J. theor. Biol. (1977) 69, 471-510

A Simple Network Thermodynamic Method for Modeling Series-parallel Coupled Flows I. The Linear Case D. C.

MIKULECKY,

Department

W. A.

WIEGAND

AYD

J. S.

SHINER

of Physiology, Medical College of Virginia, Richmond, Virginia 23298 U.S.A.

(Recieved 24 September 1976, and in recked form 28 January 1977) A linear network thermodynamic approach to coupled flows through biological structures is developed based on the techniques introduced by Peusner (1970) who developed a means by which electrical network diagrams could be used to represent the flow-force equations of non-equilibrium thermodynamics. In this work, Peusner’s methods are extended and elaborated, and a complete linear theory is presented. In further work in this series a non-linear theory is built upon the linear theory and applied to coupled flows through such tissues as epithelia. The network technique used here resembles the equivalent circuit approach used extensively to treat ion Rows in cell and epithelial membranes (Finkelstein & Mauro, 1963). The major distinction is that the equivalent circuit approach was developed for independent ionic currents while the network thermodynamic approach deals explicity with the coupling in a multicompartment system. Using an obvious analogy between Kirchhoff’s laws in electrical circuits and the conservation laws of continuum physics as applied to mass transport and chemical reactions, a purely topological argument leads to a proof of Tellegen’s quasi-power theorem. From this Onsager reciprocity follows almost trivially for this kind of linear system and can be extended into the non-linear domain for certain non-linear systems. Tellegen’s theorem also leads to a version of the minimum dissipation theorem in a rather elegant manner. Apart from the ease in obtaining these general theorems, the network approach has the advantage of giving a method for examining the effects of organizational or topological influences in any experimental design and allows for sorting out effects which arise from specific molecular properties (the properties of the hardware in the circuits) and those properties which arise from the way the systems are connected (topology). Methods are given for combining linear n-ports in series and parallel, reducing them symbolically to l-ports connected in the same topology, Making sure that the computations are compatible with matrix algebra, the result of ordinary circuit analysis is used to deduce the matrix algebraic T.“. 471 30

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analysis appropriate for the /z-port system from the solution to the I-port circuit. The methods are applied to the Curran-Macintosh model for isotonic transport in a series membrane system. The result shows how the membrane structure determines the composition of transported fluid independent

of pump

rate.

In Appendix A, the relation between the network approach and Lagrangian mechanics is developed. In Appendix B, the method for dealing with time dependencies by using capacitors is illustrated using the osmotic transient as an example. In the third and final Appendix, the relationship between network thermodynamics and the reductionist philosophy that underlies so much of modern molecular biology is briefly examined.

1. Introduction There are two somewhat different approaches to network thermodynamics. The most well known, by far, is that of Oster, Perelson & Katchalsky (1971) which was the subject of an editorial in Nature? when it first appeared in that journal. Since then a number of significant publications developing their approach have appeared, and they are listed in the references of this paper. This group uses an approach to physical system dynamics which focuses on energy flux and power flow as the fundamental flows in a dynamic system. Ordinary circuit graphs are not used. Instead a system of diagrams called “bond graphs” (Karnop & Rosenberg, 1975) is employed. Power flow is represented by a power “bond” somewhat analogous to the representation of chemical bonds in organic chemistry. The bond graph method has certain distinct advantages over ordinary electric circuit diagrams, but also has one important disadvantage which is the time one must spend to learn about the bond graph technique if it is unfamiliar. In fact, it is the likelihood that most biologists are probably not acquainted with bond graphs and probably somewhat familiar with electrical circuits that makes the second approach to network thermodynamics more appealing. For example, equivalent circuits have long been used as a representation of ion movements in biological systems (Finkelstein & Mauro, 1963). The second approach, due to Peusner (1970), is formulated in terms of the classical currents and voltages of electrical network theory. For this reason it should have didactic value to almost anyone familiar with simple electric circuits. In this series of papers, Peusner’s approach is modified and extended in a direction which leads away from electrical networks, although it resembles them very closely. Computational methods from electrical network theory will be shown to be applicable and i Networks in Nature. (1971). Nature

234, 380.

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useful in obtaining solutions as simply as possible. This first paper will deal with the linear theory which, for many systems, is completely valid in a region around equilibrium and a reasonable qualitative indicator of system behavior beyond that. ‘One particular application of this approach will be the analysis of coupled solute and volume flows through complex series-parallel membrane systems. Due to the convective contribution to solute flow (often called “solventdr,ag”) there is an intrinsic non-linearity in such a system. Fortunately, and in contrast to those systems which are only linear near equilibrium, its behavior far from equilibrium becomes linear again so that non-linearity is confined to a pair of transition regions between linear regions. In another work, it will be shown in great detail that some version of the linear theory will bc applicable in every region. ‘The second paper (Mikulecky, 19776) will introduce a non-linear theory based on the techniques of non-linear network theory (Chua, 1969). Both the linear and non-linear approaches will be applied to coupled solute and volume flow through epithelial membranes in a third paper, and in a fourth paper non-linear network theory will be applied to the problem of the equivalent circuit representation of ion flows (Finkelstein & Mauro, 1963) to extend that approach and incorporate it into the framework of network thermodynamics. The validity of the linear approach in the case of capillary wall permeability has recently been established (Thomas & Mikulecky, 1977). ‘The extension of Peusner’s approach leads to the point where this version of network thermodynamics can be merged with the bond-graph techniques of Oster, Perelson and the late Professor Katchalsky. This is an important point, for in the case of chemical reaction flows the bond-graph formalism is more versatile and powerful than the pseudo-electrical approach. This is especially true in light of the recent work by Oster & Perelson (1974a,b), which puts the entire realm of reaction-diffusion systems into the language of rational mechanics via the techniques of differential topology and related areas of modern mathematics, and ties this dynamic theory to a bond-graph type network representation. On the other hand, the methods described here are both compatible with that approach and require less new terminology and symbolism. The objective is to provide a way for physiological systems to be analyzed by biologists in a manner which incorporates information about structure-function relations in the simplest possible manner. ‘Once the advantages of such an approach have become clear, the more difficult and powerful methods should seem worth the additional effort necessary to master them. Recently, a set of more “tutorial” papers on network thermodynamics has appeared (Oster & Perelson, 1973; Perelson, 1975). These will serve as a

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good introduction for those wishing to look further. In the pseudo-electrical network representations, it is also possible to cast the formalism into the form of Lagrangrian dynamics in a relatively simple manner as is shown in Appendix A. Another recent work which is closely related to these is RenC Thorn’s theory of catastrophes (Thorn, 1972) in which he applies differential geometry and topology as a dynamic system theory to the morphogenesis of living systems. His work complements those mentioned above, as it provides a means by which “catastrophic” events can be fitted into the same framework as the more smoothly behaving events normally treated by these mathematical techniques.

2. Review of the Electrical Network Approach to Non-equilibrium Thermodynamics-Conservation Laws and Kirchhoff’s Laws

The point at which electrical networks and non-equilibrium thermodynamics merge is in the analogy between the conservation laws as they apply in mass transport and chemical reactions, and Kirchhoff’s famous current and voltage laws for electrical circuits. This is easiest to see when one realizes that the following relations obtained as approximations from electrical field theory are the basis for Kirchhoff’s laws: E=-vtj,

V*J = 0,

where E is the electrical field vector and 4 is the electrical potential and .I is the current density. The above relations obtain when the electrical and magnetic equations are uncoupled as is assumed in physical circuit theory. The similarity between the chemical potential gradient and the electric field is not a subtle one nor is the vanishing divergence of the current unlike the result of mass conservation in the steady state in cases where sources and sinks are absent. In discrete electrical circuits these two laws manifest themselves as the following statements: (1) Kirchhoff’s Voltage Law (KVL): There exists at each point in a circuit a potential, so that in any closed loop: CA4=0. 100p (2) Kirchhoff’s Current Law (KCL): If I,.k is the current leaving node k and entering node r and Ii, is the current leaving node r and entering node i then the sum over all the currents entering or leaving a given

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node, say r, is zero:

Since I,.k = - Ikr this could be written as :

where I,, now stands for currents entering or leaving node r; currents entering and leaving the node having opposite signs. The way of proceeding from this point is to abstract the notion of currents and potentials and assign their magnitudes as values to be assigned to a set of vertices connected by lines, called a network, so that the values of the potentials would be assigned to each point of intersection of those lines, the nodes, and the values of the currents to the line connecting two nodes. Once this is done, it is a simple matter to identify the abstracted currents and potentials with any quantity which obeys KCL and KVL. This is certainly possible for the flows and forces of non-equilibrium thermodynamics. Thus an abstract network can represent much more than just an electrical circuit, it can represent almost any thermodynamic flow-force system. Rigorous justifications are given for this notion in the various works cited. For this work it is sufficient that one sees the plausibility of these statements. Once the notion that a network can represent non-equilibrium thermodynamic flows and forces is accepted only a few more are needed. First of all, a continuously varying field can always be approximated by a discrete network, the approximation becoming better as the network is further subdivided until the two merge as the subdivision becomes infinitely fine. (This is the usual method of numerical solution to various partial differential equations arising from continuum analysis.) Second, the properties of ciectrical, and more so, abstract networks are both topological and dependent on the nature of the circuit elements. That is, the way the network is connected together is one of the primary properties that distinguishes it from other networks, and the other property which is important in any network is the nature of its circuit elements. The identity of these elements is usually abstracted by assigning a constitutive relation to each kind of element. The constitutive relation relates the flow through the segments connecting two nodes to the potential difference or driving force, between the nodes (across the element). The result is a network of nodes connected by circuit elements completely determined by the nature of the circuit elements (constitutive relations) and the particular way in which the circuit elements connect nodes, the network’s topology. Rather than using the electrical terminology of currents and potentials the more general approach is formulated in terms of flow and forces,

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introducing the various elements by their constitutive following list summarizes them (X = force, J = flow). Name

Constitutive -~__---~

resistance, R force source, X,

--.. X = RJ X = XE

flow source, Jp

J

-.

SHINER

relations.

relatiorl ~-~~ __

The

--

independent of the current through the element = Jp independent of the force across the element

These elements would suffice for stationary state descriptions of biological systems, but capacitances are necessary for non-stationary states. For the time being only stationary states will be considered and the above elements will suffice. A simple example of how to treat time dependent cases using capacitances will be introduced in Appendix B. 3. Tellegen’s Theorem The ability to model any non-equilibrium system by a network rests on the analogy between the laws of conservation of mass, charge, etc. and the current law of Kirchhoff (KCL) as well as the fact that thermodynamic potentials obey a form of Kirchhoff’s voltage law (KVL) along any closed path through a system. From these facts alone we can derive a version of Tellegen’s theorem applicable to non-equilibrium systems. Let us first look at Tellegen’s theorem as derived for purely electrical networks and then at the more general “quasi-power theorem” which is applicable to any set of dynamical systems obeying KCL and KVL. The following proof is as Tellegen (Tellegen, 1952) originally stated it. It is followed by a more powerful “quasi-power theorem” which uses notions from graph theory. As illustrated in Fig. 1, in any general network configuration, we will have a set of branch currents i such that for every node Ci = 0 (KCL) and a set of branch voltages ZI such that around every closed loop Xv = 0 (KVL). For every branch let the positive direction of the current be from the + to the -, then Cio = 0, where the summation is over all branches. Proof. Denote the branch between the jth and kth nodes as the jkth branch. Then the voltage across it is Vj, = Vi- V, and the current flowing from j to k is ijk, i jk Vjk = i jk( “; - Vk)

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/

I-K,. I. Scheme for assigning currents and voltages to branches and nodes in a graph. ,4 current is assigned to each branch and flow from more positive voltage to less positive or more negative voltage. A voltage is assigned to each node and a current to the branch linking them. now for the ,jth node:

or, in other words, the nodes labeled k are the various “nearest neighbor”’ nodes to the jth node which are connected to it by branches. Similarly, for the second term

c

i, L’i = V, C ijk.

all branches on the kth node

i

where the sum is over all the branches impinging on the kth node, i.e. over all the nodesj connected to k by branches. But KCL says that: 5 at the jth node

ijk =

z

i, = 0.

at tie kth node

Therefore : Q.E.D. This is the network equivalent of the well-known theorem: s 23’4 dP’ = 0, where L7is a solenoid vector and V4 is irrotational, i.e. 4 is a scalar potential function. The theorem holds for all types of networks, linear and non-linear, constant and variable, passive and active, and thus is independent of the nature of the elements in the branches. It is a topological statement which. in a sense,ignores the “molecular” composition of the circuit elements and dwells upon how they are connected. To apply the theorem to networks

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provided with input-output terminal pairs representing the points of experimental accessto the system the theorem is written: C i, V, = C ii Vi, * 1 where t indexes the terminal pairs of nodes and i the non-terminal or “internal” branches. An example of this appears in Fig. 2 where the external nodes, labeled e, correspond to the terminal nodes. Note that this is arbitrary and can introduce a kind of topological change in the network if changed. (The two terms have a sum of zero, the sign difference arising from the convention that terminal flow-force pairs have a sign opposite to those

‘. e’

! ’ I \ / \ \

:!!a$!e ..

I ! ‘l’ /’ ‘\

i

>’

e

“.... ,,

FIG. 2. Break-down of a network into external nodes, labeled e, which are accessible to experimental manipulation, and internal nodes, labeled i, which are not. The heavy broken line symbolizes the boundary of the “black box” into which the experimentalist has no access. The lighter broken lines illustrate one possible connection of sources or measuring devices to the external or terminal nodes.

inside the network or in other words, power dissipated in the system equals power fed in through the terminals.) To prove the more powerful “quasi-power theorem” it will be necessaryto introduce some simple graph and circuit theory (Seshu & Reed, 1961). Any network can be drawn as a simple graph such that each branch (edge) represents a circuit element and the points of connection are nodes (or vertices). The independent currents can be found by decomposing the graph into disjoint subgraphs which consist of a “tree” and “links”. This will be illustrated after introducing some of the language of graph theory. Definitions The degreeof a node (or vertex) is the number of branches (edges)incident upon it. The branch or edge sequenceis a set of edges ordered such that each has a vertex common with the preceding edge (in ordered sequence)and the

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ot!her vertex in common with the succeeding edge, and some edges may have vertices in common (vertices can, in general, be counted more than once). Multiplicity refers to the number of times an edge appears in an edge sequence. Edge train. If each edge in an edge sequence has multiplicity 1, the edge sequence is an edge train. We designate as initial and terminal vertices those in an edge train which are of degree 1 and are incident on the initial and final or terminal edge, respectively. A path is an edge train such that the degree of each non-terminal vertex is 2 (therefore, the degree of each terminal vertex is 1). A closed edge train has its initial and terminal vertices coincident. Such an edge train is called a circuit or loop if all its vertices are of degree 2. (A circuit is a closed path.) A circuit may be an entire graph or a subgraph. A graph G is connected if there exists a path between any two vertices of the graph. (Intuitively, a graph is connected if it is in one piece.) Alternative

dejinition of a circuit or loop

A circuit or loop is a connected graph or subgraph in which each vertex is of degree 2. Theorem A. If G is a connected graph and one of the circuit elements (an edge of G in a circuit of G) is removed, the resultant graph is connected. Proof: Let e, be a circuit element and let G, be the subgraph obtained when c1 is removed. Since there is a circuit in G containing e,, the vertices of e, are common to other edges of G. Only the paths in G which contain e, are absent in G,. Since there is a circuit in G containing e,, there is a path P, in G, between the vertices of e, (which therefore does not contain el). If in any pathp, of G containing e,, e, is replaced by the path p2, an edge sequence is obtained, which contains a path. (If there is an edge sequence with terminal vertices a and b there is a path between a and b.) Q.E.D. Now the decomposition sought for connected graphs can be defined. Dejnition A tree is a connected subgraph which contains all the vertices of the graph, but does not contain any circuits, i.c. a path containing all the vertices. Examples of trees belonging to a given graph appear in Fig. 3. The collection 01‘ all trees relating to a given graph is called a forest. It is ncccssary to further refine the terminology in the following manner: In graph theory all the lines connecting vertices are called “edges” and only the lines between vertices in a tree are “branches”. The part or subgraph

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Graph

Tree /

AND

J. S. SHINER Tree 2

FIG. 3. A graph and two different trees of its forest.

not in a given tree of a graph is called the set of chords or fitzks (dotted lines in Fig. 3). Theorem B. A connected graph of v vertices and e edges contains I’-- I branches and e-v+ 1 chords (or e edges!). For proof see (Seshu & Reed, 1961).

Furldamental circuits of a connected graph G with tree T are the e- c+ 1 circuits formed by each chord and its unique tree path. Figure 4 shows the breakdown of a simple graph into a tree and the resultant fundamental circuits. By designating j,, j, and j, as the current through the links I,, I,, I,, i.e. the currents in the fundamental circuits, the current through any tree branch can be represented as a linear combination of the three independent link currents. (Link currents are independent because they can be open circuited individually without in any way influencing each other.)

4 = c %Js

Graph

Tree

T i-i

I1 I ’ , 1 I 2 ~c

t:

3

FIG. 4. A graph, one of its trees and the corresponding set of fundamental circuits. The fundamental circuits,jl,j2,j3, are found by replacing the links (broken lines labeled l,, 12, I,) one at a time.

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or i = BT] 9 wherejP = link current, i, = branch current and BP, = loop matrix, defined below. 11

l2

i, --_____

i4

i, --~~-~_

i,

i,

jl

1

1

0

0

0

0

j2

0

1

1

0

0

0

0 _____ 1

0

j,

0 -----0

1 ~----1

-__

----

--

-1

I

In the example above: i, =j1, i2 =jl, k = .il -j2, i4 = j,.

ij = j2 -j3. is = j,, i, =j3,

lt is convenient to recognize two matrices essential in defining the topology of a given graph or network: (i) Node-branch (edge) matrix where : A, = +1 if edge j is directed out of node i, Aij = - 1 if edge j is directed into node i, Aij = 0 if edge j is not incident on node i. (ii) Branch (edge)--loop matrix (In this the subgraphs which are circuits, not just important to note that in electrical circuit fundamental circuits would be necessary for expression of Kirchhoff’s laws.) where :

representation the loops are aZ1 the fundamental circuits. It is analysis, only the independent, a solution to the circuit and the

B, = +l if branch k bounds loop i in the + direction, B, = - 1 if branch k bounds loop i in the - direction, B, = 0 if branch k does not belong to loop i. NO~WKirchhoff’s

current and voltage laws are Ai=o Ba=Ti.

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Furthermore AB’=O BA’ = 0. In the limit of infinitesimal mesh size, the topological become the familiar vector identities

identities

above

curl (i) = 0 curl *grad (4) = 0, where C#Iis the “node to datum” potential at any point in the system. The following is an example of Kirchhoff’s Laws expressed in terms of the nodebranch and branch-loop matrices. The network and its graph are shown in Fig. 5. The current equations are: node node node node

a

E 0

i, +i,

-i,+i3+i4

b d c

-i,

-i3

=0 -i,

=O

- i, + i, = 0.

d

4’

FIG. 5. A sample network and its graph as an example for node-branch and branch loop analysis. The number of vertices, v, is 4 and the number of edges, e, is 5. The number of branches or elements in the tree is v - 1 = 3. The number of links is e - v + 1 = 2. Thus only current sets 1 and 2, 1 and 3, or 2 and 3 can be independent. The incidence matrices are given in the text.

Thus

Ai =

1 I 00 1

1

0

0

0

-1 o-1

0

-1 11

0

-1 0 .

o-1

1

I

The voltage equations are (once again, it is possible to eliminate one of the relations below since only two are independent. This would be the procedure

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in practical electrical circuit analysis) : loop 1 loop 2 loop 3

-01 +“z +u, = 0 L-J--v‘$--vug = 0 -v1 +v2 +v4+vg = 0

and

where

In terms of the branch-loop

matrix, Kirchhoff’s Bo = c &v, a

voltage law is now

= 0,

and the power theorem analogous to Tellegen’s Theorem follows :

can be derived as

so that

4. “Quasi”

Power Theorem

Consider two “states” of the network-same elements, element values, currents and/or voltages. actual states of two &Brent “networks” with the one could be an electrical network and the other the of any dynamical system with the same graph which

topology but different This could even be the same topology. In fact network representation obeys KCL and KVL.

where the topology is invariant in BBz while the primed currents are “state” dependent. Also c B,, v; = 0, II

c B,, v; = 0. m

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Now:

OC

C iiv; - c izv: = 0 (negative sign due to convention) f E when terminals and non-terminal edges are distinguished. This amazing result is true as long as the topology, all of which is expressed in the matrix BB,, does not change! 5. Tellegen’s Theorem and the Onsager Reciprocal Relations In 1931 Lam Onsager provided a statistical proof for the reciprocity between the coupling coefficients of non-equilibrium thermodynamics. In a two-flow linear system, the flows and forces are related as follows: J1 = LllXl J2 = LzlXl

+L12X, +L,,X,

or or

X1 = RllJ, +R,,J,, X2 = Rzl J, +R,,J,,

where the matrices of coefficients are related by R = L-‘. Onsager showed that those matrices were symmetric or, in other words, that Rij = Rji and Lij = Lji. The validity of Onsager’s proof was restricted to the region of fluctuations around equilibrium, although experimental verification has been obtained for situations significantly further from equilibrium than this (Miller, 1969). It is now clear that Tellegen’s Theorem and reciprocity are one and the same phenomenon in most linear resistive (l-port) networks (Oster & Desoer, 1971). Tellegen’s Theorem can be used to prove that Onsager symmetry holds for all linear resistive (l-port) networks. Although it is not shown here, this result holds for non-linear l-ports as well. Consider performing two sets of measurements on the same resistive network. Let J;X~ be one set of variables and JiXi the other. As shown above, Tellegen’s Theorem can be written in the form:

c X,J, - c X,Jt = 0, I where the negative sign arises due to the fact that terminal flow-force pairs are taken in a sense opposite to those inside the network. This is in effect a statement of power conservation-power dissipated by the network is power supplied by the terminals.

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Using the quasi-power theorem, the above equation can be written in two alternative forms which apply to the two sets of flow-force measurements, one primed, and one unprimed. ~ XiJf = C X, JI I or

Figure 2 above illustrates the manner in which a network can be divided into internal nodes and terminal or external nodes. The internal nodes represent the inside of a “black box” which might be inaccessible to experimental manipulation and/or measurement. However, for the purposes of this discussion it will be assumed that it is a linear resistive network w that each circuit element obeys Ohm’s law xi = R,J,.

Xf = R,Jf.

This knowledge of the inside of the black box will be used to prove that Onsager reciprocity must hold for the terminal pairs. Introducing Ohm’s law into the sum, it takes the form: T X;Ji = F R,J;J; = 7 R,J,Jf which is equivalent

to the reciprocity

= ; XiJ;.

relation for the terminal

pairs

1 X;J, = T X,J;. f Now if the relations k

and X; = x Rtk J; k

(where in each case k is summed over all terminal pairs) are introduced, two sides of the reciprocity relation can now be written as

the

cI Xf Jz = cf ck &J; J, 1 XJ: = ct ck R,,J,J;. f Since both sets of indices run over all the terminal interchanged in the last equation so that

ct Xt J: = c -$ &,J, J;,

pairs t and k can be

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and since

c XtJ: = T XJt, Q.E.D.

&t = Rtk.

The reciprocity theorem can be extended to apply to certain systems having non-linear constitutive relations (Brayton & Moser, 1964a,b). In that case, it is the Jacobian relating the forces and flows which is symmetric. This is a very powerful result, and its implications will be discussed in the following papers on the non-linear theory (Mikulecky, 1977a,6). A further result of Tellegen’s Theorem is the principle of minimum dissipation. This proof is after Oster & DeSoer (197 1).

6. Minimum

Dissipation and Tellegen’s Theorem

The principle of minimum dissipation also follows almost trivially from Tellegen’s Theorem. Starting with the power conservation statement in the form T

%h

=

7

vkik9

where t is over all terminal branches and k is over all non-terminal branches, and all currents and voltages are assumed to be a solution to the network, a small variation, Si,, in the non-terminal currents which obeys Kirchhoff’s law is considered. This defines a new non-terminal current, i;, i; = i,+di,

which is not a solution to the network, but which does obey Kirchhoff’s so that Tellegen’s theorem still applies. Now it is true that:

law,

T v,i, = & v,i;

by Tellegen’s Theorem, so that: q vk6ik = 0.

Therefore at the steady state, the “content”

defined as

6G = cu,6i, k

is stationary. If all the resistances are monotonically increasing, then G can be shown to be strictly convex, hence the stationary point is the unique absolute minimum.

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A variation in the forces II; = v,+6v, th,e “cocontent”, G*, defined as

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leads to the minimum principle for

G* = c jk6vk = 0 k

in a similar manner. A resistive network with constant sourcesis always in the steady state. A network with dynamic (energy storage) elements and with constant sourceswill be purely resistive in the steady state. Hence, the steady state is characterized by the minimum of content and cocontent. The sum of content and cocontent is the total dissipation G+G* = P. For linear constitutive relations G = G* = 1/2P.

7. The “I-Equivalent”t

Network for a Two Port Linear System

Next a linear two flow energy transducer, or two port, will be developed a!; a network problem. The phenomenological equations for such a process from non-equilibrium thermodynamics are X, = R1,J, +R,,J,, X, = R,, J, fR,,

J,,

where arbitrarily set “one” is called the driving process or input so that IX,J,I 3 IX,J,j. To model these equations by a network we must deal with the fact that each force depends on both the conjugate and the nonconjugate flow. To accomplish this the terms involving these non-conjugate flows are replaced by a force source of equivalent magnitude so that the equations to be modeled are:

X, = R,, J, i-x,,, X, = Rx Jz +Xm which can be represented by the isolated circuits shown in Fig. 6. The problem now is to convert this representation to a connected network. A connected network can be made by replacing the force sources by the resistancesR, 2 and R,, with the non-conjugate currents injected into them .i-In earlier, preliminary

versions of this work (Mikulecky,

1974 & 1976) this network

wascalledan “H-equivalent”network.Peusner hassincepointedout that thisterminology could be misleading in that the name H-network has been used for one of the hybrid representations of this network. To avoid such confusion we will call it the “l-equivalent network, T.B. 31

488

D. C. MIKULECKY,

W. A. WIEGAND

AND

J. S. SHINER

p--+

t I

FIG.6. Disjoint networks to represent the two equations describing a 2-force, flow energy converter. The potential sources represent the effect of the coupling of the driving force of one process to the flow of the other and vice versa

FIG.7. Replacing potential sources by coupling resistors with “alien” currents injected into them. The non-coupling resistor has had its value modified to correspond to the algebraic rearrangement of the equations defining each network (see text). as shown in Fig. 7. This network corresponds to the following modifications of the set of phenomenological equations:

x, = RI,--RIZ 2 x

-Rzz-R21 ~- 2

2-

J, +(J, +Jz)&

+ RII 2-- -RIZ J,,

J2 +(J1 +J2)R,,

2 + Rx-%I

J2.

Now if R,, = R,, the circuits can be connected and the I-equiva!ent network for the transducer is obtained (Fig. 8). Notice that since any other network representation of the processes modeled by the equations above is equivalent to this one, the existence of a linear, coupled system of I-port elements and the Onsager condition are inseparable. It is possible to extend this idea to any number of coupled processes (Wiegand, 1976). The paradigm will be developed by first considering J-, 4- and Sports and then the n-port case,

LINEAR

r

NETWORK

MODEL

OF COUPLED

&

& -us-- ,

--~-I

----f-~

I I 2? I

h I2

A I I

__ * --’0

I

-

5.~. . .

h.

i

’ - ! J,

cl

51

489

FLOWS

22

t‘, ‘_

J

FIG. 8. “I-equivalent” network representing coupled two-flow, force system. The ability to represent this 2-port as a set of connected l-ports (resistors) is based on its reciprocity, Riz = &. Here R, = (RI, - R&/2 and R2 = (R22 - R,,)/2.

For a 3-port, the resistive form of the phenomenological written :

equations

Xl = RIIJ, +Rl2J2 +R13J3

= &I JI +R,, J, +&, J, X, = RMJI+R,,J~+R,,J, for R2, = R12, Rz3 = Rst, Rsl = R,3 these can be rewritten as X2

XI = CR,,-(R~~+R~~)IJ~+R~Z(J~+J~)+R~~(J~+J~) Xz =R,,(J,+J1)+CR,,-(R,2+R23)1J2+R23(JZ+J3) X, = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Cdl R =RdR,,+W 1

3

R

’

=

R22-CR12

2

+R23)

3

R = R,, +RzJ __- -(RI, 3 3 ’ and then the above equations are rewritten as:

XI = 34 J, +R,,(Jz +J,)+R,4J3 x2 = RdJ2 +JI) +~RzJz +R23(J2 X3

=R,3(53+Jl)+R23(JZ+J3)+3R3J3.

+Jd +J3)

-

is

490

D.

C.

MIKULECKY,

W.

A.

WIEGAND

AND

.I.

S.

SHINER

The corresponding resistive circuit appears in Fig. 9. The resistive circuit for a 4-port can be drawn by requiring that:

R = R22 -(R,z 2

R

+&,

+R,J

4

-

_ R,,-Ub+R24+Rd 4

4

J3

FIG.

and then the corresponding

J,

J2

9. Network for coupled 3-port.

phenomenological

Xi = 4RiJi + i R,j(Ji +Jj)

equations are (i = 1, . . .,4)

j=i

j#i

and the network is shown in Fig. 10. Before drawing the graph for the Sport case, it is convenient to introduce some notational simplifications as shown in Fig. 11: (a) in place of each Rjj the numbers q are written. (b) if there is a line between Rij and Rik there is a resistor R, between them.

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NETWORK

MODEL

OF

COUPLED

FLOWS

491

FIG. 10.Networkfor coupledCport.

FIG. 11. Simplification in notation. For each non-coupling resistor R, between two coupling resistors &, R,, a set of two connected boxes labeled gand ik is drawn.

In this notation the Sport is shown in Fig. 12. Note that there is a “pattern” to the way in which the ij’s are arranged. (a) Thej’s are the same w/in rows. (b) The i’s are the same w/in ~01s. (This is much like the manner in which one numbers the elements of a matrix.) (c) The 12 element has two “inputs” just as the 45 element has two “outputs” with all the rest on the “outside” row and column having only one. Finally, the pattern can be utilized to put together an n-port device as shown in Fig. 13. A word of caution is in order: although the Z-equivalent network is indeed an electrical circuit, the drawings above for the 3 through n-port network

elements cannot be so interpreted. The reason for this is that extraneous loops have been created in the networks. For example, take the 3-port case and examine the graph and tree network which appear in Fig. 14.

492

D.

C.

MIKULECKY,

W.

A.

WIEGAND

v -

J5 FIG.

bi J4

AND

J.

S.

SHINER

23 ‘\ 13

?

‘L -

+

J,

12. Network for coupled S-port.

.

Input

s!de

FIG. 13. General network for n coupled flows and forces (n-port).

LINEAR

NETWORK

MODEL

OF COUPLED

FLOWS

493

Although there are only three independent flows in the system being modelled there are four in the electrical circuit having the same graph. The key to this paradox is the fact that in the thermodynamic network the paths for the pseudo-currents are predetermined by the identity of the different current analogs, and in the most general electrical network, the current paths are determined not only by the resistances, but by the voltages also. Thus, these thermodynamic networks are more highly determined from the start and require less analysis than do electrical networks. In fact, for a single

Tree

Graph

ond

fundomentol clrcults

3

-I 2 @

3

l%

3

2

-I 41

2

21 ;

--iA) \

@

.---

‘22 I

,,P,

FIG. 14. Graph, tree and fundamental circuits for the 3-flow, force energy converter. Note that by graphical analysis, there are 4 fundamental circuits although only three exist in the physical system being modelled.

/z-port unit it is al1 done-in the starting equations! The greatest utility of th[e network drawings comes from the ability to visualize the syst m, particularly in situations where one wishes to analyze the n-port networks linked up in series and/or parallel combinations. One more formal qualification: th!e flows in the networks are regarded as pseudo-currents having the numerical value of the thermodynamic flows, the thermodynamic potentials are regarded as pseudo-electrical potentials having the magnitude of the thermodynamic potentials and the phenomenological coefficients are regarded as pseudo-resistances with the usual resistance units and the magnitude of the phenomenological coefficients. This allows for the combination of currents, resistances and/or potentials without worrying about the addition of quantities with different units as in the case of Ji+Jj, for example. In the bond-graph language, since energy flow is what is studied, special transducer elements are used to avoid this difficulty. It is important to realize that the

494

D.

C.

MIKULECKY,

W.

A.

WJkGANIl

AND

J.

S.

SHlNliR

pseudo-electrical networks being utilized obey KCL and KVL and therefore Tellegen’s Theorem. Next it will be necessary to see how series-parallel combinations can be analyzed. 8. Series-Parallel

Combinations of the Z-Equivalent and n x n Networks

The I-equivalent for a 2-port will be used as an example from here on, but once a paradigm for manipulating these elements in series-parallel combinations is developed, it should be clear that the same paradigm will

FIG. 15. Graph and tree of series combination of I-equivalent networks. Note that although the single I-equivalent network has only two fundamental currents, the series combination has three due to the introduction of another loop. For each Z-network v =: 6, e=7ande-~+l=2.Fortheseriescombinatione=12,v== lOande---+1==3.

work for the n-port network elements introduced above. First of all, even though the I-equivalent network is truly an electrical network, it is not possible to create simple series and/or parallel combinations of these networks and preserve the isomorphism between electrical circuits and the thermodynamic networks. The graph and tree of a series combination is shown in Fig. 15. It can be seen that there are now three independent currents where we require that there be only two. It would be possible to avoid this divergence if a “T-equivalent” network were used in place of the I, but more than two T’s could not be hooked up in series without introducing extraneous currents. Recognizing this, the I will be used throughout for it does provide a convenient graphic representation in a manner which the T fails to do. (The T does not clearly split out the two distinct pseudo-currents on the output

LINEAR

NETWORK

MODEL

OF

COUPLED

FLOWS

495

side.) The method to be used for the analysis of series-parallel combinations of n-ports is an extension of the methods introduced by Peusner (1970). His analysis is based on the idea that if 0 is an n-vector of outputs, I an n-vector of inputs, and M is the n x IZ constitutive matrix relating them, 0 = MI, in a configuration where inputs are constant (with flows as inputs this would he a series connection of elements for example) the input-output relations combine as follows:

9. Analysis of the Series Combination

of Z-Equivalent Networks

The equations modelled by the Z-equivalent (or n-port) network can be written in matrix form: X,, = R,,i,, where in the 2-port case i,, is the Rii Rij two vector of currents ‘.l , Rij is the 2 x 2 resistance matrix R,,R,, and (I 12 ( 11 JJ> X,z is the two vector of driving forces (potential differences) X,, = Xl x ( > The matrix equation is analogous to Oh’s law for a network consisting’@ single element, current and potential tf$erence. When there are two such systems connected in series, the currents are common to both, but the resistances and potentials are additive in the following manner by KVL: Agailr hation shown series

I= xl2 +x3, = (R,, +R34)i12. tlmc is a matrix equivalent to the scalar equations for tltc series doom qf single resistive elements! Thus, the series combination of ‘-port5 in Fig. 16 can be represented by a simpler network which is simply a combination of two simple resistive l-ports.

FIG.

16. Series combinationof 2-port elements.

10. Analysis of the Parallel Combination of Z-Equivalent Networks A similar analogy can be drawn for parallel combinations of the Zequivalent (or n-port) networks, but the “Ohm’s law” equations must now be written in conductance form : i 12 =

b2Xl2,

496

I).

C.

MIKULECKY,

W.

A.

WIEGAND

AND

J.

S.

SHINER

Wht=X

L,, = R;;. In parallel combination, currents divide and potential differences are common across the elements, so that the conductances and currents add in the following manner due to KCL: it = i,, +is4

= (LIZ +L,,)X,,.

Written in resistive form Xl, = (LIZ +L,,)-‘i,. This results in the same analogy between parallel combinations of the Iequivalent network and the simple parallel combination of single resistive l-ports as was obtained in the series case. The parallel combination of I or n-port elements as shown in Fig. 17 can be represented as the parallel combination of single resistive l-ports. It is now possible to analyze any network

FIG.

17.Parallelcombinationof 2-port elements.

which is an array of series-parallel combinations of I-equivalent (or n-port) networks as if it were a simple network of single resistive l-ports. In analyzing the simplified network of l-port elements it is necessary to carry out the operation of division by using the inverse of the divisor and premultiplying. This preserves the character of the matrix-algebraic manipulations. The resultant scalar equations for the simple analog will then be isomorphic with the matrix equations for the more complicated networks. This is a very convenient aid in computation which is readily utilizable in computer analysis. If one dislikes or is unfamiliar with matrix algebraic manipulations, an alternative computational scheme simply involves writing the equations directly by following all the voltage drops along loops in the complete network diagram. The resultant set of simultaneous equations will still need to be solved with Cramer’s rule and determinants. With the above paradigm one never deals with anything more than a 2 x 2 determinant,

LINEAR

NETWORK

MODEL

OF

COUPLED

FLOWS

497

ll. Application to Coupled Solute and Volume Flow through Membranes In order to provide a concrete example of the utility of this paradigm it will be applied to the analysis of coupled solute and volume flow through a membrane. In some cases the effects of an active transport pump which moves solute against an electrochemical gradient will be added. Electrical effects will be ignored at this stage of analysis, so that the pump can be viewed as a single electroneutral NaCl pump or, to be more general, a solute pump. In these networks such pumps will be represented by ideal current sources. More sophisticated sources could be utilized for more accurate models of the actual physiological system, but at this stage of analysis, the ideal current source will be adequate. To make a bridge between this paradigm and what is probably the most familiar formulation of coupled solute and volume flow through a membrane, we will begin by rearranging the Kedem-Katchalsky practical phenomenological equations into a purely resistive form.

12. The Resistive Form of the Kedem-Katchalsky Equations The “practical” phenomenological equations and volume flow through a membrane are: Volume flow: Solute flow:

Phemonenoiogical

describing coupled solute

J, = L,(Ap-aAn) J, = a?,(1 - a)J,+ oArr,.

The symbols in the above equations are defined as follows: J, is the volume flow per unit area of membrane in cm s- ‘. J, is the solute flow per unit area in mol cm-’ s-l.

Ap is the hydrostatic pressure difference across membrane pL -pR where ,uL and pR are the pressures in the left and right baths in dyn cm-‘. Arrr,is the osmotic pressure difference across membrane due to a permeable solute, AX = RT(c,-c,) where c, and c, are the concentrations in the baths. Ani is the osmotic pressure difference across membrane due to impermeable solutes. Arc = Ani+An,. L, is the filtration coefficient in cm3 dyn- ’ s-i. cr is the reflection coefficient (dimensionless). o is the solute permeability in mol dyn-’ s-l. C, is the average concentration (c,+c,)/2 (see comments on linearity which follow).

498

D.

C.

MIKULECKY,

W.

A.

WIEGAND

ANI)

J.

S.

SHINER

These equations have been applied to a single set of series membranes and a single set of parallel membranes by Kedem & Katchalsky (1963a,h.c,). In their analysis the coefficients o, L, and CTwere derived for the composite membrane in terms of those same coefficients for the elements. We have already shown that the simplest method for combining coefficients is that of adding resistances for series arrays and conductances for parallel arrays. In the case of the Kedem-Katchalsky treatment, this procedure was not followed and the result appears more complicated than necessary, although the interpretation of the elemental as well as the composite coefficients is consistent with the practical nature of their formulation of the flow equations for a single membrane. The following analysis suggests that for series-parallel systems, the purely resistive and/or conductive format is indeed more convenient at every level, elemental and composite. The equations in “resistive” form are : J” + [ 5,,)]

AP- AZ = 2=In the following

rq]-J‘,+

[&-I

analysis the resistors

J,.

will be identifiable

(a) Rij-“straight”

resistance in the volume current

(b) Rjj-“straight”

resistance in the solute current

(c) Rij--coupling

*I,

as:

resistance -(l-a) =---0

and of course, the potential differences are : Xi = AP’and the currents

A7ti,

Xj = A&

are: ii = J’,,

ij = Jj.

There is an immediate benefit of the resistive formulation. The coupling resistors, having a negative resistance, can be viewed as a dependent source in the system, with the characteristic that Vij = Rij(ii+ii). This encourages

LINEAR

NETWORK

MODEL

OF COUPLED

FLOWS

499

the redrawing of the I-equivalent network for coupled volume and solute floxwsas in Fig. 18. 1-nall the above equations, the i’s andj’s denote particular currents and/or potentials, through and/or across individual elements (Z-equivalents) in such a way that the i’s will stand only for odd integers and are to be associated on’ly with volume current and/or potential and the j’s will stand only for even integers and are to be associated only with solute current and/or potential. The r; are an average concentration for the system. The use of Fsby Kedem and Katchalsky actually constituted a linearization of the transformation between APTand AC, which they used to define T, as ?, = An,lA~c~. Obviously if a quantity is linear in Ap,, it cannot simultaneously be linear in Arc,. The problem as to which values should be used to compute Fs has

f;IG. 18. The Z-equivalent network for coupled solute and volume flow through a membrane. The coupling resistor, which has a negative value, is drawn as a controlled source to emphasize this point.

been discussedby many workers (Sha’afi, Rich, Mikulecky & Solomon, 19’70; Levitt, 1975; Caplan & Mikulecky, 1966; Mikulecky, 1977a,b), and the approximation C, = (cl + cJ/2 is generally accepted. It will be shown in the non-linear treatment that a correct non-linear analysis yields this value of C, in its linearized form. c1 and c2 are concentrations on either side of the membrane, in the caseof single membranes. In complex membrane systems the values c1 and c2 may need to be taken as concentrations on either side of the membrane system, but that causesthe approximation to be a much cruder one. In the non-linear treatment in the following paper (Mikulecky, 1977a,b), the approach will be modified in such a way as to recognize this concentration dependency in constitutive relations as the sole source of nonlinearity in the system and the limitations of the linear approach will be spelled out in some detail. For now the linear approximation will be used and it will also become the basis for a piecewise linear approach to the

500

D.

C.

MIKULECKY,

W.

A.

WIEGAND

AND

J.

S.

SHINER

actual non-linear behavior in the more refined theory. The series membrane system has been utilized as a model for isotonic transport in epithelial membranes for some time and is a good place to begin to apply the network paradigm.

12. The Curran-Macintosh

Model for Isotonic Transport

In essence, the phenomenon of isotonic transport in the absence of an external driving force is that volume flow which occurs in the gut, for example, during those periods when the gut lumen contains a volume of fluid isotonic with blood. The tissue has the capacity for moving fluid across the epithelium and finally into the bloodstream in the absence of any trans-

solute

infustcn

A

+ Mem bran? “tight”

FIG.

I

ft Membrane "1005e"

il

19. Experimental set-up for the Curran-Macintosh

model of isotonic transport.

epithelial driving force without significantly altering the composition of the fluid in the process of transport. Curran & Macintosh (1962) devised a simple laboratory model which seemed to capture the main features necessary for a system having the structure of the epithelium to exhibit the function of isotonic transport as described. The key is that a solute pump had to be so located in such a place as to “inject” solute into a compartment bounded by two membranes, one “loose” and one “tight” as shown in the device pictured in Fig. 19. As the solute is “injected”, the system becomes hypertonic in the middle compartment, B. Water is drawn into compartment B, increasing its volume and creating a driving force for volume flow. The intrinsic asymmetry (Curie’s principle) of the system dictates that most of the resultant volume flow goes through the loose membrane rather than the tight one. After a while, if the solute infusion is constant, a stationary state will occur which

LINEAR

NETWORK

MODEL

OF

COUPLED

FLOWS

501

results in fluid of a constant composition moving through the system from A to C. The composition of the fluid moving through the system will, 01 course, depend on the nature of the two membranes. The nature of that dependence and the other possible controlling factors are not necessarily intuitively obvious and will come from the network analysis. 14. Network

Analysis of the Curran-Macintosh

Model

The network for the Curran-Macintosh Mode1 is shown in Fig. 20. The presence of the current source, JP, which represents the continuous salt. infusion fixes the topology so that the system is not strictly a simple series membrane system. Also, in the isotonic transport case Vi = I’!, or X, = 0

c

Membrane

I

Membrane

II

FIG. 20. Network representation of Curran-Macintosh model using Z-equivalent networks to represent coupled solute and volume flow through each membrane, and a constant current source to represent the constant infusion of salt into the system. (Volume introduced by salt infusion is assumed negligible.)

and Vi = Vc, or X, = 0, so that for the stationary state these points are essentially connected to “ground”. In light of this and using the fact that single elements can be used in place of the I-equivalent networks and a scalar analysis can be used to take the place of matrix algebra, the system can be viewed as the simplified network shown in Fig. 21. By defining the vector 0 IP= J an analysis of the system in terms of the Kirchhoff laws yields two 0 simultaieous equations: ip = i,2 +ia‘$,

&CL)

%i12-R34f34 = 0, ww where all symbols are as defined above for seriesand/or parallel combination of the I-equivalent 2-port networks.

502

11.

C.

MIKULECKY,

W.

A.

WIEGAND

AlvD

J.

S.

SHINlzR

*

FIG. 21. Simplified current divider made up of l-port resistive elements symbolizes the scalar isomorphs of the matrix-vector relations describing the two port elements in the Curran-Macintosh model.

The solution to the set of simultaneous equations gives the usual current divider relationship: i 12 = (RI2 +R34)-1R34i,y

44 = (R12 +R34)-“R12 i,. Once more it is possible to view theseequations as simple scalar relationships for the circuit above, or alternatively as matrix-vector equations for the coupled solute and volume flows modeled by the I-equivalent networks. By performing the indicated matrix manipulations and making the identification 11 = -f,, i, = -.T,, i, = J, and i4 = J&+J,,, the solution: J, =

R,,RI~-R,,R,, A __

JP

Js= R,dR,, I +R,d J ___- +Rsd--RdR, .-~ P A

J

+ J s

=

R22@11

+R,,)-R,~(RI~+&J

P

A

J P

is obtained, where A = (R, 1 + R,,)(R,, + R,,) -(RI2 + R,,)‘. The “tonicity” of the transported fluid is the ratio of total solute flow to volume flow tonicity =

Js+J, _ R22&1 Jo

+R,,)-R,,(R,2+R,,)

RdG2-R,,R,4

a

This result illustrates the dependence of the transported fluid’s composition on the indivual membrane coefficients. By use of the relations which relate V, L, and w to the Rij’S the above result can be expressed in terms of those coefficients.

LINEAR

NETWORK

MODEL

OF

COUPLED

FLOWS

503

where the subscript 1 refers to membrane I and 2 to membrane II. A number of conclusions can be drawn at this point: (I) The composition of the transported fluid depends only on membrane parameters and is independent of the pump rate. The pump rate only governs the amount of fluid transported. The constancy of the tonicity of the transported fluid is sensitive to the variability of ESat each membrane, but since the system is in a stationary state, this effect is minimized. This point will be examined further in the subsequent papers in this series. (2) The Kedem-Katchalsky practical coefficients do not lend themselves co as compact a computational scheme as the resistive coefficients. The final criterion for which set is more useful lies in their experimental accessibility. Certainly, though, the Rij’S are more useful for computation. (3) Although the coupled 2-flow, force system follows a simple analogy with a current divider in an ordinary resistive circuit the “path of least resistance” in the coupled 2-flow, force system is determined by a number of factors, not just a single ratio of resistances. The ratio of the magnitudes of the solute flows is: =

ij-& P

R&VI

z ++J

- R&L+d~

R,,iR,,+R,J-R,dR,,+R,,) ’

or by defining r i (R, z + R3J/(R,,

+ RJ3):

This ratio demonstrates that the “current division” “modified permeabilities” having the form Rjj-

is given by a ratio of

Rijr

for a simple current division of the ratio would depend on the ratio of the Rji’s alone. Now, some of the usefulness of the network paradigm has been seen in a simple example. A fairly complete analysis of the coupled system was achieved with a minimum of computation. The analysis gave a very helpful insight into the behavior of the system and its ability to carry out a function analogous to isotonic transport. The second paper, which accompanies this one (Mikulecky, 1977b), examines the non-linearities introduced by ESin the solvent drag term for solute flow, and introduces a general technique for non-linear network analysis. In another paper in this series [a preliminary version has already appeared (Mikulecky, 1976)] these principles as well as the further refinements of the non-linear theory will be applied to the more complicated epithelial network which is a series-parallel system containing 1.8. 32

504

D.

C.

MIKULECKY,

W.

A.

WIEGAND

AND

J.

S.

SHINER

a pump performing the function of the current source in the CurranMacintosh model. The pump in the epithelial membrane functions by moving

salt from one compartment

to another within the system by active transport.

One of us (D.C.M.) has had the privilege of trying out some of these ideas before an astute audience (Mikulecky, 1976, 1977a) and thanks the organizers of the Winter Schools on the Biophysics of Membrane Transport held each February in Poland for making the papers possible through their invitations to these stimulating meetings. In particular, special thanks go to Professor B. Lindeman, whose workshop on Epithelia at the First Winter School stimulated much of the applications in this and the forthcoming papers. Also our gratitude to S. R. Thomas who, as he began to use these methods to model the countercurrent systems in kidney for his thesis project, asked many leading questions which necessitated making the distinctions between these pseudo-electric networks and real electrical networks as clear as possible. Thanks to George Oster and John Wyatt for steering us back onto the path when we were lost in the woods. And again special gratitude to L. Peusner whose paternal interest in the work kept us from missing some essential points in his thesis which were of great value in simplifying and clarifying some of the points in this extension of that work.

REFERENCES BRAYTON, BRAYTON, CAPLAN,

R. K. & MOSER, J. K. (1964u). Q. Appl. Math. XXIII I, 33. R. K. & MOSER, J. K. (19646).Q. Appl. Math. XXIII 1, 81, S. R. & MIKULECKY, D. C. (1966).In Zon Exchange (J. A. Marinsky, ed.), vol.

I, p. 1. New York: Dekker. CHUA, L. 0. (1969)Introduction to Non-linear Network Theory. New York:McGraw-Hill. CURRAN. P. F. & MACINTOSH. J. R. (1962). Nature 193. 347. DESOER,‘~. & OSTER, G. F. (1973).Iit. J. Eng. Sci. 11,’ 141. FINKELSTEIN, A. & MAURO, A. (1963).Biophys. J. 3, 215. GOLDSTEIN, H. (1950).Classical Mechanics. Cambridge:Addison-Wesley. KARNOP, D. & ROSENBERG, R. (1975).Systems Dynamics: A Unified Approach.

Wiley-Interscience.

KEDEM, 0. & KATCHALSKY, A. (1963a).Trans. Faraday KEDEM, 0. & KATCHALSKY, A. (19636). Trans Faraday KEDEM, 0. & KATCHALSKY, A. (1963c). Trans Faraday LEVITT, D. G. (1975).Biophys. J. 15, 533. MIKULECKY, D. C. (1974).In Biophysics of Membrane

New York:

Sot. 59, 1931. Sot. 59, 1941. Sot. 59, 1954.

Transport, p.120(S. Miekisz & J. Gomulkiewicz,eds.)Wreclaw,Poland:Agricultural Universityof Wroclaw. MIKULECKY, D. C. (1976). In Biophysics ofMembrane Transport, p. 182.(J. Kuczera& S. Przestalski,eds.)Wroelaw,Poland:Agricultural Universityof Wroclaw. MIKULECKY, D. C. (1977a). In Biophysics of Membrane Transport p. 31 (S. Przestalski, J. Kuczera& J. Idzior, eds.),Wroclaw,Poland:AgriculturalUniversityof Wroclaw. MIKULECKY, D. C. (19776). J. theor. Biol. 69,511. MILLER, D. (1969).In Transport Phenomena in FIuids, p. 377.(H. Hanley,ed.),NewYork: Dekker. ONSAGER, L. (1931).Phys. Rev. 37,405. ONSAGER, L. (1931).Phys Rev. 38,2665.

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G. F. & AUSLANDER, D. M. (1971b). J. Franklin Inst. 292, 77. OSTER. G. F. & DFSOER. C. A. (1971). J. theor. Eiol. 32. 219. OSTIER;G. F. PERELSON,‘A. (1973). Z&ael J. Gem. 11,4k.

OSTER,

OST~R, G. F. & PERELSON, A. (1974u). Archs. Ration. Mech. Analysis. 55, 230. OSTER, G. F. & PERELSON,A. (1974b). Archs. Ration. Mech. Analysis. 57, 31. OSTER, G. F., PERELSON, A. & KATCHALSKY, A. (1971). Nature 234, 393. OSTER, G. F., PERELSON, A. & KATCHALSKY, A. (1973). Q. Rev. Biophys. 6, 1. PERIELSON, A. (1975). Biophys. J. 15, 667. PELLSNER, L. (1970). Ph.D. Thesis entitled, “The Principles of Network Thermodynamics”

Cambridge, Massachusetts: Harvard University. M. B. (1961). Linear Graphs and Electrical Networks. Reading,-. Mass.: Addison-Wesley. SHA’AFI, R. I., RICH, G. T., MIKLJLECKY, D. C. & SOLOMON, A. K. (1970). J. Gen. Physiol. In the apoendix bv D. C. Mikuleckv. 55.427. SHINER, J.-s. & M&JI.ECKY, D. C. (lb75).~Bu/l. Am. Phys. Sot., Series N 20, 1438. SESFKJ, S. & REED,

TELLEGEN, B. D. H. (1952). Phiiips Res. Rep. 7, 259. THOM, R. (1972). Stabilite Structurelle et Morphogenesese

Essai d’une

Theorie.

New

York:

Benjamin. (French ed. 1972. Eng. ed. 1975). THOMAS, S. R. & MIKULECKY, D. C. (1977). Microvasular Research (in press). WIEGAND, W. A. (1976). Ph.D. Thesis entitled, “Topology in Biology, The Relationship Structure to Function”, Richmond, Virginia: Medical College of Virginia.

of

APPENDIX A

Lagrangian Analysis of Networks of Resistively Coupled Systems Subject to External Forces in the Steady State (Shiner & Mikulecky, 1975) J. S.

SHINER

According to the Lagrangian formulation of classical mechanics (Goldstein 1950) the equations of motion for a system of n degrees of freedom are d c?L c?L aF --_+a~,=0, dt aJj aqj

j=

1, . . . . tr,

(Al)

where L is the Lagrangian, debed as-the difference between the kinetic and potentiai energies and F is the dissipation function which is defined below. The qj, called generalized co-ordinates, may be any n parameters describing the: system as long as they are a complete description. The Jj c dqj/dt are known as generalized velocities. Since we are interested in steady-state situations, inertial effects may be neglected, and we consider L to be simply the negative of the potential energy. We only consider forces Xj which are “external”, i.e. independent

506

D.

C.

MIKULECKY,

W.

of the qj; the potential

A.

WIEGAND

AND

J.

S.

SHIN1

R

energy is then U=-~~Xjdqj=--Xjqj j

j

and the Lagrangian is L = C Xjqj.

(43

i

Since the system is resistively coupled the dissipation function has the form F=3CRjJf

j

+3CRjkJjJk=3CRjJi2

i, k

+tCRSkJjJk,

j

j#k

(A3)

j. k jfk

where Rsk = 3(Rjk + Rkj) = R;j.

(A4)

Upon neglecting inertial terms, the Lagrange equations become,

aL

aF =

4j

~j’

or, in vector form OL =vJF, (A5) where Oj signifies the same operator as the gradient except that the partials are taken with respect to the Jj instead of the qj. Also, note that L and F may be written as L = x.4 646) and F=$R:J’, (A7) where

R=

R, Ri2

R;, R,

Ri3 Ri3

. .. . .. , I

WI

I Therefore, the equations of motion for the system are 1 = R.J, (A9) which are identical to the phenomenological equations. The question now becomes: how do we connect systems described by equations (A6) and (A7) or (A9) into the networks? The Lagrangian of the network is simply the sum of the Lagrangians of the constituent systems and similarly for the dissipation function. In addition, if two systems are connected in series, we have the constraint J1 = J, (AlO) and if they are connected in parallel, we have the relation: x, = x2. .

.

.

.

.

.

.

(All)

LINEAR

NETWORK

MODEL

OF

COUPLED

E.uamples:

F:or 2 systemsconnected in series: L = Ll +L* = x’ig,+X**q* F=+R:Jf+$R,:J; J, = J, = J+q1

= q,+2,

therefore L = X,(?j, -t-c) +x,-q, F = 4(R, +Rz) : 6* al-d Xl +X2 = (R, +R,).J. For 2 systemsconnected in parallel: L = x&j, +x,-q2 F=+R,:J:++R,:J; x1 = x2 = x. The overall Lagrangian is then

L = X,*(q,

i-42).

therefore x = RI 4, 8 = R,*J, and j, = R-‘.R,*J,. this leads to the familiar result: j = J,-d,

= (R-‘~R2+I)~&

= (R-‘.R2~I)*R;1.W, or X = RI-R,-(R,

+R,)-‘-1.

For 2 systemsconnected in seriesand a pump in parallel L = xi*gi+x2q~ F=$R,$+$R,:J; J2=J,-J,-q2=(11-JpAt+(;, therefore L=Xi4,+X,.(q,-J,At+c) F=fR,:Jf+3R,:(J,-J,)*

FLOWS

507

508

D.

C.

MIKULECKY,

W.

A.

WIEGAND

AND

J.

S.

SHINER

and 8, +x2 = R, *J, +R,*(J,

-J,)

or 1, +fi;, +R,J,,

= (R, +RJ*J,.

APPENDIX

B

The Osmotic Transient as an Example of a Time Dependent Network

To model transients, the capacitive aspects of the system must be included. These are then included as network elements with constitutive relations which relate the rate of change of some driving force with a flow. In the osmotic transient, solutions of differing concentrations are placed on the two sides of a membrane. The flows relate to the rate of change of driving forces through known geometric factors which are analogous to capacitances. It is convenient to choose the set of variables J,, JD, AI-‘, An to describe the system. This choice leads to the network in Fig. 22. Notice that it is an I-network

FIG. 22. The network for the osomotic transient. Capacitances are substituted for the driving forces and a new set of flow-force variables is utilized for convenience.

with capacitors replacing the usual sources. The constitutive relations for the capacitors are dAP -$-=

ClvJ

and !!&T=CJ

2 Ds dt where the value of the capacitor C1 is determined by the area of the membranes and the cross-sectional area of a set of vertical capillary tubes used to create the pressure difference across the membrane, and C, by the membrane

LINEAR

NETWORK

MODEL

OF

COUPLED

FLOWS

509

area and chamber volumes. The combination of these constitutive relations and the conductive forms of the phenomenological equations leads to a set of coupled first order ordinary differential equations which can be solved with appropriate initial conditions to yield the usual osmotic transient. Notice that the network diagram suggests that if the system starts with a concentration difference (C, charged) and no pressure difference (C, discharged) the discharge of C2 will temporarily charge C, and then it also will discharge to equilibrium. Oscillations are prevented by the positive-scmidefiniteness of the coefficient determinant which is a result of the positivcsemidefiniteness of the dissipation function which results from the second law of thermodynamics.

APPENDIX Reductionism

vs. Wholism:

C

The Role of Network

Models

Living tissues are very complicated structures involving a variety of geometric forms and a multitude of interconnections. The jump from physical and chemical models to a living system has always seemed to involve concepts which arc beyond simple physics and chemistry. At one time this “extra ingredient” was explained by the “postulate” of vital spirits and vital forces. Gradually, as more and more experimental science replaced philosophy, some of the biochemical components were isolated and identified, fractionation techniques improved. and the question of the existence of vital forces became a joke. Eventually, the reductionist view became dominant and molecular biology began to dominate comparative anatomy, ecology, taxonomy. and the other classical subdisciplines of biology. Yet, something was still missing. Hints began to appear that breaking down structure to get at pieces to study in isolation also broke down function-often those functions which wsre under investigation. Now we can again assign a name to that missing something-organization. Gaylord Simpson once pointed out that the reason biology was higher in the hierarchical structure of science than physics or chemistry was that it dealt with levels of organization not found among the objects of study in those disciplines. Indeed, the molecules and the physical-chemical laws are the same, but things are put together in ctiffercnt ways when they are aiive! Rene Thorn (1972) formufates these two philosophical trends as reductionism and vitalism. To avoid the word but it seems that this is skirting the vitalism one might substitute “wholism” issue. Thom certainly seems correct in stating that if one of these approaches is metaphysical, it is reductionism and not vitalism.

510

D.

C.

MIKULECKY,

W.

A.

WIEGAND

AND

J.

S. SHINER

The main question of interest here is whether the network approach will be of any help in resolving this philosophical dilemma. To the extent that it provides a method for dealing with more complicated organizational patterns, it must. On the other hand, reducing a living system to a network is not far from reducing it to a collection of molecules. The networks, as models, are more models of our theories and hypotheses about how the living system works than of the living system itself. By creating the appropriate network, various notions we have about the workings of an organism can be quantitatively tested and a lot of hand-waving and speculation done away with. This, it seems, will be the role of network models in the next phase of understanding the nature of the living system. Before the network approach, the task of analyzing highly-organized systems looked hopeless. Now that part of the problem seems almost trivial. The late Aharon Katzir-Katchalsky was known to often quote, “The goal of all science is to reduce all its problems to triviality.”