J. theor. Biol. (1979) 78, 241-250

Non-equilibrium

Thermodynamics and Biological Growth and Development D. LURIfi Universidad de Barcelona, Barcelona, Spain

Dpto. de Fisica Te&ica,

AND

Dpto. de TermoIogia,

J. WAGENSBERG Universidad de Barcelona, Barcelona, Spain

(Received 12 July 1978, and in revisedform

17 November

1978)

It is shown that there exists in principle no incompatibility between the observed increase in specific dissipation during early embryogenesis and the theorem of minimum entropy production based on linear irreversible thermodynamics. As a specific illustration we exhibit a linear model for which the time evolution of the three terms in the entropy balance is parallel to that of the evolution of living systems. 1. Introduction In the application of the thermodynamics of irreversible processes to the problem of biological development a central role is played by the entropy balance equation for an open system dS = diS+ d,S, where diSis the entropy production due to irreversible processes inside the system and d,S is the entropy flow due to exchange with the surroundings. Although d,S must always be positive or zero as a result of the second law, the flux term d,S has no definite sign. Hence, if d,S is negative and greater in absolute value than diS, the entropy of the open system will decrease, thereby reflecting an evolution of the system toward a more ordered state. In physics, the effect is well known to occur in the case of a Knudsen gas (Prigogine, 1962) and more than thirty years ago Schriidinger (1944) proposed that such an effect could account for the tendency of a living system to structure itself during its growth and development. In applications of the entropy balance equation to biological phenomena one is interested in the specific (i.e. per unit mass) rates of entropy flow, entropy production and entropy variation. Correspondingly, one writes the balance equation in the form 1 dS m dt

1 d,S 1 d,S mdt+mdt’

(1)

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fZJ 1979 Academic

Press Inc. (London)

Ltd.

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Information on the entropy flow term is directly obtainable from the experimental data on specific heat dissipation in animals. The latter represents the entire caloric contribution to m- ’ d,S/dt. The specific entropy production, on the other hand, cannot be measured directly. The Prigogine-Wiame (1946) theory of biological growth and development assumes that the basic relations of the thermodynamics of linear irreversible processes can be applied to biological development. In particular it is assumed that the development, growth and aging of living organisms can be interpreted thermodynamically as a continuous approach to the adult stationary state accompanied by a decrease in the specific rate of entropy production in accordance with Prigogine’s minimum entropy production theorem (Prigogine, 1962). Traditionally (Trintscher, 1966; Zotin & Zotina, 1967) it has been considered that the specific entropy production is, to a good approximation, given by the specific heat dissipation. Figure 1 displays a typical biological (specific) thermogram as given for example by Zotin (1972). As can be seen from the figure, a characteristic feature of the thermogram is the increase in the specific heat production during the embryonic development stage. This feature has been the chief source of difficulty in the way of the PrigogineWiame theory. As Herniaux & Babloyantz (1976) state in a recent article “the inability of the minimum entropy production theorem to describe the increase in dissipation during the early stages of embryogenesis has often been interpreted as an inadequacy of irreversible thermodynamics to account for biological phenomena. Various suggestions have been made as to how this difficulty might be circumvented. Zotin has argued that the assumption of fixed external constraints, upon which Prigogine’s minimum entropy production theorem is based, must be relaxed during the embryogenetic stage. A more drastic suggestion, discussed by Hiernaux & Babloyantz (1976) is that the assumption of linear relations between thermodynamic fluxes and forces must be abandoned during the early stage of development. In this view, the embryogenetic stage is characterized by non-linear thermodynamics of systems far from equilibrium (Glandsdorff & Prigogine, 1971). Such non-

FIG. 1. General form of thermogram production in arbitrary units.

for

living

organism.

Figure

shows

specific

heat

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linear systems can evolve through a sequence of instability thresholds thereby undergoing transitions to states of increasing entropy production. The purpose of the present paper is to call attention to the fact that neither linearity nor fixed constraints need be abandoned to account for the observed increase in dissipation during embryogenesis. The thermogram data is seen to be perfectly compatible with the validity of the minimum entropy production theorem once it is realized (Lurie & Wagensberg, 1979) that the identification (to within a sign) of the heat dissipation with the entropy production may become a poor approximation precisely during embryogenesis. The breakdown of this approximation and its implications for the evolution of the three terms in the entropy balance are discussed in section 2. In section 3 we develop a linear kinetic model which exhibits similar entropy balance profiles to those exhibited by living systems. The implications of this model are discussed in the final section, section 4.

2. Heat Dissipation

and the Entropy Balance

Let us consider (Prigogine, 1962, p. 27), a set of Y simultaneous chemical reactions taking place at temperature T. The entropy production is given by d,S 1 ’ -? z Apup=+ dt P-1

i; A$ p-1

Here the 5, are De Donder’s extent of reaction parameters, up = dr,/dt are the reaction rates and the affinities A are given by

A=-EEL= -[~]p,T+T[~]p,T (3)

where H denotes the enthalpy. We can rewrite equation (2) as an entropy balance equation d,S 4s - -dt+z. dt

Here

dS (4)

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WAGENSBERG

is the entropy transfer and (6)

is the entropy variation term. If we can neglect the entropy variation term with respect to the d,S/dt term in equation (4), then

Em dt -

i1

-- 1 -dQ T

dt

(7)

P.T

and as Prigogine notes “in this approximation the entropy production of a living organism can be measured by its metabolism as recorded by calorimetry” (Prigogine, 1962). This approximation becomes more valid the closer one approaches to the adult stationary state for which dS N 0 and diS N - d,S. Conversely the further one gets from the adult stationary state, and this is particularly true for the early stages of embryogenesis, the poorer this approximation can be expected to be. Nevertheless if the approximation is imposed, i.e. if the heat dissipation is taken to measure the entropy production throughout the entire biological development including embryogenesis then one must expect to come up against contradictions of the type discussed in the introduction. If the approximation of identifying the specific heat dissipation is not made then the experimentally determined form of the specific heat dissipation curve as a function of time can only furnish information about the specific entropy transfer m - 1 d,S/dt in the specific entropy balance equation (1). This information is shown in the form of the dotted curve in Fig. 2. It represents a typical biological thermogram as given for example by Zotin (1972). It has been reflected through the x-axis to take into account the fact that heat

Time

2. Time evolution of the three terms in the d,S/dt. dotted line denotes m -’ d,S/dt and dashed

FIG.

w-l

entropy balance. line denotes me1

Solid dS/dt.

line

denotes

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dissipation represents a negative entropy transfer term for the living system. We may now also, without contradiction, assume Prigogine’s minimum entropy production theorem to be valid, and this yields qualitative information on the behaviour of the m- 1 d,S/dt term in the balance equation (I), i.e.

IdiS,() mdt

’

This is represented by the monotonic decreasing curve (solid line) in Fig. 2. The dashed line in the figure represents the general form of the m- ’ dS/df term obtained by taking the sum of m-l d,Sjdt and m-’ d,S/dt. The m-l dS/dt term is not directly measurable nor can it be supposed that this term is always negative, as a naive application of Schrodinger’s original proposal in “What is Life?” might seem to suggest. Indeed the m-l dS/dt term represents a superposition of growth and differentiation effects as may be seen by writing

where ??is the entropy contained in a unit mass of the living system and k = m- ’ dm/dt is the relative growth rate. Again neither of the two terms on the right-hand side of (9) is directly measurable but if we accept Schriidinger’s view then the differentiation term d%/dt could be expected to be negative. The first term, representing the production of new biomass is of course always positive. 3. A Linear Model Although one has no direct knowledge of the relevant thermodynamic fluxes and forces for biological systems owing to their enormous complexity one does know that the m-l d,S/dt curve has essentially the same experimental form for all living systems studied until now. This points to the existence of a common denominator in all such systems as regards their initial structural configuration. It also suggests that further insight might be gained by seeking simple physico-chemical models for which similar evolutionary profiles for the entropy flow and entropy production occur. Consider an open system undergoing a sequence of monomolecular chemical reactions

246

D.

LURIk

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J. WAGENSBERG

in which the external concentrations of Y, and Y, are kept fixed. The entropy change is given by N-l N-l dS = C dSi = C xi dni (10) i=l

i=l

where xi = --pi/T, pi is the chemical potential of Iii and the n, are the mole numbers. The time evolution of the ni is described by the system of linear differential equations dn. dt These equations can also be written as L=7C(ni+l-tZn,-l-22ni)

i=

1,2, . . . . N-l.

(11)

in terms of the reaction rates oi = Ic(ni--ni- r). The vi are thermodynamic fluxes and we shall assume linear phenomenological relations of the form i=O,l,...,N

vi=LXi

between the fluxes Vi and the thermodynamic

(13)

forces

Xi = Xi-l-Xi.

(14)

We underline the fact that the assumption of linear kinetics and linear phenomenological laws are independent assumptions (Nicolis & Prigogine, 1977). Summation of equations (11) over i and use of equation (10) yields after some rearrangement, the entropy balance equation dS diS dt=dt+dt

d,S

(15)

where d.S

N-L

* = izo

vi(Xi-

d,S-

1 -

Xi)

(16)

(17) dt - Xovo - XNVN- 1 are respectively the internal entropy production and the external entropy transfer. Equation (16) has the usual form of a sum of products of fluxes and forces. The assumption of linear phenomenological laws allows (16) and (17) to be expressed in terms of the ni and the fixed reservoir parameters x0 and XN.

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Although an analytic solution to the kinetic equations (11) can easily be written down, more insight is gained by directly performing a computer simulation to compute the time evolution of the three terms in the entropy balance for given fixed initial configurations of n,‘s. We have performed such a computer simulation for three initial configurations with N = 10. Figure 3 shows the three initial configurations as well as the corresponding evolutions toward the final stationary state. Corresponding to each of the three cases the evolution of the three terms in the entropy balance is exhibited in Fig. 4. We see that case c yields an entropy balance profile of the same form as our biological profiles of Fig. 2. In the following section we shall discuss the significance of this analogy. 4. Conclusions Comparison of entropy balance profiles for the case c in Fig. 3 with Fig. 2 indicates that the initial configuration needed to reproduce a “biological-

FIG. 3. Time evolution of ni configuration at 2-s time intervals for three initial figurations. Values of K = 1 and L = low5 have been taken in simulation ofkinetic equations x0 = 3 x 1o-3 , xN = 4 x 10m3, no = 210 and nN = 140.

conwith

248

D.

LURIfi

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J. WAGENSBERG

Time

(s)

FIG. 4. Time evolution of the three terms in the entropy balance corresponding evolution of Fig. 3. Solid lines denote entropy production, dotted lines denote entropy and dashed lines denote entropy variation.

to the transfer

type” evolution of the entropy balance are of a very special kind. Such initial configurations give rise during the subsequent evolution to two competing effects, one tending to increase S and the other tending to decrease it. To summarize the point of view developed in this paper, a consistent interpretation of the development can be given by assuming that: (i) The thermogram data provides quantitative information on the time variation of the specific entropy transfer term in the entropy balance. (ii) There exist certain linear phenomenological laws relating fluxes and forces. (iii) Certain external constraints are maintained constant from the moment of conception onward.

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These assumptions which underlie Prigogine’s minimum entropy production theorem are not, we have seen, inconsistent with the data on heat dissipation. In this view, fertilization would transmit the external constraints to the new system and establish its initial configuration as a perturbation or “fluctuation” with respect to the adult stationary state. In this sense the newly created biomass could be viewed as an unstable, “imperfect”, system which requires its subsequent evolution in order to progressively adjust itself to the externally imposed constraints. Of course the adult state cannot maintain itself indefinitely and aging could be interpreted (Johnson, 1970) as a gradual tendency of the stationary state toward the final state of thermal equilibrium or death through the parallel relaxation of the external constraints. In a sense the system forgets the orders it received during fertilization as a result of a succession of errors. A final comment may be made concerning the validity of linear thermodynamic laws (our second assumption) for the description of the macroscopic level of biological development. Experience seems to show that biological dissipative structures appear mainly at the cellular level where biochemical processes play a primary role. At higher levels and in particular at the level of population dynamics (Wagensberg et al., 1978 ; Wagensberg, 1978), dissipative structures in the form of temporal rhythms or spatial orderings tend to appear as second order effects, superimposed on a linear evolution law. We take the view that the deterministic aspect of biological evolution-which is operative at the level of the organism as a whole-may be described by linear thermodynamic laws. This deterministic evolution represents adaptation to externally imposed constraints rather than creation of unexpected structures, and is the expression of Boltzmann’s order principle. On the other hand, the principle of order through fluctuations becomes progressively more important the further one lowers the level of observation.

REFERENCES GLANSDORFF,

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P. & PRIGOGINE,

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York:

I. (1971).

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HIERNAUX, J. & BABLOYANTZ, .I. Non-Equilib. Thermo&n. 1, 33. JOHNSON, A. (1970. Science 168, 1545. ISJR&, D. & WAGENSBERG, J. (1979). J. Non-Equilib. Thermodyn. 4, 127. NICOLIS, E. & PRIGOGINE, I. (1977). Selforaanization in Non-Eauilibrium Svstems. New John Wiley & Sons. ” PRIGOGINE, I. (1962). Introduction to Thermodynamics of Irreversible Processes. New Interscience. SCHR~DINGER, E. (1944). What is Life?. New York: Cambridge University Press. TRINTSCHER, K. S. (1966). Biology and Information. New York: Consultants Bureau.

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WAGENSBERG, J. (1978). Stochastic Processes in Nonequilibrium Systems. Lecture Notes Physics, Vol. 84, p. 350. New York: Springer-Verlag. ZOTIN, A. I. (1972). Thermodynamic Aspects of Deuelopmental Biology. Basel: Karger ZOTIN, A. I. & ZOTINA, R. S. (1967). J. theor. Biology 17, 57.

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