.u.]
3 2 1 0 -1 -2 -3 -4 4 3
amplitude [a.u.]
2 1 0 -1 -2 -3 -4 4
amplitude [a.u.]
3 2 1 0 -1 -2 -3 -4 -7
-6
-5
-4
-3
-2
log time [s]
-1 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 3. The characteristic lifetimes and the discrete amplitudes for the intermediate states.
A Gaussian random noise was added to the simulated absorption changes and these simulated data were analyzed by the MEM. The images (discrete amplitudes and the amplitude distributions on the lifetime scale) are shown in Figs. 4 and 5 for the two simulated data sets (pH 7; 20 C and pH 9; 20 C, respectively). These figures show the discrete amplitudes at the characteristic lifetimes (calculated by Eqs. (6), (8) and (12) using the assumed rate constants) and the distribution of amplitudes (peaks) from the MEM analysis (Eq. (13)) at five different wavelengths. In Figs. 4f and 5f the MEM peaks for different wavelengths are presented together showing that the peaks appear approximately around the same characteristic lifetimes. To compare the discrete amplitudes with the peaks of the MEM analysis Gaussian curves were fitted to these peaks and the area under the peaks were determined. This result is shown in Fig. 5, too.
This analysis is concluded with the following remarks: Instead of discrete amplitudes we get distribution (approximately a Gaussian peak) around a given characteristic lifetime. The characteristic lifetimes (si), or eigenvalues of the kinetic model can be reproduced by MEM analysis: the peaks appear nearly at the same lifetimes for the different wavelengths. But to recover all eigenvalues absorption data are needed for more wavelengths because if an amplitude at a given wavelength is too small it may not be recovered. There may be such a case that the amplitudes for a given characteristic lifetime are very small at all measuring wavelengths. For example, in the case shown in Fig. 5 the amplitudes for the fourth characteristic lifetime (log s 3.5) are very small and the MEM hardly detects it. In those cases the MEM underestimates the number of the intermediate states. If two characteristic
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
8
10
10
410 nm
amplitude [a.u.]
(a)
(b)
500 nm
15 (d)
610 nm
5
5
0
0
-5
-5
-10
-10
30
570 nm
(c)
amplitude [a.u.]
20
10
10
5
0
0 -5
-10
-10
-20 -15
-30 20
650 nm
10 (e)
410 nm 500 nm 570 nm 610 nm 650 nm
(f)
amplitude [a.u.]
10
5 0
0 -10
-5 -20
-10 -7
-6
-5
-4
-3
-2
-1
log time [s]
0
-30
-7
-6
-5
-4
-3
-2
-1
0
log time [s]
Fig. 4. Discrete amplitudes of the kinetic analysis (dotted lines with an x at the end) and the amplitude distributions from the MEM analysis (solid lines) at different wavelengths for simulated absorption data. (For Fig. 4f see the text.)
lifetimes (eigenvalues) are too close, the MEM may not resolve them into two peaks. The MEM analysis is able to recover the main features of the kinetics behind the (simulated) absorption change. 4.2. Experimental data analysis A MEM analysis was done on experimental absorption data taken at different temperatures and pH. Data measured in our laboratory and data reported in [7] were used. Similarly to the simulated data at every pH and temperature five absorption data sets, measured at different wavelengths (410, 500, 570, 610 and 650 nm), were analyzed. Fig. 6 shows the amplitude distributions at low pH. For a better comparison in Fig. 6(f), the sum of the absolute values of the five different amplitude distributions is presented. Seven different characteristic life-
times with differing amplitudes can be distinguished and not all peaks are seen at every wavelengths. The peak at the shortest lifetime is very weak and it appears only at two wavelengths (610 and 650 nm) and its position on the lifetime scale is rather uncertain but the noise of the experimental data at these early times are rather high. All these peaks can be detected at different temperatures with shifting characteristic lifetimes. Fig. 7 shows the result at pH 7 in a similar representation. At this pH eight characteristic lifetimes are detected (Fig. 7(f)) with six stronger and two weak amplitudes. The behavior of the amplitude distributions at pH 7 is similar to that at pH 4.5 except that at pH 7 a new and weak peak (peak 3 in Fig. 7(f)) seems to appear, but only at one wavelength (570 nm). It was found that at pH 7 and at lower pH there is a very characteristic triple peak appearing at every measured temperature at the longest lifetime range and most significantly at 650 nm (Figs. 6(e) and 7(e)). Later in seeking a possible kinetics
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
410 nm
(b)
500 nm
(c)
570 nm
(d)
610 nm
20 (e)
650 nm
(f)
amplitude [a.u.]
20 (a)
9
10 0 -10 -20
20
amplitude [a.u.]
10 0
-10
amplitude [a.u.]
-20
410 nm 500 nm 570 nm 610 nm 650 nm
10 0 -10 -20 -7
-6
-5
-4
-3
-2
-1
log time [s]
0
-7
-6
-5
-4
-3
-2
-1
0
log time [s]
Fig. 5. Discrete amplitudes of the kinetic analysis (dotted lines with an x at the end), the amplitude distributions from the MEM analysis (solid lines) and the amplitudes (solid discrete lines) from the Gauss fit to the peaks at different wavelengths for simulated absorption data. (For Fig. 5f see the text.)
the presence of this triple peak will be a selecting condition. At high pH (pH 9, Fig. 8) at least eight peaks can be detected in the amplitude distribution. This is summarized in Fig. 8(f). Here the peaks are broader than at lower pH. The behavior of the characteristic lifetimes in dependence of the temperature is very similar to the Arrhenius temperature dependence of the rate constants. The MEM fit to the experimental absorption data is very good, the root mean square deviation is usually and approximately an order of magnitude less than the deviation found at kinetic model analysis (Fig. 9). Summarizing the result of the MEM analysis, we can say that there are at least eight (or seven at low pH) peaks in the amplitude distributions on the lifetime scale. These peaks appear distinctly at approximately the same characteristic lifetimes for a given pH and temperature (Figs. 6(f), 7(f) and 8(f)). This is a very important requirement for a characteristic lifetime to correspond to an eigenvalue. The consequence of that
result is that at least eight (or seven at lower pH) different intermediate states have to be assumed in the BR photocycle kinetics. The MEM result can be compared with that of the multiexponential global analysis. In global analysis, the number of exponentials is decided by the goodness of the fit. In Fig. 10, the characteristic lifetimes from MEM analysis (taken as the average of the distributions), the result of the global fits of [15, Fig. 6(b)] and [16] are compared as an Arrhenius plot (instead of lifetimes the apparent rate constants k = 1/s are shown at neutral pH). In [15], seven characteristic lifetimes were found. Six rate constants agree quite well in the two (MEM and [15]) data sets, but there are two other rapid rate constants in the MEM and the slowest one is missing. (Later in [16] it was proposed that this slow rate constant is connected with the BR 13-cis photocycle.) In [16], eight characteristic lifetimes were found. There is an approximate correspondence between seven rate constants. As mentioned above the slowest rate constant in [16] was identified as one which belongs to the BR
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
10
10
amplitude [a.u.]
10 410 nm
500 nm
(a)
(b)
5 5
0
0
-5 -10
-5
-15 20
570 nm
(c)
amplitude [a.u.]
10 0
610 nm
15
(d)
10
-10
5 -20
0
-30 -40
-5
-50
15 650 nm
(e)
7. 1.2e-2
100
(f)
amplitude [a.u.]
10
80
5 0
60
-5
-20 -7
-6
-5
-4
-3
-2
log time [s]
-1
4. 1.15e-4 3. 6.2e-5
20
-15
6. 4.8e-3
2. 1.6e-6
40
-10 1. 3.0e-7
5. 1.5e-3
0 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 6. MEM result for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 4.5, 20 C). In Fig. 6f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
13-cis photocycle, this is missing in the MEM. In summary these two results from global analysis recover at least six or seven characteristic lifetimes and the above mentioned triplet is present in both around 650 nm. 4.3. Possible photocycle kinetics With the photocycle kinetics a problem arises: both the multiexponential analysis and the MEM give more (6, 7, 8) characteristic lifetimes (eigenvalues) than the number (5) of the known, spectrally distinct intermediate states. This problem may be resolved by several ways. One can assume that there are more than five intermediate states but some of them are spectrally very similar. The reversible sequential model [7] assumes six intermediate states with two M forms. A parallel, irreversible model was proposed in [17] with Lf, Mf and Ls, Ms in the two branches. In [16] this problem is solved quite differently. Only five intermediate states (K, L, M, N, O) are assumed in a linear photocycle but fast quasiequilibration between L M M, M M N, N M O produce
more (7) characteristic lifetimes in the multiexponential global fitting. It was shown in Section 3 that there is no direct way to determine the photocycle kinetics (e.g., the K matrix) from the result of the MEM analysis. To find a kinetics which would agree with the MEM, we have to carry out a time consuming procedure similar to that what was made in the case of the simulated data. A K matrix is assumed and the rate constants are determined by fitting to the measured absorption data, then solving the eigenvalue problem the discrete amplitudes on the lifetime scale can be calculated. This is the result what can be compared with the MEM result. The MEM serves partly as a filter, it can accept or reject an assumed kinetics. From MEM it is known that there are at least seven or eight intermediate states appearing during the photocycle. One can select only from the five known, spectrally distinct intermediate states: K, L, M, N, O, and assume that some of them appear more than once, with the same spectral property. (As e.g., in the reversible model, Eq. (1), the M state is doubled M1 and M2.) Tak-
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
amplitude [a.u.]
10 500 nm
(a)
10 410 nm
11
(b)
5
5 0
0
-5
-5
-10
-10
15
15 10
610 nm
(c)
570 nm
(d)
10
amplitude [a.u.]
5 0
5
-5 -10
0
-15
-5
-20 -25
-10
amplitude [a.u.]
15
80
650 nm
(e)
8. 8.1e-3
70
10
60
5
50
0
40
-5
30
5. 1.7e-4
2. 1.45e-6
4. 7.5e-5
20
-10
1. 6.8e-7
10 -15 -7
(f)
-6
-5
-4
-3
-2
log time [s]
-1
3. 7.5e-6
7. 3.1e-3
6. 8.6e-4
0 -7
-6
-5
-4
-3
-2
-1
log time [s]
Fig. 7. MEM result (solid lines) for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 7, 20 C), the characteristic lifetimes and discrete amplitudes (dotted lines with an x at the end) of the reversible kinetic model (Eq. (1)) and the characteristic lifetimes and discrete amplitudes (solid discrete lines) of 4M kinetic model (Eq. (15)). In Fig. 7f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
ing into account this selection and the possible different kinetic connections between the intermediate states we get a huge number of possibilities. Around a hundred different kinetic models were checked. In this procedure, the following selection rules were applied: A good fit to the measured absorption change. As a standard the rms deviations were compared to that of the reversible model (Eq. (1)). The kinetic model should be applicable in a wide range of pH and temperature and the rate constants should show an Arrhenius type behavior. An acceptable similarity to the MEM result for the eigenvalues and the amplitude distribution. We followed especially the characteristic triplet eigenvalues observed at the longer lifetimes (see, e.g. Fig. 7e). Similarity of concentrations with the result of the direct concentration determinations. Fig. 7 shows partly the MEM result with the discrete amplitudes of the reversible kinetic model (Eq. (1)) with
two M states. There is a rather good similarity between them, but at least one line is missing in the kinetic model (from the triplet, see Fig. 7 at 650 nm). Though many different kinetic models were tried (with 2O, 2N, 2L, etc. intermediate states), the result shown in Fig. 7 suggests that a generalized kinetic model based on the reversible model, Eq. (1), would give better results. Finally, from all the analyzed kinetic models, a branching model, Eq. (15), with four M states gave the most satisfactory result. In this model there is a branching at the L state. K $ L $ M 1 $ M 2 $ N $ O ! BR and L $ M 3 ! M 4 ! N
ð15Þ
Fig. 7 shows the discrete amplitudes of the 4M kinetic model, Eq. (15), too, at pH 7. This model reproduces the characteristic triplet (Fig. 7, 650 nm) but the agreement is not so good as with the simulated data. For example, one discrete amplitude of the kinetic model at 500 nm (Fig. 7) is not seen by the MEM.
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
12
10 410 nm
6
(a)
500 nm
(b)
610 nm
(d)
amplitude [a.u.]
4
5
2 0
0
-2
-5
-4
-10
-6
570 nm
8
(c)
amplitude [a.u.]
6
10
4 2
5
0 -2 0
-4 -6
-5
-8
5
amplitude [a.u.]
4
650 nm
(e)
2. 1.8e-6
30
5. 1.9e-4
(f)
25
3
6. 3.2e-3
20
2
8. 5.9e-2 7. 1.4e-2
15
1
10
0
4. 3.05e-5 1. 7.5e-7
5
3. 7.16e-6
-1 0
-2 -7
-6
-5
-4
-3
-2
-1
0
-7
-6
log time [s]
-5
-4
-3
-2
-1
0
log time [s]
Fig. 8. MEM result for the amplitude distributions by analyzing experimental absorption data measured at different wavelengths (pH 9, 20 C). The characteristic lifetimes and discrete amplitudes (dotted lines with an x at the end) of the reversible kinetic model (Eq. (1)). In Fig. 8f the sum of the absolute values of the amplitude distributions is presented (solid lines) with the estimated characteristic lifetimes (dotted lines).
410 nm
100
500 nm
absorption change [a.u.]
570 nm 610 nm
50
650 nm
0 -50 -100 -150 -200
-6
-5
-4 -3 log time [s]
-2
-1
Fig. 9. Experimental absorption change data measured at five different wavelength (symbols, pH 7, 20 C) and the MEM fit to the data (continuous line).
Apparent rate constants log k [1/s]
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
13
6 5 4 3 2 1 3.20
3.30 3.40 3.50 1/Temperature [1000/K]
3.60
Fig. 10. Comparison of the seven apparent rate constants (up triangles) taken from [15] (pH 7), the eight rate constants (circles) from [16] (pH 7.2) and the result of MEM (eight rate constants, filled squares, pH 7).
100
80
410 nm
500 nm
60
amplitude [a.u.]
40
50
20 0
0
-20 -40
-50
-60 -80
-100
100
570 nm
610 nm
100
amplitude [a.u.]
50 50
0 0
-50 -50
-100 -100
-7
-6
-5
-4
-3
-2
-1
0
-7
-6
amplitude [a.u.]
70 60 50 40 30 20 10 0 -10 -20 -7
-5
-4
-3
-2
-1
0
log time [s]
log time [s]
650 nm
-6
-5
-4
-3
-2
-1
0
log time [s] Fig. 11. The characteristic lifetimes and the discrete amplitudes for the Gauss fitted MEM result (solid lines) and for the 4M kinetic model (Eq. (16), dotted lines with an x at the end).
14
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
Concentration [a.u.]
250 200
200
150
150
100
100
50
50
0
0
250
Concentration [a.u.]
250
K
M
N
200
200
150 150
100 100
50
50 0
0
5
300
O Concentration [a.u.]
L
BR
4
250
3
200
2
150 100
1
50
0
0
-1
-7
-6
-5
-4
-3
-2
-1
-7
0
-6
-5
-4
-3
-2
-1
0
log time [s]
log time [s]
Fig. 12. Comparison of the concentrations for the intermediate states. Direct concentration calculations (x), concentration for the 4M model (solid line), concentration for the reversible kinetic model (square).
It was found that at pH 4.5 the L M M3 is a stand alone transition and the M4 intermediate state does not appear at all. This may be the reason that at low pH the MEM detects only seven peaks (Fig. 6). At high pH a 4M kinetic model with branching gave the best result, but the branching is at the K state (Eq. (16)). This branching is markedly preferred at high pH against the branching at the L state.
7
KL LK
ð16Þ
Fig. 11 shows the discrete amplitudes after the Gauss fits to the MEM result (Fig. 8) together with the characteristic lifetimes and discrete amplitudes of the 4M kinetic model, Eq. (16). As can be seen there is a high similarity between the kinetic model and the MEM result. The kinetic model of Eq. (16) reproduces all eigenvalues and amplitudes of the MEM analysis, though there are some shifts in the eigenvalues. For comparison Fig. 8 shows the eigenvalues and amplitudes of the reversible kinetic model (Eq. (1)) together with the MEM result. As can be seen some of the eigenvalues are missing in this kinetic model. There is some difference between the 2M and 4M kinetic models in the
Rate constants log k [1/s]
K $ L $ M 1 ! M 2 $ N ! O ! BR and K $ M 3 ! M 4 ! N
goodness of the fit to the experimental absorption data, too. The 4M model gives at least the same (or better) goodness of the fit as the reversible 2M model. A comparison between the concentrations determined from the different kinetic models and from the direct concentration calculation (Section 3) is shown in Fig. 12.
6
LM1 M1L M1M2
5
M2N NM2
4
NO OBr KM3
3
M3K M3M4 M4N
2
M4O
1 0
3.30
3.40 3.50 3.60 1/Temperature [1000/K]
3.70
Fig. 13. Arrhenius plot of the rate constants of the 4M kinetic model at pH 9.
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
As it was emphasized, from the possible kinetic models we accepted only that which gave approximately linear reciprocal temperature dependence in the Arrhenius plot of the rate constants. Such a plot is given in Fig. 13 for pH 9. It is interesting to note, that the branching at the N state (N ! BR transition) in the reversible model (Eq. (1)) was not supported by the rate constants fitting on the basis of the 4M kinetic models.
5. Conclusion In this and a previous paper [9] a MEM procedure was applied to analyze absorption kinetic data for the BR photocycle. Though this method, the MEM, is not a kinetic analysis in the sense that it does not give a specific kinetic model, but the information resulting from MEM is very useful for selecting a possible kinetic model and it is free from any assumption. The most important result following from the MEM analysis is that at least eight intermediate states are present in the BR photocycle. Many kinetic models with eight intermediate states were analyzed and compared with the MEM result. A branching kinetic model with four different M states gave the best agreement. (Perhaps, it is interesting to remark that in an earlier paper [18] it was reported that within a simplified and local analysis taking into account only two M states a branching model gave better result than a sequential model.) The proposed 4M kinetic model (Eqs. (15) and (16)) includes most of the properties of the reversible 2M model given by Eq. (1) [7] as one part of the branching. The branching is most pronounced at high pH. This may be the reason that the cooperativity in the BR photocycle is most pronounced at high pH, too [19]. A MEM analysis results in an amplitude distribution (peaks) on the lifetime scale instead of discrete lines as the case in a kinetic model analysis. The entropy, Eq. (14), is determined by the number of the peaks and by the width of the peaks. The broadening of the peaks will increase the entropy. There may be two sources of the broadening of the peaks and the increase of the entropy [9]: a noise generated entropy and what was called in [9], the inherent entropy. The inherent entropy may be connected with the protein dynamics and heterogeneity. In that case the kinetic processes are described by rate distributions instead of rate constants [20]. With simulated absorption kinetic data there is only noise generated broadening and the peaks are rather narrow (Figs. 4 and 5). For real absorption data the peaks are broader and there is an increase of entropy with increasing pH (Figs. 6–8), though the noise level is approximately the same. It was remarked earlier that the MEM fit to the experimental absorption change is much better than the discrete exponential fit (kinetic model analysis). This
15
is probably because the MEM takes into account the rate distribution. The molecular background for the 4M kinetic model is not clear. The branching occurs probably at the proton release side of the Schiff-base. The M3 ! M4 transition may be similarly connected to the proton switch as in the case of the M1 ! M2 transition, but it is pH dependent. At low pH there is no M3 ! M4 transition and the M4 intermediate state does not appear. To find a kinetics which would agree with the MEM result, it was assumed that eight intermediate states exist. But the MEM result does not exclude the existence of more intermediate states functionally important in the photocycle. If there are more such intermediate states the MEM may not detect them either because the corresponding peaks are too small or the corresponding eigenvalues are degenerate. In this work there was no attempt to analyze photocycles with more than eight intermediate states.
Acknowledgements We are grateful to Dr. G. Va´ro´ (Biophysics Institute of the Hungarian Academy of Sciences) to provide the experimental absorption data set and to allow to use the RATE program for kinetic analysis. We thank Dr. Z. Dancsha´zy (Biophysics Institute of the Hungarian Academy of Sciences) for providing us with purple membrane preparations and for valuable discussions.
References [1] W. Stoeckenius, R.H. Lozier, R.A. Bogomolni, Biochem. Biophys. Acta 505 (1979) 215–278. [2] W. Ku¨hlbrandt, Nature 406 (2000) 569–570. [3] S. Subramaniam, R. Henderson, Nature 406 (2000) 653–657. [4] C. Gergely, L. Zima´nyi, G. Va´ro´, J. Phys. Chem. B 101 (1997) 9390–9395. [5] J.K. Lanyi, G. Va´ro´, Isr. J. Chem. 35 (1995) 365–385. [6] J.F. Nagle, Biophys. J. 59 (1991) 476–487. [7] K. Ludmann, C. Gergely, G. Va´ro´, Biophys. J. 75 (1998) 3110– 3119. [8] J. Skilling, R.K. Bryan, Mon. Not. R. Astron. Soc. 211 (1984) 111–124. [9] Zs. Ablonczy, A. Luka´cs, E. Papp, Biophys. Chem. 104 (2003) 249–258. [10] C. Gergely, C. Ganea, G.I. Groma, G. Va´ro´, Biophys. J. 65 (1993) 2478–2483. [11] C. Gergely, C. Ganea, G. Va´ro´, Biophys. J. 67 (1994) 855–861. [12] G. Va´ro´, J.K. Lanyi, Biochemistry 34 (1995) 12161–12169. [13] Tyson J.J., The Belousov–Zhabotinskii reaction, in: Levin S. (Ed.), Lecture Notes in Biomathematics, vol. 10, Springer, Berlin , Heidelberg, New York, MA, 1976. [14] I. Szundi, J.W. Lewis, D.S. Kliger, Biophys. J. 73 (1997) 688– 702. [15] A.H. Xie, J.F. Nagle, R.H. Lozier, Biophys. J. 51 (1987) 627–635.
16
A. Luka´cs, E. Papp / Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1–16
[16] I. Chizov, D.S. Chernavskii, M. Engelhard, K.-H. Mueller, B.V. Zubov, B. Hess, Biophys. J. 71 (1996) 2329–2345. [17] R.W. Hendler, R.I. Shrager, S. Bose, J. Phys. Chem. B 105 (2001) 3319–3328. [18] E. Papp, V.H. Ha, Biophys. Chem. 57 (1996) 155–161.
[19] Zs. Ablonczy, V.H. Ha, E. Papp, Biophys. Chem. 71 (1998) 235– 243. [20] P.J. Steinbach, K. Chu, H. Frauenfelder, J.B. Johnson, D.C. Lamb, G.U. Nienhaus, T.B. Sauke, R.D. Young, Biophys. J. 61 (1992) 235–245.