fulltext

Some optimization criteria for biological systems

in linear irreversible thermodynamics (*)

M. SANTILLA´N(**), L. A. ARIAS-HERNA´NDEZand F. ANGULO-BROWN Departamento de Física, Escuela Superior de Física y Matemáticas Instituto Politécnico Nacional

Edificio 9, Unidad Profesional Zacatenco, México D.F. 07738, México

(ricevuto il 10 Maggio 1996; revisionato il 12 Ottobre 1996; approvato il 19 Novembre 1996)

Summary. — In the present work, some optimum working regimes for linear energy converters are analyzed from a thermodynamical point of view. The regimes studied are minimum entropy production, maximum power output and another one which represents a good compromise between high power output and low entropy production. The analysis is made within the domain of linear irreversible thermodynamics. Finally, the possibility that some biological systems satisfy these criteria is discussed.

PACS 87.10 – General, theoretical, and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodyamics, information theory, cybernetics, and bionics).

1. – Introduction

Energy conversion processes are of major importance in the biological world and have been studied from a thermodynamical point of view by several authors (see for example [1-3]). Caplan and Essig [1] developed a theory, based on linear irreversible thermodynamics, for the study of linear energy converters working in steady states, where they introduce definitions of power output and efficiency, besides the well-known notion of entropy production rate. Such a theory was successfully employed by Stucki [2] in the study of oxidative phosphorylation, provided the linearity of that process, which had been tested experimentally. In his work, Stucki analyzes, with the aid of Caplan and Essig’s definitions of power output and efficiency, some optimum working regimes different from that of minimum entropy production studied before by Prigogine [4].

In the present work, we study once more the minimum entropy production regime

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: moyoHesfm.ipn.mx

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and some others, not previously analyzed by Stucki, like that of maximum power output and another which represents a good compromise between high power output and low entropy production. The advantages and disadvantages of the mentioned regimes are discussed from a thermodynamical point of view. The feasibility of using these regimes for the study of some biological systems is shown for the case of ATP synthesis by anaerobic glycolysis and respiration, where there are enough evidences to assume the linearity of the process. Finally we discuss the possibility of extending the results obtained to the study of muscle contraction, even when the linearity of this process has been strongly criticized [3]. We suggest that the results obtained for the optimum regimes of linear energy converters can explain, at least qualitatively, some empirical aspects of muscle contraction given the evidences reported [5-9].

2. – Linear irreversible thermodynamics and optimal regimes

Of major interest in irreversible thermodynamics are steady states, which are characterized by the constancy in time of the system’s thermodynamical variables, even when they are not necessarily homogeneous inside the system’s bulk. To maintain these inhomogeneities there are required some fluxes like mass, energy, charge, or some others. Let us consider, following Prigogine [4] and Yourgrau et al. [10], a system in a steady state consisting of two coupled processes with generalized fluxes J1, J2and their corresponding forces X1, X2. As is well known from the theories of near-equilibrium irreversible thermodynamics, the system’s entropy production rate is given by

s 4J1X11 J2X2. (1)

If the system is an energy converter, one of the two coupled processes is spontaneous and has a positive contribution to s, while the other has a negative entropy production and therefore cannot occur spontaneously. Without loss of generality, we can choose J1X1E 0 to be the driven process, and J2X2D 0 to be the driver process [1]. Obviously J1X11 J2X2D 0, in accordance with the second law of thermodynamics. To have a steady state, we shall consider X2constant [1, 10]. Although steady states can also be obtained by setting J2or J2X2constant [1], X2constant seems to be the case in many biological systems [1,4].

Caplan and Essig [1] introduced definitions of power output P and efficiency h for energy converters working in steady states at constant pressure and temperature, in the following manner:

P 42TJ1X1 (2) and h 42J1X1 J2X2 . (3)

With the above definitions, optimum regimes like those which maximize the power output, minimize the entropy production, and in general optimize proper functions of these variables, can be studied. The importance of having thermodynamical optimum regimes like the above mentioned has been widely analyzed for several energy transforming processes; see for example ref. [11] where optimizations methods for

cryogenics systems, heat transfer processes, solar power plants, nuclear and fossil plants, and refrigerations plants, are presented. As for biological processes, the optimal performance regimes have been studied by several authors like Prigogine [4] with his minimum entropy production theorem, Odum and Pinkerton [12] who analyze the maximum power output regime for various physical and biological systems, Stucki [2] that introduces some functions of interest, besides the above mentioned, to figure out what is the optimum regime oxidative phosphorylation is carried out in, or Torres [13] and Angulo-Brown et al. [14] who study the possibility for ATP synthesis to be carried out in regimes close to the one of maximum power output and another one which represents a good compromise between high power output and low entropy production. In the present work, the thermodynamical properties of the minimum entropy production (MEP) regime are studied. As is shown below, this regime requires J1to be zero, so from eq. (2) its power output is also null which makes it useless in some cases. The maximum power output (MPO) on the contrary, produces a lot of entropy and has little efficiency as can be seen below and in some works of finite-time thermodynamics (FTT) applied to thermal engines [15, 16]. According to Maddox [17], for biological systems a high degree of fitness in the Darwinian sense implies not only a high power output, but also a high efficiency, a low entropy production, and a low energy-consumption rate, requirements which are fulfilled by none of the above regimes. Thus, in order to have a regime that satisfies all these requirements, and in particular to have a good compromise between high power output and low entropy production, we analyze a regime called ecological, first defined in the context of FTT [15], which maximizes the ecological function E defined as

E 4P2Ts. (4)

In the context of thermal engines the compromise between high power output and low entropy production reached by this regime is excellent since its power output is around 0.8 times that of the MPO regime, while its entropy production reduces down to 0.3 times the one of the MPO regime [15, 16]. The usefulness of this ecological regime has been tested in the analysis of electrical power plants [15,16], and even of the superconducting transition [18]. An ecological-like regime has been also analyzed by de Vos [19] for power plants in terms of the relative cost of investment and fuel consumption. Although the ecological regime represents a good compromise between high power output and low entropy production, the function E is not necessarily the only objective function which can be defined in this direction. In the present work, we analyze whether the performance regimes of some biological processes resemble the ecological one, not to conclude that its working regime is the ecological, but some one intermediate between those of MPO and MEP. It is important to notice that although we take the optimum performance regimes usually studied by FTT, the approaches of this theory and LIT (the one here employed), are quite different. FTT studies thermal engines working in cycles and considers them as internally reversible taking into account only external contributions to the system’s entropy increments. On the other hand, LIT locally studies general systems in non-equilibrium states and considers both internal and external contributions to the entropy increments, S._{4 S}.int1 S

.

ext[4]. S .

int is usually called the entropy production and is denoted as s, while S.ext is related to the entropy outflux JKsby S

.

ext4 2˜lJ K

s[10]. As in refs. [1, 2, 4, 14], in the present work we

consider only internal irreversibilities since they are responsible for the whole entropy increments of the universe. To see that, let us take into account that both the system’s

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S.S_{, and the environment’s S}.E_{, entropy increments, have internal end external} contributions, S.S 4 S . int S 1 S . ext S _{and S}.E 4 S . intE 1 S . ext E _{, respectively, that S}. int E 4 0, since only the system suffers the process under study, and that S.extS 4 2 S

. ext

E _{, given that the} entropy flux from the environment equals the entropy flux toward the system; so S.₄ S.S_{1 S}.E_{4 S}.intS. Anyway, both the theories that consider only external irreversibilities and those that consider internal and external irreversibilities led in general to similar results as regards the thermodynamical characteristics of the performance regimes studied [11].

By definition, for linear energy converters the relations between fluxes and forces, or phenomenological relations, are as follows [1]:

Ji4

!

j 41

2

LijXj, i 41, 2

(5)

with constants Lijcalled phenomenological coefficients. It is also assumed the validity

of Onsager’s reciprocity theorem which establishes the symmetry of the matrix of phenomenological coefficients , i.e. Lij4 Lji. The possibility of some biological energy

converters to be linear has been extensively discussed, see for example Caplan and Essig [1], Prigogine [4], and Stucki [2]. However, there is no general agreement in that respect and each system under study deserves a verification of its linear nature in order to make use of the following results.

From eq. (5) and Onsager’s reciprocity theorem, equations (1)-(4) become s 4L22X22

[

(

1 2q2

)

1 q2(1 2x) 2

_]

, (6) P 4TL22X22q2x (1 2x) , (7) h 42q 2 x(1 2x) 1 2q2_x (8) and E 42TL22X22

[

q2(1 2x)(2x21)2

(

1 2q2

)

]

, (9) with q2 4 L12 2 kL11L22

a coefficient that measures the degree of coupling between the driven and driver processes [1] , x 4X1O X10, and X104 2 (L12O L11) X2 the driven force at which J1_{4 0}

(

see eq. (5)

)

.

From eqs. (6)-(8), the MEP regime is obtained for x 41, from which it follows that X14 X10and that J14 0. In this case the entropy production, the power output, and the efficiency are given by

smin4 L22X22

(

1 2q2

)

, (10) P_{s4 0} (11) and hs4 0 . (12)

From eq. (7) the MPO regime is obtained for x 40.5. Substituting into eqs. (6)-(9) the following values of entropy production, power output, and efficiency in this regime are obtained: sP4 L22X22

g

1 2 3 4q 2

h

_, (13) Pmax4 TL22X22

g

1 4q 2

h

_, (14) hP4 1 2

g

q2 2 2q2

h

. (15)

As expected, this regime has a positive power output and produces more entropy than the MEP regime

(

eqs. (10) and (13)

)

.

Finally, from eq. (9) the condition for the E function to be maximum is x 40.75. So from eqs. (6)-(9) the entropy production, the power output, and the efficiency of the

Fig. 1. – Plots of a) entropy production, b) power output, and c) efficiency as a function of NqN for the three criteria studied. The solid lines correspond to the MEP regime, the dashed lines to the MPO regime, and the dot-dashed lines to the ecological regime. The vertical axis in fig. 1a) is in units of L22X22, while the corresponding axis in fig. 1b) is in units of TL22X22. Notice that the

efficiency for the MPO and ecological regimes are very similar until NqNB0.7 where they separate rapidly from each other.

. , . . - and . -104

ecological regime are given by

sE4 L22X22

g

1 2 15 16q 2

h

_, (16) PE4 TL22X22

g

3 16q 2

h

(17) and hE4 3 4

g

q2 4 23q2

h

. (18)

In fig. 1, there are shown the plots of entropy production, power output, and efficiency as functions of NqN for the MEP, MPO, and ecological regimes. It is important to notice that when the processes are not coupled at all (q 40), the three regimes are completely equivalent, i.e. they have the same power output P 40, the same efficiency h 40, and the same entropy production s4L22X22. In fig. 1a), it can be seen in all cases that the entropy production rate decreases as NqN increases, reaching the optimum values when NqN41. In the case of the MEP regime, this optimum value corresponds to zero entropy production and thus, the steady state reached is an equilibrium one, while for the MPO and ecological regimes the optimum values are related by sE4 0 .25 sP.

From fig. 1b), it follows that the power output of the MPO and ecological regimes is a growing function of NqN, and that PE4 0 .75 Pmaxfor all values of NqN different from zero

(

see eqs. (14) and (17)

)

. Finally, fig. 1c) shows that the efficiency of the MPO and ecological regimes increases with NqN up to NqN41, where their efficiencies become respectively hP4 0 .5 and hE4 0 .75. In conclusion, all quantities of interest are

optimized as NqN41, or in other words, this is a very advantageous modus operandi from a thermodynamical point of view.

3. – Application to biological systems

In order to employ the results of the previous section in the study of some biological energy converters, it must first be verified whether they satisfy all the hypotheses we have made, i.e.

– The system is in a non-equilibrium steady state consisting of two coupled processes with generalized fluxes and forces J1, J2, and X1, X2, respectively.

– The driver force X2is constant.

– The relations between fluxes and forces are linear.

The steady-state condition of many biological systems including the ones studied here, has been extensively discussed before, see for example refs. [1, 2, 4, 20, 21]. And, although there is not general agreement in that respect, we will consider, following the above cited references, that this is an acceptable approximation for the cases here studied.

Let us consider now the synthesis of ATP by anaerobic glycolysis and by respiration. In such systems, chemical energy conversion can be described by the

following reactions [21]:

]glucose( 1 [ 2 ADP 1 2 P1]4 ]2 lactate(1 [2 ATP ] (19)

and

] 6 O21 glucose( 1 [ 36 ADP 1 36 P1]4 ]6CO21 6 H2O(1 [36 ATP ] . (20)

In both cases, the driver reaction is indicated with curly brackets, and the driven reaction by square brackets. For this system the generalized fluxes and forces are thus as follows [4, 10]: J14 v1, J24 v2 (21) and X14 2 DG1 T , X24 2 DG2 T , (22)

where v1 and v2 are the reaction velocities of the driven and driver reactions respectively, DG1, DG2the molar Gibbs energy changes of the corresponding reactions, and T the temperature. Given that the driver reaction is spontaneous and that the driven reaction is not spontaneous, the free energy changes are DG1D 0 and DG2E 0. Then from eq. (22), X_{1E 0 and X2D 0, in accordance with the sign of the contributions} to the global entropy production rate from the driven and driver processes.

As for the reaction velocities, since the reactions are coupled, their velocities should be the same [22],

v14 v24 v . (23)

Thus, from this fact and from eqs. (21) and (22), the formulas for the entropy production rate, the power output, and the efficiency

(

eqs. (1), (2), and (3)

)

become

s 42 v T(DG11 DG2) , (24) P 4DG1v (25) and h 42DG1 DG2 . (26)

These equations are completely concomitant with those employed by Torres [13] and Angulo-Brown et al. [14] in recent studies on the same system.

Furthermore, Angulo-Brown et al. [14] demonstrated for ATP synthesis that the reaction velocity v may be approximated by

v B vmfb R

g

2 DG1 T 2 DG2 T

h

, (27)

with vmfthe maximum forward velocity, b a positive constant that takes into account the enzymatic processes [13], and R the ideal gas constant. From eqs. (21)-(23), it follows that eq. (27) establishes not only that the relations between fluxes and forces are linear,

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but that the phenomenological coefficients are L114 L224 L124

vmfb R . (28)

The above assertion implies then that the systems under consideration are linear energy converters, and also that the coupling coefficient NqN is in this case unity.

In order to use the results of the previous section, it remains only to check the constancy of the driven force. Nevertheless as Torres [13] asserts, this fact comes out from the experimental results of Sacktor et al. [23, 24]. Once all the hypotheses have been verified, and knowing that the coupling coefficient is equal to unity, if the reactions under study are carried out by regimes close to the MPO or ecological ones, they ought to have efficiencies close to 0.5 and 0.75, respectively

(

eqs. (15) and (18)

)

. As a matter of fact, and as Angulo-Brown et al. [14] assert, it can be concluded that ATP synthesis by anaerobic glycolysis works close to the MPO regime and that respiration works near to the ecological regime, since their efficiencies have measured values of about 0.53 and 0.7, respectively [13, 21, 26]. Furthermore, since NqN41 and from eqs. (13) and (16), the fraction of the entropy produced by respiration sR and anaerobic glycolysis sGis sRO sG4 sEO sP4 0 .25 (see [25]).

The treatment here presented is completely equivalent to the one presented in a previous work [14]. However the present treatment is not exclusive for chemical reactions as the previous one [14], and thus could permit us to study some other biological energy converters as will be shown later.

4. – Muscle contraction

Another biological energy converter which could in principle be studied by this method is muscle contraction, which converts chemical energy from ATP into mechanical energy. At this point the experimental data reported by Volkenstein result are very suggestive since according to him the measured efficiencies of the frog’s sartorius muscle and tortoise muscle are respectively around 0.5 and 0.75 [8]. We could also mention the data reported by Woledge [9], who presents plots of h vs. F, where F is muscle’s tension, with maximum values of h B0.5 and hB0.75 for frog and tortoise muscles, respectively. From the above results it could naively be concluded that frog’s muscle works in a regime close to the MPO, while turtle’s muscle does it close to the ecological regime. However, before doing that, it must be verified whether muscle as an energy converter satisfies the requirements listed at the beginning of sect. 3, i.e. a) it works in a steady state and driven and driver processes with corresponding fluxes and forces can be identified, b) the driver force keeps constant, and c) the relations between fluxes and forces are linear. In what follows, these questions will be discussed.

The feasibility of studying muscle contraction as a steady phenomenon has been extensively discussed by Caplan and Essig [1], so we will suppose this approximation is adequate. On the other hand, muscle contraction consists of a mechanical movement (driven process) coupled to a chemical reaction (ATP hydrolysis) as the driver process. Then, the driver force is proportional to Gibbs’ free energy change of the ATP hydrolysis, the driver flux is the reaction’s velocity, the driven force results to be proportional to the mechanical force, and the driven flux is nothing else but the velocity of contraction [1].

Finally there is a controversy between the constancy of the driven force and the linearity of the relations between fluxes and forces. Assuming the validity of both of them gives rise to a contradiction with Hill’s force-velocity relations for muscle [27]. Caplan and Essig [1] assumed the linearity of phenomenological relations and supposed that the driver force depends on the driven one via a feedback mechanism. With this assumption, they found a function for the driver force in terms of the driven one with which Hill’s force-velocity relation is recovered. Nevertheless, Wilkie and Woledge [3] demonstrate that if Caplan and Essig treatment were correct, the driver force should vary in a way incompatible with sarcomere’s internal quasi-steady conditions. They suggest that the evidences imply the constancy of the driver force and the non-linearity of the relations between fluxes and forces.

Despite muscle does not satisfy all the necessary requirements, we believe that the optimum working regimes obtained for linear energy converters could explain, at least qualitatively, some of the empirical observations of muscle contraction described by several authors. In the following paragraphs we extend this discussion.

If muscle were a linear energy converter, the MEP regime would require its contraction velocity to be zero. This regime in fact exists and is called isometric contraction. Furthermore, according to Hill and Sec’s results [27], it is in this regime that the muscle delivers the smallest amount of heat and then produces the smallest amount of entropy.

On the other hand, the existence of various ways of performance for muscle contraction, has been pointed out by several authors. Szent-Györgyi [7], for example, states that three working regimes for muscle can be recognized: “the one that causes the wing muscle of some insects to contract several times a second, the one that causes the slow, regular beat of the heart, and the one that keeps a clam shell closed for hours, despite the actomyosin machine seems to be identical in all cases”. We have already seen that the third case could correspond to the MEP regime, but in the first two cases we could expect them to be similar to the MPO and ecological regimes respectively, since there is a qualitative agreement between their performance and the description given by the optimum regimes mentioned. Another qualitative example on the feasibility of the MPO and ecological regimes for muscle contraction is given by Alpert et al. [6] who conclude that the tortoise and rat soleus muscles, which are slow, “perform with a high economy and low power”, while in contrast, “the rat extensor digitorum longus is fast, has low economy, and has high power output”. Hill [5] mentions on his own that every muscle has two optimum speeds: “One for maximum efficiency and one for maximum power output”. For this kind of systems, the highest efficiency possible is unity, which is reached at equilibrium (s 40)

(

see eqs. (1) and (3)

)

, so the first regime Hill talks about is not the one of maximum efficiency but only one of very high efficiency which could correspond to the ecological regime h 40.75.

5. – Concluding remarks

From what has been seen in the present work, it can be concluded that the MEP regime, or Prigogine’s principle, is just one of the different optimization criteria that could be elected by nature. It has the advantage of producing very few entropy or, in other words, wasting very little energy. However it has no power output and for some applications it could be useless. In order to have power output, more entropy must be produced, but the MPO regime produces it in excess, and its efficiency is only 0.5 in the

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best case. Looking for a regime with a power output a little smaller, but with a highly reduced entropy production, the function E was studied. The maximum E or ecological regime satisfies not only these objectives (it was seen above that PE4 0 .75 Pmax and

sE4 0 .25sP), but also can increase the efficiency up to 0.75, and from the definition of

efficiency

(

eq. (3)

)

and the above relations, its consuming energy rate is related to that of the MPO regime by UE4 0 .5 UP. Furthermore, as regards the feasibility of

analyzing muscle performance regimes with the results obtained for the working regimes of linear energy converters (despite we know muscle is not linear), it turns out to be interesting to look at the following statements of recognized researchers who have studied muscle contraction: “The election of a working regime depends on the needs or evolutionary advantages it provides, which could differ among species” [9]. In such a way, “it could be asserted that the muscles of frog and tortoise are different since they have reached different compromises between high power output and greater economy of maintaining tension” [5]. Of course, similar conclusions can be expressed for ATP synthesis by glycolysis and respiration. Finally, we want to call attention to the global efficiency reported for photosynthesis in higher plants which is around 0.75 [20]. This fact makes this biological energy converter a good candidate to work in a regime like the ecological one.

* * *

We acknowledge fruitful discussion with J. ORTIZ-LÓPEZ. This work was partially supported by COFAA-IPN and IMP.

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[25] In a previous work [14] it was stated that sRO sP4 0 .58 in apparent contradiction with the

above result. However, our previous paper contains a mistake which comes from a misprint in eq. (14) that must be s 4 (cOT) (12h)2_.

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