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Diffusion kinetics in phospholipid vesicles
Volume
37, number
DIFFUSION
CHEMICAL
3
1 February
LETTERS
1976
KINETICS IN PHOSPHOLIPLD VESICLES
M. TOMKIEWICZ
and G.A. CORKER
IBM Tllotnas J. Warsoti Rcscarch Cenrer, Yorkrowtr Received
PllYSICS
7 August
Heights, New York IO598, USA
1975
When a chemical reaction takes place in il heterogeneous environment, one can foresee circumstances in which diffusion of one of the participants will have pronounced effects on the observed kinetics of the reaction. This paper presents equations which describe the time dependence of the reactants for the specific case where a chemical reaction takes place on the surface of phospholipid vesicles.The model is implemented on experimental results which consist of the kinetics of the chlorophyll radical which was produced by photoexcitntion of chlorophyll. which w;1s dissolved in the phospholipid vesiclcs and which, as a result of the excitition. rrsnsferred its electron to an acceptor which was located outside the vesicle.
I. Introduction Recent experimental results from experiments involving the photochemical production of the chlorophyll cation radicn! in cl~lorophyll-containing phospholipid vesicles su,,Ooest that, under certain experimental conditions, diffusion of the electron acceptor in the aqueous phase in which the vesicles are suspended, intluences the time behavior of the chlorophyll radical. We present here a theoretical treatment which was developed to explain the time behavior of the cation radical but is general enough to be applicable to other heterogeneous systems in which chemical reaction kinetics are influenced by the diffusion of one of the chemical species involved. Possible systems in which this situation might arise are water suspensions of vesicles, rnicelles and artiticial or biologicti membranes. We will limit our discussions to sonicated phospholipid vesicles but the arguments could easily be estended to include other forms of aggregation. We can imagine a homogeneous suspension of phospholipid vesicles with unif0r.m external radius: r. and a generalized chemical reaction: A+B-+P.
(11
If we postulate that A is soluble only in the phospholipid and B only in the water, then the reactiofis will occur only on the surface of the vesicles. Assume further that B is present only on the exterior of the vesi-
cles, not the interior, and that A is homogeneously distributed in the phospholipid. This, combined with the spherical shape of the vesicles, will make the problem spherically symmetric. In addition, only the concentration of B will be both time and zpace dependent. Althougjl electrostatic interactions and hydrophobi=: forces might have an influence in a real system, we will also assume that the dominant; force effecting the diffusion of B will be concentration gradients; that is, the diffusion obeys Fick’s laws [1] with a constant diffusion coei’ficient. Our purpose is to develop an equation which describes the time (and space) dependence of the concentration of B in terms of experimentally measurabie quantities. In the next secrion we develop such an equation and in the last section we present an example of the use of this equation as applied to some experimental data from our work with chlorophyll.
2. 6eneral
formulation
As mentioned above, the diffusion of B will obey the simp!e diffusion law which in spherical coordinates has the following forrn [I] :
at/at= D[a2clar2f (2/i-)ac/af], with the following
boundary
(2)
conditions: 537
Vclumc c
37, number 3
CHENICAL
=C(r,r),
GvO’
t>cJ;
=X1(r),
f =ro,
tao;
=fbh
r>,rO,
r=o,
PHYSICS
[-(f-r')2j4Dr]
>o +-,,1/2 J
Defined this way, y is the amount of I3 per unit volume of 4~r/3 (neglecting the volume occupied by the vesicle) in the system containing a single vesicle in an infmite medium. We require the following relationship in order to relate]’ to the macroscopic concentration of B in a real system, one containing more than one vesicle. = Co + [3u/4n(l
where
r.
- exp [-(r+r’-2r0)‘/4Dr]}
'0
PI
J 2f(TAf)‘i2 f'f(f'){exp
1976
(9) (3)
where D is the diffusion coefficient of B in the aqueous phase, ro is the external radius of the vesicle, C is the concentration of B, x,(tj is the concentration of B at the surface of the vesicles and j(r) describes the distribution of B at r =O. Assuming a single vesicle in an infinite medium we can solve eq. (2) to obtain [2]: C=
1 February
LETTERS
-x3)r;](jyln-
[B] = concentration
Cc),
of B, given in units
(10) of
moles/ml, and Co = lim,%,,e C(r, t). We can substitute tain:
dr’
C from eq. (4) into eq. (9) to ob-
X, [I- (r-r,)2/4DP21
(r-r&21ur)1’2
X exp(--p’)dp.
(4)
Our purpose is to develop equations which are aplicable to experimental data. However, C as defined by eq. (4) is not a macroscopically measurable quantity. In addition, the single vesicle assumption puts
an upper limit on t for which (4) remains valid. In a real syStem containing a large number of vesicles, if diffusion distances become as large or larger than the average distance between vesicles, then eq. (4) is not valid. In such a system, the average distance between vesicles is given by: rd = 2ro [( 1 - X3)/V] y3 )
(5)
where u is the partial volume of the phospholipid and x is the ratic between the internal and external vesicie diameters. If we take (Dt)lI? as an approximate distance that a molecule wilI travel, then es_. (4) is only valid for:
t=C@D)[(l -x’)/uly3,
d&,
(11)
Oi Y = q
+ 5,
where rI is dependent upon the initial spatial distribution,f(r), while lY2 is not. By changing variables and integrating by parts, I’2 can be evaluated to give
+Q
(6)
or t
X exp (-I’)
‘Xl(r-S)(il-li’erC7_7e-~~)d;l, s G
(12)
where -&/“]~3/,,
where a, a more useful parameter, G = TO/,;.
(7) is defined
as;
7 = [(I--r3)/2ro]
(i+7E)“2
2nd (8)
In order to relate C of eo,. (4) to a macroscopic concentration, we must define Some new functions. First we dcime a function y such tllat
erfy = 2i7-‘12
= (I&)
(sr/a#?
Y s
exp (-x2) dx .
0
However, eq. (5) defines an upper limit for at which means that for f. Q 1, -y> 1 and under these circum-
Volume 37, member 3
1 Fcbruxry 1976
CHEMICAL PHYStCS LETTERS
stances eq. (12) can be approximated
by
f b(1 +ar/b2)-“2-
arctg](af)Y2/b]l
The evaluation of PI requires an explicit knowledge of the initial distributicn function, f(;,. We will ana-
lyze two specific initial distributions. (a) Homogeneous For this case,
It is not possible to extend this treatment
initial distribution
f(r) = F= constant.
114)
For such a situation, with increasing time, the regions near the surfaces of the vesicles Ml! be depleted of B relative to the bulk. It simplifies the problem if, instead of treating the diffusion of B toward the vesicles, we consider the diffusion of “the lack of 5” away from the vesicle surfaces and that at time equal zero, the concentration of these “lack of B’s” or “vacancies” is also zero. For this case we obtain rl as:
(b) Initial gaussian distribution For this case f(r)=Fexp[-11.6[(r-r0)/dJ2},
(16)
where d is the width of the distribution (the distance from the center to the maximum slope). Extending the integration in cq. (11) to infinity, we obtain: l-j = abr;F{$ -t b(1
f b +b2
+or/b*)-u2 - arctg[(nr)1/2/b] ),
t 1 + [u(r-$)/sr]V 0
3. Example To illustrate the use of the theory, we treat some in which chlorophyll a was dissolved ti phospholipid vesicles, K3 Fe(CN)h was addad after sonication as an electron acceptor, and the sample was illuminated with red light. & a result of the illumination, an electron was transferred from the chlorophyll to the acceptor. The chlorophyl radical thl;s formed was detected and its time behavior followed by using EPR. Preliminary results concerning this work were presented [3] and will be pub lished in greater detail elsewhere. The results are tl,eated in terms of the following mechansim:
results from an experiment
Chl
---.%
chi*
-
Chl” + ACCM+ +A$
Using r2, eqs. (10) and (11) and the NO fomls of I’l [eqs. (15) and (17)], we can obtain the dependence of the concentration of B on time for the two initial distributions: For the homogeneous initial distribution
-s
next section we show a specific example that demonstrates the use of eq. (18) or (19)
[
F[2P
tt(o/n)“2]
XIG)dB]
Q-#J2
and for the gaussian initial distribution
CM*,
(20a
ml,
(20b
Chl+ +A;,
(XC
CM f A,.
(7,Od
kl
kz
(17)
where b = d/3.8r0.
9uau2 [B] =F-------4a2( 1 -x3)
without
knowing the explicit form ofX1(k). However, in the
(18)
The acceptor, in this case KJ Fe(CN), , was soluble only in the aqueous phase, while the chlorophyll was soluble only in the membrane. They interacted.on tht surface of the vesicles. The acceptor was present in large excess, so its concentration is assumed not to change with time. The’generalized chemical reaction that was presenl in the introduction
is equivalent
to the back reaction
between the chlorophyll rrdica! and the reduced acceptor [eq. (20d)], The ia@ of change of the local concentration of the chlor&phyIl’radical is given by: 53s
‘dAdla _
d[Ci-d+l_ ---
dt
k, fk2 [A,]
(t)
1
3
’
= [AJ.
4x2(1-&
(21)
+ b(l+af/b2)-“’ + u1/2
(k3u)“2
exp[2t(Luk3)‘12] -
[Chl+],,,
exp [2t(Lvk,)1’z
j +1
= [Chl+],,,l(l+~,ur[Ci~+J~~~)~
(23)
(24)
+ k2 [A,].
For the case where eq. (6) is fulkIled, namely at times which are shorter than the one specified there, X,(t) will not be equal to [A,]. Thus, eqs. (23) and (24) are not valid. For such a case we are obliged to find the correct relation between X,(t) and [A;]. The equations which were deve!oped in the previous section provide such relations. For this particular case, a better agreement of the calculated behavior with the experimentally observed one could be achieved through the use of eq. (19), than with (18). With this equation, when I, f 0, F= 0. d [Chl+]/dr
the following
= L -. k, [Chlf] X,(I),
=
1,
(26b)
relation
(25a)
b(&+%+P).
9F 27?(1-X3)
(27)
.1/Z
J/7
9
=
2[2]
-;
4iG( l-x3)
where L = k, [A,] IJk,
X,G)dE
(t-t)“’
}
Eqs. (25) and (26) have to be solved numerically. We found the midpoint product integration method [4] to be a satisfactory way to proceed. However, the calculated results were not sensitive to the individual parameters, but rather to the ratios of these parameters. Thus, we defined the following parameters:
- 1
and for Ia = 0: [%I+]
1+ [,(r--$)/*p
J 0 where F obeys
Z[l] &_!!?
f
-- arctg[(ar)1/2/b]
(22)
Two extreme cases will be considered. In the first one the rate of diffusion is faster than the rate of the reaction. In this case a very short time after the onset of the reaction, eq. (6) will not be fulfilled and the reduced acceptor will attain a nearly uniform distribution. In that event, X,(r) will equal [A;] and eq. (21) will reduce to an equation which describes a simple second order reaction with the following solutions. For ia e 0:
[CM’]
1976
9
[Chl+] = [chl+lx
k
where X,(C) is the concentration of the reduced acceptor at the surface of the vesicle. If the set of reactions given in (20) fully describes the system, then charge conservation requires that u[Chlf]
1 February
CHEMICAL PHYSICS LETTERS
Volume 37, number 3
=L.;
Z[3]
=7(28)
k,
We have found also that under conditions where the time approaches the upper limit allowed by the theory, neglecting [o(r-.$)/n] ‘I3 in comparison with 1, will produce at most a 20% vkation in the parameters. This last approximation was found to be very convenient, because, without it, absolute values of 0 are required. Using this approximation and eqs. (28), eqs. (25) and (26) reduce to the following forms. 1n f 0:
[all+] = Z[l] Ia
J 0
‘Z[2]
-d[Chl+]/dt
do.
(29)
[Chl+](t-E)“”
=o:
[Chl+]
= 2Z[1]
CZ[2] - (d[Chl+]idr),,lO_$
x ~n+b+b2+b{1+~(Z[3])2}-“2Z[3]
arctg(Z[3]tii3)
Pq~,off
d W+l/dE --z[ll j 0 [al+]&$)Y’
dE .
(30)
b was found from eqs. (27) and (25a) to obey the following equation: and for I,=O,F+O,L
d [Chl+]/dr = -k3 [Chl+& 540
._I _.:.--
[c~+l’2=roff= =[I1 IZ[21 -@[Chlil/d~),,,o~I
=O we obtain
(r),
(26a)
x ($n+26+b2)/Z[3].
(31)
Volume 37, number 3
CHEXI!CAL PHYSICS LETTERS
TIME isecl FIN. 1. Change in the EPR amplirude of the chlorophyll
cation radical, during and after red light illumination at - 20°C. 1o-2 hl K3Fe(CN)6 was used as the electron acceptor. + = experimental results; q = least mean squares fit of eqs. (29) and (30) using the following parameters: Z[ I ] = 4,Z[ 21 = 0.7, Z[3] q 0.3; o = second order kinelics 3s calculated by using cqs. (23) and (24) with k3 = 2.4 X 10m3 and f_ = 0.14 which were calculated from the least mean squares fit of the decay portion of the data given in fig. 2 and rhe amplitude of the plateau attained during the illumination. All the parameters arc given in arbitrary units.
The maximum time resolution in our measurement was used as a constant grid spacing in the integration routine, this gave an accuracy of around 1% in the calculations. A typical experiment consists of turning the light on, following the radical during the illumination period and then, at time equal to toff, turning the light off and monitoring the radical decay. Formulas (29) and (30) were calculated in such a way that the 3 parameters were found by a least mean squares fit of the experimental and theoretical curves. The Fletcher and Powell [5] minimization routine was used for that purpose. For the reason that no special assumptions were made in the theory concerning the spatial distribution of the reduced acceptor during the illumination, we chose to fit Z[ l] and 2[2] with the portion of the curve obtained when the light was on and then held them constaRt and fit the decay by varying Z[3]. Fig. 1 presents an example of such a fit of the theoretical (open squares) to experimental data (the i’s) obtained with 1 min of illumination. When the illumination period was extended from 1 min to 4 min, the data (the +‘s) shown in fig. 2,were obtained. Also included in fig . 2 are points (open squares) calculated using eqs. (29) and (30) with the parameters found for the fit presented in fig. 1. As can be seen, these points deviate quite early from the experimen-
I February 1976
Fig. 2. The same as fig. 1 but with a longer period of illumination.
tal points. The principal cause of this difference is that the reduced acceptor approaches a homogeneous distribution and the basic assumption of a single vesicle in an infinite medium is not valid. Eq. (6) is not satisfied at the longer times. Thus, eqs. (29) and (30), which were used to calculate the theoretical points, arc not valid at the longer times. The signal in fig. 2 reaches a plateau value after about 3 min of illumination. Furthermore, the decay does not obey eq. (30) but instead obeys eq. (24) which is valid for homogeneous second order kinetics. From the plateau value and from the decay, we could find Fcj and L, assuming that the reaction was second order at shorter times as well. We used these values of X-, and L and plotted eqs. (23) and (34) (open circles) in figs. 1 and 2. In both figures the diffusion model does not fit the experimental data at times greater then 75 s. When the illumination was stopped after this time (fig. 2), we observe a smooth transition to second order kinetics. We do not observe such a transition when the iliumination was terminated at shortei time (fig. 1j. In a future publication, in which a more detailed account of the chlorophyll-phospholipid work will be presented, we will show that as we lower the temperature the time evolution of the system follows the diffusion kinetics for longer times.
Acknowledgement We would like to thank Drs. Jane K. Cullum and Ralph A. Willoughby for suggesting ref. [4] as suitable for the numerical
solution
of eqs. (29) and (30).
541
Volume 37, number 3
CHEMICAL PHYSICS LETTERS
References
[3] hi.
1 February
Tomkiewicz
and G.A.
Third International
[ 11 W. lost, Diffusion in solids, liquids, gases (Academic Press, New York, 1952) ch. 1. [ 21 H.S. Carslaw and J.C. jaeger, Cocductkn (Clarendon Press, Oxford, 1959) p. 247.
of heat in solid
Corker,
in: Proceedings
Congress on Photosynthesis,
ed. hl. Avron (Elsevier, Amsterdam,
1976 of the
Vol. 1,
1975) p. 265; Photo-
chcm. F’hotobiol. (197.5), to be published. [3] R. Weiss and R.S. Anderson, Numer. Math. 18 (1972)
[5] R. Fietcher
and M.J.D. Powell, Computer
442.
J. 6 (1963) 163.
37, number
DIFFUSION
CHEMICAL
3
1 February
LETTERS
1976
KINETICS IN PHOSPHOLIPLD VESICLES
M. TOMKIEWICZ
and G.A. CORKER
IBM Tllotnas J. Warsoti Rcscarch Cenrer, Yorkrowtr Received
PllYSICS
7 August
Heights, New York IO598, USA
1975
When a chemical reaction takes place in il heterogeneous environment, one can foresee circumstances in which diffusion of one of the participants will have pronounced effects on the observed kinetics of the reaction. This paper presents equations which describe the time dependence of the reactants for the specific case where a chemical reaction takes place on the surface of phospholipid vesicles.The model is implemented on experimental results which consist of the kinetics of the chlorophyll radical which was produced by photoexcitntion of chlorophyll. which w;1s dissolved in the phospholipid vesiclcs and which, as a result of the excitition. rrsnsferred its electron to an acceptor which was located outside the vesicle.
I. Introduction Recent experimental results from experiments involving the photochemical production of the chlorophyll cation radicn! in cl~lorophyll-containing phospholipid vesicles su,,Ooest that, under certain experimental conditions, diffusion of the electron acceptor in the aqueous phase in which the vesicles are suspended, intluences the time behavior of the chlorophyll radical. We present here a theoretical treatment which was developed to explain the time behavior of the cation radical but is general enough to be applicable to other heterogeneous systems in which chemical reaction kinetics are influenced by the diffusion of one of the chemical species involved. Possible systems in which this situation might arise are water suspensions of vesicles, rnicelles and artiticial or biologicti membranes. We will limit our discussions to sonicated phospholipid vesicles but the arguments could easily be estended to include other forms of aggregation. We can imagine a homogeneous suspension of phospholipid vesicles with unif0r.m external radius: r. and a generalized chemical reaction: A+B-+P.
(11
If we postulate that A is soluble only in the phospholipid and B only in the water, then the reactiofis will occur only on the surface of the vesicles. Assume further that B is present only on the exterior of the vesi-
cles, not the interior, and that A is homogeneously distributed in the phospholipid. This, combined with the spherical shape of the vesicles, will make the problem spherically symmetric. In addition, only the concentration of B will be both time and zpace dependent. Althougjl electrostatic interactions and hydrophobi=: forces might have an influence in a real system, we will also assume that the dominant; force effecting the diffusion of B will be concentration gradients; that is, the diffusion obeys Fick’s laws [1] with a constant diffusion coei’ficient. Our purpose is to develop an equation which describes the time (and space) dependence of the concentration of B in terms of experimentally measurabie quantities. In the next secrion we develop such an equation and in the last section we present an example of the use of this equation as applied to some experimental data from our work with chlorophyll.
2. 6eneral
formulation
As mentioned above, the diffusion of B will obey the simp!e diffusion law which in spherical coordinates has the following forrn [I] :
at/at= D[a2clar2f (2/i-)ac/af], with the following
boundary
(2)
conditions: 537
Vclumc c
37, number 3
CHENICAL
=C(r,r),
GvO’
t>cJ;
=X1(r),
f =ro,
tao;
=fbh
r>,rO,
r=o,
PHYSICS
[-(f-r')2j4Dr]
>o +-,,1/2 J
Defined this way, y is the amount of I3 per unit volume of 4~r/3 (neglecting the volume occupied by the vesicle) in the system containing a single vesicle in an infmite medium. We require the following relationship in order to relate]’ to the macroscopic concentration of B in a real system, one containing more than one vesicle. = Co + [3u/4n(l
where
r.
- exp [-(r+r’-2r0)‘/4Dr]}
'0
PI
J 2f(TAf)‘i2 f'f(f'){exp
1976
(9) (3)
where D is the diffusion coefficient of B in the aqueous phase, ro is the external radius of the vesicle, C is the concentration of B, x,(tj is the concentration of B at the surface of the vesicles and j(r) describes the distribution of B at r =O. Assuming a single vesicle in an infinite medium we can solve eq. (2) to obtain [2]: C=
1 February
LETTERS
-x3)r;](jyln-
[B] = concentration
Cc),
of B, given in units
(10) of
moles/ml, and Co = lim,%,,e C(r, t). We can substitute tain:
dr’
C from eq. (4) into eq. (9) to ob-
X, [I- (r-r,)2/4DP21
(r-r&21ur)1’2
X exp(--p’)dp.
(4)
Our purpose is to develop equations which are aplicable to experimental data. However, C as defined by eq. (4) is not a macroscopically measurable quantity. In addition, the single vesicle assumption puts
an upper limit on t for which (4) remains valid. In a real syStem containing a large number of vesicles, if diffusion distances become as large or larger than the average distance between vesicles, then eq. (4) is not valid. In such a system, the average distance between vesicles is given by: rd = 2ro [( 1 - X3)/V] y3 )
(5)
where u is the partial volume of the phospholipid and x is the ratic between the internal and external vesicie diameters. If we take (Dt)lI? as an approximate distance that a molecule wilI travel, then es_. (4) is only valid for:
t=C@D)[(l -x’)/uly3,
d&,
(11)
Oi Y = q
+ 5,
where rI is dependent upon the initial spatial distribution,f(r), while lY2 is not. By changing variables and integrating by parts, I’2 can be evaluated to give
+Q
(6)
or t
X exp (-I’)
‘Xl(r-S)(il-li’erC7_7e-~~)d;l, s G
(12)
where -&/“]~3/,,
where a, a more useful parameter, G = TO/,;.
(7) is defined
as;
7 = [(I--r3)/2ro]
(i+7E)“2
2nd (8)
In order to relate C of eo,. (4) to a macroscopic concentration, we must define Some new functions. First we dcime a function y such tllat
erfy = 2i7-‘12
= (I&)
(sr/a#?
Y s
exp (-x2) dx .
0
However, eq. (5) defines an upper limit for at which means that for f. Q 1, -y> 1 and under these circum-
Volume 37, member 3
1 Fcbruxry 1976
CHEMICAL PHYStCS LETTERS
stances eq. (12) can be approximated
by
f b(1 +ar/b2)-“2-
arctg](af)Y2/b]l
The evaluation of PI requires an explicit knowledge of the initial distributicn function, f(;,. We will ana-
lyze two specific initial distributions. (a) Homogeneous For this case,
It is not possible to extend this treatment
initial distribution
f(r) = F= constant.
114)
For such a situation, with increasing time, the regions near the surfaces of the vesicles Ml! be depleted of B relative to the bulk. It simplifies the problem if, instead of treating the diffusion of B toward the vesicles, we consider the diffusion of “the lack of 5” away from the vesicle surfaces and that at time equal zero, the concentration of these “lack of B’s” or “vacancies” is also zero. For this case we obtain rl as:
(b) Initial gaussian distribution For this case f(r)=Fexp[-11.6[(r-r0)/dJ2},
(16)
where d is the width of the distribution (the distance from the center to the maximum slope). Extending the integration in cq. (11) to infinity, we obtain: l-j = abr;F{$ -t b(1
f b +b2
+or/b*)-u2 - arctg[(nr)1/2/b] ),
t 1 + [u(r-$)/sr]V 0
3. Example To illustrate the use of the theory, we treat some in which chlorophyll a was dissolved ti phospholipid vesicles, K3 Fe(CN)h was addad after sonication as an electron acceptor, and the sample was illuminated with red light. & a result of the illumination, an electron was transferred from the chlorophyll to the acceptor. The chlorophyl radical thl;s formed was detected and its time behavior followed by using EPR. Preliminary results concerning this work were presented [3] and will be pub lished in greater detail elsewhere. The results are tl,eated in terms of the following mechansim:
results from an experiment
Chl
---.%
chi*
-
Chl” + ACCM+ +A$
Using r2, eqs. (10) and (11) and the NO fomls of I’l [eqs. (15) and (17)], we can obtain the dependence of the concentration of B on time for the two initial distributions: For the homogeneous initial distribution
-s
next section we show a specific example that demonstrates the use of eq. (18) or (19)
[
F[2P
tt(o/n)“2]
XIG)dB]
Q-#J2
and for the gaussian initial distribution
CM*,
(20a
ml,
(20b
Chl+ +A;,
(XC
CM f A,.
(7,Od
kl
kz
(17)
where b = d/3.8r0.
9uau2 [B] =F-------4a2( 1 -x3)
without
knowing the explicit form ofX1(k). However, in the
(18)
The acceptor, in this case KJ Fe(CN), , was soluble only in the aqueous phase, while the chlorophyll was soluble only in the membrane. They interacted.on tht surface of the vesicles. The acceptor was present in large excess, so its concentration is assumed not to change with time. The’generalized chemical reaction that was presenl in the introduction
is equivalent
to the back reaction
between the chlorophyll rrdica! and the reduced acceptor [eq. (20d)], The ia@ of change of the local concentration of the chlor&phyIl’radical is given by: 53s
‘dAdla _
d[Ci-d+l_ ---
dt
k, fk2 [A,]
(t)
1
3
’
= [AJ.
4x2(1-&
(21)
+ b(l+af/b2)-“’ + u1/2
(k3u)“2
exp[2t(Luk3)‘12] -
[Chl+],,,
exp [2t(Lvk,)1’z
j +1
= [Chl+],,,l(l+~,ur[Ci~+J~~~)~
(23)
(24)
+ k2 [A,].
For the case where eq. (6) is fulkIled, namely at times which are shorter than the one specified there, X,(t) will not be equal to [A,]. Thus, eqs. (23) and (24) are not valid. For such a case we are obliged to find the correct relation between X,(t) and [A;]. The equations which were deve!oped in the previous section provide such relations. For this particular case, a better agreement of the calculated behavior with the experimentally observed one could be achieved through the use of eq. (19), than with (18). With this equation, when I, f 0, F= 0. d [Chl+]/dr
the following
= L -. k, [Chlf] X,(I),
=
1,
(26b)
relation
(25a)
b(&+%+P).
9F 27?(1-X3)
(27)
.1/Z
J/7
9
=
2[2]
-;
4iG( l-x3)
where L = k, [A,] IJk,
X,G)dE
(t-t)“’
}
Eqs. (25) and (26) have to be solved numerically. We found the midpoint product integration method [4] to be a satisfactory way to proceed. However, the calculated results were not sensitive to the individual parameters, but rather to the ratios of these parameters. Thus, we defined the following parameters:
- 1
and for Ia = 0: [%I+]
1+ [,(r--$)/*p
J 0 where F obeys
Z[l] &_!!?
f
-- arctg[(ar)1/2/b]
(22)
Two extreme cases will be considered. In the first one the rate of diffusion is faster than the rate of the reaction. In this case a very short time after the onset of the reaction, eq. (6) will not be fulfilled and the reduced acceptor will attain a nearly uniform distribution. In that event, X,(r) will equal [A;] and eq. (21) will reduce to an equation which describes a simple second order reaction with the following solutions. For ia e 0:
[CM’]
1976
9
[Chl+] = [chl+lx
k
where X,(C) is the concentration of the reduced acceptor at the surface of the vesicle. If the set of reactions given in (20) fully describes the system, then charge conservation requires that u[Chlf]
1 February
CHEMICAL PHYSICS LETTERS
Volume 37, number 3
=L.;
Z[3]
=7(28)
k,
We have found also that under conditions where the time approaches the upper limit allowed by the theory, neglecting [o(r-.$)/n] ‘I3 in comparison with 1, will produce at most a 20% vkation in the parameters. This last approximation was found to be very convenient, because, without it, absolute values of 0 are required. Using this approximation and eqs. (28), eqs. (25) and (26) reduce to the following forms. 1n f 0:
[all+] = Z[l] Ia
J 0
‘Z[2]
-d[Chl+]/dt
do.
(29)
[Chl+](t-E)“”
=o:
[Chl+]
= 2Z[1]
CZ[2] - (d[Chl+]idr),,lO_$
x ~n+b+b2+b{1+~(Z[3])2}-“2Z[3]
arctg(Z[3]tii3)
Pq~,off
d W+l/dE --z[ll j 0 [al+]&$)Y’
dE .
(30)
b was found from eqs. (27) and (25a) to obey the following equation: and for I,=O,F+O,L
d [Chl+]/dr = -k3 [Chl+& 540
._I _.:.--
[c~+l’2=roff= =[I1 IZ[21 -@[Chlil/d~),,,o~I
=O we obtain
(r),
(26a)
x ($n+26+b2)/Z[3].
(31)
Volume 37, number 3
CHEXI!CAL PHYSICS LETTERS
TIME isecl FIN. 1. Change in the EPR amplirude of the chlorophyll
cation radical, during and after red light illumination at - 20°C. 1o-2 hl K3Fe(CN)6 was used as the electron acceptor. + = experimental results; q = least mean squares fit of eqs. (29) and (30) using the following parameters: Z[ I ] = 4,Z[ 21 = 0.7, Z[3] q 0.3; o = second order kinelics 3s calculated by using cqs. (23) and (24) with k3 = 2.4 X 10m3 and f_ = 0.14 which were calculated from the least mean squares fit of the decay portion of the data given in fig. 2 and rhe amplitude of the plateau attained during the illumination. All the parameters arc given in arbitrary units.
The maximum time resolution in our measurement was used as a constant grid spacing in the integration routine, this gave an accuracy of around 1% in the calculations. A typical experiment consists of turning the light on, following the radical during the illumination period and then, at time equal to toff, turning the light off and monitoring the radical decay. Formulas (29) and (30) were calculated in such a way that the 3 parameters were found by a least mean squares fit of the experimental and theoretical curves. The Fletcher and Powell [5] minimization routine was used for that purpose. For the reason that no special assumptions were made in the theory concerning the spatial distribution of the reduced acceptor during the illumination, we chose to fit Z[ l] and 2[2] with the portion of the curve obtained when the light was on and then held them constaRt and fit the decay by varying Z[3]. Fig. 1 presents an example of such a fit of the theoretical (open squares) to experimental data (the i’s) obtained with 1 min of illumination. When the illumination period was extended from 1 min to 4 min, the data (the +‘s) shown in fig. 2,were obtained. Also included in fig . 2 are points (open squares) calculated using eqs. (29) and (30) with the parameters found for the fit presented in fig. 1. As can be seen, these points deviate quite early from the experimen-
I February 1976
Fig. 2. The same as fig. 1 but with a longer period of illumination.
tal points. The principal cause of this difference is that the reduced acceptor approaches a homogeneous distribution and the basic assumption of a single vesicle in an infinite medium is not valid. Eq. (6) is not satisfied at the longer times. Thus, eqs. (29) and (30), which were used to calculate the theoretical points, arc not valid at the longer times. The signal in fig. 2 reaches a plateau value after about 3 min of illumination. Furthermore, the decay does not obey eq. (30) but instead obeys eq. (24) which is valid for homogeneous second order kinetics. From the plateau value and from the decay, we could find Fcj and L, assuming that the reaction was second order at shorter times as well. We used these values of X-, and L and plotted eqs. (23) and (34) (open circles) in figs. 1 and 2. In both figures the diffusion model does not fit the experimental data at times greater then 75 s. When the illumination was stopped after this time (fig. 2), we observe a smooth transition to second order kinetics. We do not observe such a transition when the iliumination was terminated at shortei time (fig. 1j. In a future publication, in which a more detailed account of the chlorophyll-phospholipid work will be presented, we will show that as we lower the temperature the time evolution of the system follows the diffusion kinetics for longer times.
Acknowledgement We would like to thank Drs. Jane K. Cullum and Ralph A. Willoughby for suggesting ref. [4] as suitable for the numerical
solution
of eqs. (29) and (30).
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Volume 37, number 3
CHEMICAL PHYSICS LETTERS
References
[3] hi.
1 February
Tomkiewicz
and G.A.
Third International
[ 11 W. lost, Diffusion in solids, liquids, gases (Academic Press, New York, 1952) ch. 1. [ 21 H.S. Carslaw and J.C. jaeger, Cocductkn (Clarendon Press, Oxford, 1959) p. 247.
of heat in solid
Corker,
in: Proceedings
Congress on Photosynthesis,
ed. hl. Avron (Elsevier, Amsterdam,
1976 of the
Vol. 1,
1975) p. 265; Photo-
chcm. F’hotobiol. (197.5), to be published. [3] R. Weiss and R.S. Anderson, Numer. Math. 18 (1972)
[5] R. Fietcher
and M.J.D. Powell, Computer
442.
J. 6 (1963) 163.