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Biological energy-coupling in terms of irreversible thermodynamics
Biochemistry and Molecular Biology Education 28 (2000) 301}303
Biological energy-coupling in terms of irreversible thermodynamics John L. Howland*, Matthew Needleman Department of Biology and Committee on Biochemistry, Bowdoin College, Brunswick, ME 04011-8465, USA
Abstract Irreversible thermodynamics provides an apt description of bioenergetic processes, of which oxidative phosphorylation is taken as an example. 2001 IUBMB. Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Classical, or equilibrium thermodynamics (ET) yields information about the spontaneity of events (such as mechanical or chemical processes). Its scope is limited to closed systems, in which equilibrium states, and in"nitely small perturbations thereof, may be considered without reference to particular molecular structures or mechanisms. In such `ideal processesa, the #ow of time, and therefore, rates may not be considered. However, there are kinetic features of systems, such as rates of free energy dissipation, that are germane to the description of processes, and, in particular, to life processes. Moreover, in biochemistry, many reactions are far from equilibrium and biological systems are never closed, continuously receiving and discharging energy and matter. Indeed, biological entities that are closed (or at equilibrium) may be considered to be dead. Thus, ET must be applied to biological systems with some care.
2. Irreversible thermodynamics Open systems, being generally remote from equilibrium, are necessarily irreversible, and the branch of thermodynamics suited to their description is known as nonequilibrium or irreversible thermodynamics (IT). In the language of IT, the static (equilibrium) states of ET are replaced by #uxes (generalized rates of #ow or reaction). Moreover, the central importance of equilibrium in ET is replaced, in IT, by a focus on the steady state, where
* Corresponding author. Tel.: #1-207-725-3585; fax: #1-207-7253405. E-mail address: [email protected] (J.L. Howland).
all state attributes are constant in time. It is possible to summarize the utility of IT through a small collection of equations, of which the "rst generalizes the common observation that #uxes are often linear functions of the forces that drive them. An electrical example is Ohm's law, where current (the #ux) is proportional to the voltage (the driving force), the proportionality constant being the conductance. Other examples of forces include transmembrane potentials (that drive ion transport) and the di!erence in chemical potential, *k, which drives chemical reactions. The general `phenomenologicala equation describing these #uxes and forces is J"¸(X), where J represents the #ux, X the driving force, and L a proportionality constant, often termed as the coupling coe$cient [1,2]. Such linearity is the case only when events are close to equilibrium; otherwise, the equation becomes, at best, an approximation. To accommodate multiple #uxes and forces, the equation may be generalized as: J "&¸ X . G GH H Notice that this equation lends itself to matrix notation, a strategy particular suited to complex systems. Coe$cients may be zero (no coupling) or "nite. The sign of the coe$cient re#ects either positive or negative coupling (as in instances of co- and counter-transport processes). Also, when there are multiple #uxes and forces in a single system, any force may, in principle, be coupled to any #ux (and vice versa) and the matrix of coupling coe$cients (L's) can be an extremely useful summary. Notice also that the phenomenological equations are kinetic in nature, and coupling coe$cients may incorporate attributes of chemical rate constants. For instance, the occurrence
1470-8175/01/$20.00 2001 IUBMB. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 4 7 0 - 8 1 7 5 ( 0 0 ) 0 0 0 4 5 - X
302
J.L. Howland, M. Needleman / Biochemistry and Molecular Biology Education 28 (2000) 301}303
of catalysis can dramatically in#uence the proportionality between driving force (e.g., *k) and chemical reaction rate, as re#ected in the coupling coe$cient, ¸ GH [3]. IT is also able to evaluate entropy production in open systems. Increase in entropy, *S, mirrored by a comparable free-energy diminution, !*G, occurs in any irreversible process, and can be considered an increase in randomness or loss of information. In biochemistry, the rate of dissipation of free-energy sets limits to e$ciency, and is clearly required for a complete bioenergetic description. IT de"nes the dissipation function * the rate of entropy increase * as the sum of products of all #uxes (J 's) and relevant driving forces (X 's): U"RJ X G G In the case of chemical reactions, this takes the form U"k *n, k denoting the chemical potential and n, the number of moles reacting [1,2]. For reasons of history and convenience, one often employs the `a$nity of the reactiona, A, as the chemical-potential-based driving force for chemical reactions [4]. A$nity, taking into account the individual participants in a reaction, is de"ned as A"! l k , G G G where m is the stoichiometric coe$cient of the ith species G in the reaction (positive if a product, negative if a reactant). As the a$nity is derived from the chemical potential, it is a function of state and can thus be expressed under standard conditions as A.
3. Application to bioenergetics With emphasis on force-#ux coupling, together with the ability to analyze e$ciency via the dissipation function, IT is particularly suited to application to bioenergetics [3,4]. Indeed, its language, often resembling that of hydrodynamics, seems a natural one to describe the energetics of open systems as observed in biochemistry. For example, mention of the steady state commonly brings to mind a reservoir with an input and output such that the two #ows are constant and equal, and all features of the reservoir * volume, pressure, temperature, etc. * constant in time. The heuristic value of this image is enhanced by the observation that, just as entropy production becomes zero at equilibrium, it approaches a minimum, "nite value in the steady state. Accordingly, evolution has tended to produced organisms for which the steady state is the norm. In this connection, it should be noted that evolution has also generally maintained forces and #uxes (i.e. rates) to be as small as possible, again minimizing entropy production.
Bioenergetics chie#y describes energy converters, systems that couple input force}#ux pairs (J X ) with the corresponding output (J X ). Under steady-state condi tions, experimental or natural setting of the forces, X and X will lead to unique values for the #uxes, according to the phenomenological equations. However, if one force or #ux is set at zero, the system will also evolve in a characteristic fashion. For instance, if the input force (X ) is "nite and the output force (X ) is zero, the system is said to exhibit `level #owa, a condition observed in many transport systems, where a large opposing concentration gradient, which would create an output force, is minimal. Conversely, in the state of `static heada, J , the output #ow is zero, but X is "nite. This is observed, for example, in uptake transport by suspended cells, where a concentration gradient is created, opposing further transport. One encounters both level #ow and static head in the measurement of oxidative phosphorylation [5].
4. Oxidative phosphorylation Most aerobic cells synthesize ATP (or inorganic pyrophosphate) by oxidative phosphorylation, in which oxidation of a substrate, such as succinate, is coupled via a hydrogen ion circuit, to the phosphorylation of ADP [6,7]. In eukaryotic cells, oxidative phosphorylation occurs in mitochondria, with hydrogen ion #uxes across the inner membrane; in prokaryotic cells, it occurs across the plasma membrane, with ATP synthesis in the interior. In IT terms, the input force is the a$nity for the oxidation and the input #ux, the oxidation rate. Likewise, the output force is the a$nity for the phosphorylation (the free energy cost for ATP synthesis) and the output #ux, the rate of ATP formation. Since hydrogen ion #uxes couple input and output, three #uxes must be taken into account and a single, general equation describes entropy production for the entire system: U"J X #J X #J X . . & & - "J A #J *k #J A , . . & & - where subscripts P, H, and O indicate phosphorylation, net proton #ow, and oxidation, respectively [5,8]. Also *k is the electrochemical potential di!erence across the & membrane and A's are a$nities, as de"ned above. This equation describes a steady-state system, in which the system includes both the intra- and extramitochondrial regions, and a$nity determinations from both locations are equivalent. The three-#ux system is described by the following phenomenological equations: J "¸ A #¸ *k #¸ A , . . . .& & .- J "¸ A #¸ *k #¸ A , & .& . & & -& J "¸ A #¸ *k #¸ A . .- . -& & - -
J.L. Howland, M. Needleman / Biochemistry and Molecular Biology Education 28 (2000) 301}303
Here, the coupling coe$cients with a single subscript (e.g., L ) are `straighta coe$cients that connect #uxes . with their conjugate forces, as the phosphorylation rate (J ) with the a$nity for that reaction (A ). The double . . subscripts (as in L ) denote `cross coe$cientsa, in that .instance connecting the oxidation #ux with the phosphorylation a$nity. Cross coe$cients, in particular, may be zero: according to the chemiosmotic view, at a "nite value for *k , J and A are independent, as are J and & . . A [5]. In practice, complete information about #uxes may be obtained by straightforward measurements with an oxygen polarographic electrode, an approach often employed in teaching laboratory exercises (see Ref. [5], p. 189). Here, one measures J under various coupling conditions, e.g., static head, level #ow, or collapse of *k & using a proton ionophore. For example, inclusion of glucose plus hexokinase in the reaction medium traps ATP in the form of glucose phosphate and leads to the level #ow situation. In contrast, omission of ADP, by eliminating the phosphorylation reaction (J "0), leads . to an approach to the static head situation, also known as `state 4a. State 3 is J in the presence of ADP (when the oxidation is coupled to ATP synthesis) and the ratio of state 3 to state 4 is called the `respiratory control ratioa * an approximate measure of the `tightness of couplinga. Likewise, an approximate measure of the e$ciency of the net process is the ATP/O ratio; this is estimated by adding a small amount of ADP to the reaction mixture, producing a brief stimulation of oxygen consumption (state 4 to state 3 transition). In the ratio, the ATP is the amount of ADP added, and the O, the extra oxygen consumed (beyond state 4). These measurements may be carried out at di!erent values of *l , as & controlled by small additions of uncoupling proton ionophores. Finally, IT employs values of the coupling coe$cients, forces, and #uxes to provide explicit measures of degree of coupling and overall e$ciency of energy transducing systems [5,8]. For instance, in the case of coupling of phosphorylation to hydrogen ion #ux (the "rst two of the phenomenological equations, above) the degree of coupling, q, is q"¸ /(¸ ¸ ) . .& . &
303
This represents coupling intrinsic to the mechanism, and ranges from zero (no coupling) to one (stoichiometry). It thus di!ers from respiratory control, which is a!ected, among other things, by the accidents of extraneous ionic leakage across the coupling membrane. The e$ciency,c, of the same coupling process (output power/input power) is g"!J A /J *k . . . & & This di!ers from the ATP:O ratio, the ratio of the two #uxes, by also taking driving forces into account. Finally, the maximum attainable value of g may be obtained from the degree of coupling (above): g "q/[1#(1!q) ].
Thus, based upon the phenomenological equations, irreversible thermodynamics provides a framework for considering biological energy coupling, including analysis of force}#ow relationships, the rate of free-energy dissipation, and explicit measures of e$ciency (g) and the degree of coupling (q).
References [1] I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 2nd Edition, Wiley, New York, 1955, pp. 76}78. [2] L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931) 405}426. [3] H. Westerho!, K. van Dam, Thermodynamics and Control of Biological Free-Energy Transduction, Elsevier, Amsterdam, 1987. [4] R. Patterson, Irreversible thermodynamics as applied to biological systems, in: E.E. Bittar (Ed.), Membranes and Ion Transport, Vol. 1 , Wiley, New York, 1970, p. 128. [5] H. Rottenberg, S.R. Caplan, A. Essig, A thermodynamic appraisal of oxidative phosphorylation with special reference to ion transport by mitochondria, in: E.E. Bittar (Ed.), Membranes and Ion Transport, Vol. 1 , Wiley-Interscience, New York, 1970, p. 165. [6] P. Mitchell, Chemiosmotic Coupling and Energy Transduction, Glynn Research Ltd., Bodmin, 1968. [7] P. Mitchell, Nature 191 (1961) 144. [8] S.R. Caplan, Nonequilibrium thermodynamics and its application to biophysics, in: D.R. Sanadi (Ed.), Current Topics in Bioenergetics, Vol. 4 , Academic Press, New York, 1971, pp. 2}79.
Biological energy-coupling in terms of irreversible thermodynamics John L. Howland*, Matthew Needleman Department of Biology and Committee on Biochemistry, Bowdoin College, Brunswick, ME 04011-8465, USA
Abstract Irreversible thermodynamics provides an apt description of bioenergetic processes, of which oxidative phosphorylation is taken as an example. 2001 IUBMB. Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Classical, or equilibrium thermodynamics (ET) yields information about the spontaneity of events (such as mechanical or chemical processes). Its scope is limited to closed systems, in which equilibrium states, and in"nitely small perturbations thereof, may be considered without reference to particular molecular structures or mechanisms. In such `ideal processesa, the #ow of time, and therefore, rates may not be considered. However, there are kinetic features of systems, such as rates of free energy dissipation, that are germane to the description of processes, and, in particular, to life processes. Moreover, in biochemistry, many reactions are far from equilibrium and biological systems are never closed, continuously receiving and discharging energy and matter. Indeed, biological entities that are closed (or at equilibrium) may be considered to be dead. Thus, ET must be applied to biological systems with some care.
2. Irreversible thermodynamics Open systems, being generally remote from equilibrium, are necessarily irreversible, and the branch of thermodynamics suited to their description is known as nonequilibrium or irreversible thermodynamics (IT). In the language of IT, the static (equilibrium) states of ET are replaced by #uxes (generalized rates of #ow or reaction). Moreover, the central importance of equilibrium in ET is replaced, in IT, by a focus on the steady state, where
* Corresponding author. Tel.: #1-207-725-3585; fax: #1-207-7253405. E-mail address: [email protected] (J.L. Howland).
all state attributes are constant in time. It is possible to summarize the utility of IT through a small collection of equations, of which the "rst generalizes the common observation that #uxes are often linear functions of the forces that drive them. An electrical example is Ohm's law, where current (the #ux) is proportional to the voltage (the driving force), the proportionality constant being the conductance. Other examples of forces include transmembrane potentials (that drive ion transport) and the di!erence in chemical potential, *k, which drives chemical reactions. The general `phenomenologicala equation describing these #uxes and forces is J"¸(X), where J represents the #ux, X the driving force, and L a proportionality constant, often termed as the coupling coe$cient [1,2]. Such linearity is the case only when events are close to equilibrium; otherwise, the equation becomes, at best, an approximation. To accommodate multiple #uxes and forces, the equation may be generalized as: J "&¸ X . G GH H Notice that this equation lends itself to matrix notation, a strategy particular suited to complex systems. Coe$cients may be zero (no coupling) or "nite. The sign of the coe$cient re#ects either positive or negative coupling (as in instances of co- and counter-transport processes). Also, when there are multiple #uxes and forces in a single system, any force may, in principle, be coupled to any #ux (and vice versa) and the matrix of coupling coe$cients (L's) can be an extremely useful summary. Notice also that the phenomenological equations are kinetic in nature, and coupling coe$cients may incorporate attributes of chemical rate constants. For instance, the occurrence
1470-8175/01/$20.00 2001 IUBMB. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 4 7 0 - 8 1 7 5 ( 0 0 ) 0 0 0 4 5 - X
302
J.L. Howland, M. Needleman / Biochemistry and Molecular Biology Education 28 (2000) 301}303
of catalysis can dramatically in#uence the proportionality between driving force (e.g., *k) and chemical reaction rate, as re#ected in the coupling coe$cient, ¸ GH [3]. IT is also able to evaluate entropy production in open systems. Increase in entropy, *S, mirrored by a comparable free-energy diminution, !*G, occurs in any irreversible process, and can be considered an increase in randomness or loss of information. In biochemistry, the rate of dissipation of free-energy sets limits to e$ciency, and is clearly required for a complete bioenergetic description. IT de"nes the dissipation function * the rate of entropy increase * as the sum of products of all #uxes (J 's) and relevant driving forces (X 's): U"RJ X G G In the case of chemical reactions, this takes the form U"k *n, k denoting the chemical potential and n, the number of moles reacting [1,2]. For reasons of history and convenience, one often employs the `a$nity of the reactiona, A, as the chemical-potential-based driving force for chemical reactions [4]. A$nity, taking into account the individual participants in a reaction, is de"ned as A"! l k , G G G where m is the stoichiometric coe$cient of the ith species G in the reaction (positive if a product, negative if a reactant). As the a$nity is derived from the chemical potential, it is a function of state and can thus be expressed under standard conditions as A.
3. Application to bioenergetics With emphasis on force-#ux coupling, together with the ability to analyze e$ciency via the dissipation function, IT is particularly suited to application to bioenergetics [3,4]. Indeed, its language, often resembling that of hydrodynamics, seems a natural one to describe the energetics of open systems as observed in biochemistry. For example, mention of the steady state commonly brings to mind a reservoir with an input and output such that the two #ows are constant and equal, and all features of the reservoir * volume, pressure, temperature, etc. * constant in time. The heuristic value of this image is enhanced by the observation that, just as entropy production becomes zero at equilibrium, it approaches a minimum, "nite value in the steady state. Accordingly, evolution has tended to produced organisms for which the steady state is the norm. In this connection, it should be noted that evolution has also generally maintained forces and #uxes (i.e. rates) to be as small as possible, again minimizing entropy production.
Bioenergetics chie#y describes energy converters, systems that couple input force}#ux pairs (J X ) with the corresponding output (J X ). Under steady-state condi tions, experimental or natural setting of the forces, X and X will lead to unique values for the #uxes, according to the phenomenological equations. However, if one force or #ux is set at zero, the system will also evolve in a characteristic fashion. For instance, if the input force (X ) is "nite and the output force (X ) is zero, the system is said to exhibit `level #owa, a condition observed in many transport systems, where a large opposing concentration gradient, which would create an output force, is minimal. Conversely, in the state of `static heada, J , the output #ow is zero, but X is "nite. This is observed, for example, in uptake transport by suspended cells, where a concentration gradient is created, opposing further transport. One encounters both level #ow and static head in the measurement of oxidative phosphorylation [5].
4. Oxidative phosphorylation Most aerobic cells synthesize ATP (or inorganic pyrophosphate) by oxidative phosphorylation, in which oxidation of a substrate, such as succinate, is coupled via a hydrogen ion circuit, to the phosphorylation of ADP [6,7]. In eukaryotic cells, oxidative phosphorylation occurs in mitochondria, with hydrogen ion #uxes across the inner membrane; in prokaryotic cells, it occurs across the plasma membrane, with ATP synthesis in the interior. In IT terms, the input force is the a$nity for the oxidation and the input #ux, the oxidation rate. Likewise, the output force is the a$nity for the phosphorylation (the free energy cost for ATP synthesis) and the output #ux, the rate of ATP formation. Since hydrogen ion #uxes couple input and output, three #uxes must be taken into account and a single, general equation describes entropy production for the entire system: U"J X #J X #J X . . & & - "J A #J *k #J A , . . & & - where subscripts P, H, and O indicate phosphorylation, net proton #ow, and oxidation, respectively [5,8]. Also *k is the electrochemical potential di!erence across the & membrane and A's are a$nities, as de"ned above. This equation describes a steady-state system, in which the system includes both the intra- and extramitochondrial regions, and a$nity determinations from both locations are equivalent. The three-#ux system is described by the following phenomenological equations: J "¸ A #¸ *k #¸ A , . . . .& & .- J "¸ A #¸ *k #¸ A , & .& . & & -& J "¸ A #¸ *k #¸ A . .- . -& & - -
J.L. Howland, M. Needleman / Biochemistry and Molecular Biology Education 28 (2000) 301}303
Here, the coupling coe$cients with a single subscript (e.g., L ) are `straighta coe$cients that connect #uxes . with their conjugate forces, as the phosphorylation rate (J ) with the a$nity for that reaction (A ). The double . . subscripts (as in L ) denote `cross coe$cientsa, in that .instance connecting the oxidation #ux with the phosphorylation a$nity. Cross coe$cients, in particular, may be zero: according to the chemiosmotic view, at a "nite value for *k , J and A are independent, as are J and & . . A [5]. In practice, complete information about #uxes may be obtained by straightforward measurements with an oxygen polarographic electrode, an approach often employed in teaching laboratory exercises (see Ref. [5], p. 189). Here, one measures J under various coupling conditions, e.g., static head, level #ow, or collapse of *k & using a proton ionophore. For example, inclusion of glucose plus hexokinase in the reaction medium traps ATP in the form of glucose phosphate and leads to the level #ow situation. In contrast, omission of ADP, by eliminating the phosphorylation reaction (J "0), leads . to an approach to the static head situation, also known as `state 4a. State 3 is J in the presence of ADP (when the oxidation is coupled to ATP synthesis) and the ratio of state 3 to state 4 is called the `respiratory control ratioa * an approximate measure of the `tightness of couplinga. Likewise, an approximate measure of the e$ciency of the net process is the ATP/O ratio; this is estimated by adding a small amount of ADP to the reaction mixture, producing a brief stimulation of oxygen consumption (state 4 to state 3 transition). In the ratio, the ATP is the amount of ADP added, and the O, the extra oxygen consumed (beyond state 4). These measurements may be carried out at di!erent values of *l , as & controlled by small additions of uncoupling proton ionophores. Finally, IT employs values of the coupling coe$cients, forces, and #uxes to provide explicit measures of degree of coupling and overall e$ciency of energy transducing systems [5,8]. For instance, in the case of coupling of phosphorylation to hydrogen ion #ux (the "rst two of the phenomenological equations, above) the degree of coupling, q, is q"¸ /(¸ ¸ ) . .& . &
303
This represents coupling intrinsic to the mechanism, and ranges from zero (no coupling) to one (stoichiometry). It thus di!ers from respiratory control, which is a!ected, among other things, by the accidents of extraneous ionic leakage across the coupling membrane. The e$ciency,c, of the same coupling process (output power/input power) is g"!J A /J *k . . . & & This di!ers from the ATP:O ratio, the ratio of the two #uxes, by also taking driving forces into account. Finally, the maximum attainable value of g may be obtained from the degree of coupling (above): g "q/[1#(1!q) ].
Thus, based upon the phenomenological equations, irreversible thermodynamics provides a framework for considering biological energy coupling, including analysis of force}#ow relationships, the rate of free-energy dissipation, and explicit measures of e$ciency (g) and the degree of coupling (q).
References [1] I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 2nd Edition, Wiley, New York, 1955, pp. 76}78. [2] L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931) 405}426. [3] H. Westerho!, K. van Dam, Thermodynamics and Control of Biological Free-Energy Transduction, Elsevier, Amsterdam, 1987. [4] R. Patterson, Irreversible thermodynamics as applied to biological systems, in: E.E. Bittar (Ed.), Membranes and Ion Transport, Vol. 1 , Wiley, New York, 1970, p. 128. [5] H. Rottenberg, S.R. Caplan, A. Essig, A thermodynamic appraisal of oxidative phosphorylation with special reference to ion transport by mitochondria, in: E.E. Bittar (Ed.), Membranes and Ion Transport, Vol. 1 , Wiley-Interscience, New York, 1970, p. 165. [6] P. Mitchell, Chemiosmotic Coupling and Energy Transduction, Glynn Research Ltd., Bodmin, 1968. [7] P. Mitchell, Nature 191 (1961) 144. [8] S.R. Caplan, Nonequilibrium thermodynamics and its application to biophysics, in: D.R. Sanadi (Ed.), Current Topics in Bioenergetics, Vol. 4 , Academic Press, New York, 1971, pp. 2}79.