The binding of quinone to the photosynthetic
reaction centers. Kinetics and thermodynamics
of reactions occurring at the QB-site in
zwitterionic and anionic liposomes.
Supplementary material
(A) Table S1. Start guess kinetic constants for PC and PG liposomes at 25°C (see Figure 1);

is obtained using KQWAM .
kout
Kinetic Constant
PC start guess
Δk/k %
PG start guess
Δk/k %

(s−1)
kout
154
3
44
3
kBA (s−1)
487
0.4
97
0.3
kout (s−1)
500
17
500
16
RMSD
1.49%
1.60%
R2
0.992
0.992
1
(B) Figure S1. Experimental decays (black lines) of D+ for PC (upper panel) and PG (lower
panel) at 25°C contrasted with the ODE set calculated data (grey lines)
2
In this figure the experimental traces at 25°C for PC and PG and the best fit curves obtained
from ODE set calculation are shown. Considering that all the traces at the different Q/RC ratios
are simulated simultaneously and that the optimization is performed with only two parameters,
the agreement with the experimental data is satisfactory.
3
*
(C) Figure S2. R2 as function of optimized kout
for PC and PG at the temperatures indicated
in the legend.
A set of independent optimizations were performed using twenty values between 1 and 100 s−1

as fixed values for kout
. For each run, new optimized values of k*in and kin are obtained. The
4


dependence of the R2 values on kout
is shown for PC and PG. By varying kout
in the above
mentioned range, the exchange regime is forced to change from slow to fast and the simulated
decays correspondingly show a different agreement with the experimental CRR traces.

The shape of the obtained curves is similar at all temperatures: R2 is lower at low kout
values,
then increases and reaches a plateau for higher values, indicating that the fast quinone exchange

is always a good mathematical description of the system. However, the value of kout
at which the

R2 value starts to decrease can be taken as the kout
lower limit.
Table S2. Estimated upper (having set kin at its higher value) and lower (from figure S2) limits

of kout
in PC and PG liposomes
t(°C)

(s–1)
kout
lower PC

(s–1)
kout
upper PC

(s–1)
kout
lower PG

(s–1)
kout
upper PG
6.5
20
35
3
3.5
10
20
33
3
4.5
15
20
53
5
10
20
20
80
8
15
25
20
146
8
36
30
20
238
8
72
35
20
450
8
150
5
(D) Correlation among kinetic parameters.
We studied the Hessian matrix of the sum of quadratic deviations between experimental data
and ODE set numerical solutions (χ2) as a function of the optimized physical parameters of our
model. This study provides different and useful kinds of information about the actual
dependency of the goodness of the fit upon the parameters and therefore about their relevancy to
the model. However, the parameter variations will most probably result coupled by the set of
equations describing the model and the coupling coefficients will often have a physical
significance and provide additional insight into the working of the model.
To this effect, the Hessian matrix is diagonalized. True minimization respect to the Hessian
variables is ensured by the emergency of all positive eigenvalues. The smallest eigenvalues are
those to be watched upon with more attention since they signify the possibility of greater
deviance from the minimum values within the same confidence interval when moving along the
associated eigenvector.
Since the value of parameters fixed along the optimization was the result of independent
experiments, their confidence should be separately assessed. However the impact of variation of
these parameters upon the results of optimization is of obvious interest as well. We therefore
evaluated how RMSD is affected by the above said variation.
Both the gradient with respect to the fixed parameters and the Hessian matrix with respect to
the optimized ones were obtained by means of finite difference using a displacement of 1% from
the reference value.
The following formulas were used for evaluating the gradient components:
 2 (kopt  k ) -  2 (kopt )
 2

kopt (S2.1)
 ln k
k
6
and the Hessian matrix:
a  a2  2a0 2
2  2
 1
kopt (S2.2)
2
2
 ln k
kopt
a  a00  a01  a10
2  2
 11
ka kb (S2.3)
 ln ka  ln kb
ka kb
where a0 = χ2(kopt), a1 = χ2(kopt + Δk), a2 = χ2(kopt - Δk) and a00 = χ2(ka, opt, kb, opt), a10 = χ2(ka, opt
+ Δka, kb, opt), a01 = χ2(ka, opt, kb, opt + Δkb), and a11 = χ2(ka, opt, + Δka, kb, opt + Δkb).
Since logarithmic derivatives are used, gradient and Hessian are describing χ2 change against
relative variation of the parameters.
The obtained matrix is diagonalized and the eigenvalues are extracted.
The Hessian matrix represents a parabolic approximation of the χ2 dependence on the deviation
of parameters from their optimal values. The eigenvalues D are associated to the convexity of the
χ2 paraboloid along the V column vector; a high D value indicates that χ2 grows faster along the
associated V direction and that the parameter V, linear combination of the original parameters, is
obtained with good accuracy, P(χ2)/√𝐷 being the actual probability distribution for deviation
along the direction V.
To ease the interpretation of the data we represent here only 2x2 sections of the complete
hessian matrix.
As already mentioned, k*in and k*out are expected to be strongly correlated since their ratio
gives K*Q. This is mathematically demonstrated by the eigenvalues (D) and the eigenvectors (V)
of the Hessian.
7

Table S3.1 Correlation table for kin and kout
V1
V2
kin
-0.714
-0.700

kout
-0.700
0.714
√𝐷
0.04
0.5
% err
12
1.0

Table S3.1 shows the hessian analysis results for kin and kout
and evidences that the two

parameters are strongly correlated. kin and kout
can move in the same direction (V1), leaving K Q
unaltered, without significant χ2 change; on the other side vector V2 representing the fastest
direction for K Q change has a large associated D value. This shows that K Q is a meaningful
parameter for the present set of experiments and that it can be obtained with a much better
precision (about 10 times) than the individual dividends.
On the other hand, the contributions of LAB and K Q can be considered well separated as

demonstrated by the relevant V and D, calculated on the basis of kout
and kBA, shown in Table
Sb.

Table S3.2 Correlation table for kout
and kBA.
V1
V2

kout
-0.977
0.213
kBA
0.213
0.977
√𝐷
0.25
1.1
8
1.8
% err
0.4
The V eigenvectors are approximately directed along the parameter directions and are in both
cases determined within a fair error. This finding is important since in previous works done in
detergent it was not possible to obtain reliable thermodynamic parameters for the long chain
quinone binding constant to the QB-site due to the temperature dependence of the micellar
properties.1

The other parameters analyzed for correlation are kout
and kout:

Table S3.3 Correlation table for kout
and kout.
V1
V2

kout
-0.184
-0.983
kout
-0.983
0.184
0.028
0.35
17
1.3
√𝐷
% err

and kout are only weakly correlated since V1 and V2 are directed along the main axes. As a
kout
consequence, the values of K Q and KQ are completely independent, K Q being well determined
and KQ being less relevant for the model.
1.
McComb, J. C.; Stein, R. R.; Wraight, C. A. Investigations on the influence of headgroup
substitution and isoprene side-chain length in the function of primary and secondary quinones of
bacterial reaction centers. Biochim Biophys Acta 1990, 1015 (1), 156-71.
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