ESERGETICS O F TR.%KSIEST A S D STE.1DY STATES
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T H E ESERGETICS OF TRAXSIEST ASD STEADY STATES, K I T H SPECLiL R E F E R E S C E TO BIOLOGICAL SYSTEMS J. 11. REISER
ASD
6 . SPIEGELlIAS
Department of P h y s i o l o g y . Cniierszty o j Minnesota, .lfinneapolis, Minnesota, and Department o j Bacteriolog!l and Immunology, l h e School o j Xedicane, 11-ashington C7nziersity,St. Louis, .lIzssourz
Receiced October IO, 1944 ISTRODUCTIOS
Tn-o characteristics of living systems vitiate the indiscriminate use of the elementary l a m of physics and chemistry for the theoretical description of biological phenomena. One of them is the obvious structured nature of living cells. This unquestionably imposes geometrical constraints on the reactants ; and so complicating factors are introduced into the interpretation of those physicochemical measurements (e.g., osidation-reduction potential, equilibrium constants) which are usually taken to cypress tendencies of reactants in homogeneous systems (6). The other characteristic, the one n-hich the present paper n-ill consider, is the “open” nature of living cells. The recent ~ o r kof Schonheimer (10) and his collaborators has emphasized that the maintenance of structural integrity in most biological systems involves a continuous flon- of matter and energy through them. Thus, the time-independent state in biological systems is a “steady state” rather than an equilibrium (the distinction will be made precise in what follows) * This fact has received increasing recognition in modern biological literature, but has as yet not been implemented by a sufficiently detailed analysis to be of much value to biological theory or experiment. For example, one would like to knov- n-hat, properties distinguish the steady $tate, and, if such exist, whether they present satisfactory models for the known properties of living tissue. Some answers to these questions have been provided by the researches of Burton (3), who was able to shov among other things that “overshoot” and “undershoot” phenomena, so commonly observed in stimulus-response curves, are derivable from very simple models of open systems approaching the steady state. Bertalanffy (1) has recently used essentially the same type of analysis t o explain the relative independence from initial conditions reported in embryological and regeneration esperiments. Both of these authors, hoivever, confined their attention solely to the question of mas3 transport and its time variations , A closer approach to biological reality ~ o u l dalso consider the question of the energetics of such systems, and the properties Trhich stem from the fact that their maintenance requires an energy flus as well as mass transport. Such an inrestigation, in addition to its greater generality, could also hope properly to delineate the r61e of thermodynamics in biology,-a question which has received much
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attention of late. This branch of physics has been among the most frequently employed in biological theory. Thermodynamically derived laws of a special character, such as the law of mass action and the van’t Hoff law, are in regular use. Among the most popular applications in recent years ( 5 , 7 ) has been the calculation of free-energy changes in attempts to elucidate the mechanism of energy transport. Unfortunately, thermodynamics and most of the associated field of statistical mechanics are governed implicitly and explicitly by the assumptions TThich denote the equilibrium state. It is true that thermodynamics takes cognizance of non-equilibrium states and of irreversible changes. But for these cases it has only inequalities to offer instead of equations. These inequalities are not sufficiently restricted to permit derivation of a host of special empirical laws such as those mentioned in the preceding paragraph, which are derived under the assumption of equilibrium and strictly reversible processes. Thus all the most usual applications of thermodynamics are strictly not available for nonequilibrium states or irreversible processes. Since, however, biological states and processes are almost always of this latter character, it follows that the usual laws and results of classical thermodynamics offer little hope to biological theory. It is the aim of the present paper to render the needed formulations-at least the more important ones-explicit. The authors have worked out many consequences of this formulation, and these n-ill appear in future communications. In what follom u-e propose to analyze the problem in as general a way as possible-to indicate the paths of detailed exploration which are made available by the fundamental formulae. CHANGES O F STATE
Let us consider a system composed of elements such as molecules or ions. These elements are capable of existing in various states (e.g., those characterized by given energy values or a given kind of chemical combination). Let the states be numbered, and denote the number of particles in the ithstate by N, , 1T-e knoly that in general a particle does not always remain in one state. Collisions, movement from one part of a force field t o another, combination and dissociation produce changes of the states of the individual particles. As such changes occur, the set of numbers -Ir$ may or may not change likewise. Denote the change in -Yt during a given standard time interval by A N , , Denote the number of particles leaving the state z for the state j in the unit of time by T,, . Using the symbol B to denote summation over the possible values of j , n-e may 2
write:
Ci Tij
That is, the increase in the number in state i is the number entering it from all other states less the number leaving it for any other state. If the right side of equation 1 does not vanish, S,is changing, and we have a non-stationary state. If the right side of equation 1 vanishes, S,does not change. If this is true for every i, the system is in a steady state.
ESERGETICS O F TRASSIENT AKD STEADY SThTES
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If for every 1-alue of i and j ive have :
T , , = T,, (2) then also the right side of equation 1 will vanish. But this condition is much more special and restrictive t.han the vanishing of equation 1, and it is this condition which corresponds to an equilibrium state. This is what is usually known as the principle of detailed balancing. Let us illustrate by a simple example which is closely related to the biological problems with which we shall be concerned. Suppose a system is characterized by the concentration of one or more substances, which may diffuse into or out of the system and which may undergo chemical reaction in the system. For any substance, with concentration I L , the rate of change of the concentration is given by :
* dt
=
( R , - R:)
+ (Rd - Ri) = ( R , + Rd) - (RL + Ri)
(3)
where R , is the rate a t which the substance is produced through chemical reaction, and R6 the rate at nhich it disappears through chemical reaction; while Rd and Rl have the corresponding significance for the rate of diffusion into and out of the system, The steady state n-odd be characterized by the vanishing of the right side of equation 3; physically this would mean that the net rate at which the substance flows into the y-stem equals the net rate a t which it is consumed by chemical reaction, or that the net rate of production equals the net rate of outflow. The equilibrium state would demand that R , = RL and Rd = Ri ; that is, the chemical reaction is in equilibrium, and there is no diffusion. If there were more than one chemical reaction into which the substance could enter, each reaction must be in equilibrium separately: e.g., Rcl = RLl, Rcz = RL2,etc. It is obvious from thiq that the steady state, as here defined, corresponds most closely to the char'icteristics oi a biological system Tvhenever it is not changing with time, while the equilibrium qtate is one which is likely to characterize such a system only when it is dead. living system is marked hy a continual chemical reactivity, and :i continual diffusion of metabolites. A \
T H E DYS.IMICS O F PHYSICAL SYSTEMS
The systems in which we are interested may be characterized chiefly by specifying their chemical composition, their structure, and their motion. The first two may be treated together by giving the chemical composition a t every point of space. By motion TI e shall understand every form from the slow motion of gron-th through protoplasmic streaming and local deformation and shape change t o gross motion thiough .;pace. This may likevise be dealt with by specifying the vector velocity of every point of the system. Especially significant from the biological point of view are the energy relations of the system. -in organism is the site of a continuous flow of energy, and of chemical, transformation of energy lietn-een its various forms,-thermal, mechanical, electrical, etc. &Anytreatment in physical terms must give some
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account of this and indicate lion it is related to the a,-ociated motion- and chemical transf ormations. -1system of equations of the qort \\ hich is iequired may be found in the recent work of Eckart (4). However, the equations of Eckart are so expressed as to be most appropriate to a situation where no special distinction is made among the various constituents of the system, and where either no phase boundarieq are observed, or such boundaries move on the average \\ith all the constituent?. SOITthe experimental finding5 of the biologist reveal a somewhat different situation. He always finds a phaie boundary. Moreoi.er, he finds that one set of components does not diffuqe through this boundary, qo that the movement of the boundary is the average movement of these .ub*tances alone. These non-diffusible coniponentc: are in a qen3e permanent, and the region nithin the boundary they determine is called the cell (or the organism). Thus the equations of Eckait mu-t he
