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NETWORK THERMODYNAMICS
AN OVERVIEW
ALAN S. PERELSON
From the Theoretical Division, University ofCaiforia Los Alamos
Scientific Laboratory,^ Los^ Alamos,^ New^ Mexico^87544
INTRODUCTION
Thermodynamics was one of Aharon Katzir-Katchalsky's great loves. To^ the^ physi-
cists of the late nineteenth and early twentieth centuries thermodynamics epitomized
the simplicity, generality, and symmetry one expected from a physical theory. Aharon,
however, was well aware of its limitations when applied in the biological realm. He
saw a need to extend the scope of thermodynamics in a new direction, in hopes of
addressing the puzzle of biological organization and complexity from a phenomeno-
logical viewpoint. George Oster and I had the great privilege of collaborating with
Aharon on this, his last^ major research effort. In this memorial symposium on thermo-
dynamics and^ life^ it^ seems^ particularly appropriate^ to^ recount^ the^ history^ of what
Aharon christened "network thermodynamics," and to discuss its relevance and
potential in modern biophysics.
In 1968 Aharon brought to Berkeley news of Prigogine's work on dissipative struc-
tures (Prigogine, 1969). His enthusiasm for discovering the basis of life in temporal
and spatial structuring was, as many of you know, quite contagious; George Oster,
then a postdoctoral student, and I, then a graduate student, became infected. We
understood how temporal oscillations in electrical and mechanical systems worked. If
one had two energy storage devices, such as an inductor and capacitor, one could
transfer energy between the devices periodically. Therefore, we began looking for
chemical analogs of^ capacitors and inductors. We also realized that the mere presence
of capacitors and inductors was not enough; their interconnection had to be precisely
right. Thus, we also began searching for ways to quantitate the notion of topology in
chemical systems. Graphical and network techniques naturally came to mind. The
idea of applying concepts in electrical network theory to thermodynamics was not new,
Meixner (1963, 1964, 1965, 1966 a,b) had already proposed a nonequilibrium thermo-
dynamic theory based on the general theory of linear passive systems. Out of a syn-
thesis of concepts from thermodynamics, circuit theory, graph theory, and differential
geometry, network thermodynamics was developed.
Unfortunately, Katchalsky did^ not^ live^ to see^ the^ work^ grow to^ fruition.^ During his
lifetime we spent much time "putting old wine in new bottles" so to speak; much of
nonequilibrium thermodynamics as applied to biophysics was^ reformulated^ in^ terms^ of
BIOPHYSICAL JOURNAL VOLUME 15 1975 667
network thermodynamics, theoretical results such as the Glansdorff-Prigogine dx P in-
equality and b,P stability criterion were given rigorous and elegant network formula-
tions (Glansdorff and Prigogine, 1971; Oster and Desoer, 1971), and the framework for
a nonlinear far-from-equilibrium theory of thermodynamic processes was created
(Oster et al., 1971). Frankly, however, this period produced no significant new results.
Before discussing more recent work it is first worthwhile to explain in some detail
what network thermodynamics is and why Aharon thought it worthwhile to translate known results into network thermodynamic terms.
WHAT IS NETWORK THERMODYNAMICS?
Great progress had been made in understanding simple thermodynamic systems using
equilibrium thermodynamics, and nonequilibrium thermodynamics as developed by
Onsager, Prigogine, deGroot, and^ Meixner. However, Aharon could see clearly that
the conventional tools of irreversible thermodynamics were not suitable for treating
the complex nonlinear biological processes which occur far from thermodynamic
equilibrium. As an analogy consider the problem of analyzing or designing a radio.
The basic physics is completely summarized in Maxwell's equations. One could, in
principle, integrate these equations to obtain an overall system description. In prac-
tice, however, too^ much irrelevant^ information^ is^ required to^ integrate the^ equations
over such a complex object. As the engineers have discovered, a lumped parameter
approximation to Maxwell's equations-network theory is the most convenient tool
to use. Since the complexity of a biological organelle such as a membrane or chloro-
plast is more akin to a radio than an isotropic continuum, a lumped parameter ap-
proximation to the field equations or irreversible thermodynamics and continuum
mechanics would seem to be the preferred description. Network thermodynamics is
precisely such^ an^ approximation. By establishing an^ isomorphorism between the
underlying mathematical structure of network theory and thermodynamics, problems
in thermodynamics can be formulated and solved using network methods.
The advantages of a network approach are numerous. Engineering experience and
network methods of solving large nonlinear systems can be brought to bear on thermo-
dynamic problems. For example, complex systems can commonly be thought of as
composite systems made up of interconnected subsystems. In many instances such
systems can be "torn" or decomposed into their subsystems, the subsystems solved,
and these solutions used to generate the solution to the whole system. Such a de-
composition technique can be applied to network models of diffusion-reaction systems.
Besides (^) specific method of (^) analysis, there are (^) many network theorems that can be used
to solve thermodynamic problems. One of the most important is Tellegen's theorem.
This is a powerful theorem whose generality derives from the fact that it is independent
of the nature of the network elements-that is, whether they are linear or nonlinear,
reciprocal or (^) nonreciprocal. All that matters is the network topology; that is, the way in which the elements are interconnected (Penfield et al., 1970).
Graphical representations similar to engineering circuit diagrams can be constructed
for thermodynamic systems. Although the proverb that a picture is worth a thousand
668 BIOPHYSICAL JOURNAL VOLUME 15 1975
however, we assume that one can conceptually separate these processes into dissipative
and nondissipative parts.
EQUILIBRIUM THERMODYNAMICS
In order to demonstrate the application of network thermodynamics we consider some
examples. Let us start with equilibrium thermodynamics. Equilibrium thermo-
dynamics regards a (^) system as a (^) "black box" that can be described by a finite number of
external measurements. For example, the piston and cylinder shown in Fig. 1 a can
interact with the environment through the mechanical movement of the piston or by
exchanging mass and heat. We can represent this system schematically as shown in
Fig. 1 b, where each mode of interaction has been represented by a line called a port.
In circuit parlance this object is called an n-port. In classical thermodynamics one
usually postulates the existence of a real valued function, the internal energy U, which
depends on a set of thermodynamic displacements q, which include the entropy, S, the
volume, V, and the mole numbers, Ni. Then one usually defines a set of conjugate
thermodynamic potentials by differentiating U. For example the chemical potential
Ai a^ U/ONi. We^ shall^ denote thermodynamic potentials^ e1^ and^ call^ them efforts. In
network thermodynamics we follow an approach to thermodynamics initiated by
Bronsted in the 1930s. We focus on measureable quantities not on energy functions
for reasons which become apparent when we treat irreversible processes. Thus e1 and
qi are^ taken^ as^ primitive^ variables^ in^ network^ thermodynamics^ and^ we^ postulate^ that
the properties of any n-port can be described by relations between these port variables,
called constitutive relations or "equations of state" (i.e., the local equilibrium postu-
late). An example is the relation g - g,(T,p,N,, * * N.). In general, a constitutive
Diathermol
Semi - pormeable
Adiabotic (a)
T Is
N V
(b) FIGURE 1 (a) Piston and cylinder. (b) Its n-port representation.
670 BIOPHYSICAL JOURNAL VOLUME 15 1975
relation is a map F which assigns to a set of independent port variables the set of con-
jugate port variables:
F: RXR^ RI
x ry (1)
where x and y are vectors of port variables.
Next, consider two simple thermodynamic systems surrounded by rigid adiabatic
walls and separated by a rigid semipermeable membrane as shown in Fig. 2. Conven-
tionally, the equilibrium of this composite system is found by minimizing the total
energy of the system subject to the constraint N1 + N2 = constant. One finds U is a
minimumwhenA^1 =^ js2-
In network theory equilibrium is generally calculated by imposing the requirements
of Kirchhoffs laws at the interconnection of two systems. This method, which does
not require the introduction of an energy function, is equivalent to the usual thermo-
dynamic method. The continuity of potential at system boundaries is Kirchhoff's
voltage law (KVL). Conservation of thermodynamic displacements is a form of
Kirchhoff's current law (KCL). For the system illustrated in Fig. 2
Al =^ M2 (2)
is KVL, while
N, + N2 =^ constant, (3A)
or
NI+N2=0 (3B)
is KCL. Since S and V are constant for systems 1 and 2
dU =^ AIMdN + A2dN2- (4)
Eqs. 2 and 3 imply
dU = (Al - A2)dNI = 0, (5)
i.e., the system energy is at a minimum. In general, utilizing Kirchhoff's laws one can
obtain all of the standard equilibrium conditions and, in fact, one can always show
T (^) I.0 T (^) SSO
PI Al I~~~~jL2 _P V1-O N (^) I N 2 V2.
FIGURE 2 Two (^) thermodynamic systems surrounded (^) by rigid adiabatic walls and (^) separated by a semipermeable membrane.
ALAN S. PERELSON Network Thermodynamics 671
J
JI~ ~ ~ ~ ~ ~ A
n JI +^ _ J,l
J4LJ
J2 (^) M J? L J" 2 (b) FIGURE 3 (a) The steady-state transport of a single species across a^ membrane^ viewed as a 1-port. (b) The steady-state transport of two species.
Irreversible thermodynamics is generally concerned with coupled flow phenomena.
The obvious generalization is an n-port resistor defined by a constitutive relation
between the flows of n solutes and the set of n chemical potential differences (Fig. 3 b).
We postulate that, whatever its internal transport mechanism, the observed behavior
of the membrane in steady state can be completely characterized^ by^ a^ finite^ set^ of^ port
variables. If this is not the case we^ must^ look^ for^ additional^ variables^ or^ assume^ that
the membrane has^ not^ yet reach^ steady^ state.^ (This^ is^ the^ nonequilibrium equivalent
of the "local state" postulate.)
We place no restriction on the form of resistive constitutive relations; they may be
linear or nonlinear. Analogous to equilibrium systems we say that an n-port is recipro-
cal if the Jacobian matrix of its constitutive equation is^ symmetric. For^ linear^ constitu-
tive equations this is the usual Onsager reciprocity.^ In^ the nonlinear^ case^ it^ is the
obvious generalization. However,^ in^ general,^ we^ need^ make^ no^ assumption^ of
reciprocity.
If all the resistors in the systems are 1-ports then it is easy to show that the system
must be reciprocal (Brayton, 1971). Thus as we whall see in order to^ model the^ non-
reciprocal nature of far-from-equilibrium chemical reactions one^ cannot^ use^ 1-ports.
In cases where the^ system is^ reciprocal,^ there^ always exists^ a^ potential^ function from
which the constitutive relation^ can^ be derived and^ which^ is extremal^ at^ a^ steady^ state
(Brayton and Moser, 1964, Stern, 1971; Oster and Desoer, 1971). For^ linear^ systems the (^) potential is simply the entropy productions, while for nonlinear systems the poten-
tial function is called the "content" (Millar, 1951).
ALAN (^) S. PERELSON Network Thermodyanmics 673
(a)
B 2~~~~
+_ ~~~~~~~~~~~~~~~~~~~~(b) FIGURE 4 FIGURE (^5) FIGURE 4 Transport across a composite membrane. FIGURE 5 (a) 0-junction. (b) 1-junction.
INTERCONNECTION OF RESISTIVE AND CAPACITIVE N-PORTS
Reaction and transport processes involve only resistive and (^) capacitive n-ports. Thus all that remains is to specify how to interconnect these devices. As before, at points of interconnection thermodynamic potentials are continuous and there is no loss of
flow. These restrictions are just generalized statements of Kirchhoff's laws. What is
needed is some systematic way of writing these laws for complex interconnections. For example, consider the composite membrane shown in Fig. 4. At point p the membranes A, B, and C are interconnected. Observe that since the membranes are all in contact the chemical potential of the substrate must be the same at the right- most surface of A and the left surfaces of B and C, and whatever flows out of A must flow into B and C. This type of (^) parallel interconnection can be represented by a
special graphical symbol, the 0-junction or parallel junction, shown in Fig. 5 a. The
lines incident on the junction are called bonds and represent perfect lossless con-
nectors. Denoting generalized thermodynamic potentials or efforts by e, and flows
by f, a parallel junction is defined by
* e, = e2* *= = eN ~~(8)
N E u,f1 =0O (9) A-I
where
674 BIOPHYSICAL JOURNAL VOLUME 15 1975
(a)
R R C C R (^) h%000ft I I (^0)
(b)
R2 C4 R6rC 2 4 6 E, _ 3 5 7 I- _0^ - -I _E (c) FIGURE 6 (a) Membrane transport system. (b) Network representation.
A CA A
XA J^ A rXA
TD( (^) r) ( rB)TD
TD(r) TD^ B
El A _-A 1 _ 0 O1 -^ - E2A
FIGuRE 7 COUPled nonstationary transport of two species.
676 BIOPHYSICAL^ JOURNAL^ VOLUME 15^1975
CES
(+1) (+1) (+1) 2 (+1)
CY CS,TD-1TD^ R^ ITD^ --^ RR TD^ CpI T
TD( +1) (^) TD 1 (TD I (+1) (+2) CX_1- TD^ R^ R^ RR^ TD1 E1I
FIGURE 8 FIGURE 9 FIGURE 8 The autocatalytic reaction X^ + Y^ 2X. FIGURE (^9) AnenzymicreactionS + E L ES 2 E + P.
where Di, 1Ji, and c, are the diffusion coefficient, diffusive flux, and concentra-
tion of species i, respectively.
Fig. 7 shows the^ coupled transport of^ two^ species. The^ chemical^ potential difference
that causes a flow ofA also influences B and can cause its transport.
Chemical reactions can also be represented in network terms. Here we assume that
the isothermal reaction mixture is well mixed and maintained homogeneous so that
spatial considerations are unimportant. The network represents purely topological
relationships between the dissipative and storage aspects of a reaction. Fig. 8 shows
the representation of an autocatalytic reaction. The stoichiometry of a reaction repre-
sent various scalings that are occurring e.g., 2 mol of X must appear every time 1 mol
of X and 1 mol of Y (^) combine. In (^) mechanical systems this type of scaling is performed
by gears, while in electrical systems transformers are used. Thus, in representing reac-
tions one must use scaling transducers to represent stoichiometry. Observe that Fig. 8
contains a positive feedback loop, indicating that autocatalytic reactions could con-
tribute to the instability of a reaction system. Fig. 9 represents an enzymic reaction.
Notice that the network diagram clearly illustrates that the enzyme cycles back and
forth between free and combined forms. Although these diagrams look complex they
can be generated algorithmically from conventional biochemical diagrams as shown in
Fig. 10 (Morowitz, 1973). Other workers have used standard network representa-
tions for reactions (Hess et al., 1972; Busse and Hess, 1973).
Reaction networks contain only resistors, capacitors, and transducers, making the
prediction of oscillations a difficult task. However, Atlan and Weisbuch (1973) have
shown that the effects of time delays in the reaction process can be approximated by
adding inductors.
It should be apparent that the network representations for reaction and transport
processes can be combined to form complex models of chemico-diffusional systems.
2Network analogs of chemical systems based on kinetic rather than thermodynamic models have been devel-
oped by Seelig(1970, 1971), Seelig and Gobber (1971), (^) Seelig and Denzel (^) (1972), and (^) Rossler(1974, 1975).
ALAN S. PERELSON Network Thermodynamics 677
C1 RR, c2C.
R1. cl^ C2^ R, TD(1) (1)TD^ C lils (1)^ (1)^12 E1._ 1 TD_^1 1-TD^ B _O^1 E^.
A:1 RR^ J JR iR 2 2 TD(1) (1)TD
o 1 _ C2,
FIGURE 1 Facilitated transport.
cesses governing the operation of the system. The model thus totally reflects the
thermodynamic description.
CHEMICAL REACTION NETWORKS
Besides extending the range of linear irreversible thermodynamics, network ideas can
be applied to^ theoretical^ problems in^ chemistry to^ provide new^ insights into^ the^ struc-
ture of large chemical systems.
Chemical kinetics deals with the problem of determining rates of chemical reac-
tions. Assuming temperature and pressure are maintained constant, and that the
volume change due to reaction is negligible, the "state" of a chemical system is deter-
mined by the number of moles of each component. The equations of chemical kinetics
describe the vector field that propels the state point through concentration space.
One frequently assumes that this vector field is given by the law of mass action. Thus
one is faced with the very difficult problem of solving large systems of nonlinear dif-
ferential equations. In each set of reactions a different system of equations results and
thus it has been very difficult to establish any general properties for reaction systems.
I like to think of this approach as analogous to using Newton's equations in mechan-
ics-a given problem may be easy to^ solve but general theorems are difficult to come by.
By using network thermodynamic methods one can formulate a generic set of differ-
ential equations to describe reactioh dynamics that are in some sense analogous to
Hamilton's (^) equations in mechanics. (^) Although these (^) equations are valid for nonlinear
far-from-equilibrium reactions I shall illustrate how they are derived by examining
the simpler near equilibrium case.
For each reaction we can define an extent or advancement of reaction, tk. If we
ALAN (^) S. PERELSON Network (^) Thermodynamics 679
can determine how the reaction extents change with time then the law of definite proportions
n,(t) =^ n,(0) +^ E (^) VPkik (15) k
determines how^ the mole numbers^ of^ each^ species^ change^ in time. Thus^ it suffices^ to find a differential equation in the extents.
For a set of M chemical reactions occurring among N^ chemical species, let^91 =^ RN
be the species space and let 911 =^ RM be the reaction space. (In Oster and^ Perelson
[1974 a, b], a more general treatment is^ given in^ which^91 and^9 can^ be^ nonlinear
spaces; i.e. differentiable manifolds.) The^ mole numbers^ n,, n2,^ ...^ nN^ can^ be^ assem-
bled into^ a^ vector^ n^ e 91, and the^ reaction^ extents^ {,, 422 ...^ 4M collected into^ a^ vec-
tor t E M. The law of definite proportions provides a map between the species and
reaction spaces. Let v^ be the N x^ M stoichiometric matrix, then
v: M l-91 (16)
t(t) H+n(O) +^ vt(t) =^ n(t). (17)
By differentiating this map we obtain a^ relation between^ the^ rate^ of^ change of^ mole
numbers, n(t), and the rate^ of reaction^ j =^ d4/dt^ -^ given^ by
n(t) =^ vi(t) =^ vj. (18)
To construct the equations of motion on the reaction space 9am^ we^ must^ introduce
the constitutive equations for the^ species capacitors and^ reaction^ resistors.^ The^ former
are defined by the^ map
n > A(n) (19)
on 1, where g(n) =^ [g (n), s2(n),. .,N(n)]A is the^ vector^ of chemical^ potentials. The
reactions are characterized by a^ nonlinear constitutive^ map A^ defined^ on^9
a I->A(a) = j, (20)
where a is the vector of chemical affinities (a =^ -iTj) and^ j is the^ reaction^ rate^ vec-
tor. The driving forces in species space g and the^ driving forces^ in^ reaction^ space a
are related by the law of definite proportions. In fact a is uniquely determined, given
.u and the^ law^ of definite^ proportions,^ by^ the^ operation^ of^ pulling^ back^ a^ covector
field along a map (see Oster and Perelson, 1974 b for the^ details of this^ technical^ point).
By composing the constitutive relation and law of definite proportion maps the
equations of motion can be constructed. The law^ of^ definite^ proportions (Eq. 15)
determines n as a function of t. The capacitive constitutive relation (Eq. 19) assigns
a unique chemical potential vector (^) ;s to n, and then as indicated above the law of
680 BIoPHYSICAL^ JOURNAL^ VOLUME 15^1975
CN TD^ R'm
FIGURE 12 Generalized bond graph representation of a reaction system.
gradient of the Gibbs free energy, G, the equations of motion of a chemical system can
be written
= (^) AV+)(t), (23)
where (t) - G[n(O) + vt] is a scalar potential defined on C, i.e., 4: = --+R.
The form of the two sets of equations is superficially similar. They are identical only
if 9= is even dimensional and A = J.
The generic equations (21) can be given a network interpretation (Perelson and
Oster, 1974). Fig. 12 shows a general bond graph representation of^ any reaction
system. With^ the capacitor constitutive relation^ given by^ ,u()^ and the^ resistor^ con-
stitutive relation^ A(.),^ the^ equations describing^ this network^ are^ precisely^ Eqs.^ 21.
The capacitors represent species space, the resistors represent reaction space and the
set ofjunctions and transformers representing the network topology correspond to the
reaction stoichiometry v (Oster and Perelson, 1974a).
Using the network representation as a guide the equations of chemical dynamics
can be extended in^ many ways. First, the^ generalization to^ open reaction^ systems can
be derived^ by simply adding sources to^ the^ network^ (Perelson and^ Oster, 1974).^ Next,
the biologically interesting situation in which there are a collection of cells com-
municating with each other and their environment via the transport of material across
their outer boundaries can be modeled by a network which is formally identical to a
reaction network. Here however, the resistors exhibit the dissipation due to transport
and the capacitors represent the storage of chemical species in^ N^ cells instead of the
storage of^ N^ species in^ one^ cell. This model^ can^ then be further^ generalized to^ include
multiple species undergoing chemical reaction within each^ cell.^ The^ generic^ differential
equations for this^ complex transport-reaction system^ can^ be^ easily^ derived^ from^ the
network (Perelson and Oster, 1974); they are a generalization of the equations dealt
with by Othmer and Scriven (1971) in studying instabilities and dynamic patterns in
cellular networks.
FUTURE OF NETWORK THERMODYNAMICS
The analysis of large chemical systems is far from complete. Eq. 21 which described
the dynamics of reaction systems needs to be thoroughly examined under a variety
682 BIOPHYSICAL JOURNAL VOLUME 15 1975
of constitutive assumptions. For example, by fixing the capacitive and resistive
constitutive equations one can study the effects of changes in reaction topology. Are
there certain topologies such as the positive feedback loop of autocatalytic reactions
that make oscillations more probable? Alternatively, one can hold the topology and
the reaction constitutive relation fixed and vary the capacitance. (^) Some effects of
capacitance variation on system stability and bifurcations to periodic solutions have
been investigated (Luss, 1974).3 Given a system of known reactions and equilibrium
properties one can study the qualitative behavior of systems in which the non- linearity of the reaction constitutive relations are (^) restricted. For example monotone, passive, or quasi-linear (Duffin, 1946), resistive constitutive relations (^) can be studied. Also the asymptotic behavior and the number of maxima or minima in the constitutive relation (^) can be specified. Perelson and Oster (1974) consider the reciprocal, passive, and locally passive (^) cases.
The graphical techniques developed to represent thermodynamic systems need to
be exploited. Since the graphs are just another notation for the dynamic equations
one may be able to find graphical criteria for stability and oscillations. The decompo-
sition or "tearing" of a complex system into subsystems has been utilized in treating
one reaction-transport system (Perelson and Oster, 1974). This technique should prove
useful in analyzing a variety of large systems.
The ease of interfacing bond graph models with computer systems can be exploited
in a classroom situation. Experience at Berkeley in teaching both undergraduates
and graduates bas shown that students with no previous computer experience can
formulate and solve complex physiological modeling problems. Thus an intuitive
understanding of complex system behavior can quickly be attained by students with
little or no experience in nonlinear systems. The biophysics can be emphasized while
submerging the^ formal^ mathematics.
Let me end by mentioning an area in which I foresee future growth of network
thermodynamics. The goal of biophysics is to understand how biological systems
work. Traditionally we have approached (^) this problem through reductionist analyses. However, when we have isolated every enzyme and catalogued every reaction that occurs in a cell will we understand how the system (^) works? I think not, for there are
complex dynamic interactions that impart to matter the property that we call life.
However, if (^) we can design and synthesize systems which have these dynamic charac-
teristics we will have made significant progress towards understanding them. Engi-
neers have enormous experience in synthesis and design, and it is my hope that through
network thermodynamics, these techniques can be applied to synthesize chemical net-
works with prescribed behaviors.
Network thermodynamics is based upon an idea of great simplicity-that the logical
foundation of finite-dimensional thermodynamic models is formally identical to that
of (^) network theory. It was Aharon's hope that this similarity could be exploited to
3Perelson, A.^ 1975. A^ note on the qualitative theory of lumped parameter systems. Chem. Eng. Sci. In press.
ALAN S. (^) PERELSON Network Thermodynamics 683
OSTER, G. and A. PERELSON. 1974 b. Chemical reaction dynamics: geometric structure.^ Arch.^ Rational Mech. Anal. 55:230. OSTER, G., A. PERELSON, and A. KATCHALSKY. 1971.^ Network^ thermodynamics.^ Nature^ (Lond.).^ 234:393. OSTER, G., A. PERELSON, and A. KATCHALSKY. 1973. Network thermodynamics: dynamic modelling of bio- physical systems. Q. Rev. Biophys. 6:1. OTHMER, H. G., and L. E. SCRIVEN. 1971. Instability and^ dynamic pattern^ in^ cellular^ networks.^ J.^ Theor. Biol. 32:502. PENFIELD, P., R. SPENCE, and S. DUINKER. 1970. Tellegen's Theorem and Electrical Networks. MIT Press, Cambridge, Mass. PERELSON, A., and G. OSTER. 1974. Chemical reaction dynamics: network structure. Arch. Rational Mech. Anal. 57:31. PRIGOGINE, I. 1969. Structure, Dissipation, and Life. In Theoretical Physics and Biology,^ Proceedings of^ the First International Conference, Versailles, 1967.^ M.^ Marios,^ editor.^ North-Holland^ Publishing^ Co.,^ Am- sterdam. ROSENBERG, R. C.,^ and^ D.^ C. KARNOPP.^ 1972.^ A^ definition^ of the bond^ graph^ language.^ J.^ Dynamic Sys- tems, Measurement and Control. Trans. ASME. 94:179. ROSSLER, 0. E. 1974. A synthetic approach to exotic kinetics (with examples). Lecture Notes in Biomathe- matics, Vol. 4. Physics and Mathematics of the Nervous System. M. Conrad, W. Guttinger, and M. Dal Cin, editors. Springer-Verlag, Berlin. ROSSLER, 0. E. 1975. A multivibrating switching network in homogeneous^ kinetics.^ Bull.^ Math.^ Biol.^ In press. SEELIG, F. F. 1970. Undamped sinusoidal oscillations in linear chemical reaction systems. J. 7heor. Biol. 27:197. SEELIG, F. F. 1971. Activated enzyme catalysis as a possible realization of the stable^ linear chemical^ oscilla- tor model. J.^ Theor. Biol.^ 30:497. SEEUG, F. F., and^ B. DENZEL. 1972.^ Hysteresis^ with^ autocatalysis:^ simple^ enzyme^ systems^ as^ possible^ binary memory elements. FEBS Lett. 24:283. SEELIG, F. F., and F. GOBBER. 1971. Stable linear reaction oscillator. J. Theor. Biol. 30:485. SMALE, S. 1972. On the mathematical foundations ofcircuit theory. J. Differ. Geom. 7:193. STERN, T. E. 1971. On reciprocity in nonlinear networks. In Aspects of Network and Systems^ Theory. R. E. Kalman and N. DeClaris, editors. Holt, Rinehart and Winston, Inc., New York.
ALAN S. PERELSON Network Thermodynamics 685
