coevolution: new thermodynamic theorems

J. theor. Biol. (1990) 147, 205-212

Coevolution: New Thermodynamic Theorems

ANTONIO LEON

Seminario de Ciencias Naturales, lnsti tuto Luis de Oongora, Cordoba, Spain

(Received on 21 November 1989, Accepted in revised form on 14 M a y 1990)

The formal incorporation of the concept of interaction in thermodynamic analysis makes it possible to deduce, in the broadest terms, new results relating to the coevolution of interacting systems, irrespective of their distance from equilibrium. We have demonstrated the existence of privileged coevolution trajectories, characterized by the way in which systems progress along them in exchange for the production of a minimal amount of entropy within the systems that do so.

Introduction

One of the primary objectives of irreversible thermodynamics is the study of open systems, systems which--l ike biological systems--maintain a continuous exchange (flow) of matter and energy with their environment. Through these flows, open systems organize themselves in space and time.

However, the behaviour of these systems differs considerably according to whether they are close to or far from equilibrium. Close to equilibrium, the phenomenological relationships which bind flow to the conjugate forces responsible for it are roughly linear, i.e. o f the type:

Ji = • LoXj, i , j = 1, 2 , . . . , n, J

where Ji is the flow, Xj the force and L o the so-called phenomenological coefficient giving the Onsager (1931) reciprocal relations ( L o = Lji, i ~ j ) .

In these conditions, Prigogine's theorem (1945) asserts the existence of steady- states characterized by minimum entropy production. Systems can assimilate their own fluctuations, and be self-sustaining.

Far from equilibrium, on the other hand, the phenomenology may become clearly non-linear, permitting the development of certain fluctuations which will re-configure the system (Glansdorff & Prigogine, 1971). Irreversible thermodynamics can now only provide information regarding its stability. There is nothing approaching an evolution potential (Vokenshtein, 1983).

The purpose of this study, therefore, is to draw certain conclusions regarding the future of these systems.

Definitions

The following paper is based on two prior assumptions: (i) Certain interesting open systems exist whose available resources are limited.

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206 A. LEON

(ii) Owing to these l imi ta t ions , such systems compe te with one a n o t h e r to ma in ta in

the i r flow. Both a s s u m p t i o n s l ead d i rec t ly to the concep t o f in terac t ion . But be fo re p r o p o s i n g

a formal def in i t ion o f in t e rac t ion let us examine Fig. 1, in o r d e r to ob t a in an intui t ive idea o f the type o f p r o b l e m be ing dea l t with. F igure 1 p rov ides a schemat i c r ep re sen t a t i on o f three open systems ou t s ide t h e r m o d y n a m i c s equ i l ib r ium. In one case (i) the sys tem if not sub jec t ed to in te rac t ion with o the r systems, and ma in ta ins a th rough- f low of ma t t e r and energy which d e p e n d s s o l e l y - - w i t h o u t going into p h e n o m e n o l o g i c a l d e t a i l s - - o n its own degree o f imba lance . In the o the r case (ii) the sys tems in terac t with each other , and the i r respec t ive flows d e p e n d on the degree o f i m b a l a n c e o f bo th sys tems s imul t aneous ly . It is this s i tua t ion which will be e x p l o r e d here as b r o a d l y as poss ib le , pa r t i cu l a r ly f rom the s t a n d p o i n t o f the j o in t

evo lu t ion o f the two systems.

I "

SOURCE ,,

SINK

FIG. 1. Circles represent open systems outside thermodynamics equilibrium. Rectangles represent significant parts of the systems" environment. Flows are shown as large arrows crossing systems from source to sink. Black arrows represent the degree of imbalance of the system. Curved broken lines of direction represent interaction. (a) In the absence of interaction with other systems, flow depends only on the degree of imbalance of the system itself, X, a relationship generally expressed here as Jx =f(X) (see text). (b) Here systems use the same, limited source of resources, which forces them to compete in order to maintain their flows (a situation strongly inspired by biology). Flow in each system now depends on the degree of imbalance of both systems simultaneously. Flows are expressed by expressions of the type: Jx =f(X)-g( Y); Jv = f ( Y ) - g ( X ) (see text). In order to maintain the same flow as case (b), a greater degree of imbalance is required, expressed schematically by larger black arrows.

Let f a n d g be any two type C 2 func t ions ( con t inuous func t ions with first and s econd der iva t ives which are also con t inuous ) , def ined on R (real n u m b e r s ) such that :

(1) f and g are s t r ic t ly mcreas ing , with f ( 0 ) = g ( 0 ) = 0, (2) ( f - g ) is s tr ict ly inc reas ing and its first de r iva t ive ( f - g ) ' is increas ing . f may be t e r m e d the p h e n o m e n o l o g i c a l func t ion and g the in te rac t ion funct ion .

The flow in a sys tem will be given by Jx = f ( X ) , where X is the gene ra l i zed force, a m easu re o f the degree o f the sys tem's i m b a l a n c e or d i s t ance f rom equ i l ib r ium; hence , X --- 0 ( X = 0 at equ i l ib r ium) . X will a lso be used to des igna te the systems.

G i v e n two sys tems with the same p h e n o m e n o l o g y f and forces X and Y respec- t ively, an in t e rac t ion can be sa id to exist be tween t hem if the i r flow can be desc r ibed

C O E V O L U T I O N : N E W T H E R M O D Y N A M I C S T H E O R E M S 207

by the following expressions:

J x = f ( X ) + C x y g ( Y ) ; - l<-Cxw<-I

J y = f ( Y ) + C y x g ( X ) ; - 1 - < C g x - < l

where Cxy, Cyx are the types (positive or negative) and degrees of interaction between the systems. We shall deal here only with the double negative interaction, assuming--while still speaking in general terms-- that Cxy = Cyx = - 1 . In other words:

Jx = f ( X ) - g ( Y)~[ (1)

J g = f ( Y ) - g ( X ) J " f

The only (X, Y) pairs permitted will be those which give a positive value for expressions (1) above. (X, Y) pairs giving f ( X ) - g ( Y ) = O shall be termed the points of extinction of system X. The same applies to system Y.

The interaction thus defined may be seen as an external constraint imposed by the systems on each other, forcing them to reach steady-states simultaneously.

Let us now justify the constraints on f and g. Firstly, f (0) = g(0) = 0 indicates that in the absence of forces (thermodynamics equilibrium), there is neither flow nor interaction, but both increase with X, in other words, as we move away from equilibrium. The second constraint implies that the systems are progressively as sensitive to their own forces as they are to the forces of the systems interacting, or more so.

Entropy Production

The purpose of the following discussion is to obtain information relating to the coevolution of two open systems, X and Y, which interact negatively with each other. It is recognized that the entropy balance for an open system can be expressed a s "

dS = d i S + d e S

where dis represents entropy production within the systems due to flow, and des is the entropy exchange with its surroundings. The second law of thermodynamics dictates that always diS >-0 (zero only at equilibrium).

One of the most interesting aspects of irreversible thermodynamics is the incorporation of time in its equations; thus

d,S/dt = ~ XjJ i. J

In our case, for system X we have

d,xS/dt = X [ f ( X ) - g( Y)] -> 0,

and for system Y

diyS/dt = Y[ f ( Y) - g (X ) ] -> 0,

(2)

(3)

208 A. LEON

and the joint production of internal entropy

d i x S / d t + d ~ v S / d t = S ( X , Y ) = X [ f ( X ) - g ( Y ) ] + Y [ f ( Y ) - g ( X ) ] . (4)

Graphically, S(X, Y) is a surface in the space defined by the axes S X Y (Fig. 2). A curve in the plane X Y represents a possible coevolutionary history for both systems [as long as X and Y satisfy (2) and (3)]. Its projection on S(X, Y) shows the cost of this history in terms of internal entropy production.

The analysis of surface S(X, Y) is thus of great interest. Let us examine its intersection with planes parallel to SX, in other words, the joint production of internal entropy for each Y constant. For example if Y = b, we have

S ( X , b) = X [ f ( X ) - g(b)] + b [ f ( b ) - g(X)] , (5)

if we derive (5) with respect to X, calling the derivative h ( X ) b :

OS(X, b ) / O X = h ( X ) b = f ( X ) - g ( b ) + X f ' ( X ) - b g ' ( X ) ,

where h ( X ) b is strictly increasing. Moreover:

h(O)b = -- g (b ) - bg'(O) < 0,

and

h(b)b = f ( b ) - g ( b ) + b [ f ' ( b ) - g'(b)] > 0.

According to Bolzano's theorem, therefore, there is a single point Xb at which

h(Xb)b =0; O<Xb < b.

A¢

FIG. 2. Curve (1-2) represents any coevolutionary history between system X and Y. Curve (1'-2'), its projection on .~(X, Y), represents the joint production of internal entropy throughout that history.


Furthermore, since h(X)b is strictly increasing we have

a2$(X, b) /aX 2 = h'(X)b > O,

Xb is therefore a minimum of S(X, b) (Fig. 3). We have shown that for each Y constant there is a single value of X which

minimizes the joint production of internal entropy. The (X, Y) pairs which satisfy this condition form a curve which will henceforth be termed the coevolution trajectory. Its equation is

f ( X ) - g ( Y ) + X f ' ( X ) - Yg'(X) = 0. (6)

For Y = 0 we have

f ( X ) + X f ' ( X ) = O,

which is only true for X -- 0. The point (0, 0) thus belongs to the trajectory. All its points thus give:

0 < _ X < Y.

Further information can still be obtained regarding this. Deriving (6) with respect to X we obtain

2 f ' ( X ) + Xf" (X ) - g'( Y) Y ' - g '(X) Y ' - Yg' (X) = 0,

whence

Y'= [2 f ' (X)+ X f " ( X ) - Yg"(X)]/[g ' (X)+g'( Y)]

= h'(X) y /[g ' (X) + g'( g) ] > 0.

We are therefore dealing with a strictly increasing curve, contained in the region defined by axis Y and the bisector Y- -X. This means that system Y is always ahead of system X with respect to equilibrium. A point of extinction may exist for system X if the rate of increase of Y with respect to X exceeds the rate of increase

of ( f - g ) .

Y=X

F=b h(X)b

G

FIG. 3. Intersection of surface .~(X, Y) with plane Y = b, parallel to SX. X h is a minimum of S(X, b).

210 A. LE6N

Theorems

The foregoing discussion is based on those sections of ,~(X, Y) parallel to plane ,~Y. A new coevolution trajectory would be obtained, symmetrical to the first with respect to the bisector Y = X, and sharing with it the point (0, 0) of the thermodynamics equilibrium.

All this allows us to plot the surface S(X, Y) as shown in Fig. 4, and reflect the results obtained so far in the form of the following theorems:

T H E O R E M 1

For each pair of systems interacting negatively with each other, two alternative and symmetrical coevolution trajectories exist, formed by a succession of steady- states characterized by minimum joint entropy production within the systems.

I t I I I I

Y x

fc tc

FIG. 4. S(X, Y) plotted in four orthogonal section two by two. Curves tc represent two coevolution trajectories.

T H E O R E M 2

Systems progressing along these trajectories do so in such a way that one is always ahead of the other with respect to thermodynamics equilibrium.

Generalization

Our analysis has been restricted to the double negative interaction ( - , - ) ; but the same method can be applied to other types of interaction (+, +), (+, 0), (+, - ) ,


( - , 0) where " + " indicates positive interaction and "0" the absence of interaction. Though it is not our intention to analyse these here, it should be stressed that:

T H E O R E M 3

In order for coevolution trajectories to exist, at least one of the interactions must of necessity be negative.

For the case (+, +), instead of (1) we have

• Ix = f ( X ) + g( Y) ] ,

Jr = f ( Y ) + g ( X ) J

and instead of (4)

S(X, Y) = X [ f ( X ) + g ( Y)]+ Y[f( Y)+g(X)],

whence

aS(X, b)/oX = h(X)b = f ( X ) + g(b) + Xf ' (X) + bg'(X) (7)

aS(a, Y) /aY= h( Y),, = f ( Y)+g(a)+ Yf'( Y)+ ag'(Y). (8)

But both derivatives are always positive, and cancel out simultaneously and exclus- ively at the point (0, 0) of equilibrium. There are therefore no critical points for S(X, b), S(a, Y) which in accordance with (7) and (8) are strictly increasing. The same results are obtained for (0, +) and (+, 0).

If we considered n systems instead of 2, the following expressions would be obtained for flow:

Jx, = f (Xi ) +Y~ Cijg(Xj) J

-1--< C,~-< 1; i , j = l , 2 , . . . , n ,

where C U represents (as the binary case) the type and degree of interaction between system X~ and Xj. The joint production of internal entropy could thus be expressed

S(X,, X2 , . . . , X,) is a hypersurface on N "÷~, in which we can consider ~ ( n - 1) three-dimensional subspaces SXiXj in order to analyse the interaction between system X~ and system Xj using the same method.

Discussion

The most relevant feature of an open system is its exchange boundary, and particularly the set of processes which arise and are maintained at the expense of flow. These processes provide the system with a certain degree of organization, i.e. a given space-time configuration or arrangement from which the maintenance of the flow is derived.

212 A. LEON

But not all processes are equally efficient in flow maintenance. More efficient processes have a greater capacity for assimilating their own fluctuations, and thus a greater chance of being self-sustaining. To a certain extent, the degree of organization of an open system may be interpreted as a measure of its ability to maintain the flow.

We have shown that coevolution trajectories are the only histories of coevolution compat ible with a maximum level of organization of systems; in other words, they are histories of maximum efficiency in flow maintenance. Simply in view of the imposit ion of external constraints like mutability and selective pressure suffered by certain types of systems (biological, economic, etc), it is evident that these trajectories are to be taken as true references in the coevolution of these types of systems.

In addition to the existence of the trajectories, perhaps the most striking result is their progression along them referred to in theorem 2. There is compatibil i ty with the empirical data inspiring what many biologists have referred to as St Matthew's principle (1980).

We should like to thank Dr Ram6n Margalef, Professor of Ecology at the Autonomous University of Barcelona, for kindly agreeing to read this paper in draft form.

REFERENCES

GLANSDORFF, P. & PRIGOGINE, 1. (1971). Structure Stabilit~ et Fluctuations. Paris: Mason MARGALEF, R. (1980). La Biosfera. Barcelona: Omega. ONSAGER, L. (1981). Phys. Reo. 37, 405. PRIGOGINE, I. (1945). Bull. Acad. R. Belg. 31, 600. VOLKENSHTEIN, H. V. (1983). Biophysics. Moscow: Mir.

coevolution: new thermodynamic theorems

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