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Co-operativity in biological machines
J. theor. Biol. (1973) 39, 653-657
Co-operativity in Biological Machines D. MARKOWITZ~
AND
R. M.
NISBET$
School of Biological Sciences, University of Sussex, Falmer, Brighton BN1 9QG, England (Received 17 October 1972)
McClare has recently discussed the properties of machines which operate too fast for there to be appreciable thermalization between components. We argue that co-operative behaviour is likely in those machines and that if there is co-operativity, the machine cannot be treated as the superposition of a large number of “molecular energy machines”. This point may be relevant to models of muscle contraction. 1. Introduction This paper follows the lead of three recent papers by McClare (1971, 1972a,b) in which he models processes such as muscle contraction where the necessary energy is provided by the breakdown of ATP to ADP. McClare points out that biological machines operate far from thermodynamic equilibrium and yet operate rapidly and with remarkable efficiency. He argues that these features are incompatible in a conventional thermodynamic machine and introduces the “molecular energy machine” where lossless energy transfer is effected by a molecular resonance. In the absence of sufficient spectroscopic data, we must reserve judgement on whether muscle contraction or active transport involves this type of molecular mechanism; our own attitude is that at least a prima facie case has been made and that it is thus worthwhile to explore the physical properties of these systems. Our main point will be that a largely macroscopic description of a machine with resonant energy transfer is perfectly possible and that it is neither necessary nor useful to treat individual molecules as components of such a machine. Nor do we accept the view (McClare 1971) that the internal energy is, of necessity, a sum of separate “molecular energies”. Rather, we emphasize ? Permanent address: Department of Physics, University of Connecticut, Storrs, Connecticut 06268, U.S.A. $ Present address: Department of Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland. 653
654
D. MARKOWITZ
AND
R. M. NISBET
the possibility of co-operative behaviour in which the dynamical behaviour of individual molecules is synchronized by a field which, in turn, is maintained by this very synchronized motion of the molecules. At no point do we discuss “thermodynamic” properties of individual molecules-an approach which at best is only valid in the absence of co-operativity. 2. Resonance Machines A system which transforms one form of energy into another is called a rransducer.A machineis a transducer that performs useful work, i.e. converts
one form of stored energy into another. Energy is said to be stored if it is available for conversion to another form within a time 2; thus a characteristic operating time z is involved in the definition of a specific machine, If the operation of the machine is naturally described in terms of parameters attributable to certain parts of the machine, then these parts are called componentsof the machine. McClare (1971) has distinguished between two types of efficient machine: (i) Conventional chemical machines (also known as constrained equilibrium machines or thermodynamic machines) operate sufficiently slowly for the entropy production per cycle to be (in a given context) acceptably small. (ii) Molecular energy machines operate too fast for thermal equilibration to occur. Energy transduction is effected by a suitable resonance and not by variation in a thermodynamic parameter. Molecular energy machines thus operate in a regime far from thermodynamic equilibrium; their behaviour is essentially dynamical in nature and entropy production per cycle is negligible. We retain the name conventional machine for the first type, but for reasons to be given later we rename the second a resonancemachine.
It is the cycle time, r, that determines the regime in which a machine operates and for both types of machine there exist upper and lower bounds, z,, and zt, on r, z, being the cycle time at which all the energy is wastefully dissipated and z, the response time of the slowest component of the machine. For a conventional machine z, characterizes the time necessary for thermal leaks to the environment while zI is the time required for thermalization between components. For a resonance machine z, is the thermalization time between components and z,is the largest response time of a single component.? We can illustrate the two regimes and the bounds on the cycle time with t We are regarding a heat reservoir as a component of the machine and not of its environment. A leak is a transfer of heat from machine to environment.
CO-OPERATIVITY
IN
BIOLOGICAL
MACHINES
655
two well-known phenomena involving magnetic spins in solids-the production of low temperatures by adiabatic demagnetization, and spin echoes. The former process involves a conventional refrigeration cycle consisting of successive isothermal magnetization and adiabatic demagnetization of a paramagnetic salt; the bound z, is the spin-lattice relaxation time while r, is the thermal leak time. Spin echoes also involve the cyclic appearance and disappearance of the magnetization but through a resonant transfer of energy ; in this case z, is the spin-lattice relaxation time while T, is the quantummechanical spin dephasing time-a non-thermal quantity. The reader will find more detail on adiabatic demagnetization in Zemansky (1957) and on spin echoes in Abragam (1961) or Schumacher (1970). 3. Co-operative Resonance Machines There is nothing in the previous discussion of resonance machines to suggest that individual molecular events in a resonance machine are of necessity autonomous. Indeed Newman (1970) has pointed out that when a system becomes “organized” in some sense, then new “basic entities” emerge as the natural components in terms of which to describe the system. Newman’s concept of “organization” corresponds to the concept of “cooperativity”-a laser and a normal lamp are examples of co-operative and non-co-operative machines producing light-and we now argue that the presence of a co-operative effect in a machine leads to a complete change in what can usefully be considered a component of the machine. The foremost physical example of a co-operative resonance machine is the laser where energy is transferred from the excited molecules to the radiation field through stimulated emission of photons. For a very readable introduction to the principles of laser action we refer the reader to Allen (1969), and we have confined ourselves here to illustrating in Fig. 1 the atomic cycle of an idealized three-level laser. As the laser is a resonance machine, the criteria of the previous section must be satisfied and we find z, = A;:‘, and TV= max(W,,-‘, S,,-r). (It is noteworthy that the idea of largely autonomous molecules may already break down at this point, e.g. in a solid state laser, as the electronic energy levels shown in Fig. 1 refer to the entire lasing medium and not to individual molecules.) The bounds on the cycle time are in fact of limited value for an analysis of the machine cycle as they fail to exploit the coherence in phase of the emitted photons. Because of this coherence, it can be shown (Allen, 1969) that if there are n molecules emitting radiation then the energy of this coherent radiation is proportional to n2 and thus the resonance machine is not a superposition of a large number of molecular energy machines. In addition, unless a certain minimum number of molecules
656
D. MARKOWITZ
AND
R. hi. NISBET
Energy 4 Bond Level 3
-i
of excited
states
I ,
I I
Level
2
43
Level
I
FIG. 1. Energy level diagram for an idealized three-level laser. W,, denotes the stimulated transition probability per second from level i to levelj. AIJ denotes the spontaneous transition probability per second from level i to level j. Sa2 denotes the radiationless transition probability per second from level 3 to level 2.
are excited, the machine will not start to lase. In short, rather than treat molecules and photons as components of the machine, we can and should analyse it in terms of populations of energy levels and properties of the radiation field. It is possible that these considerations are relevant to the problem of muscle contraction. A model of muscle must involve a mechanism for rapid, efficient utilization of energy at each actin-myosin bridge (the energy utilization problem) but in addition, after the release of Ca’+ ions into the sarcoplasm, each bridge must utilize ATP at approximately the same rate and time as other bridges (the synchronization or co-operation problem). McClare (19726) has emphasized the importance of the latter problem and in his muscle model has introduced a mechanism whereby all the bridges in a given thin filament (and perhaps in an entire sarcomere) are coupled by the a-helix of tropomyosin which acts as a “wave-guide” for the transmission of phonons. Thus “once one ATP has hydrolysed spontaneously the reaction could be made to spread autocatalytically along the thin filament like a controlled explosion” (McClare 1972b). An idea remarkably akin to co-operativity has crept into a “molecular energy” muscle model. If there is strong co-operative coupling between bridges it does not seem to involve the environment; at least the concentration of Cazf and of ATP remains essentially constant throughout a contraction, the ATP used being replaced by the rapid breakdown of phosphocreatine (Lehninger, 1970, and references therein). A co-operative mechanism like McClare’s seems much
CO-OPERATIVITY
IN
BIOLOGICAL
MACHINES
657
more likely and in fact there is a notable similarity between this mechanism and the laser, our typical co-operative, resonance machine. The output field of a laser-the radiation field whose coherence is also responsible for the efficiency of the machine-is the same field as is responsible for the cooperativity. The corresponding “output field” for contracting muscle is the mechanical strain in the myofilaments and it thus seems natural to consider co-operative muscle models with the dynamical behaviour of individual bridges coupled by phonons in the thin filaments. 4. Conclusions We have tried to demonstrate that resonance machines are theoretically respectable and that a largely macroscopic description of such machines is possible. We have supported the idea that the regime in which a machine operates is determined by the time required for thermalization between components. We have argued that co-operative phenomena are likely in a resonance machine and that where they occur the machine cannot be treated as the superposition of a number of “molecular energy machines”. We have speculated on a type of co-operativity which may occur in muscle. Only experiments will ultimately determine whether biological energy transduction involves a resonance machine, and if so, whether co-operativity is present ; what is clear is that no controversial thermodynamic considerations arise in the description of resonance machines. We acknowledge valuable discussions with Drs B. C. Goodwin, A. J. Leggett and S. A. Newman. REFERENCES ABRAGAM, A. (1961). l?te Principles of ALLEN, L. (1969). Essentials of Lasers. LEHNINGER, A. L. (1970). Biochemistty.
Nuclear Magnetism. Oxford: Clarendon. London: Pergamon Press. New York: Worth Publishers.
MCCLARE,C. W. F. (1971).J. theor. Biol. 30, 1.
MCCLARE, C. W. F. (1972u).J. theor. Biof. 35,233. MCCLARE, C. W. F. (1972b). J. theor. Biol. 35, 569. NEWMAN, S. A. (1970). J. theor. Biol. 28,411. SCHUMACHER, R. T. (1970). Magnetic Resonance. New ZEMANSKY, M. W. (1957).Heat and i%ermodynamics.
York: Benjamin. New York: McGraw-Hill.
Co-operativity in Biological Machines D. MARKOWITZ~
AND
R. M.
NISBET$
School of Biological Sciences, University of Sussex, Falmer, Brighton BN1 9QG, England (Received 17 October 1972)
McClare has recently discussed the properties of machines which operate too fast for there to be appreciable thermalization between components. We argue that co-operative behaviour is likely in those machines and that if there is co-operativity, the machine cannot be treated as the superposition of a large number of “molecular energy machines”. This point may be relevant to models of muscle contraction. 1. Introduction This paper follows the lead of three recent papers by McClare (1971, 1972a,b) in which he models processes such as muscle contraction where the necessary energy is provided by the breakdown of ATP to ADP. McClare points out that biological machines operate far from thermodynamic equilibrium and yet operate rapidly and with remarkable efficiency. He argues that these features are incompatible in a conventional thermodynamic machine and introduces the “molecular energy machine” where lossless energy transfer is effected by a molecular resonance. In the absence of sufficient spectroscopic data, we must reserve judgement on whether muscle contraction or active transport involves this type of molecular mechanism; our own attitude is that at least a prima facie case has been made and that it is thus worthwhile to explore the physical properties of these systems. Our main point will be that a largely macroscopic description of a machine with resonant energy transfer is perfectly possible and that it is neither necessary nor useful to treat individual molecules as components of such a machine. Nor do we accept the view (McClare 1971) that the internal energy is, of necessity, a sum of separate “molecular energies”. Rather, we emphasize ? Permanent address: Department of Physics, University of Connecticut, Storrs, Connecticut 06268, U.S.A. $ Present address: Department of Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland. 653
654
D. MARKOWITZ
AND
R. M. NISBET
the possibility of co-operative behaviour in which the dynamical behaviour of individual molecules is synchronized by a field which, in turn, is maintained by this very synchronized motion of the molecules. At no point do we discuss “thermodynamic” properties of individual molecules-an approach which at best is only valid in the absence of co-operativity. 2. Resonance Machines A system which transforms one form of energy into another is called a rransducer.A machineis a transducer that performs useful work, i.e. converts
one form of stored energy into another. Energy is said to be stored if it is available for conversion to another form within a time 2; thus a characteristic operating time z is involved in the definition of a specific machine, If the operation of the machine is naturally described in terms of parameters attributable to certain parts of the machine, then these parts are called componentsof the machine. McClare (1971) has distinguished between two types of efficient machine: (i) Conventional chemical machines (also known as constrained equilibrium machines or thermodynamic machines) operate sufficiently slowly for the entropy production per cycle to be (in a given context) acceptably small. (ii) Molecular energy machines operate too fast for thermal equilibration to occur. Energy transduction is effected by a suitable resonance and not by variation in a thermodynamic parameter. Molecular energy machines thus operate in a regime far from thermodynamic equilibrium; their behaviour is essentially dynamical in nature and entropy production per cycle is negligible. We retain the name conventional machine for the first type, but for reasons to be given later we rename the second a resonancemachine.
It is the cycle time, r, that determines the regime in which a machine operates and for both types of machine there exist upper and lower bounds, z,, and zt, on r, z, being the cycle time at which all the energy is wastefully dissipated and z, the response time of the slowest component of the machine. For a conventional machine z, characterizes the time necessary for thermal leaks to the environment while zI is the time required for thermalization between components. For a resonance machine z, is the thermalization time between components and z,is the largest response time of a single component.? We can illustrate the two regimes and the bounds on the cycle time with t We are regarding a heat reservoir as a component of the machine and not of its environment. A leak is a transfer of heat from machine to environment.
CO-OPERATIVITY
IN
BIOLOGICAL
MACHINES
655
two well-known phenomena involving magnetic spins in solids-the production of low temperatures by adiabatic demagnetization, and spin echoes. The former process involves a conventional refrigeration cycle consisting of successive isothermal magnetization and adiabatic demagnetization of a paramagnetic salt; the bound z, is the spin-lattice relaxation time while r, is the thermal leak time. Spin echoes also involve the cyclic appearance and disappearance of the magnetization but through a resonant transfer of energy ; in this case z, is the spin-lattice relaxation time while T, is the quantummechanical spin dephasing time-a non-thermal quantity. The reader will find more detail on adiabatic demagnetization in Zemansky (1957) and on spin echoes in Abragam (1961) or Schumacher (1970). 3. Co-operative Resonance Machines There is nothing in the previous discussion of resonance machines to suggest that individual molecular events in a resonance machine are of necessity autonomous. Indeed Newman (1970) has pointed out that when a system becomes “organized” in some sense, then new “basic entities” emerge as the natural components in terms of which to describe the system. Newman’s concept of “organization” corresponds to the concept of “cooperativity”-a laser and a normal lamp are examples of co-operative and non-co-operative machines producing light-and we now argue that the presence of a co-operative effect in a machine leads to a complete change in what can usefully be considered a component of the machine. The foremost physical example of a co-operative resonance machine is the laser where energy is transferred from the excited molecules to the radiation field through stimulated emission of photons. For a very readable introduction to the principles of laser action we refer the reader to Allen (1969), and we have confined ourselves here to illustrating in Fig. 1 the atomic cycle of an idealized three-level laser. As the laser is a resonance machine, the criteria of the previous section must be satisfied and we find z, = A;:‘, and TV= max(W,,-‘, S,,-r). (It is noteworthy that the idea of largely autonomous molecules may already break down at this point, e.g. in a solid state laser, as the electronic energy levels shown in Fig. 1 refer to the entire lasing medium and not to individual molecules.) The bounds on the cycle time are in fact of limited value for an analysis of the machine cycle as they fail to exploit the coherence in phase of the emitted photons. Because of this coherence, it can be shown (Allen, 1969) that if there are n molecules emitting radiation then the energy of this coherent radiation is proportional to n2 and thus the resonance machine is not a superposition of a large number of molecular energy machines. In addition, unless a certain minimum number of molecules
656
D. MARKOWITZ
AND
R. hi. NISBET
Energy 4 Bond Level 3
-i
of excited
states
I ,
I I
Level
2
43
Level
I
FIG. 1. Energy level diagram for an idealized three-level laser. W,, denotes the stimulated transition probability per second from level i to levelj. AIJ denotes the spontaneous transition probability per second from level i to level j. Sa2 denotes the radiationless transition probability per second from level 3 to level 2.
are excited, the machine will not start to lase. In short, rather than treat molecules and photons as components of the machine, we can and should analyse it in terms of populations of energy levels and properties of the radiation field. It is possible that these considerations are relevant to the problem of muscle contraction. A model of muscle must involve a mechanism for rapid, efficient utilization of energy at each actin-myosin bridge (the energy utilization problem) but in addition, after the release of Ca’+ ions into the sarcoplasm, each bridge must utilize ATP at approximately the same rate and time as other bridges (the synchronization or co-operation problem). McClare (19726) has emphasized the importance of the latter problem and in his muscle model has introduced a mechanism whereby all the bridges in a given thin filament (and perhaps in an entire sarcomere) are coupled by the a-helix of tropomyosin which acts as a “wave-guide” for the transmission of phonons. Thus “once one ATP has hydrolysed spontaneously the reaction could be made to spread autocatalytically along the thin filament like a controlled explosion” (McClare 1972b). An idea remarkably akin to co-operativity has crept into a “molecular energy” muscle model. If there is strong co-operative coupling between bridges it does not seem to involve the environment; at least the concentration of Cazf and of ATP remains essentially constant throughout a contraction, the ATP used being replaced by the rapid breakdown of phosphocreatine (Lehninger, 1970, and references therein). A co-operative mechanism like McClare’s seems much
CO-OPERATIVITY
IN
BIOLOGICAL
MACHINES
657
more likely and in fact there is a notable similarity between this mechanism and the laser, our typical co-operative, resonance machine. The output field of a laser-the radiation field whose coherence is also responsible for the efficiency of the machine-is the same field as is responsible for the cooperativity. The corresponding “output field” for contracting muscle is the mechanical strain in the myofilaments and it thus seems natural to consider co-operative muscle models with the dynamical behaviour of individual bridges coupled by phonons in the thin filaments. 4. Conclusions We have tried to demonstrate that resonance machines are theoretically respectable and that a largely macroscopic description of such machines is possible. We have supported the idea that the regime in which a machine operates is determined by the time required for thermalization between components. We have argued that co-operative phenomena are likely in a resonance machine and that where they occur the machine cannot be treated as the superposition of a number of “molecular energy machines”. We have speculated on a type of co-operativity which may occur in muscle. Only experiments will ultimately determine whether biological energy transduction involves a resonance machine, and if so, whether co-operativity is present ; what is clear is that no controversial thermodynamic considerations arise in the description of resonance machines. We acknowledge valuable discussions with Drs B. C. Goodwin, A. J. Leggett and S. A. Newman. REFERENCES ABRAGAM, A. (1961). l?te Principles of ALLEN, L. (1969). Essentials of Lasers. LEHNINGER, A. L. (1970). Biochemistty.
Nuclear Magnetism. Oxford: Clarendon. London: Pergamon Press. New York: Worth Publishers.
MCCLARE,C. W. F. (1971).J. theor. Biol. 30, 1.
MCCLARE, C. W. F. (1972u).J. theor. Biof. 35,233. MCCLARE, C. W. F. (1972b). J. theor. Biol. 35, 569. NEWMAN, S. A. (1970). J. theor. Biol. 28,411. SCHUMACHER, R. T. (1970). Magnetic Resonance. New ZEMANSKY, M. W. (1957).Heat and i%ermodynamics.
York: Benjamin. New York: McGraw-Hill.