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Ann. Rev. Biophys. Bioeng. 1977. 6:111-33 Copyright re) 1977 by Annual Reviews Inc. All rights reserved
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
REACTION RATE THEORY
+9089
IN BIOLUMINESCENCE AND OTHER LIFE PHENOMENA Frank
H.
Johnson
Biology Department, Princeton University, Princeton. New Jersey 08540
Henry Eyring
Chemistry Department, University of Utah. Salt Lake City, Utah 84112
Betsy Jones Stover
Department of Ph a rmacology. University of North Carolina, Chapel Hill,
North Carolina 27514 and Department of Anatomy, College of Medicine,
University of Utah, Sal t Lake Ci ty , Utah 84112
The modern theory of absolute reaction rates
(l, 2) provides the basis, in principle,
for calculating the absolute rate of any chemical reaction from a knowledge of the properties of the reactants and the well-established laws governing their behavior. This theory rests upon, and is in fact the outgrowth of, modern quantum theory together with the older disciplines of thermodynamics and mechanics, classical .and statistical as well as quantum. Although the rate theory cannot be reviewed in detail within the scope of this review, it should be useful in the interests of clarity to briefly state the conceptual nature and quantitative formulation. According to this theory, every elementary chemical rate process, whatever its
nature (e.g . decomposition, synthesis, hydrolysis, condensation, diffusion, lubrica tion), involves the formation of an unstable intermediate complex between the
. reactants, a complex known as the activated state or transition state (or more properly states) often designated by a double-dagger superscript
(t).
The activated
state is characterized by a certain amount of energy above that of the reactants in the normal state. The activated state has a lifetime of less than the period of a single vibration of an oscillator, i.e. a lifetime of the order of 10-13 sec. Decomposition of
the activated complex occurs with a universal frequency that is the same for all
reactions and is given by the expression kT/h, where k is the Boltzmann constant
R/N. the gas constant over Avogadro's number, or the gas constant per molecule,
III
JOHNSON, EYRING & STOVER
112
T is the absolute temperature, and
h
is Planck's constant. The quantum mechanical
chance that the decomposition of the complex will be in the direction of products rather than reconstitution of the reactants is designated by the constant
K,
which
for many if not most reactions is unity or nearly so. Calculating the absolute rate of a reaction from first principles requires determin ing the height of the energy barrier of a potential energy surface, i.e. the activation
energy AE�. Calculating the potential energy surface itself involves calculating the
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
energies for all possible configurations of the reactants; this involves as many dimen sions as are required to fix the potential energy-a formidable task for even the
1 . 3 kcal that the calculated value
simplest chemical reaction. Moreover, for each
differs from the actual value of the activation energy, the predicted rate of reaction will be off by a factor of
10 at blood temperature.
Even with rates that are actually
measured in experiments, it is sometimes difficult to arrive at activation energies that are correct to within better than
1
or
2 kcal.
Because it is the free energy of activation rather than simply the activation energy that determines the reaction rate, it is' necessary to know the entropy of activation,
ASt, as well as the heat of activation, AH�. In principle, a value for ASt can be
determined with the aid of well-known expressions for partition functions applicable to any well-defined, relatively simple system, such as a diatomic molecule with degrees of freedom, i.e. as any molecule has
6
3 of translation, 2 of rotation, and I of vibration. Inasmuch
3n
degrees of freedom, where
n
is the number of atoms, the
problem obviously becomes enormously more complicated with increasing com plexity of the molecule. Thus, it is clearly not feasible at present to calculate from first principles any typical biological reaction, and this seems all the more evident in view of the circumstan,?e that even a simple enzyme reaction involves participa tion of all the many bonds that take part in conformation changes of the protein moiety (cf. discussion of the elastomeric rack) Accepting the concept of
kT/h,
(3).
which stems from quantum and classical physical
theory, it follows that the overall rate of the reaction is determined by the rate of attaining the activated state(s), which requires a certain free energy of activation; this notion goes back to the activation energy of Arrhenius but, in its modern version, it is susceptible to precise and rational interpretation. It is usually pictured as a region at the top of an energy barrier on a potential energy surface that represents the diffen:nt energies pertaining to different distances apart of the atoms involved. The chances of reaching the activated state are given by the expression exp(-A Gt/ R T), where fj. Gt is the free energy of activation and is analogous to the
Gibbs free energy of reaction of an equilibrium. Despite some conceptual differ ences, the free energy of activation is defined in the same way as the free energy of reaction, thus:
�G+
=
�H:j:
-
T�S:j:
=
�E* + p�V+
-
T�S
=
:j: �Gp=l
+
P
a�G+
j-----:rp-
p=l
dp
1.
MODERN REACTION RATE THEORY
1 13
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
where P is in atmospheres, V in cubic centimeters, which requires R to equal 82.1. Equation 1 obviously defines an equilibrium constant of activation K! that differs conceptually from an equilibrium constant K of reaction in that the latter is a ratio between the rate in the forward direction and the rate in the backward direction when these rates become equal at equilibrium. Each rate involves the crossing of the same energy barrier and the formation of the same, intermediate, unstable activated complex. The expression for the specific rate k' in either direction is of the form
k'
=
(KkT/h)(K+)
=
(KkT/h) exp (-b..G+/RD.
2.
For all practical purposes the various quantities of free energy of activation, en thalpy H. volume V. and entropy S, behave the same as the analogous quantities in thermodynamic equilibria and may be treated in exactly the same way, thus making available the invaluable methods of thermodynamics and statistical mechan ics for coping with reaction rates. Fortunately, useful applications of the theory of absolute reaction rates do not depend on such an esoteric approach as was necessary to arrive at the theory in the first place. Furthermore, despite the innate complexity of biological processes and the intricate chemical environment in which reactions take place in living celIs, there is every reason to believe that the same natural laws apply to reactions in both test tubes and cells, with the single basic modification that the cellular environment frequently involves a selective influence through the action of templates, or through limitations imposed by certain features of organization such as the structure of membranes and subcellular units of various sorts, e.g. mitochondria. Ideally, the rates of intracellular reactions should be measured in a way that does not harm the structure or integrity of the cell. Even more ideally, the method of measurement should yield an accurate and instantaneous value. The phenomenon that comes closest to fulfilling these desiderata seems to be the one that is familiarly known as bioluminescence, wherein the intensity of the light is proportional to the reaction velocity of the system. One of the simplest examples is the luminescent system of the small ostracod crustacean, Cypridina, involving the aerobic oxidation of a substrate, referred to by the general term luciferin, catalyzed by a specific enzyme, referred to by the general term luciferase (3a). The chemical structure of Cypridina luciferin is now known (4), as well as the chief properties of Cypridina luciferase (5, 6), the main features of the reaction pathway (7-9), the identity of the light-emitting complex (10), and the action of important factors influencing the quantum yield (II). Although the activity of Cypridina luciferase has a nonspecific requirement of small concentrations of cations (5, 1 2, 1 3), the minimum components otherwise needed for a light-emitting reaction consist of Cypridina luciferin, Cy pri d i na luciferase, molecular oxygen, and water. In the present context, the impor tant feature is that the reaction velocity of this luciferin-Iuciferase system, under given favorable conditions, determines the intensity of the light, and over a wide range in concentrations of the substrate and enzyme this velocity v is equal to the product of substrate and enzyme conce�trations ( 1 4). In partially purified or pure
JOHNSON, EYRING & STOVER
114
extracts the total amount of light emitted when luciferin is added to luciferase is proportional to the amount of luciferin initially present, and the actual intensity I at any moment is proportional to the amount of enzyme. The reaction is first order
with . regard to luciferin (Figure I ). The overall reaction thus can be written:
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
1= bv
=
bk' (LH2)(A)
=
bK (kTjh)(LH2)(A)K:j: b(kTjh)(LH2)(A) exp (-!lCt jR T), =
3.
where b is a proportionality constant, LH2 represents luciferin, A represents luci ferase, and the other symbols have the same meanings as in equation 2.
In living cells of luminous bacteria, under favorable physiological conditions, the" intensity of the light may remain constant over considerable periods of time (i.e. for many minutes) thus olfering a convenient process for getting instantaneous measure ments of the effect of various factors on the system involved. Figure 2 illustrates the reversible effects of an inhibitor, diisopropylfiuorophosphate, which acts upon the light-emitting system both in vitro and in vivo
(IS).
Figure 2 also illustrates the
essentially constant intensity of the light over the 60-min observation period. Biochemically, the luminescent system of bacteria is comprised of bacterial luci ferase which is a flavin enzyme that catalyzes the oxidation of any of several long-chain, aliphatic aldehydes according to the following stoichiometry (16-21):
R-CHO + FMNH2 + 02
b acterial •
1UCI'ferase
R-COOH + FMH + H20 + light.
4.
The quantum yield is 0.17 ± om (21), and the light-emitting species of molecules
or photagogikon (22) is an oxygenated form of reduced flavin mononucleotide (23).
The metabolic pathway by which the aldehyde is generated is unknown, but there is evidence that during steady-state luminescence the amount of aldehyde present at any moment is only enough to sustain the process of light emission for less than
1
sec
(21). A
limited excess of FMNH2 accumulates when dissolved oxygen is
removed from a suspension of luminous bacteria, resulting in a momentary excess
light intensity, referred to as a
flash,
when oxygen is suddenly readmitted.
A
spectacular increase in magnitude of the flash, as well as in the steady-state intensity afterwards, occurs if glucose is added either aerobically or anaerobically
(24),
sug
gesting that the production of aldehyde is somehow related to dissimilation of glucose (cf. 25). The constant level of light intensity, as illustrated in Figure 2, is evidence that under the conditions involved the constants in equation
3 do not change with time
• .
and neither do the concentrations of the reactants luciferin and luciferase. On changing the temp(!rature, the steady-state level of luminescence changes virtually at once (26). Moreover, on raising the temperature from cold towards the familiar
Figure 1
(A) Total light emitted as a function of time after mixing a constant small amount
of luciferase with valious amounts of luciferin, as indicated in the figure, in a constant final
volume (93).
(0) Firs,t-order rates of the luminescence reaction with a constant initial amount
of luciferin and various amounts ofluciferase, indicated in the figure. in a constant final volume of reaction mixture [after (94»).
MODERN REACTION RATE THEORY
115
300 r-------,r�,_--�
Further
Quick links to online content
Ann. Rev. Biophys. Bioeng. 1977. 6:111-33 Copyright re) 1977 by Annual Reviews Inc. All rights reserved
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
REACTION RATE THEORY
+9089
IN BIOLUMINESCENCE AND OTHER LIFE PHENOMENA Frank
H.
Johnson
Biology Department, Princeton University, Princeton. New Jersey 08540
Henry Eyring
Chemistry Department, University of Utah. Salt Lake City, Utah 84112
Betsy Jones Stover
Department of Ph a rmacology. University of North Carolina, Chapel Hill,
North Carolina 27514 and Department of Anatomy, College of Medicine,
University of Utah, Sal t Lake Ci ty , Utah 84112
The modern theory of absolute reaction rates
(l, 2) provides the basis, in principle,
for calculating the absolute rate of any chemical reaction from a knowledge of the properties of the reactants and the well-established laws governing their behavior. This theory rests upon, and is in fact the outgrowth of, modern quantum theory together with the older disciplines of thermodynamics and mechanics, classical .and statistical as well as quantum. Although the rate theory cannot be reviewed in detail within the scope of this review, it should be useful in the interests of clarity to briefly state the conceptual nature and quantitative formulation. According to this theory, every elementary chemical rate process, whatever its
nature (e.g . decomposition, synthesis, hydrolysis, condensation, diffusion, lubrica tion), involves the formation of an unstable intermediate complex between the
. reactants, a complex known as the activated state or transition state (or more properly states) often designated by a double-dagger superscript
(t).
The activated
state is characterized by a certain amount of energy above that of the reactants in the normal state. The activated state has a lifetime of less than the period of a single vibration of an oscillator, i.e. a lifetime of the order of 10-13 sec. Decomposition of
the activated complex occurs with a universal frequency that is the same for all
reactions and is given by the expression kT/h, where k is the Boltzmann constant
R/N. the gas constant over Avogadro's number, or the gas constant per molecule,
III
JOHNSON, EYRING & STOVER
112
T is the absolute temperature, and
h
is Planck's constant. The quantum mechanical
chance that the decomposition of the complex will be in the direction of products rather than reconstitution of the reactants is designated by the constant
K,
which
for many if not most reactions is unity or nearly so. Calculating the absolute rate of a reaction from first principles requires determin ing the height of the energy barrier of a potential energy surface, i.e. the activation
energy AE�. Calculating the potential energy surface itself involves calculating the
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
energies for all possible configurations of the reactants; this involves as many dimen sions as are required to fix the potential energy-a formidable task for even the
1 . 3 kcal that the calculated value
simplest chemical reaction. Moreover, for each
differs from the actual value of the activation energy, the predicted rate of reaction will be off by a factor of
10 at blood temperature.
Even with rates that are actually
measured in experiments, it is sometimes difficult to arrive at activation energies that are correct to within better than
1
or
2 kcal.
Because it is the free energy of activation rather than simply the activation energy that determines the reaction rate, it is' necessary to know the entropy of activation,
ASt, as well as the heat of activation, AH�. In principle, a value for ASt can be
determined with the aid of well-known expressions for partition functions applicable to any well-defined, relatively simple system, such as a diatomic molecule with degrees of freedom, i.e. as any molecule has
6
3 of translation, 2 of rotation, and I of vibration. Inasmuch
3n
degrees of freedom, where
n
is the number of atoms, the
problem obviously becomes enormously more complicated with increasing com plexity of the molecule. Thus, it is clearly not feasible at present to calculate from first principles any typical biological reaction, and this seems all the more evident in view of the circumstan,?e that even a simple enzyme reaction involves participa tion of all the many bonds that take part in conformation changes of the protein moiety (cf. discussion of the elastomeric rack) Accepting the concept of
kT/h,
(3).
which stems from quantum and classical physical
theory, it follows that the overall rate of the reaction is determined by the rate of attaining the activated state(s), which requires a certain free energy of activation; this notion goes back to the activation energy of Arrhenius but, in its modern version, it is susceptible to precise and rational interpretation. It is usually pictured as a region at the top of an energy barrier on a potential energy surface that represents the diffen:nt energies pertaining to different distances apart of the atoms involved. The chances of reaching the activated state are given by the expression exp(-A Gt/ R T), where fj. Gt is the free energy of activation and is analogous to the
Gibbs free energy of reaction of an equilibrium. Despite some conceptual differ ences, the free energy of activation is defined in the same way as the free energy of reaction, thus:
�G+
=
�H:j:
-
T�S:j:
=
�E* + p�V+
-
T�S
=
:j: �Gp=l
+
P
a�G+
j-----:rp-
p=l
dp
1.
MODERN REACTION RATE THEORY
1 13
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
where P is in atmospheres, V in cubic centimeters, which requires R to equal 82.1. Equation 1 obviously defines an equilibrium constant of activation K! that differs conceptually from an equilibrium constant K of reaction in that the latter is a ratio between the rate in the forward direction and the rate in the backward direction when these rates become equal at equilibrium. Each rate involves the crossing of the same energy barrier and the formation of the same, intermediate, unstable activated complex. The expression for the specific rate k' in either direction is of the form
k'
=
(KkT/h)(K+)
=
(KkT/h) exp (-b..G+/RD.
2.
For all practical purposes the various quantities of free energy of activation, en thalpy H. volume V. and entropy S, behave the same as the analogous quantities in thermodynamic equilibria and may be treated in exactly the same way, thus making available the invaluable methods of thermodynamics and statistical mechan ics for coping with reaction rates. Fortunately, useful applications of the theory of absolute reaction rates do not depend on such an esoteric approach as was necessary to arrive at the theory in the first place. Furthermore, despite the innate complexity of biological processes and the intricate chemical environment in which reactions take place in living celIs, there is every reason to believe that the same natural laws apply to reactions in both test tubes and cells, with the single basic modification that the cellular environment frequently involves a selective influence through the action of templates, or through limitations imposed by certain features of organization such as the structure of membranes and subcellular units of various sorts, e.g. mitochondria. Ideally, the rates of intracellular reactions should be measured in a way that does not harm the structure or integrity of the cell. Even more ideally, the method of measurement should yield an accurate and instantaneous value. The phenomenon that comes closest to fulfilling these desiderata seems to be the one that is familiarly known as bioluminescence, wherein the intensity of the light is proportional to the reaction velocity of the system. One of the simplest examples is the luminescent system of the small ostracod crustacean, Cypridina, involving the aerobic oxidation of a substrate, referred to by the general term luciferin, catalyzed by a specific enzyme, referred to by the general term luciferase (3a). The chemical structure of Cypridina luciferin is now known (4), as well as the chief properties of Cypridina luciferase (5, 6), the main features of the reaction pathway (7-9), the identity of the light-emitting complex (10), and the action of important factors influencing the quantum yield (II). Although the activity of Cypridina luciferase has a nonspecific requirement of small concentrations of cations (5, 1 2, 1 3), the minimum components otherwise needed for a light-emitting reaction consist of Cypridina luciferin, Cy pri d i na luciferase, molecular oxygen, and water. In the present context, the impor tant feature is that the reaction velocity of this luciferin-Iuciferase system, under given favorable conditions, determines the intensity of the light, and over a wide range in concentrations of the substrate and enzyme this velocity v is equal to the product of substrate and enzyme conce�trations ( 1 4). In partially purified or pure
JOHNSON, EYRING & STOVER
114
extracts the total amount of light emitted when luciferin is added to luciferase is proportional to the amount of luciferin initially present, and the actual intensity I at any moment is proportional to the amount of enzyme. The reaction is first order
with . regard to luciferin (Figure I ). The overall reaction thus can be written:
Annu. Rev. Biophys. Bioeng. 1977.6:111-133. Downloaded from www.annualreviews.org by University of Guelph on 06/16/12. For personal use only.
1= bv
=
bk' (LH2)(A)
=
bK (kTjh)(LH2)(A)K:j: b(kTjh)(LH2)(A) exp (-!lCt jR T), =
3.
where b is a proportionality constant, LH2 represents luciferin, A represents luci ferase, and the other symbols have the same meanings as in equation 2.
In living cells of luminous bacteria, under favorable physiological conditions, the" intensity of the light may remain constant over considerable periods of time (i.e. for many minutes) thus olfering a convenient process for getting instantaneous measure ments of the effect of various factors on the system involved. Figure 2 illustrates the reversible effects of an inhibitor, diisopropylfiuorophosphate, which acts upon the light-emitting system both in vitro and in vivo
(IS).
Figure 2 also illustrates the
essentially constant intensity of the light over the 60-min observation period. Biochemically, the luminescent system of bacteria is comprised of bacterial luci ferase which is a flavin enzyme that catalyzes the oxidation of any of several long-chain, aliphatic aldehydes according to the following stoichiometry (16-21):
R-CHO + FMNH2 + 02
b acterial •
1UCI'ferase
R-COOH + FMH + H20 + light.
4.
The quantum yield is 0.17 ± om (21), and the light-emitting species of molecules
or photagogikon (22) is an oxygenated form of reduced flavin mononucleotide (23).
The metabolic pathway by which the aldehyde is generated is unknown, but there is evidence that during steady-state luminescence the amount of aldehyde present at any moment is only enough to sustain the process of light emission for less than
1
sec
(21). A
limited excess of FMNH2 accumulates when dissolved oxygen is
removed from a suspension of luminous bacteria, resulting in a momentary excess
light intensity, referred to as a
flash,
when oxygen is suddenly readmitted.
A
spectacular increase in magnitude of the flash, as well as in the steady-state intensity afterwards, occurs if glucose is added either aerobically or anaerobically
(24),
sug
gesting that the production of aldehyde is somehow related to dissimilation of glucose (cf. 25). The constant level of light intensity, as illustrated in Figure 2, is evidence that under the conditions involved the constants in equation
3 do not change with time
• .
and neither do the concentrations of the reactants luciferin and luciferase. On changing the temp(!rature, the steady-state level of luminescence changes virtually at once (26). Moreover, on raising the temperature from cold towards the familiar
Figure 1
(A) Total light emitted as a function of time after mixing a constant small amount
of luciferase with valious amounts of luciferin, as indicated in the figure, in a constant final
volume (93).
(0) Firs,t-order rates of the luminescence reaction with a constant initial amount
of luciferin and various amounts ofluciferase, indicated in the figure. in a constant final volume of reaction mixture [after (94»).
MODERN REACTION RATE THEORY
115
300 r-------,r�,_--�
