23
Bioscience Reports ~, 23-31 (1984)
Printed in Great Britain
Two ( c o m p l e t e l y ) rate-limiting steps in one m e t a b o l i c p a t h w a y ?
The resolution of a paradox
using bacteriorhodopsin liposomes and the control theory
Hans V. WESTERHOFF* and 3os C. ARENTS
Laboratory of Biochemistry, B.C.P. Jansen Institute, University
of Amsterdam, P.O. Box 20151, 1000 HD Amsterdam, The Netherlands
(Received 1 November 1983)
Proton pumping by bacteriorhodopsin and chargecompensating ion movement can both and simultaneously
behave as the rate-limiting step in light-driven proton
uptake into bacteriorhodopsin
liposomes.
This
apparently excessive control exerted on the net proton
influx is possible because of the negative (-1) 'control
coefficient' of the net proton influx with respect to the
proton leaks.
Furthermore, the property of bacteriorhodopsin that it is inhibited by the membrane potential
is responsible for the transfer of part of the control on
the net proton influx from the first, irreversible, step
in the pathway (i.e. bacteriorhodopsin) to the second,
r e v e r s i b l e , step ( i . e . , c h a r g e - c o m p e n s a t i n g ion
movement).
It is a not uncommon view that metabolic pathways generally
consist of one 'pacemaker' enzyme (Krebs, 1957), one 'rate-limiting'
enzyme (Bi3cher & RiJssmann, 1963), one 'flux-generating' enzyme
(Newsholme, 19g0), or one 'key' enzyme (Weber, 1975), followed by
enzymes that do not control the flux through that pathway.
The
concept of the flux-control coefficient [the new systematic name for
'control strength' (Higgins, 1965; Heinrich & Rapoport, 1974) and
' s e n s i t i v i t y ' (Kacser & Burns, 1973); see the accompanying review
(Westerhoff et al., 19g#)] established an operational definition that
could replace the q u a l i t a t i v e term 'rate-limiting'.
In terms of
flux-control coefficients the abovementioned view can be rephrased as
follows:
metabolic pathways generally have one ('rate-limiting')
enzyme (usually at the beginning) with respect to which the fluxcontrol coefficient is equal to l, while the flux-control coefficients
with respect to the other enzymes are zero.
In this paper we demonstrate experimentally that this view is too
simplistic:
one pathway can contain two reactions that are simultaneously ' r a t e - l i m i t i n g ' .
Under representative conditions, proton
*'Present address: Laboratory of Molecular Biology, National
Institute of Arthritis, Diabetes, and Digestive and Kidney
Diseases, National Institutes of Health, Building 2, Room 319,
Bethesda, MD 20205, U.S.A.
01984
The Biochemical
Society
24
WESTERHOFF AND ARENTS
pumping by bacteriorhodopsin and charge-compensating ion movement
simultaneously behaved as the rate-limiting step in light-driven net
proton uptake into bacteriorhodopsin, i.e., the flux-control coefficients
with respect to the two processes almost equalled I.
Consequently, the simple picture of metabolic control outlined
above is inadequate for the discussion of control in metabolism.
We
show t h a t the thermodynamic v a r i a n t (Westerhoff et al., 1993;
Westerhoff & Van Dam, 1994) of the control theory developed by
Kacser and Burns (1973) and Heinrich and Rapoport (1974) can
account for the observations.
This and other recent quantitative
approaches to metabolic control are reviewed in the accompanying
paper (Westerhoff et al., 1984).
M a t e r i a l s and M e t h o d s
The experiments were carried out as described by Westerhoff et al.
(1981).
Proton uptake into bacteriorhodopsin liposomes suspended at
0.07 or 0.3 mg bacteriorhodopsin/ml was measured with a pH electrode.
One single preparation of liposomes was used containing 25
mg/ml egg phosphatidyl choline, 0.5 mg/ml stearyl amine, 2.5 mg/ml
b a c t e r i o r h o d o p s i n in 0.25 M K~SO~, 1.0 mM (K)EDTA, pH 5.8,
sonicated t~0 x f5 s. The suspension at 0.07 mg bacteriorhodopsin/ml
was illuminated with a slide projector:
' i 0 0 % ' = 0.65 kW/m2; the
s u s p en s io n at 0.3 mg bacteriorhodopsin/ml was illuminated with a
xenon lamp: '25%' = 0.17 kW/m2. Off and on switching was done by
(dis)connecting the lamp's power supply in the former case and with a
shutter in the l a t t e r . Valinomycin was a gift from Dr. V. C. Pettinga,
Eli Lilly & Co.; chlorpromazine was purchased from Serva, Heidelberg.
The light intensity was varied with neutral density filters.
R e s u l t s and D i s c u s s i o n
Two completely 'rate-limiting' steps
When the light-driven proton pump bacteriorhodopsin is incorporated
into small unilamellar liposomes (Oesterhelt and Stoeckenius, 1973;
Hellingwerf et al., 197g, 1979; Westerhoff et al., 19gl; cf. Fig. 1),
i l l u m i n a t i o n c a u s e s alkalinization of the suspending medium.
We
focussed on the initial net proton flux (3 H) into the liposomes, i.e.
between 3 and 20 s a f t e r switching on the light (cf. Hellingwerf et
al., 1979). Under these conditions, the pH, the log [ C l - ] , and the log
[K +] g r a d i e n t s a r e ( s t i l l ) c l o s e to zero, whereas the el ect ri cal
potential d i f f e r e n c e across the membrane (A0) has reached a significant magnitude and is slowly decreasing (Hellingwerf et al., 1979).
The rate at which Aq~ decreases is negligible compared to the rat e at
which it increased just a f t e r the light was switched on; net proton
movement is almost completely electrically compensated for by the
movement of other ions.
Hence the system's condition is close to
t h a t of a ' s t a t i c - h e a d ' steady state in which the transmernbranec h e m i c a l - p o t e n t i a l differences of all ions (including H +) would be
fixed and A~ would have adjusted until net e l e c t r i c flow had become
zero.
In t h e Iollowing we shall t r e a t our system as if it were
completely satisfying these s t e a d y - s t a t e conditions. The system can be
considered as a simple model for a f r e e - e n e r g y transduction pathway:
FLUX CONTROL IN BACTERIORHODOPSIN LIPOSOMES
4
25
K
H*teak
illumin~ation ~
val'in~
chlor pr omazin
m
Fig. I.
Bacteriorhodopsin liposome as free-energy
transducer.
free energy in the form of the thermodynamic potential of absorbed
photons is converted to free energy in the form of an electrical
potential across the membrane.
The degree to which an enzyme E k (or a reaction) controls a flux
has been defined as the flux-control coefficient of the flux (3) with
respect to that enzyme:
Cj def (dlnIJ[/dln[Ek])ss
=
(dJ/J)/(d[Ek]/[Ek]). (1)
In the s y s t e m we c o n s i d e r h e r e , k may stand for v (bacteriorhodopsin), i (ion leaks, i.e. charge-compensating ion movement), or ~,
(proton leak).
A flux-control coefficient with respect to an enzyme
equals l if the f r a c t i o n a l change in the flux (d3/3) equals the
f r a c t i o n a l c h a n g e in the a c t i v i t y of the enzyme (dE/E).
The
f l u x - c o n t r o l c o e f f i c i e n t is mathematically equivalent to the In/In
derivative of the flux with respect to the activity of the enzyme.
Consequently, the slope in double logarithmic plots of the dependence
of the net proton influx on the activity of one of the reactions in the
s y s t e m equals the f l u x - c o n t r o l c o e f f i c i e n t with respect to that
reaction.
We v a r i e d c h a r g e - c o m p e n s a t i n g ion movement through
titration with valinomycin, proton leakage by adding the protonophorous
uncoupler chlorpromazine, and the activity of bacteriorhodopsin by
varying the light intensity (at the illumination intensities employed,
the c a t a l y t i c a c t i v i t y of bacteriorhodopsin has been shown to be
proportional to the light intensity [Hellingwerf et al., 1979]).
26
WESTERHOFF AND ARENTS
Fig. 2 shows the e f f e c t of valinomycin on the illumination-induced
initial proton influx into bacteriorhodopsin liposomes. Because we are
i n t e r e s t e d in the c o n t r o l by the t o t a l c h a r g e - c o m p e n s a t i n g ion
m o v e m e n t ( i . e . , the t o t a l leakage of K+ and Cl-), the abscissa
indicates the concentration of added valinomycin plus the concentration
of v a l i n o m y c i n t h a t was e q u i v a l e n t to the e n d o g e n o u s chargecompensating ion movement (in this case 0.04 pg/mg phospholipid).
The l a t t e r c o n c e n t r a t i o n was d e t e r m i n e d as the e x t r a p o l a t e d
abscissa-intersection point of the bilinear plot of initial proton influx
versus the concentration of added valinomycin (Westerhoff et al.,
1981; Groen et al., 1982; Westerhoff & Van Dam, 1984).
The
potential error in this determination will only significantly a f f e c t the
part of Fig. 2 that represents the low valinomycin concentrations.
At v a l i n o m y c i n c o n c e n t r a t i o n s b e t w e e n 0.05 and 0.3 pg/mg
phospholipid, the angles between the abscissa and the tangents to the
line drawn in Fig. 2 were approx. 45 ~ i.e., the flux-control coefficient
with r e s p e c t to the charge-compensating ion movement (C~iH was
a l m o s t 1 (tg 45~
At higher concentrations of valinomycin the
flux-control coefficient with respect to the charge-compensating ion
movement decreased to approximately zero: with 'excess' valinomycin
the charge-compensating ion movement no longer controls ('limits')
the flux.
Comparison of the flux-control coefficients with respect to the
d i f f e r e n t enzymes in one system is meaningful only if those control
coefficients have been determined under identical conditions.
Here,
A
C
E
= 0.5
tY
~0.2
x
:O 01
~r - .05
0105
i
01
012
i
0s
i
I
2
0.04 +[valinom~in] (/Jg/mgPL)
Fig. 2. The effect of increased ion permeability on
the initial proton influx.
Light intensity was 25%
(cf. Westerhoff et al., 1981).
FLUX CONTROL IN BACTERIORHODOPSIN LIPOSOMES
27
05
r-
E
O
%
0.2
0
E
3
x- 1
0.1
tO
0.0s
o-~:-:~77J 12_" _ _ _
10
'
2'0
'
50
[ighf infensify (~
Fig. 3.
The effect of light intensity on the
initial proton influx.
@ , no valinomycin;
9 0.20
Ng valinomycin/mg phospholipid.
we shall f o c u s on t h e ' s t a n d a r d ' c o n d i t i o n of 25% illumination
intensity, no added protonophore, and 0.20 pg/mg phospholipid of added
valinomycin (i.e. the point for which the angle was drawn in Fig. 2).
From Fig. 2 we c o n c l u d e t h a t under t h e s e c o n d i t i q n s chargecompensating ion movement is completely 'rate-limiting' (C~iH is almost
1). In Fig. 3 we determined the flux-control c o e f f i c i e n t with respect
to bacteriorhodopsin itself (C~H). Under the standard conditions (open
circles, 25% light intensity) the proton pump is almost com pel et el y
' r a t e - l i m i t i n g ' (C~H is about 0.9 = tg #3~
Here we arrive at the central paradox: simultaneously two steps in
our metabolic system seem to be the ' r a t e - l i m i t i n g ' ('pace-maker')
step.
This p a r a d o x can also be f o r m u l a t e d
in t e r m s of the
flux-control coefficients:
the sum of the flux-control c o e f f i c i e n t s of
bacteriorhodopsin and the ion leaks is 1.9 and this exceeds i, the
magnitude of the sum of all flux-control c o e f f i c i e n t s according to the
summation theorem (Kacser & Burns, 1973; Heinrich & Rapoport, 197#;
for a review see Westerhoff et al., 19g#).
We also determined the flux-control c o e f f i c i e n t with respect to the
proton-leakage reaction (Fig. #, open circles) by t i t rat i on with the
uncoupler chlorpromazine (cf. Westerhoff et al., 19gl):
tg(-#9 ~ =
-l.1. The f a c t that this value is negative simply r e f l e c t s the decrease
in the net proton influx upon increasing the c a t a l y t i c activity of the
proton back-leak. However, the very finding of a substantial negative
flux-control c o e f f i c i e n t in the system resolves the paradox (cf. Kacser
& Burns, 1981; Flint et al., 198l):
the sum of all flux-control
2g
WESTERHOFF AND ARENTS
rQ-&-
e-
so
\
-6
E
g_
.~_
r-o
h
k
=
5
io
sb
s6o
72 + [ch[orpromazin](s
Fig. 4.
The effect of increasing proton permeability on the intial proton influx.
0.20 pg
valinomycin/mg
phospholipid.
O , 25% light
intensity; 9
7% light intensity.
The abscissa
gives the concentration of added chlorpromazine plus
the concentration of the protonophore that is
equivalent to the endogenous proton leak.
coefficients (0.g) approximately equals the value of i predicted by the
summation theorem (Kacser & Burns, 1973; Heinrich & Rapoport,
1974).
Apparently, two steps in a system can both be completely
flux-limiting (have flux-control coefficients close to 1), provided that
there is also a third reaction with the property that the flux through
the system is increased when that reaction is inhibited.
Not all control
in the irreversible step
In view of the alleged irreversibility of photochemical reactions
(Keizer, 1976; Hill, 1977; see also Westerhoff and Van Dam, 1994),
one might expect all control in the system to reside in bacteriorhodopsin.
However, under the standard experimental conditions, the
control c o e f f i c i e n t of net p~,oton influx with respect to chargecompensating ion movement (CiH) did not conform to this expectation:
it was closer to ] than to 0 (see above).
Also, the control
coefficient with respect to bacteriorhodopsin can be much less than 1
(e.g. tg(12 ~ = 0.2 in Fig. 3, closed circles).
The enzymic property
that determines the extent to which the individual enzymes control the
flux through a metabolic pathway (defined exactly as the elasticity
coefficient of the enzyme, see Westerhoff et al., 19g#)~ is its relative
'readiness' to change its rate following changes in concentrations of
the intermediary metabolites or free-energy differences in the system.
For the absolute e l a s t i c i t y c o e f f i c i e n t of bacteriorhodopsin with
FLUX CONTROL IN BACTERIORHODOPSIN LIPOSOMES
29
U
respect to the membrane potential A0, (acA~p), the definition would
work out as follows:
(a~.)l~!~ = RT/F. 3J~t/SAlb
(2)
This elasticity c o e f f i c i e n t would equal the slope of the line relating
th e pumping r a t e (;7~t) to the transmembrane el ect ri cal potential
difference.
For t h e two other processes in the system, chargec o m p e n s a t i n g ion m o v e m e n t and p r o t o n b a c k - l e a k a g e , a b s o l u t e
elasticity coef f i ci ent s are defined in an analogous fashion by replacing
by i or ~, respectively. The summation and connectivity theorems
of the control theory (Kacser & Burns, 1979; Heinrich & Rapoport,
197~; for a review see Westerhoff et al., 1984) allow us to express
the flux-control c o e f f i c i e n t with respect to bacteriorhodopsin CasH in
the absolute elasticity coefficients:
C~Ha
=
(J~IJH) I (I + {(as)Aol(a~)A~}
x)
i
+ {-(ae)A~} I (am)A~)
(3)
In this equation only the parameters that are usually negative have
been m u l t i p l i e d by - i and t h e n placed between brackets.
This
equation demonstrates that the control c o e f f i c i e n t of the initial net
proton influx with respect to bacteriorhodopsin depends on the ratio of
the flux of the protons pumped (J~t) and the net proton influx ( J H ) ,
as well as on ratios of the di f f e r ent absolute elasticity coeffi ci ent s in
the system.
Apparently, one reason for the flux-control c o e f f i c i e n t
with respect to bacteriorhodopsin being lower than 1 (notwithstanding
the irreversibility of the reaction it catalyses) could be that there is
a significant proton back-leakage in the system .that would be strongly
a f f e c t e d by changes in A~h (i.e. {am}~tb >> {ar
A second reason
could be that the absolute e l a s t i c i t y c o e f f i c i e n t of bacteriorhodopsin
with respect to A0 would be negative: l i abi l i t y. of bacteriorhodopsin to
be i n h i b i t e d by A0 ( i' . e . -{aE}~.~..>> {ae}Atp,
1
the so-called backpressure e f f e c t ; for a review see Westerhoff and Dancshazy, 1994),
should give rise to a decrease in the control c o e f f i c i e n t of the net
proton influx with respect to bacteriorhodopsin.
The l a t t e r p r o p o s a l , viz. t h a t t h e fact that the flux-control
co ef f icien t with respect to bacteriorhodopsin is often lower than 1,
can be explained by the back-pressure e f f e c t , suggests the following
p r e d i c t i o n s which can be subjected to experimental test: (i) The
flux-control c o e f f i c i e n t with respect to the charge-compensating ion
movement should increase with increasing membrane potential (hence
with i n c r e a s i n g light i n t e n s i t y ) at t he cost of the flux-control
c o e f f i c i e n t with r e s p e c t to b a c t e r i o r h o d o p s i n , because the backpressure e f f e c t would be expect ed to become more relevant at higher
membrane potentials; and (it) the flux-control c o e f f i c i e n t with respect
to bacteriorhodopsin should increase with the concent rat i on of added
valinomycin, which would lower A0 and thus take away the basis for
the back-pressure e f f e c t .
Fig. 5 shows that both predictions are confirmed by the corresponding experimental test.
From the differences between 15 and
100% light intensity in Fig. JA and between no valinomycin and excess
valinomycin in Fig. 5B, it may be estimated that the back-pressure
30
WESTERHOFF AND ARENTS
1
=,
i
-
I
i
i
I
I
!
B
cO
"
C
0
"6
cO
~
0
I
0
I
I
I
50
Light intensity (%)
~|
100
-0
I
i
05
1.0
vaUno'nycin(/.g/mgPL)
Fig. 5.
The effect (A) of light intensity on the
flux-control
coefficient with respect to the
charge-compensating
ion movement and (B) of
increased ion permeability on the flux-control
coefficient with respect to bacteriorhodopsin.
From
experiments like those in Fig. 2, at several light
intensities.
e f f e c t is strong enough to be responsible for a d e c r e a s e of at l e a s t
0.6 in the f l u x - c o n t r o l c o e f f i c i e n t with r e s p e c t to b a c t e r i o r h o d o p s i n
and an i n c r e a s e of 0.3 in the f l u x - c o n t r o l c o e f f i c i e n t with r e s p e c t to
the c h a r g e - c o m p e n s a t i n g ion m o v e m e n t .
Acknowledgements
We t h a n k B e r t Groen, K a r e l van D a m , K l a a s
Wanders, and Rob van der Bend for discussions.
Hellingwerf,
Ron
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IN B A C T E R I O R H O D O P S I N LIPOSOMES
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