- Email: [email protected]
Photodissociation of carbon monoxy myoglobin: Kinetics of carbon monoxide rebinding
CHEMICAL
Volume 153, number 5
PHYSICS LETTERS
23 December
1988
PWOTODISSOCIATION OF CARBON MONOXY MYOGLOBIN: KINETICS OF CARBON MONOXIDE REBINDING Andrzej PLONKA,
Jerzy KROH
Institute ofApp!ied Radiation Chemistry, Technical University of Lode Wroblewskiego 15, 93-590 Lodz, Poland
and Yurii A. BERLIN Instituteof Chemical Physics, USSR Academy ofSciences, Kosygina 4. I1 7977Moscow. USSR Received 28 September
1988
Carbon monoxide rebinding following quately described using the second-order from the Kohlmusch relaxation function.
It was found function s(t) =cxp I-
[ 1 ] that the Kohlrausch (tl7oYl
,
laser photodissociation of carbon monoxy myoglobin in frozen solutions can be adeequal concentration kinetic equation with a time-dependent rate constant deducible
relaxation (1)
where z. and IX(0 < LYd 1) are constants, is adequate to describe the non-exponential relaxation of the protein following laser photodissociation of carbon monoxy myoglobin in frozen solutions. In compa tition with protein relaxation towards the new equilibrium state of deoxyMb, CO rebinding, restoring the MbCO state [ 2 1, also occurs. Assuming that the kinetic pattern of rebinding is dominated by protein dynamics one is prompted to use in the kinetic equation for recombination the time-dependent specific reaction rate k(t) in a form analogous to that in the Kohlrausch relaxation function (1 ), i.e. dd(t) W)=-dtS(t)=<
1
(Y 0
t r,
a-’
and indeed it fits well the experimental data on CO rebinding, kindly supplied by Professor H. Frauenfelder (cf. fig. 1 and table 1 collecting the rate parameters). Eq. (3) rewritten in the form 1 1+ (1/Tb)a ’
C
-= CO
0
”
n -1
4
(2)
-2
For recombination, the second-order equal concentration kinetic equation with k(t) given by eq. (2) reads [3] 1 c-c,
1
=-
tCI 0 ro
0 009-2614/88/$ (North-Holland
(3) 03.50 0 Elsevier Science Publishers Physics Publishing Division )
Fig. 1. Recombination of CO with Mb ( 14 pM) in 75% (v/v) glycerol/water solution in the temperature range 6 I -7-l 6 1.2 K. Observation wavelength 430 nm.
B.V.
433
Volume 153, number
CHEMICAL
5
Table 1 Rate parameters from fitting the experimental fig. 1 using eq. (3)
data presented
in
1y
%I (s)
61.7 81.3 101.3 121.3 141.3 161.2
0.25 0.31 0.36 0.43 0.49 0.55
2.94 1.39x1o-2 4.34x 1o-4 3.40x 10-5 4.91 x10-6 1.43x10-6
1
0
1 = l+(t/T;)“’
Using Mellin integral transformation sides of eq. (6 ) one gets
I
7’-‘f(7)
0
dr=
’
(8)
’
(9)
where A =a (E- Eo) /RT, for which the first moment is equal to
density
(10)
where E. is the activation energy determined from the Arrhenius plot of corb, and the second moment is equal to aZ,=~(~RT/a)*(1--a2).
(11)
Because co is virtually constant under the reported experimental conditions of laser photodissociation [ 21 one can directly plot tb in the Arrhenius coordinate system, cf. fig. 2, to obtain the effective activation energy E. equal to 12.13 kJ/mol. Fig. 3 depicts the densities calculated according to relation (9) at 6 1.7 and 161.2 K. Surprisingly they almost coincide. This is because the first moment, cf. eq. (lo), is temperature independent and the increase of the second moment with temperature, cf. eq. ( 11)) is evidently compensated by the increase with temperature of the numerical values of a, cf. table 1, which has the opposite effect. Also included is the enthalpy density distribution g(H) used in Frauenfelder’s laboratory to describe the same set of data with the following rebinding function
(6)
[9] on both
sb’-‘sin[x(s-l)]
cusin[irr(s-l)]
from which f(7) can be obtained as the inverse Melhn transform of the right-hand side. The result is [ 81 434
(r/7&)” (t/7~)2”+2(~/7~)acosxcu+1
g’E)=~expW+2~~~expA+1
(5)
where 7, = c&,. Introducing the distribution f(r) one finds for the overall process -f(r)dr I -1+2/r
sin
1988
Assuming that the Arrhenius equation holds for any r one can express the distribution of reactivity in terms of the activation energy
where rb = r,,c; I@, closely resembles the power law (1 +t/t,,)“. Over a wide range it is hard to distinguish this from the first-order kinetic equation with a time-dependent rate constant of the same form: LY= 0.5 yields the so-called diffusion-controlled geminate recombination kinetics; CY=1 corresponds to classical recombination kinetics, and the lower the numerical value of (Ythe better eq. (4) mimics classical first-order kinetics over appreciable intervals. Thus eq. (4) may rationalize the reported kinetic behaviour of the above system in different temperature regions [ 4-7 1. The Kohlrausch relaxation function was regarded in ref. [ 1] as the superposition of simple exponential decays with probability density f(z). It is possible [ 8 ] to interpret eq. (4) in a similar way, i.e. as the superposition of recombinations proceeding in a large number of isolated systems. In each of them the solution of the kinetic equation can be written as c, -=E7 COi
f(z)=
XT
T(K)
23 December
PHYSICS LETTERS
Fig. 2. Arrhenius
plot of sb from eq. (4).
Volume 153, number
5
CHEMICAL
PHYSICS LETTERS
23
December I988
We are grateful to Professor Hans Frauenfelder for providing us with experimental data from his Laboratory and informative correspondence, and to Professor Vitalii I. Goldanskii for many fruitful discussions. YuAB would like to acknowledge the hospitality during his stay in the Institute of Applied Radiation Chemistry. This work was supported in parts by contrasts CPBP 01.19 and RP-II- 11. 30
20 E.
Fig cq. for the
References
kJ/mole
3. Densities of reactivity distributionsg(E) calculated from (9 ) for the rate parameters from table 1 (solid lines, the lower 6 1.7 K, the higher curve for I6 1.2 K), and g(H) used with rebinding function ( 12) (dashed line).
flmnnoz
N(t) -= N(O)
s
g(H) exp [ -WC WI W,
(12)
ffmm
which fits the experimental data equally well. So it is not the goodness of the tit which advocates the use of eq. ( 3). It is rather its sound basis of common dynamics for rebinding and relaxation in the myoglobin pocket and its ability to rationalize the reported kinetic patterns for rebinding.
[ 1 ] A. Plonka, Chem. Phys. Letters 15 I ( 1988) 466. [2] A. Ansari, J. Berendzen, D. Braunstein, B.R. Cowen, H. Frauenfelder, M.K. Hong, I.E.T. Iben, J.B. Johnson, P. Ormos, T.B. Sauke, R. Scholl, A. Sehulte, P.J. Steinbach, J. Vittitow and R.D. Young, Biophys. Chem. 26 ( 1987) 337. [ 3 ] A. Plonka, Lecture notes in chemistry, Vol. 40. Timedependent reactivity of species in condensed media (Springer, Berlin, 1986). [ 41 R.H. Austin, K. Beeson, L. Eisenstein, H. Frauenfelder, I.C. Gunsalus and V.P. Marshall, Science 18 1 ( 1973) 541. [ 51R.H. Austin, K. Beeson, L. Eiscnstein, H. Frauenfelder, I.C. Gunsalus and V.P. Marshall, Phys. Rev. Letters 32 ( 1974) 403. [6] B.B. Haainoff, J. Phys. Chem. 82 (1978) 2630. [ 71 L. Lindqvist, S. El Mohshi, F. Tfibel, B. Alpert and J.C. Andre, Chem. Phys. Letters 79 ( I98 1) 525. [ 81A. Plonka, Yu. A. Berlin and N.I. Chekunaev, to be published. [9] A. Erdtlyi, ed., Tables of integral transforms, Vol. 1 (McGraw-Hill, New York, 1954).
435
Volume 153, number 5
PHYSICS LETTERS
23 December
1988
PWOTODISSOCIATION OF CARBON MONOXY MYOGLOBIN: KINETICS OF CARBON MONOXIDE REBINDING Andrzej PLONKA,
Jerzy KROH
Institute ofApp!ied Radiation Chemistry, Technical University of Lode Wroblewskiego 15, 93-590 Lodz, Poland
and Yurii A. BERLIN Instituteof Chemical Physics, USSR Academy ofSciences, Kosygina 4. I1 7977Moscow. USSR Received 28 September
1988
Carbon monoxide rebinding following quately described using the second-order from the Kohlmusch relaxation function.
It was found function s(t) =cxp I-
[ 1 ] that the Kohlrausch (tl7oYl
,
laser photodissociation of carbon monoxy myoglobin in frozen solutions can be adeequal concentration kinetic equation with a time-dependent rate constant deducible
relaxation (1)
where z. and IX(0 < LYd 1) are constants, is adequate to describe the non-exponential relaxation of the protein following laser photodissociation of carbon monoxy myoglobin in frozen solutions. In compa tition with protein relaxation towards the new equilibrium state of deoxyMb, CO rebinding, restoring the MbCO state [ 2 1, also occurs. Assuming that the kinetic pattern of rebinding is dominated by protein dynamics one is prompted to use in the kinetic equation for recombination the time-dependent specific reaction rate k(t) in a form analogous to that in the Kohlrausch relaxation function (1 ), i.e. dd(t) W)=-dtS(t)=<
1
(Y 0
t r,
a-’
and indeed it fits well the experimental data on CO rebinding, kindly supplied by Professor H. Frauenfelder (cf. fig. 1 and table 1 collecting the rate parameters). Eq. (3) rewritten in the form 1 1+ (1/Tb)a ’
C
-= CO
0
”
n -1
4
(2)
-2
For recombination, the second-order equal concentration kinetic equation with k(t) given by eq. (2) reads [3] 1 c-c,
1
=-
tCI 0 ro
0 009-2614/88/$ (North-Holland
(3) 03.50 0 Elsevier Science Publishers Physics Publishing Division )
Fig. 1. Recombination of CO with Mb ( 14 pM) in 75% (v/v) glycerol/water solution in the temperature range 6 I -7-l 6 1.2 K. Observation wavelength 430 nm.
B.V.
433
Volume 153, number
CHEMICAL
5
Table 1 Rate parameters from fitting the experimental fig. 1 using eq. (3)
data presented
in
1y
%I (s)
61.7 81.3 101.3 121.3 141.3 161.2
0.25 0.31 0.36 0.43 0.49 0.55
2.94 1.39x1o-2 4.34x 1o-4 3.40x 10-5 4.91 x10-6 1.43x10-6
1
0
1 = l+(t/T;)“’
Using Mellin integral transformation sides of eq. (6 ) one gets
I
7’-‘f(7)
0
dr=
’
(8)
’
(9)
where A =a (E- Eo) /RT, for which the first moment is equal to
density
(10)
where E. is the activation energy determined from the Arrhenius plot of corb, and the second moment is equal to aZ,=~(~RT/a)*(1--a2).
(11)
Because co is virtually constant under the reported experimental conditions of laser photodissociation [ 21 one can directly plot tb in the Arrhenius coordinate system, cf. fig. 2, to obtain the effective activation energy E. equal to 12.13 kJ/mol. Fig. 3 depicts the densities calculated according to relation (9) at 6 1.7 and 161.2 K. Surprisingly they almost coincide. This is because the first moment, cf. eq. (lo), is temperature independent and the increase of the second moment with temperature, cf. eq. ( 11)) is evidently compensated by the increase with temperature of the numerical values of a, cf. table 1, which has the opposite effect. Also included is the enthalpy density distribution g(H) used in Frauenfelder’s laboratory to describe the same set of data with the following rebinding function
(6)
[9] on both
sb’-‘sin[x(s-l)]
cusin[irr(s-l)]
from which f(7) can be obtained as the inverse Melhn transform of the right-hand side. The result is [ 81 434
(r/7&)” (t/7~)2”+2(~/7~)acosxcu+1
g’E)=~expW+2~~~expA+1
(5)
where 7, = c&,. Introducing the distribution f(r) one finds for the overall process -f(r)dr I -1+2/r
sin
1988
Assuming that the Arrhenius equation holds for any r one can express the distribution of reactivity in terms of the activation energy
where rb = r,,c; I@, closely resembles the power law (1 +t/t,,)“. Over a wide range it is hard to distinguish this from the first-order kinetic equation with a time-dependent rate constant of the same form: LY= 0.5 yields the so-called diffusion-controlled geminate recombination kinetics; CY=1 corresponds to classical recombination kinetics, and the lower the numerical value of (Ythe better eq. (4) mimics classical first-order kinetics over appreciable intervals. Thus eq. (4) may rationalize the reported kinetic behaviour of the above system in different temperature regions [ 4-7 1. The Kohlrausch relaxation function was regarded in ref. [ 1] as the superposition of simple exponential decays with probability density f(z). It is possible [ 8 ] to interpret eq. (4) in a similar way, i.e. as the superposition of recombinations proceeding in a large number of isolated systems. In each of them the solution of the kinetic equation can be written as c, -=E7 COi
f(z)=
XT
T(K)
23 December
PHYSICS LETTERS
Fig. 2. Arrhenius
plot of sb from eq. (4).
Volume 153, number
5
CHEMICAL
PHYSICS LETTERS
23
December I988
We are grateful to Professor Hans Frauenfelder for providing us with experimental data from his Laboratory and informative correspondence, and to Professor Vitalii I. Goldanskii for many fruitful discussions. YuAB would like to acknowledge the hospitality during his stay in the Institute of Applied Radiation Chemistry. This work was supported in parts by contrasts CPBP 01.19 and RP-II- 11. 30
20 E.
Fig cq. for the
References
kJ/mole
3. Densities of reactivity distributionsg(E) calculated from (9 ) for the rate parameters from table 1 (solid lines, the lower 6 1.7 K, the higher curve for I6 1.2 K), and g(H) used with rebinding function ( 12) (dashed line).
flmnnoz
N(t) -= N(O)
s
g(H) exp [ -WC WI W,
(12)
ffmm
which fits the experimental data equally well. So it is not the goodness of the tit which advocates the use of eq. ( 3). It is rather its sound basis of common dynamics for rebinding and relaxation in the myoglobin pocket and its ability to rationalize the reported kinetic patterns for rebinding.
[ 1 ] A. Plonka, Chem. Phys. Letters 15 I ( 1988) 466. [2] A. Ansari, J. Berendzen, D. Braunstein, B.R. Cowen, H. Frauenfelder, M.K. Hong, I.E.T. Iben, J.B. Johnson, P. Ormos, T.B. Sauke, R. Scholl, A. Sehulte, P.J. Steinbach, J. Vittitow and R.D. Young, Biophys. Chem. 26 ( 1987) 337. [ 3 ] A. Plonka, Lecture notes in chemistry, Vol. 40. Timedependent reactivity of species in condensed media (Springer, Berlin, 1986). [ 41 R.H. Austin, K. Beeson, L. Eisenstein, H. Frauenfelder, I.C. Gunsalus and V.P. Marshall, Science 18 1 ( 1973) 541. [ 51R.H. Austin, K. Beeson, L. Eiscnstein, H. Frauenfelder, I.C. Gunsalus and V.P. Marshall, Phys. Rev. Letters 32 ( 1974) 403. [6] B.B. Haainoff, J. Phys. Chem. 82 (1978) 2630. [ 71 L. Lindqvist, S. El Mohshi, F. Tfibel, B. Alpert and J.C. Andre, Chem. Phys. Letters 79 ( I98 1) 525. [ 81A. Plonka, Yu. A. Berlin and N.I. Chekunaev, to be published. [9] A. Erdtlyi, ed., Tables of integral transforms, Vol. 1 (McGraw-Hill, New York, 1954).
435