Bioehimica et BiophysicaActa, 683 (1982) 181-220 181

Elsevier Biomedical Press

BBA86090

THERMODYNAMICS OF GROWTH

NON-EQUILIBRIUM THERMODYNAMICS OF BACTERIAL GROWTH

THE PHENOMENOLOGICAL AND THE MOSAIC APPROACH

HANS V. WESTERHOFF b,*, JUKE S. LOLKEMA a ROEL OTTO a and KLAAS J. HELLINGWERF a

" Laboratory for Mierobiolog)', University of Groningen, Biological Centre, Kerklaan 30, 9751 NN Haren and h Laboratory of

Biochemistry, University of Amsterdam, B.C.P. Jansen Institute, Plantage Muidergracht 12, 1018 TV Amsterdam (The Netherlands)

(Received January 28th, 1982)

Contents

I. Summary ............................................................................. 184

II. Introduction ........................................................................... 184

1II. Description of growth by phenomenological non-equilibrium thermodynamics ............................. 186

IV. Correspondence between thermodynamic and microbiological growth parameters ........................... 190

V. Interpretation of microbial growth by phenomenological non-equilibrium thermodynamics .................... 192

A. Relations between growth rate parameters ................................................... 192

B. Optimization ........................................................................ 193

VI. Description of microbial growth by mosaic non-equilibrium thermodynamics .............................. 197

VII. Correspondence between microbiological growth parameters and mosaic non-equilibrium thermodynamic parameters . 200

VIII. Coupling, stoicheiometry and efficiency in the mosaic non-equilibrium thermodynamics interpretation ............ 204

IX. The interpretation of yield deficiencies ......................................................... 205

X. Futile cycles ........................................................................... 207

XI. Efficiency and optimization of microbial growth: alternative growth models ............................... 209

XI I. Discussion ............................................................................. 215

Appendix A: Calculation of efficiencies from growth yields ............................................... 216

1. Calculation of the carbon/carbon yield at equilibrium ....................................... 216

2. Calculation of the thermodynamic efficiencies ............................................. 217

* To whom reprint requests should be addressed.

0304-4173/82/0000-0000/$02.75 © 1982 Elsevier Biomedical Press

182

Appendi x B. Numeri cal exampl es of the cal cul at i on of efficiencies .......................................... 217

1. Anaerobi c growt h on met hanol ....................................................... 217

2. Aerobi c growt h on met hanol ......................................................... 217

Appendi x C. Deri vat i on of Eqn. 89 ................................................................ 218

Acknowl edgement s ............................................................................ 218

References .................................................................................. 218

Gl ossary

Symbol s Meani ng Uni t s Usual sign Equat i on

first used

Latin

A r affinity ( --- Gi bbs free energy) of J-t ool I 6

reaction r

c i concent rat i on of subst ance i + 7

cM C-mol ar weight; weight in g of an amount + 17

that cont ai ns 1 tool carbon

G free ent hal py; Gi bbs free energy J 2

J,, rate of anabol i sm C-tool. g - 1. h i - 5

J~ rate of cat abol i sm C- mol.g ]-h ~ + 5

Jp total rate of ATP hydrolysis mol.g i.h i 0 34

Jp~ rate of ATP hydrolysis in anabol i sm mol.g 1.h i + 27

Jp (mi nus the) rate of ATP synthesis in cat abol i sm mol .g - l.h i - 26

l

Jp rate of ATP hydrolysis due to ATP leak mol - g -I. h i + 30

Jpr rate of cat abol i c product format i on C- mol.g ].h i + 80

J,~ reduced heat flow J 6

J, rate of total subst rat e utilization C- mol.g ~-h ~ + 80

J,~ rate of subst rat e utilization connect ed to C- mol.g a. h- t + 81

cat abol i sm

J,2 rate of subst rat e utilization connect ed to anabol i sm C- mol.g- 1. h i + 82

J~ flow of subst rat e x C- mol.g l.h i + 5

In(x) nat ural l ogari t hm of x - 2

L x el ement al linearity coefficient of process x tool 2.h i.g i.j t + 28

L~ linearitycoefficientrelatingJ~toAGy mol 2.h I.g I.j I 10

n~p mi nus the (theoretical) number of ATP molecules - - 27

hydrol yzed in anabol i sm per C-tool bi omass

synthesized

n~ mi nus the (theoretical) number of ATP molecules - 26

synthesized in cat abol i sm per C-mol

cat abol i c subst rat e consumed

n" x st oi chei omet ry at which x is consumed per unit - - 18

of cat abol i sm

R gas const ant (8.3 J.mol i.K l) J.mol - r - K ~ + 2

T absol ut e t emperat ure K + 2

t time h + 7

q coupl i ng coefficient 14

Qx product i on or consumpt i on rate of x tool .g- i. h - i 18

V vol ume 1 + 5

Y yield in g bi omass per unit cat abol i c g.C- mol i + 19

subst rat e

Y~ yield in g per C-tool x g- C- mol i + 19

C-tool. 1 i

g- C-tool i

Glossary (continued)

Symbols Meaning Units Usual sign Equation

first used

YATP yield in g biomass per mol ATP g. mol - z + 32

calculated to be formed in catabolism

YAXP,:~I .... YATP at infinite AG c, J~, and J~, but g-mol I +

finite AG,

YAxP,~st .... YATP at infinite - AG a, ,L,~ and J~, but g. mol i +

finite AG~

Y~"~'xth~or YATp calculated from known anabolic pat hways g. mol I + 52

and biomass composition

YAnTtI~" YATP,csl or YAxP.asl g" mol I + 62

Z phenomenological stoicheiometry - 15

Z~. d reduced phenomenological stoicheiometry - + 65

Greek

A 2

V' 6

,#,, 5

5

183

difference

grad ( O~ Ox, O~ Oy, O~ & )

the negligible part of J-h I = W/3600

volume-integrated free enthalpy dissipation J.h ~ = W/3600

function

X force ratio AGa/AG ~ 16

"O efficiency

Cth efficiency as defined by Roels [35] + 87

~t~ f efficiency as defined by Rods and Van Suydam [36] - + 85

T]tot efficiency of microbial growth - 9

7It ff = ~tot 86

~. (electro)chemical potential of substance i J. mol -1 4

fi'~ concentration- and electric potential-dependent J. mol - I 6

part of >i

inverse of maxi mum yield (i.e., yield corrected C-mol. g- i

for growth rate-independent maintenance)

fl growth rate-independent maintenance C-mol - g- i, h - i + 2 I

/~ growth rate h i + 17

,/~ coefficient weighing the relative effects of + 28

two t hermodynami c forces, e.g., AG x vs. AGs

fraction of stoicheiometry lost due to

futile cycling

- e?21 78

Index

a anabolism 4

afc anabolic futile cycling 71

as anabolic substrate 4

asl under anabolic substrate limitation 38

b biomass 80

c catabolism 8

cs catabolic substrate 19

csl under catabolic substrate limitation 3"/

ds double substrate 90

fc futile cycling 72

gd growth rate dependent 43

1 leakage 30

p phosphorylation 26

pr product 80

s substrate 80

ss single substrate 89

ts total substrate 19a

# reference 10

+ 21

184

1. Summary

Microbial growth is analyzed in terms of mosaic

and phenomenological non-equilibrium thermody-

namics. It turns out that already existing parame-

ters devised to measure bacterial growth, such as

YATP, /*, and Q~ub~t~e, have as thermodynamic

equivalents flow ratio, output flow and input flow.

With this characterisation it becomes possible to

apply much of the already existing knowledge of

phenomenological non-equilibrium thermody-

namics to bacterial growth. One of the conclusions

is that the frequent observation that YATP is only

50% of its theoretical maximum does not mean

that the microbe corresponds to a thermodynamic

system that has been optimized for maximal out-

put power, as has been suggested. Rather, at least

in some cases, it corresponds to a system that has

been optimized towards maximum growth rate.

When the degree of reduction of the (single) carbon

source is significantly smaller than that of the

biomass produced, the efficiency of biomass

synthesis has been kept as high (i.e., about 24%) as

is consistent with maximization of the growth rate

at optimal efficiency.

Mosaic thermodynamics allows an analysis of

processes which in microbial metabolism may be

responsible for any particular growth behaviour.

Equations are derived that predict the effect of

uncoupling through leaks, futile cycling, or 'slip'

on microbial growth. It turns out that uncoupling

is expected to affect both the growth rate-indepen-

dent and the growth rate-dependent 'maintenance

coefficient'. The effect on the latter is different

when catabolic substrate limits growth than when

anabolic substrate limits growth. In the latter case,

the growth rate-dependent maintenance coefficient

is negative. It is concluded that mosaic non-equi-

librium thermodynamics will be a powerful theo-

retical tool especially in future experimental

analyses of the metabolic basis for microbial

growth characteristics and growth regulation.

11. Introduction

It was not until the development of continuous

flow culture [1,2] that the factors affecting micro-

bial growth * yield could be thoroughly studied. In

batch cultures growth yields could be determined,

but had to be considered as average values, since

the bacteria pass through a series of metabolic

states (from excess substrate to limiting substrate).

The continuous flow culture makes it possible to

keep the culture in a steady state. One of the first

aspects to be elucidated was the dependence of the

growth rate on the substrate concentration [1-4].

Next, the relations between rate of substrate con-

sumption, product formation and growth rate were

found to be quite often linear [4-9], though not

proportional ** [4-9]. Substrate consumption at

zero growth rate was proposed to be associated

with so-called maintenance processes [4,5,10 12].

Pirt [12] derived that the very occurrence of main-

tenance metabolism would lead to a dependence

of the growth yield (i.e., the amount of biomass

produced per amount of substrate consumed) on

the growth rate. With the additional assumption

that the maintenance metabolism would be growth

rate independent (an assumption doubted by

Dawes and Ribbons [11]), he predicted that the

inverse of the yield would vary linearly, though

generally not proportionally, with the inverse of

the growth rate. This prediction could be con-

firmed experimentally [8,12]. This finding con-

flicted with the ideas and results of Monod [13]

who had introduced the yield as a biological con-

stant. Another source of variation of the yield, i.e.,

the dependence of the free enthalpy- or Gibbs free

energy-yielding catabolic route on the growth rate

(e.g., see Ref. 14) had been eliminated by the

introduction of the YA'IP concept [15]. YAII' iS the

amount of biomass produced per amount of ATP

calculated to be produced from the catabolism of

the substrate. For some time [16-18] Y~,Tp was

considered to be a constant. Indeed, some studies

(e.g., see Refs. 19 and 20) were based on this

consideration.

De Vries et al. [14] started to consider the effect

of maintenance metabolism on ]/Ta, T v, Following

Pirt [12], they plotted I/YAr p VS. l/di l ut i on rate

* In this paper we confi ne di scussi on to gr owt h in the sense

of cellular mul t i pl i cat i on as opposed to i ncrease in cellular

size (for the latter, see, e.g., Ref. 103).

** The vari at i on of y with x is l i near when y = ,~x + ,8, pr opor -

tional when y = o~x.

for glucose-limited cultures of Lactobacillus casei

and observed a good fit to a straight line. The

slope of this line was identified as the amount of

ATP hydrolysis needed for maintenance, the inter-

cept with the ordinate as the reciprocal of y~r}~,

the YAve corrected for maintenance utilization of

ATP. A survey of the literature [6,21] shows that

the amount of ATP used for maintenance depends

strongly on the type of organism, the type of

medium, and which of the substrates actually limits

growth, yAmv~ X was originally proposed [6] to be a

constant for different microorganisms of normal

average cell composition. For glucoseqimited

chemostat cultures of L. casei (24.3 g/t ool [14])

and for tryptophan-limited cultures of Aerobacter

aerogenes (25.4 g/t ool [6])) (both anaerobically),

Y~'~-~e x values were found that were close to the

values (Y,~n~-apXtheor) that were calculated [18,21-23]

from the known anabolic pathways in micro-

organisms. As more and more determinations of

y2,~,~x appeared in the literature, Y2~Pr~e ~ turned out

to be usually much lower than ymax

ATP.lheor, to vary

from organism to organism, and to depend on

which of the substrates is growth limiting [21].

Neijssel and Tempest (Ref. 8, see also Refs. 21

and 24) eliminated this dilemma by generalizing

the concept of maintenance. They pointed out that

the observed doubly reciprocal relations between

yield and growth rate do not dictate that mainte-

nance processes are growth rate independent. A

growth rate-dependent maintenance, proportional

to growth rate, was proposed [8,24] which could

supposedly equal zero, or even be negative [8].

Thus, the difference between experimental Y~"~b ~

and Y~'!}~pX,, .... as well as the variability of the

former can be attributed to the occurrence of

growth rate-dependent maintenance consuming

part of the cell yield.

Although growth kinetics of microorganisms can

now be largely described and therefore, in that

sense, understood, this understanding is still un-

satisfactory. Theory can just fit the results of

growth kinetic studies, but it cannot yet pinpoint

the reason for certain types of growth behaviour as

resulting from a specific alteration in the microbe's

metabolism, let alone prescribe which metabolic

reaction should be manipulated in order to have a

microbial culture produce the desired output. At

present, the metabolic causes of growth rate-

185

dependent and growth rate-independent mainte-

nance are amply discussed (e.g., see Refs. 21, 25

and 26). These discussions are sometimes hampered

by the lack of possibility to relate quantitatively

proposed variations in metabolism to effects on

the two types of maintenance. Consequently, there

is a need for a growth kinetic theory that quantita-

tively relates growth to metabolic processes.

The existing kinetic theories for microbial

growth (e.g., see Ref, 3-5, 8, 12, 13, 24 and 27) are

all phenomenological in design. They describe

growth without detailed reference to the underly-

ing metabolism. An exception is the comparison of

Y2%~ X to Y~n~-~,Xtheo r [6]. In view of the complexity of

the metabolism of bacteria it seems impossible to

write down all kinetic equations of all enzyme

reactions and integrate them [28]. An alternative

method is to consider non-equilibrium processes

such as microbial growth [17,29] in terms of non-

equilibrium thermodynamics [30,31]. In its more

general formulations it can be used to describe

heat evolution during growth (e.g., see Refs. 9, 17,

18, 32 and 33). More recent work on the subject

[34-36] uses non-equilibrium thermodynamics

more formally, thereby avoiding misinterpreta-

tions by earlier authors [17,32]. It turns out that in

aerobic processes heat production is closely related

to the rate of oxygen consumption. Another topic

addressed by this approach is the efficiency of

microbial growth, the discussion of which is

hampered by the phenomenon that many different

definitions of efficiency are in use (Refs. 32, 35, 37

and 38; see also below).

Central equations in non-equilibrium thermo-

dynamics are the balance equations which state

that any increase in an extensive quantity can only

be due to production of the quantity in excess of

what flows out of the system. For energy (first law

of thermodynamics) but also for mass and chemi-

cal elements, the production term is absent. This

principle makes a number of flows in the growth

systems dependent on one another. It has been

elaborated for microbial growth by several authors

[21,32,33,35,39-41], the most generalized treat-

ment being given in Ref. 35. An important aspect

that follows quite naturally from these treatments,

as already suggested earlier [42], is that microbial

growth on highly reduced compounds is neces-

sarily inefficient. For growth on these compounds

186

the excess of free enthalpy or Gibbs free energy

must be dissipated. Here also the importance of

free enthalpy dissipation not coupled to biomass

production comes into play but the treatments are

unable to give a more thorough analysis of the

mechanistic basis of this uncoupling,

One paper [34] steps down from the macro-

scopic viewpoint to the microscopic one: it consid-

ers a simple model for the metabolism of the

microbial cell, takes for granted the relation be-

tween Y~,TP and growth rate as determined by the

group of Stouthamer [6,14], takes into account the

mutual dependence of the different flows and pre-

dicts trends in the relation between growth yields

and the degree of reduction of the substrate. How-

ever, again this paper stops at the point of describ-

ing the effects on growth of specific uncoupling

reactions in metabolism.

Non-equilibrium thermodynamics has a subdi-

vision [43-45] that has been called near-to-equi-

librium thermodynamics [46] because it makes the

assumption of closeness to equilibrium when de-

riving proportional relations between rates of

processes ('flows') and the free enthalpy gradients

('forces') that drive them. Also, it derives recipro-

cal relations [43] between the proportionality coef-

ficients. None of the above-mentioned treatments

of microbial growth in terms of non-equilibrium

thermodynamics has applied these linear and re-

ciprocal relations, probably because some reac-

tions, which are essential for growth, are known to

be far from equilibrium [29]. On the other hand,

such relations would turn a non-equilibrium ther-

modynamic description of microbial growth into

growth kinetics. There would no longer be the

need to embody empirical relations between yield

and growth rate [34]. Instead, such relations would

be predicted and then compared to the experimen-

tally determined relations. Moreover, in quite a

few cases of far-from-equilibrium systems [47-58a]

the potential limitation to systems that are close to

equilibrium turns out to be absent, or removable

by substituting linear equations for the originally

[43] proportional ones [47,49,59]. Therefore, it

seemed worthwhile to examine whether indeed this

other, 'further-from-equilibrium' part of non-equi-

librium thermodynamics is applicable to microbial

growth.

Yet, if it were applicable, this still would seem

not to provide us with a theory that would link

metabolic characteristics to growth behaviour.

'Phenomenological' non-equilibrium thermody-

namics [60] has been stigmatized as being uninfor-

mative about mechanisms [46]. However, there

exists another branch of non-equilibrium thermo-

dynamics, i.e., mosaic non-equilibrium thermody-

namics [53,60], that does explicitly consider the

effect of the individual metabolic processes on the

macroscopic performance of biological systems.

This paper will not be a review in the tradi-

tional sense. Rather than assemble the literature

data on one specific subject, we shall review the

existing non-equilibrium thermodynamics and the

existing microbial energetics to find out to what

extent the former may be useful for the latter. We

shall show that the kinetic relations that have been

observed in microbial growth are in strict accor-

dance with predictions by non-equilibrium ther-

modynamics. The parameters that are most fre-

quently used in the quantitation of microbial

growth correspond to parameters that are central

in a non-equilibrium thermodynamic description

of microbial growth. Furthermore, we develop a

mosaic non-equilibrium thermodynamic descrip-

tion of microbial growth. It will become clear what

effects on overall growth behaviour (maintenance

coefficients, maximal yields) are predicted for

changes in stoicheiometries, leakage processes,

futile cycles and changes in the growth-limiting

substrate. Also, new light will be shone on the

criteria, following which microbial growth may

have optimized, and on its resulting efficiency. We

conclude that (mosaic) non-equilibrium thermody-

namics may become an important theoretical tool

in future experimental analysis of how metabolic

factors determine microbial growth.

II1. Description of growth by phenomenological

non-equilibrium thermodynamics

Mosaic non-equilibrium thermodynamics is in

fact an elaboration of (phenomenological [60])

non-equilibrium thermodynamics, which, in turn,

is a method to describe energy-transducing sys-

tems more generally. Since phenomenological (but

not mosaic) non-equilibrium thermodynamics

treats such systems as a black box, its conclusions

are independent of the internal structural char-

J~

Fig. 1. The black box model of microbial growth. The input is

catabolism (arc) across a free enthalpy differences of AGc The

output is anabolism (biomass production, - J~) against a free

enthalpy difference Z~G a.

acteristics and quantities of the system and are

therefore generally valid. This important ad-

vantage is obtained at the cost of the possibility of

using phenomenological non-equilibrium thermo-

dynamics to retrieve information about the inter-

nal properties of the black box from experimental

data. To the latter purpose mosaic non-equi-

librium thermodynamics is better suited [60a].

Applying either approach on the optimal occasion

we hope to benefit from the advantages of both.

Therefore, we shall first develop an interpretation

of microbial growth according to the method of

phenomenological non-equilibrium thermody-

namics, and then incorporate the consequences of

the actual mechanisms by which the processes

occur (cf. Ref. 45).

Phenomenological non-equilibrium thermody-

namics describes systems as a black box that con-

verts an input flow to an output flow. Fig. 1

depicts the microbial system as such a black box

into which a substrate flows (at rate Jc) and out of

which biomass flows (at rate -a~). Jc will be

expressed in C-tool substrate/h per g dry weight,

Ja in C-mol [39] biomass produced/h per g dry

weight. Phenomenological non-equilibrium ther-

modynamics considers the input and output flows

to be caused by their two 'conjugated' forces. An

appropriate way to identify the force that is con-

jugated to a certain flow is to inspect the 'dissipa-

tion function' (i.e., the function that describes the

amount of free enthalpy, or Gibbs free energy that

is destroyed ('dissipated') as a result of the irre-

versibility of the process) and especially the part of

it that describes the free enthalpy dissipation (or

entropy production) that is intimately linked to

that flow [44]. For every mole of glucose that is

187

degraded by the organism to carbon dioxide and

water, an amount of free enthalpy occurs in the

dissipation function that is equal to the free en-

thalpy difference of the reaction:

C6H 1206 + 602 ~ 6CO 2 + 6H20

(1)

This free enthalpy difference depends on the con-

centration of the substrates and products. For

glucose oxidation, for instance:

[glucose]( Po2 )6 ]

AGgluc ..... idation = AG~j ...... ~, + RT In (2)

,co2i J

Under ambient conditions, AGglc , equals about 2.9

MJ/mol glucose. Similarly, for every Einstein

(mole of photons) that is absorbed by a photosyn-

thetic organism, a free enthalpy of 90-190 kJ (for

light with wavelengths between 900 and 450 nm)

[61,62] appears in the dissipation function (cf. Ref.

63). Generally, the 'thermodynamic force' con-

jugated to catabolism is identical to the free en-

thalpy of the catabolic reaction (calculated as if it

were not coupled to the synthesis of ATP). For the

general case we shall call this free enthalpy dif-

ference of the catabolic reaction AGc.

All substances of which a cell consists have a

certain free enthalpy and, therefore, also for the

anabolic reactions (e.g., defined in Ref. 46:

410CO 2 +47N 2 +316H20+3H2SO 4

+ 7H3PO 4 --* 104 g biomass + 50702

(3))

a free enthalpy of biomass synthesis can in princi-

ple be defined:

def

AG, = AG~ + RT ln([biomass]/[anabolic substrate]) =/i b -/~.~

(4)

AG, has been estimated (e.g., see Ref. 46). It can

be split into an enthalpy (Z~H) and an entropy

(AS) term. AH can be measured by subtracting

the heat of combustion (or formation) of the

anabolic substrates from the heat of combustion

(or formation) of the biomass. In practice, the

188

entropy term cannot as readily be measured. It has

been estimated [46] through, in part statistical,

calculations. The resulting values for the free en-

thalpy of biomass synthesis (474 and - 30 kJ/C-

mol for synthesis from NH 3, H2SO 4, H3PO 4, and

CO 2, or glucose, respectively [46] and /~b (69

k J/C-t ool [35]) should be considered as approxi-

mations. It will, however, turn out that a lot can be

said without knowing the exact value of AG~ and

fib, as long as the latter can be assumed to be

constant. This assumption is correct for a wide

range of growth rates in carbon-limited continuous

cultures [39]. With this set of flows and forces the

dissipation of free enthalpy can be written as

[30,31]:

cb = J,.AG,. + J AG~ + f/odV

(5)

with:

def

+o = ZJ,~7( - ~) + ZJ,.h ...... Ar +CJ;/T) wC- T)

i r

(6)

q~o can usually be left out of consideration. It is

only mentioned here as a reminder of the assump-

tions that are necessary to obtain a simple descrip-

tion. The first term in ~o represents the diffusion

processes. The second term designates all chemical

reactions that are not summarized by Eqns. 1 and

3. Together these two terms can be rewritten as

[44]:

q~o- ( J~/T) V( - T) = ~ - ( dci/dt ) ~ ~, (7)

i

In this equation the summation is to be carried out

over those substances that are not mentioned in

the reaction equations of catabolism and anabo-

lism (cf. Eqns. 1 and 3). This part of the dissipa-

tion function is zero when the concentrations of

these substances are constant (i.e., time indepen-

dent). This means that the systems should be in

steady state. Such steady, states are generally ob-

tained in continuous culture. In batch culture the

early exponential growth phase may also exhibit

the essential steady-state properties. In this paper

we shall confine ourselves to the description of

microbial growth in continuous culture. For the

disappearance of the second term of Eqn. 7, the

culture has to be well stirred, so that temperature

gradients are small. The fact that the 'reduced heat

flow' J~ is usually poorly coupled to the diffusional

fluxes allows the separation of this term from the

rest of the dissipation function [45]. We conclude

that for the description of a continuous culture,

fOodVin Eqn. 5 can be left out of consideration. It

may be useful to define the sign convention here.

J~. is taken as positive for flow in the direction of

degradation of the catabolic substrate to its prod-

ucts. Ja is taken as negative for flow in the direc-

tion of biomass synthesis.

In an isothermal, isobaric system, the free en-

thalpy ( = Gibbs free energy) describes the amount

of useful (i.e., non-volumetric) work a system can

perform. It follows that the dissipation function

keeps track of the amount of destruction of energy

that could potentially have been used to perform

work. The second law of thermodynamics states

that this dissipation function should always be

positive. Only because the black box sees to cou-

pling between the catabolic reaction (with a posi-

tive term in the dissipation function) and the

anabolic reaction (which usually has a, smaller,

negative term in the dissipation function), can the

anabolic reaction take place without violating the

second law of thermodynamics. Of course there is

a limit to the rate of the anabolic reaction:

- Jo < L~Gc/6O° (S)

An 'is equal' sign would be valid only if the

system were in equilibrium, which for a growing

microorganism is not the case [29]. In view of the

relation between free enthalpy and useful work,

the 'overall' efficiency of the process is defined

[37,38] as the ratio between the rate of free en-

thalpy output and the rate of free enthalpy input:

def

ntot = - J,,AGa/J~AG~ (9)

In terms of efficiency, the above-mentioned limi-

tation to the microbial growth rate is that the

efficiency must be lower than 100%.

It should be noted that currently different no-

tions of (thermodynamic) efficiency exist [35]. Not

infrequently, 'efficiency' of a reaction or enzyme

(e.g., a proton pump) is discussed in terms of the

number of product molecules (pumped protons)

which appear (are pumped) per molecule of sub-

strate (oxygen, photon) that is consumed. Such a

parameter should be called yield (apparent

stoicheiometry) and not efficiency. Similarly,

growth 'yields' are distinct from efficiencies; and

efficiency is exclusively free enthalpy yield (in an

isothermal, isobaric system). Often the ratio of

enthalpy output over enthalpy input is considered

as the efficiency (or yield in kcal) of the process

[9,33,40,64]. However, such an efficiency is not

bound by the second law of thermodynamics to

remain below 100%, a fact that has not always

been realized [32]. On the other hand, it has been

shown that for aerobic growth, the contribution of

the entropy term to the free enthalpy is often

negligible, so that the enthalpic and the free en-

thalpy efficiency become almost equal [35,46].

Even the free enthalpy efficiency cannot be

defined unequivocally for all systems (cf. Section

XI). However, for the system defined by us in

Fig. 1, it can, because anabolism and catabolism

can be clearly distinguished, each with their own

free enthalpy difference and their own flow.

Apart from the efficiencies of systems, non-

equilibrium thermodynamics also considers the re-

lations that exist between the forces and flows in

the system. For near-equilibrium systems it can be

shown [65] that for many types of reactions, every

flow depends proportionally on all forces. In mi-

crobial growth a number of processes are involved

that are quite far from equilibrium [29]. Conse-

quently, there is no guarantee that the relations

between the catabolic and anabolic flows and the

conjugated forces are linear. On the other hand, a

number of far-from-equilibrium systems have now

been described in which, for a large range of

values of the flows, the relation between flows and

forces is linear, though not always proportional

[47-53,66]. Consequently, it seems attractive to try

to describe the relations between the flows and the

forces in this system by linear, non-proportional

relations:

J¢ = Lcc( AG,: - AGc~)+ Lca(AGa -- AGa~ )

Ja = Lac( AG¢ -- glGc~) + Laa( AGa - AGa~)

(lO)

in which the parameters carrying the symbol

are constants (i.e., independent of AG~ and AGe).

189

(For more details on AGff and AG~ see Section

VI.) The linear relation between rate of catabolism

and anabolism and free enthalpy difference of the

reactions may seem to be in contradiction to the

generally accepted hyperbolic dependence of

growth on substrate concentration (Ref. 13; for a

discussion of alternatives see Refs. 4, 67 and 68).

Here, a linear dependence of the rate on the

logarithm of the substrate concentration is pro-

posed. One of the consequences would be that at

very low substrate concentration, growth rate

would be predicted to be negative, whereas the

generally used kinetic description [13] always gives

positive growth rates. Van Uden [27], however,

already indicated that this aspect of commonly

used kinetics is bound to be incorrect, since at very

low substrate concentrations the substrate's con-

centration gradient across the bacterial membrane

would be too high to allow influx of the substrate.

Schulze and Lipe [4] tested the hyperbolic depen-

dence of growth rate on substrate concentration.

1.0

0.5

D

0.0

I

2

[gtucose] (raM)

Fig. 2. The dependence of growth rate on substrate concentra-

tion: Comparison of experimental results to ( ) logarith-

mic and ( - - - - - - ) hyperbolic dependence. Experimental data

of Schulze and Lipe (Table I in Ref. 4), ( - - - - - - ) the fit by a

hyperbolic dependence calculated by Schulze and Lipe (Km =

0.41 mM, Dma x = 0.92 h -1 ) and ( ) a fit by a logarithmic

dependence (D = 2.37+0.24 In (S)) are shown. The fit to yet

another equation also calculated by Schulze and Lipe (not

shown) was approximately as good as the fit to the logarithmic

dependence.

190

They concluded that the correspondence of the

experimental results with the hyperbolic depen-

dence was insufficient. In Fig. 2 we have plotted

their data together with two curve fittings. The

dashed line is the fitting with the supposed hyper-

bolic dependence, as calculated by them, the full

line the result of a fit we carried out using the

logarithmic equation. Except for the two lowest

substrate concentrations (but see above), the loga-

rithmic fit turns out to be closer than the hyper-

bolic fit. We conclude that there is no reason to

consider the logarithmic dependence of growth

rate on substrate concentration as an approxima-

tion to reality that is inferior to the more com-

monly used hyperbolic dependence.

For near-equilibrium processes phenomenologi-

cal non-equilibrium thermodynamics predicts [43]

that:

c.~ = L, (11)

There are some suggestions [54,55,69] that even far

from equilibrium this 'Onsager reciprocal relation'

[43] still applies, but until now this has only been

proven for almost strictly coupled processes [54]

and for at least one system (Ref. 69a, cf. Ref. 51)

absence of reciprocity has been observed experi-

mentally. Therefore, we leave the possibility open

that L~c and L~, differ. In the mosaic approach

(see below) this possibility will be elaborated. If in

the system the relations between flows and forces

were to prove to be proportional after all, the

values of G c and AG,, would equal zero. Only if

both proportionality and reciprocity were to apply

would the equations become as simple as:

J~ = L, AG c + La~AQ,

In phenomenological, non-equilibrium thermo-

dynamics studies Eqn. 12 is often replaced by:

J~./( L~AG~ ) = 1 + qZX

J~/( L~aG~) =qz + Z2x

( 13)

with:

def

q = LJCL,,L~c (la)

and:

def

z : f~,,,/c~c (is)

Furthermore:

def

x - ac~/Ja~ (16)

This description offers the advantage that the sys-

tem has been normalized with respect to the activ-

ity of the input system, q represents the degree of

coupling and can vary between - 1 and + 1. Z is

the 'phenomenological stoicheiometry'. Only when

the degree of coupling equals one does Z equal the

mechanistic stoicheiometry. The value of Z is

hardly a priori defined, and contrary to what has

been suggested by Rottenberg and co-workers

[56,70], it cannot be assumed to be approximately

equal to the theoretical stoicheiometry between

output and input flow. On the other hand, Z is a

mathematically very practical parameter, because

at a certain degree of coupling a plot of any of the

output functions vs. Z times the ratio of output

force over input force is completely defined,

whereas the plot of the same output function vs.

the 'force ratio' (i.e., the ratio between output and

input force, in this case AGJAGc) only would still

depend on the mechanism of uncoupling [58].

We wish to stress that the use of Eqns. 12-14 as

well as of conclusions from earlier [38,551 phenom-

enological non-equilibrium thermodynamic con-

siderations based on these equations depends on

the assumption that AG~, AG[ and L~- L~,

equal zero. General proof for this is limited to

near-equilibrium systems, but there are interesting

considerations that evolutionary pressure would

have favoured proportionality and symmetry [70a].

Unless explicitly mentioned, we shall refrain from

assuming proportionality or symmetry.

IV. Correspondence between thermodynamic and

microbiological growth parameters

In previous studies [1-3,5,8,10,12-15,37,39] on

the relationship between the biomass production

of a growing microbial culture and the consump-

tion of substrate, a number of parameters have

been defined. In this section we shall investigate

which of these parameters correspond to the ther-

modynamic parameters that were introduced

above. The first parameter is the specific growth

rate '/~' (cf. Ref. 12). It is equal to ln2 times the

reciprocal of the doubling time of the cell number

in a growing culture [71]. Its connection to the

above-defined anabolic flow is given by:

= - LcM (17)

In this equation cM represents the C-molar [39]

weight of the average cell (in g/mol carbon). A set

of Q parameters is often used to keep track of the

consumption of substrates and the production of

products other than biomass; Q~ is then defined as

the specific rate of consumption of x, or produc-

tion of x (in mol x/g dry weight per h) [21]. If, for

instance, the oxygen consumption by a cell sus-

pension is measured simultaneously with its dry

weight, the Qo~ can be calculated. In terms of

non-equilibrium thermodynamics, the Q values are

191

equivalent to flows:

Q~ =J~= n~J~ (18)

In this equation n~ is the stoicheiometric constant

that relates the rate of substrate consumption to

the rate of catabolism. If the flow of catabolism

were to be expressed in terms of the rate of oxygen

consumption during catabolism, then Q% would

be identical to J~. In general we shall define:

2Q% =Jo-

Possibly the most frequently used parameter to

express the growth of a bacterial suspension is the

growth yield Y~ [13]. It has been defined as the

amount of biomass produced (usually in g dry

weight) per mol substrate consumed. During

steady-state growth, this equals the ratio of the

rate at which biomass is produced and the rate at

which a certain substrate (x) is consumed. If the

rate of catabolism is identified with the rate of

consumption of this substrate, then Y~ is simply

TABLE I

CORRESPONDENCE BETWEEN MI CROBI OLOGI CAL GROWTH PARAMETERS AND NON- EQUI LI BRI UM THERMO-

DYNAMI C PARAMETERS

Mi crobi ol ogy Equat i on

relating the two

Non-equi l i bri um t hermodynami cs

Symbol Name Name Symbol

Q~ Rat e of subst rat e

consumpt i on Q~ = n~J~ Input flow J~

~a Gr owt h rate /~ = - J.cM Out put flow J~,

Y Yi el d Y = - .I.cM/J c Flow ratio J,,/J..

YATP Mol ar growt h yield YATP = cMJa/J~ Flow ratio Ja/Jp

based on cal cul at ed

amount of ATP gen-

erated in cat abol i sm

c st at i chead

m e Gr owt h rate- rn¢ = npJ~ Input flow at

i ndependent main- static head

t enance coefficient

mg Gr owt h rate-depen- mg= - nCpflgd/cM

dent mai nt enance = - nCp((Jc/- Ja)levelflow- np/np))/cM

coefficient

c

Y,~'~-~ Mol ar growt h yield Y .... = ( - J./J~)levemowCM/- rtp Level-flow flow ratio

on ATP corrected

for growt h rate-

i ndependent main-

t enance

yrnaxATP.theor Theoretical growt h ymaXATP.theor = cM/- n ap ATP st oi chei omet ry n ~p

yield on ATP in anabol i sm

j £st atic head

(J~/J~)Levol0o~

192

the ratio between the output and the input flow of

the system:

def def

~.~ -- J.cM/J~ = - J~cMn~/J~ = Y~n~ (19)

where Yc~ signifies the growth yield on the basis of

consumption of catabolic substrate.

It should be noted that often growth yields are

expressed as biomass produced/mol total sub-

strate, i.e., that no correction is made for the part

of the substrate carbon that ends up in the bio-

mass. Hence, such a yield is defined as:

def - J~cM cM/( l + c~, )

Y,. J.--J,, (19a)

where ts represents total substrate. In this paper

we shall focus on growth yields on the basis of

catabolic substrate consumption.

In Table I we list the thermodynamic parame-

ters we have introduced next to the already exist-

ing growth parameters they correspond to.

V. Interpretation of microbial growth by phenome-

nologicai non-equilibrium thermodynamics

VA. Relations between growth rate parameters

The (experimental) relation between the rate of

growth and the rate of substrate consumption has

been investigated for a number of organisms (for a

review, see Ref. 21). This offers the possibility of

comparing predictions concerning this relationship

made by the phenomenological non-equilibrium

thermodynamic description with experimental ob-

servations. It is relevant to consider what the

variable is that induces the variations in growth

rate and substrate consumption in a particular

case examined. In papers on continuous culture

studies this variable is usually called the growth-

limiting factor. Although, technically, in such stud-

ies the primary experimental variable is the rate at

which the culture medium is replaced with cell-free

growth medium (the in situ concentrations of

growth-limiting factor and cells adjusting until the

cellular multiplication just matches the wash-out

of cells), it is, of course, only the concentration of

this growth-limiting factor that is the biologically

relevant variable [3,4]. In the phenomenological

non-equilibrium thermodynamics formalism the

catabolic substrate concentration appears in the

free enthalpy of catabolism (cf. Eqn. 2). Therefore,

if we wish to consider the relationship between the

catabolic rate and the anabolic rate under condi-

tions in which the catabolic substrate is growth

limiting, it is useful to eliminate AG~ from Eqn. 10

to obtain:

L~.. La. l

,I~=(L~./L,,~.)J~+ L~,, L,~' (AG"-AG"=) (20)

It turns out that at constant or saturating con-

centration of the anabolic substrates, phenomeno-

logical non-equilibrium thermodynamics predicts

(be it a posteriori) a linear relationship between

the rate of catabolic substrate consumption and

the growth rate. Inspection of published relation-

ships between substrate consumption and growth

rate (cf. Refs. 5, 7-9, 12, 21 and 72-74) confirms

this prediction.

When the anabolic substrate is rate limiting, it

is best to eliminate AG~ from Eqn. 10. The result is

again a linear relation betwen J,., J,, and AG,..

Consequently, also when the anabolic substrate is

growth limiting, a linear relation between the rate

of substrate consumption and the rate of growth is

expected. This is again in accordance with experi-

mental observations [6,8,73,75].

We can indicate these two cases of linear rela-

tion between catabolic and growth rate by writing:

J.. = ,~(- J.)+# (21)

It should be noted that depending on the condi-

tion being anabolic substrate limitation versus

catabolic substrate limitation, the meanings of a

and fl differ, although the phenomenological non-

equilibrium thermodynamic approach does not

show how. Also, when things other than anabolic

or catabolic substrates (e.g., cofactors) are limiting,

the linearity given by Eqn. 21 does not have a

basis. Translation of this expression into an ex-

pression in terms of growth parameters results in

(using Eqns. 17 and 18):

Q~ = nC~a~P./"M + n~l~x (22)

Such an equation was arrived at earlier [5], both

through experimentation and from theoretical con-

siderations. More often, however, the growth of a

culture is expressed in terms of yield with respect

to the amount of substrate consumed, its Y~ value

[13]. It was shown above that this yield corre-

sponds to the flow ratio of the system (Eqn. 19).

We may use Eqns. 19 and 20 to evaluate the

predicted relation between growth yield and growth

rate:

Lc ~ L~a )

Using the shorthand notation of Eqn. 22:

(23)

cM/Y~ = a +/~/- J~ (24)

This equation is similar to that derived by Pirt

[12]. The novelty in his derivation was that he took

into account that the organism expends a certain

amount of free enthalpy for 'maintenance'. Com-

paring Eqn. 21 with his Eqn. 6, it is concluded that

our term /3 corresponds to the (growth rate-inde-

pendent) maintenance coefficient [12]. At ever in-

creasing rates of growth, the yield tends to become

equal to the inverse of c~. Consequently, a corre-

sponds to the inverse of the 'yield corrected for

endogenous metabolism' [4,12], or [6,12] t he'mol ar

growth yield per mole of substrate taken up during

growth', or 'Y ma× ,. To compare the predictions of

the equations derived by phenomenological non-

equilibrium thermodynamics with experimental re-

suits we conclude from Eqn. 24 that the growth

yield should depend hyperbolically on the growth

rate:

r= ( t,/,)/( ~ + -~ ) (2s)

This is again in accordance with experimental

observations [6,8,12,14] in a number of systems.

Observations of an apparently linear relation be-

tween the yield and growth rate ~ [71] can be

explained by very high maintenance coefficients.

We conclude that the phenomenological non-

equilibrium thermodynamic description of micro-

bial growth developed here is substantiated by a

comparison with the experimental data that are

presently available. We note that this does not

include experimental tests of 'symmetry' (Eqns. 11

and 12) and proportionality (Eqn. 12).

193

VB. Optimization

After it has been shown that microbial growth

rates seem to obey relations derived by phenom-

enological non-equilibrium thermodynamics, it is

of interest to consider whether other conclusions

from phenomenological non-equilibrium thermo-

dynamics may be transferred to microbial growth.

A promising item may be the optimization of

microbial growth [76].

The available phenomenological non-equi-

librium thermodynamic studies of biological en-

ergy-converting systems in general [38,48,55,56,78]

all assumed flow-force relations (like Eqn. 10) to

be symmetrical and proportional (like Eqn. 12)

before they drew any conclusions concerning opti-

mization. As we shall try to apply these conclu-

sions to microbial growth, in their quantitative

sense our conclusions in this section will have to

be taken with a pinch of salt; they too depend on

symmetry and proportionality not yet proven to

hold for microbial growth. Yet, we expect (West-

erhoff and Van Dam, unpublished observations)

that in their qualitative sense, the conclusions will

not depend on these assumptions. Kedem and

Caplan [38] and more recently Stucki [55] have

discussed which states of biological energy con-

verters would be optimal with respect to any of a

set of specific output functions. The essential point

of their argument was that an energy converter is

not always 'optimal' when it performs at maximal

efficiency. The most simple example is given in

Fig. 3. This figure shows a voltage source with

electromotive force V, and internal resistance R~.

( - - - V )

~load

Fig. 3. Electrical analogue for fully coupled energy conversion.

Input is the electromotive force of the voltage source (V, with

internal resistance R i) and the current delivered by it. The load

placed or, the voltage source is represented by the resistance

R load'

194

The free enthalpy of this source is converted into

electrical work done by the current flowing through

the resistence R load. The figure [62] can be re-

garded as a model system for a simple microbial

cell [77] in which growth and catabolism are com-

pletely coupled, or a very realistic model for en-

ergy conversion by the light-driven proton pump

bacteriorhodopsin [63]. As already stressed by a

number of authors [38,55,77], the efficiency of

energy conversion of this system is maximal (100%)

only when no current flows through the system.

When the resistance R~o,d is considered to repre-

sent some item of the system that performs useful

work, then the state of maximal efficiency is not

identical to the state of optimal performance of

the energy converter. If, for instance, the energy

converter were to serve the purpose of the provi-

sion of power (V times I), then the optimal state

of the system would be when V equals 0.5 times

V~. In this state the efficiency is only 50%.

The example of Fig. 3 is one of a system that is

devoid of slips: it is a fully coupled system. How-

ever, to incompletely coupled systems (e.g., Fig. 3

with an extra resistance in parallel with R~o~,d),

r 1

r ,,J t

i / IS iS'

t

t

~/AGc0' fS~ \, \,l~p

"<- ~AG % ", ,U

Fig. 4. The efficiency (cf. Eqn. 9) as a function of the force

ratio (free enthalpy of the anabolic reaction divided by the free

enthalpy of the catabolic reaction) at different degrees of

coupling q (indicated above the lines). Symmetry and propor-

tionality of flow-force relations have been assumed (cf. Ref.

38). For every degree of coupling also the state of maximal

efficiency ( + +) and maximal power output ( - - ) are indi-

cated, Calculations as indicated in Ref. 38. The degrees of

coupling for which the calculation has been carried out are

relevant for the optimization discussions below.

similar arguments apply. Also, in those cases, for

almost every output function of the energy con-

verter, it can be calculated that the optimal state

does not correspond to the state with maximal

efficiency. Kedem and Caplan [38] have carried

out this calculation for the output function of

output power. Fig. 4 shows the results of that

calculation translated into microbial growth

parameters. The crossed line connects the states at

the various degrees of coupling with the maximum

efficiency, the dashed line connects the states of

maxi mum power output. Obviously the two lines

differ. Consequently, at any degree of coupling the

state of maximal efficiency is not identical to the

state in which the output power is maximal. As an

example, one may consider the case for q = 0.953.

For this degree of coupling the force ratio

ZAGa/AG ~. should be about 0.74 for the efficiency

to be maximal, but only 0.48 for the power output

to be maximal (cf. Fig. 4). Rottenberg [56] and

Stucki [78] noted that as a consequence of this

conclusion it becomes relevant that biological en-

ergy converters may have been optimized (through

evolution) with respect to some output function

other than efficiency. In the course of optimi-

zation, the load (Rjo~d in Fig. 3) would have been

adjusted until the ratio between output and input

force ( V/V i in Fig. 3) just suited the necessary

output function. We shall try to examine which

Fig. 5. The rate of biomass synthesis as a function of the force

ratio at different degrees of coupling. Calculated from Eqn. 13;

( + + ) and ( - - ) as in Fig. 4. Symmetry and proportionality of

flow-force relations have been assumed (cf. Ref. 38).

output function this may have been in the case of

microorganisms.

For the case of microbial growth, it is tempting

to suggest that the environment has selected mi-

croorganisms for most rapid growth, i.e., the

highest possible -3~, for the reason that the fastest

growing species will outgrow its competitors. Fig. 5

(full lines) shows the dependence of the output

flow J~d on Z times the ratio between output force

AG~ and input force AGc at different degrees of

coupling (indicated above the lines in the figure).

It turns out that the optimum force ratio with

respect to maximum output flow would be the

condition of zero, or rather as negative as possible,

free enthalpy content of the biomass produced.

The optimal efficiency would thus be zero or even

negative. When the anabolic substrate is modelled

as having a low free energy content (e.g., CO 2

[35]), this solution to the problem is trivially cor-

rect, philosophically intriguing, but may seem non-

sense biologically (see, however, Ref. 104 and Sec-

tion XI), as the biomass synthesized would not

have the free energy content essential for its own

existence [55]. Although it may be concluded that

to obtain a rate of biomass production as high as

possible, the organism would be inclined to keep

the free enthalpy content of its progeny and thus

~ "01 , ""

Fig. 6. The flow ratio (i.e., yield, or rate of biomass synthesis

divided by the rate of consumption of the catabolic substrate)

as a function of the force ratio at different degrees of coupling.

Calculations as in Ref. 38. Symmetry and proportionality of

flow-force relations have been assumed (cf. Ref. 38).

195

of itself as low as possible, there might be another

restraint that prevents it from lowering the free

enthalpy content of the biomass produced too

much.

The argument might be put forward that opti-

mization of the rate of biomass synthesis only,

would be very unrealistic in view of possible limi-

tations of substrates for growth. To some extent, a

choice to optimize the ratio of the output to the

input flow ('the flow ratio', or yield) would cope

with this objection. Fig. 6 shows the flow ratio

(yield) divided by Z (cf. Eqn. 15) as a function of

the force ratio (i.e., ratio between output and input

force) multiplied by Z at different degrees of cou-

pling. Once again optimization of the selected

output function (i.e., flow ratio) dictates that the

free enthalpy of the biomass produced with re-

spect to the free enthalpy of the anabolic sub-

strates is zero, or even negative.

At this point one seems to be left with a di-

lemma: although it has been concluded that maxi-

mization of the efficiency of growth leads to an

unrealistic result (no growth at all), maximization

of growth rate leads to unrealistic results as well

(growth of worthless biomass). It was Stucki [78[

who most elegantly presented the idea that not

only the force ratio may be varied in the optimi-

zation procedure, but also and even simulta-

neously the degree of coupling. This notion broke

up the traditional commonsense that nature will

always try to prevent slips of free enthalpy. (A

precedent of a function of slips had already been

analyzed in great detail: the brown adipose tissue

mitochondria that contain a regulated proton-

leakage pathway that generates heat when the

organism is in need of it [79].) The existence of

two variables that determine the relationship be-

tween output parameters and force ratio opens up

the possibility for a system to optimize for two

output parameters one after the other. Stucki

[55,78] calculated the optimal degrees of coupling,

the efficiency and several other parameters for the

situation in which the system first adjusts its force

ratio (by 'conductance matching') so that at any

degree of coupling the efficiency of the system is

maximal. This resulted in a relation between the

degree of coupling and the force ratio (cf. Fig. 4,

the + line). Then he determined the degree of

coupling that would lead to a maximal value of

196

any of a set of four output functions, all the time

keeping the condition of conductance matching

with regard to optimal efficiency intact. Translated

into microbial growth kinetics, it turned out that

after optimization of,IJ(ZL~AG~) the force ratio

multiplied by Z would amount to 0.486, the degree

of coupling would be 0.786 (cf. Fig. 5, the point

where the + line is at its maxi mum). The ef-

ficiency at this condition of maximal rate of bio-

mass synthesis would be 24% (cf. Fig. 4, read at

TABLE II

THE EFFECT OF THE COURSE OF OPTI MI ZATI ON ON THE OUTCOME OF OPTI MI ZATI ON CALCULATI ONS

The out put f unct i ons consi der ed wer e ( n = 1) t he rat e of bi omass synt hesi s ( - J,/(ZL~AG~)), ( n = 2) t he power out put of bi omass

synt hesi s ( - J AG,,/(L~ AGf ) ), ( n = 3) t he maxi mal economi c r at e of bi omass synt hesi s - J,,r//( ZL~AG~ ) and ( n = 4) t he maxi mal

def

economi c power out put of bi omass synt hesi s - ,~JG~rI/(L,:~AG f ) [55]. For x = ZAG~,/AG¢ t hese out put f unct i ons wer e wr i t t en as

f i n, x, q). Al so, "q = rt (x, q) = - ( x + q)/q + x I was consi der ed. Dependi ng on t he col umn in t he t abl e, first ei t her ~ or f was

maxi mi zed usi ng ei t her x or q as a vari abl e. The r emai ni ng degr ee of f r eedom was t hen used to maxi mi ze t he ot her funct i on. For t he

t hi rd col umn, for i nst ance, first t he r el at i onshi p bet ween q and x for whi ch jr(n, q, x) was maxi mal was det er mi ned: x - g( q). Usi ng

this rel at i onshi p, ~/(q, x) was r ewr i t t en as rl(q, g(q)) and t he val ue of q at whi ch rl(q. g( q) ) was maxi mal was cal cul at ed. Fi nal l y,

x = g( q), ~1 = ~l(q, x). f = f i n, q, x) and ot her f unct i ons wer e cal cul at ed, x was onl y consi der ed to var y bet ween 0 and 1. Symmet r y

and pr opor t i onal i t y of f l ow- f or ce rel at i ons was assumed (cf. Eqns. 12- 16) [38].

Par amet er n Fi rst rl opt i mi zed Fi rst out put f unct i on

wi t h as var i abl e opt i mi zed wi t h as

var i abl e

ZA G a/A G~ q

ZAG,,/AG, q

q 0.786 - 1 ( 1 )l i mi t -- l

ZAGa/aG c 0.486 0 0 I

,/ 0.236 0 0 1

J,,/( ZL¢cAG ~ ) ~' 0.300 -- 1 -- ( 1 ) l i mi t 0

Ja/(J¢Z) - 0.486 - 1 --(l )l l m, t -- I

q - 0.910 - 1 - 1

ZAGa/AG ~ 0.644 0.5 0.5

0.414 0.5 0.5

J.,/(ZL~:,.AG,:) - 0.644 - 0.5 - 0.5

J~AG a/( L,:,:AG 2 ) ~ O. 172 - 0.25 0.25

Ja/(J~Z) - 0.644 1 1

q - 0.953 - 1 1

ZAG,,/AG~ 0.732 0.5 0.5

0.535 0.5 0.5

J,,/( ZL~ AG~ ) - 0.732 0.5 0.5

71j,,/(ZL~AGc ) a 0.118 0.25 - 0.25

Ja/(JcZ) 0.732 - 1 1

q 0.972 1 1

ZAG~/AG~ 0.786 0.67 0.67

r/ 0.618 0.33 0.67

Ja/(ZL~,.AG~) 0.786 - 0.33 - 0.33

"qJ,,AG~/( L~JG 2 ) a 0.090 - 0.08 - 0.15

Jd/(J,.Z) - 0.786 - 1 - l

- 1

1

1

0

0

- I

- 1

1

1

0

0

- I

- 1

1

1

0

0

I

a Out put f unct i on t hat is opt i mi zed.

ZAGa/ZIG c = 0.486 and q = 0.786), the flow ratio

would be 0.49 times Z (cf. Fig. 5). Analogous

calculations were carried out for three additional

output functions. The results are shown in the

left-hand column of Table II.

Although the principle that both the force ratio

and the degree of coupling may be varied in order

to achieve optimal performance of a system is

attractive, it is not evident why the force ratio

should always be the parameter that is varied

when optimizing the efficiency, and the degree of

coupling the parameter that is varied to optimize

for the second output function. Also, the order of

optimization may make a difference. The right-

hand side of Table I! shows the results of calcula-

tions in which either the order of optimization was

reversed, or the parameters with respect to which

the functions were optimized were exchanged. It

turns out that both the order of optimization and

the combination of function and optimization

variable strongly determine the outcome. All opti-

mization procedures employed that differ from

that used by Stucki [55,78] give as result that the

optimal degree of coupling equals +_ 1. A direct

consequence of Iql being equal to one is that the

flow ratio is equal to the theoretical stoicheiometry

and constant. As the flow ratio JJJ~ is propor-

tional to the yield, this would mean that the yield

would be independent of the growth rate: the

maintenance coefficient (fl in Eqn. 24) would have

to be zero.

The observation that for most microorganisms

the maintenance coefficient has a significant value

may be taken to indicate that nature has not

followed any of the right-hand optimization proce-

dures of Table II. If nature has followed any

optimization procedure at all, then for the four

patterns in Table II, the one farthest to the left

would be the pattern concerned. This is the opti-

mization pattern proposed by Stucki [78]. Possibly,

microbes, or at least those which have a non-zero

maintenance coefficient, have evolved so that they

first optimized their efficiency, by adapting the

free enthalpy content of their biomass, and then

optimized their output function by adapting their

degree of coupling [104]. Below (Sections IX and

XI) we shall discuss which output function may

have been optimized.

197

VI. Description of microbial growth by mosaic

non-equilibrium thermodynamics

The preceding sections have shown that micro-

bial growth can indeed be treated in terms of

phenomenological non-equilibrium thermody-

namics. Thus, a set of equations seems to be

available that may be used to describe growth in a

quantitative sense. Such a description would mean

that in Eqn. 10 the coefficients L~. c, Lac, L~a, L~

and AGc as well as zIG, should be measured for a

particular microbe. Then at different substrate

concentrations growth can be described by Eqn.

10. Some authors (e.g., see Refs. 80 and 80a) have

stressed that non-equilibrium thermodynamics is

uninformative. It would merely be a curve-fitting

procedure, but can never give clues with respect to

the mechanisms that are responsible for a certain

observed behaviour of a system. This objection

against non-equilibrium thermodynamics is correct

for phenomenological non-equilibrium thermody-

namics, but not for non-equilibrium thermody-

namics in general [60a]. An example of how non-

equilibrium thermodynamics can have mechanistic

implications is that measurement of the coeffi-

cients in the description of mitochondrial oxida-

tive phosphorylation was shown to yield a direct

test of the chemiosmotic hypothesis [57,81]. A

more recent elaboration of non-equilibrium ther-

modynamics towards a tool that may yield infor-

mation about mechanisms was named 'mosaic

non-equilibrium thermodynamics' [53]. Network

thermodynamics [81 a] has similar assets, even more

general validity, but is strongly big-computer de-

pendent [60a].

In studies of microbial growth, it is probably

more interesting to study why, metabolically

speaking, growth takes place at a certain rate for a

certain microbe, than how rapid it is. With the

above-developed description of microbial growth

in terms of phenomenological non-equilibrium

thermodynamics, one hardly can get any further

than to answer the second type of question. Also,

concerning the efficiency of growth only a very

qualitative conclusion could be tentatively ob-

tained. We shall now develop a mosaic non-equi-

librium thermodynamic description of microbial

growth and show that it is additionally informa-

tive.

198

To develop an interpretation of microbial

growth in terms of mosaic non-equilibrium ther-

modynamics, it is helpful to devise first the sim-

plest model for a microbe that would still contain

much of the essentials. Such a model is depicted in

Fig. 7 (cf. Ref. 82). The metabolism of the organism

has been split up into three parts. The first part

(J~., J~) is catabolism, metabolizing fuel supplied

from the outside to generate ATP. The last part

(Jd, Jpd) is anabolism which uses ATP supplied by

catabolism to generate biomass through biosyn-

thetic reactions. The middle part (jpl) contains

those reactions that consume ATP, but do not give

rise to biomass. Some of these reactions may be

called maintenance reactions (Refs. 12 and 21; see

below). As an example of the middle part one may

consider the passive proton flux through the

bacterial membrane that results from the high

proton-motive force, but is not coupled to any

biosynthetic (or transport) reaction. Although the

model may seem to be reasonably consistent with

microbial growth more in general, it is pointed out

that Fig. 7 is meant to imply that the three

processes are independent of one another, except

through the concentrations of ATP, ADP and

inorganic phosphate. Growth of Escherichia coli

on a complex medium in a glucose-limited chemo-

star [17], or of Streptococcus cremoris on a complex

medium plus lactose as the energy source [83], are

examples. Furthermore, the model would be rele-

vant for light-limited growth of phototrophic

bacteria, the interrelating parameter between

catabolism and anabolism being either the phos-

phate potential, or the proton-motive force. How-

ever, there are numerous instances of microbial

growth that do not comply with this mutual inde-

pendence of the three processes. Yet, we shall first

glucose~

COz ~

Fig. 7. Model of microbia! growth (see text).

/....~ biomass

T

"6a

anaboilc

substPafe

consider this simple model system and then later

(see Sections X and XII) add some modifications

and analyze how these modifications affect the

conclusions.

In Fig. 7 some of the relevant flows in the

system are given a symbol. To the left there are the

flow of catabolism, Jc, and the rate of ATP synthe-

sis coupled to this catabolic flow, Jp. The rate of

ATP synthesis is to be expressed in /,mol/g dry

weight per rain. The catabolic flow will be ex-

pressed in terms of tool substrate used.

Jp=npJ~ (26)

np represents minus the number of mol ATP

formed per mol catabolic substrate that is used.

We shall assume that in catabolism ATP synthesis

is strictly coupled to substrate consumption. Near

the end of this paper we shall consider the effects

of the breakdown of this assumption if so-called

futile cycles occur.

Also, for anabolism a strict relation between

rate of ATP consumption and rate of biomass

production is assumed to occur:

J¢' = npJ,, (27)

Although changes in composition of the biomass

produced can be quite extensive [5], they are rarely

a a

large enough to affect significantly rtp [84]. rtp does

depend strongly on the anabolic substrate used.

For glucose and inorganic salts as substrates, val-

a

ues for --/'tp of 34 mmol/g biomass have been

calculated [18,21 23]. With carbon dioxide rather

than glucose as the anabolic substrate this value

increases up to 74 mmol/g biomass [21]. When the

anabolic substrate and biomass composition re-

main the same, we can assume n; to be a constant

and treat all slips as occurring within the middle

part of the system (cf. Section X). This middle part

has been represented by the leakage flow of ATP,

jpt, the rate of ATP hydrolysis due to leakage and

slip reactions.

Just like phenomenological non-equilibrium

thermodynamics, mosaic non-equilibrium thermo-

dynamics treats systems in terms of flows and

thermodynamic forces [53]. In mosaic non-equi-

librium thermodynamics the forces are defined as

the total free enthalpy differences across the 'ele-

mental' [60] processes. The flows are the rates of

these elemental processes. An elemental process

has been defined as a process that is independent

of all other processes except through the explicit

free enthalpy gradients. Every elemental process

possesses conjugated flow-force couples. The next

step in mosaic non-equilibrium thermodynamics is

[45,53,60] to write down a relation between the

flow and the force of every elemental process. For

near-equilibrium systems these relations can be

assumed to be proportional [45], but farther-

from-equilibrium deviations from proportionality

may become large [47,49,85]. In a definitive mosaic

non-equilibrium thermodynamic description, first

the actual flow-force relation for every single pro-

cess must be measured, so that the actual relation

can be further used in the calculations. Many

systems, however, have not been examined well

enough to allow the formulation of the elemental

flow-force relations (for developing exceptions, see,

Refs. 51, 52 and 86). However, elemental flow-force

relations have been considered more in general

[47-51,59,66,87] and it appears that their plots are

generally S-shaped (cf. the dashed line in Fig. 8):

at low and at high free energies of a reaction the

reaction rate is independent of the force (Vm~ ~

effect), whereas at intermediate force values the

AG

Fig. 8. The rate of a simple enzyme-catalyzed reaction as a

function of the free enthalpy of reaction. The dashed line gives

the dependence calculated (cf. Ref. 49) from enzyme kinetics.

The dotted line gives the dependence of aG n on aG. the full

line gives the dependence of aG - aG ~ on AG. Accordingly,

the latter corresponds to the approximation used for the actual

flow-force relationship.

199

flow-force relation is to a good approximation

linear. Generally, however, this linear region does

not extrapolate to the origin of the plot: the re-

lation is not proportional, as would be the case in

the near-equilibrium domain. To develop a simple

mathematical description of the S-shaped flow-

force curve in Fig. 8, it is useful to define a param-

eter ZIG = in the following way: when ZIG is so high

(low) that the enzyme(s) considered are at their

maxi mum rates, ZIG = equals ZIG minus a constant,

the effect being that AG - ZIG = is constant. When

ZIG is in the range in which it influences the flows.

ZIG = is constant so that ZIG - ZIG = varies linearly

with ZIG. The dotted and full lines in Fig. 8 show

the behaviour of AG = and ZI G- ZI G =, respec-

tively. It can be seen that the multiplication of

ZIG - ZIG = by a constant gives a good approxima-

tion of the flow. Accepting the limitations of such

an approximation we propose the following flow-

force relations:

4 = L ((ac - act )+ -

(28)

= L,{(Ac. - ac:) - acg) )

(29)

= L' (acp - ac ') (30)

Some remarks should be made with respect to

Eqns. 28-30. Firstly, the factors ~, are introduced

to indicate that, unlike the situation near equi-

librium, the relative sensitivities of the fluxes to

changes in the two forces may not be equal to the

stoicheiometric number of the flows [51,86,87].

Such a situation arises when a reaction becomes

saturated with respect to one substrate-product

couple, but not with respect to another. An exam-

ple is succinate oxidation by the mitochondrial

respiratory chain. Under the usual conditions, this

reaction is sensitive to changes in the proton-mo-

tive force [51], but not to changes in the oxygen

tension [88]. The second remark is that we have

chosen the forces to be positive. Consequently, np

and np are negative. Thirdly, the values of AGp

need not be the same for the catabolic, anabolic

and leakage processes. This problem can, however,

be solved mathematically by adding the difference

between AGp = and AG~ = to AG~ and the dif-

ference between AGpa= and AGp TM to AGff.

The third step in mosaic non-equilibrium ther-

200

modynamics is to sum up all elemental flows that

concern the same substance. The total flow of

ATP then reads:

4 = GJ. + G4 + J¢ = GL~(a6~ - aGy)

4- ( ( Y/p)2~pt c -b a 2 a

Let us consider the amount of ATP necessary to

supply the daughter cells with the normal intracell-

ular concentration of ATP as belonging to the

biomass produced, then, because the average con-

centration of ATP remains constant during growth,

the sum of all elemental flows of ATP must be

zero:

+ GL.( aG- aG,?) (31) J.=0 (34)

Together, Eqns. 28, 29 and 31 are the mosaic

non-equilibrium thermodynamic description of

microbial growth insofar as it corresponds to the

simple model considered here.

microbiological

non-equilibrium

VII. Correspondence between

growth parameters and mosaic

thermodynamic parameters

The elaboration of the phenomenological non-

equilibrium thermodynamic description of micro-

bial growth into the mosaic non-equilibrium ther-

modynamic description makes it possible to define

YATP in terms of non-equilibrium thermodynamics.

YATP has been defined [15] as the growth yield per

tool ATP synthesized in catabolism. In the present

model (Fig. 7) this means that:

def

YATP = cMJJJp (32)

Recalling our assumption that per mol catabolic

substrate the number of ATP molecules synthe-

sized (np in Eqn. 26) is constant (for futile cycles

see Section X) or/and that all leakage and slip

processes can be grouped together in the middle

part of metabolism (cf. Fig. 7), this equation can

be rewritten as:

(33)

M,, c c

YATP = C J,,/npJc = Ycs/- np

YA'rP appears to be proportional to the 'flow ratio'

(output flow divided by input flow).

To obtain more insight into the dependence of

YATP on the activities of the various elemental

processes and to investigate how maximum growth

rate and the maintenance coefficient can be ex-

pressed as functions of these activities, we shall

now develop the mosaic non-equilibrium thermo-

dynamic analogue of Eqn. 20.

From Eqns. 29-31 it then follows that:

Li ) LI

p

( ,~G, -

Jc=( nap/np) 1+ a, ~2 ( - Ja) + P , AG~?

yp( Hp) L a J t/pHp~p

(35)

Using Eqn. 28 instead of Eqn. 29, we obtain:

L=

L~

1+

( s,)

LIpL¢

+ (AG~ - AG~) (36)

We now possess two different relationships be-

tween the rate of catabolism (J~.) and the rate of

anabolism ( - J~), both of which must be valid at

all times. Both relationships seem to suffer from

the same drawback, i.e., that J~ depends on two

variables so that it is not a (single-valued) function

of Ja. The properties of AGa = and AGf (cf. Fig. 8)

eliminate this problem for two important growth

conditions. When the catabolic substrate limits

growth (the 'csl' condition), the anabolic substrate

concentration is usually without effect on the

growth rate so that AG~ -AG~, = is constant. Con-

sequently, under csl conditions the right-hand term

in Eqn. 35 becomes constant. Using the shorthand

notation introduced in Eqn. 21, we write for Eqn.

35:

csl

Jc = °~csl(- Ja ) -¢-/~cs[ (37)

and, analogously, for Eqn. 36:

as]

(38)

Again l/a is the maximum growth yield (on the

basis of the catabolic substrate) and/3 the mainte-

nance coefficient. The subscripts csl and asl de-

note 'catabolic substrate limiting' and 'anabolic

substrate limiting', respectively.

In contrast to what resulted from the phenome-

nological non-equilibrium thermodynamic ap-

proach (Eqn. 21), the maximum growth yield and

the maintenance coefficient are now related to

metabolic intimacies. Thus, in the case of growth

limitation by the catabolic substrate:

(39)

and

__ 1 a a c

fl¢~ - (Lp/(ypnpnp))(AV.- AG a )

(40)

It turns out that leakage processes (Lip, cf. Fig. 7)

increase the maintenance coefficient, but decrease

the maximum yield.

If anabolic substrates limit growth, the result is

different:

1

(41)

and:

fl~ LIpL" (AGc - AG) (42)

The maximum growth yield (on the basis of cata-

bolic substrate) now is expected to increase with

increased leakage. The maintenance coefficient is

still expected to increase with increasing leakage

coefficient, but this increase now is hyperbolic

rather than proportional.

These predictions solve a phenomenon that has

been known for some time in microbiology, i.e.,

that the dependence of the rate of catabolism on

the growth rate is much stronger when catabolic

substrates are growth limiting than when anabolic

substrates are growth limiting [7,25]. In the case of

limitation by anabolic substrates the (apparent)

maximum yield even exceeds the theoretical value

(Eqn. 41).

201

The observation that leakage reactions affect

the maximum yield ( I/a) can be taken to indicate

that maintenance reactions cannot only be repre-

sented by the coefficient of growth rate-indepen-

dent maintenance (/3). A coefficient for the growth

rate-dependent maintenance (cf. Refs. 8, 21, 24

and 84) can be defined:

fl ~f L~ (43)

c a a

n pYpnpga

In the case of growth limitation by anabolic sub-

strates:

1

flagdd I def -- Lp ( 44)

I c 2

Lp + ~;(.;) L~

Defining the theoretical yield 1/a o

def

1/% = G/n;

as:

(45)

Then:

a= ao + flgd (46)

so that:

L = "o( - Jo) + t~ + #~( - Jo)

(47)

The most striking part of this result is that (cf.

Refs. 8 and 74) in cases of growth limitation by

anabolic substrates, the coefficient for growth

rate-dependent maintenance is predicted to be

negative. As a consequence, the theoretical yield

1/% is expected to lie in between the maximum

yield obtained in a catabolic substrate-limited cul-

ture and the maximum yield obtained in a culture

limited by anabolic substrate:

O%sl

1

Lp

1+

~;(~;):Lo

=(l/aa~')/( 1+ y~(np)L~)c 2L~

(48)

Another important feature is the comparison of

the condition of catabolic substrate limitation to

that of anabolic substrate limitation, which can be

derived from Eqn. 36. In this equation, AG c - AGff

202

is at its maxi mum when the anabolic substrate is

growth limiting. When the catabolic substrate is

growth limiting, this term has varying, but always

lower (effective) values. Consequently, at the same

anabolic rate, the rate of catabolism is predicted to

be higher when the anabolic substrate is growth

limiting than when the catabolic substrate is growth

limiting. Since also the maximum growth yield (on

the basis of catabolic substrate) is highest under

conditions of anabolic substrate limitation, also

the growth rate-independent maintenance must be

higher under these conditions. Likewise, the actual

yields in the case of anabolic substrate limitation

are predicted to be always lower than in the case

of catabolic substrate limitation.

These mosaic non-equilibrium thermodynamic

relations for microbial growth offer many possibil-

ities to relate observed growth behaviour to meta-

bolic peculiarities. An example is the apparent

anomaly in the experimental results obtained by

Neijssel and Tempest [89,90]. They showed that in

certain K + -, Mg 2+ -, or phosphate-limited cultures

the relationship between oxidation rate and dilu-

tion rate is non-linear. The explanation for this

anomaly is found in the above equations (Eqns. 35

and 36): especially Mg 2+, but also K +, is a

cofactor for many enzymic reactions. Conse-

quently, limitations by such substances make the L

terms in the equations dilution rate dependent.

Light [91] reviewed evidence that even the

stoicheiometric numbers may be altered when

cofactor concentrations change. More important

than the fact that as a consequence of such a

phenomenon the relationships between catabolic

and anabolic rates may not be linear may be the

warning that theoretical yields cannot be obtained

by taking the slopes of such relationships when a

cofactor limits growth [24].

The prediction that, in cases of growth limi-

tation by the catabolic substrate, leakage processes

cause a decrease in the maximal yield ( l/a), as

well as an increase in the maintenance coefficient,

is as yet only for the second part supported by the

few available experimental data. For aerobic glu-

cose-limited A. aerogenes, Neijssel [73] did find

that the uncoupler of oxidative phosphorylation

2,4-dinitrophenol, causes a drastic increase in the

maintenance coefficient. In the same experiment,

however, the maximal yield seemed to be unaf-

fected (Ref. 73, cf. Ref. 92) or slightly decreased

[74]. Stouthamer [21] presents an explanation, con-

tending that dinitrophenol causes a certain rate of

dissipation of the energized membrane state that is

compensated by an increased maintenance respira-

tion. The remaining substrate would then be used

for biomass formation with the same efficiency as

in the case of the absence of dinitrophenol. Apart

from the term efficiency being inappropriate (flow

ratio would be more in place, see below), it seems

unlikely that the uncoupler would not diminish the

ATP/subst rat e ratio. Hence, a decrease in the

maximal yield upon addition of uncoupler would

still be expected.

We propose one explanation that is similar to

that of Stouthamer [21] but that does not suffer

from the lack of clarity of the latter. Eqns. 39 and

40 predict that both the maximum growth yield

( I/a) and the maintenance coefficient should be

affected by the protonophore, more precisely, by a

change in Zp. However, closer inspection of these

two equations shows that the maintenance coeffi-

whereas the cient depends proportionally on Lp,

maxi mum growth yield depends only by a small

l Thus, the equations predict a large

fraction on Lp.

effect of protonophore on the maintenance coeffi-

cient, but a smaller, potentially undetectable effect

on the maximum growth yield. Another explana-

tion may even be more relevant. In the experi-

ments by Neijssel [73], the growth-limiting factor

is glucose, which is both a catabolic and an

anabolic substrate. Consequently, the effect of un-

coupling on the maxi mum growth yield I/c~ is

hard to predict (see Eqns. 39 and 41). It will be

interesting to carry out more experiments under

conditions of growth limitation by an unequivo-

cally catabolic substrate to see whether the uncou-

pler does cause some decrease in maximum growth

yield.

It will be clear that also the relationships be-

tween growth yield and metabolism can be made

explicit by combining Eqns. 25, 39-42. Thus, it

can be explained why in tryptophan-limited cul-

tures in the presence of a large excess of glucose,

Ygl ..... became [71] almost independent of the

growth rate. Also, the effects of different meta-

bolic inhihitors on growth-linked processes (for a

review, see Refs. 6 and 21) can be understood

[91a].

The experimentally confirmed prediction of the

present description of a change in value of the two

maintenance coefficients upon a shift from a

catabolic to an anabolic growth limitation seems

to suggest a study of the growth kinetics of an

organism with anabolic and catabolic substrates

present simultaneously but at varying concentra-

tion ratios. Such a study has been carried out by

the group of Stouthamer [92-94]. They varied the

ratio of methanol to mannitol in the growth

medium for Paracoccus denitrificans. They ob-

served that as this ratio increased from zero to

about 0.8 the maxi mum yield and the growth

rate-independent maintenance coefficient (both on

the basis of oxygen) increased. Upon further in-

crease in this ratio the latter remained constant,

whereas the former dwindled. We interpret these

observations as follows. Growth of Paracoccus on

mannitol has been concluded (Ref. 93, see also

Ref. 74; cf. growth of Klebsiella aerogenes [74]) to

be energy rather than carbon limited. Therefore, in

the absence of methanol the maxi mum yield and

the maintenance coefficient are best described by

Eqns. 39 and 40, respectively. Methanol, however,

is a substrate with a high free energy content per

carbon atom. Therefore, the addition of methanol

to the growth medium will shift the condition from

catabolic substrate limitation to anabolic substrate

limitation. The maxi mum growth yield and the

maintenance coefficient increase accordingly (see

above). Once this shift has been accomplished, a

further increase in the ratio of methanol to man-

nitol induces ribulosebisphosphate carboxylase

[93], the key enzyme for CO 2 assimilation. Thus,

synthesis of biomass from CO 2 starts to occur in

addition to biomass synthesis from mannitol.

Anabolism from CO 2 takes more ATP per C-mol

biomass so that - np increases. Also, the reported

[93] increase in the number of 'sites' of oxidative

phosphorylation with mannitol as substrate, occur-

ring when methanol is added, increases - np. Eqns.

41 and 42 predict that the maxi mum yield on the

basis of oxygen (lflaa~ l) should decrease, but that

the maintenance should remain constant or de-

crease slightly. These predictions (a posteriori)

agree with the experimental observations [92,93].

It is only a small step towards the expression of

YA~vp in terms of the parameters of the elemental

processes. Combination of Eqns. 32 and 35 yields:

203

LI

(49)

A hyperbolic dependence of YAVP on the growth

rate is predicted, provided that AG~ is constant

(i.e., growth limitation by catabolic substrate). As

characteristic for the relation between YATP and

the growth rate, Y~p~c~l and the growth rate at

half-maximal yield may be taken:

aATP";~I = cM/Y'~TP'c~I = napll + a . 2

yp(np) La

(50)

.half ( AGa - AGa~ )

- -a.c~ = (5 r )

It would be interesting to see whether indeed

uncoupler decreases y max and increases the

ATP.cs|

growth rate at which the yield is half maximal. At

any rate, Eqn. 49 shows that uncoupler is expected

to strongly decrease YATt', both through its effect

on y~x and through its increase of the growth

rate at which the half-maximal YAvp is reached.

Such a strong decrease in YATp by protonophore is

heavily documented in the literature (for a review,

see Ref. 21).

The prediction that YaTV depends on the degree

of coupling between anabolism and catabolism

does not mean that it can assume all values de-

pending on how much leakage there is in the

system. In Eqn. 50, aAVV.csI can again (cf. above)

be split into two parts, of which only one is

leakage dependent. Stouthamer and Bettenhausen

[24] defined YA~theor as the YA~p x that has been

corrected for the effect of uncoupling. It follows

from Eqn. 50 that:

Y~%ap×.th~or = cM/- nap (52)

i.e., Y .... is the inverse of the number of ATP

ATP.theor

molecules needed to synthesize 1 g of new cells.

For a cell of known composition, this value can be

calculated through inspection of the anabolic path-

ways that are used by the organism. For a micro-

bial cell growing on glucose and inorganic acids,

204

YA~-~m .... was calculated to equal [18,21-23,27] 32

(expressed in g cells/mol ATP, formally equal to

Y ...... times the molecular weight of the cell).

ATP.t heor

The ratio of YATP to Y~n-~apxth .... is:

" - " '+( np)',pL~. (54b) L,- = ( r i p) VpLa + Lp

For the phenomenological stoicheiometry Z (cf.

Eqn. 15) it then follows:

max _ c

}'ATP/ YATP.theor -- jpa/_ jp

(53)

A consequence is that the flow performance of

microbial growth can be rewritten in such a way

that it is normalized with respect to the

stoicheiometries with which metabolism takes

place:

csl

YATP//YAnq'al~.lheor =

1+

I

I 1

,;(.;)~c. 'p"~ 7;

(54)

/ L. L'~ + ,;(.;)%

]// l a at 2

z

Lc Lp + ,p( Hp) L a

(54c)

Defining:

def

Zm, = Znp/np (54d)

F

V I ' a 2 aL

,; , + ,.

(5ae)

Owing to the asymmetry of the matrix in Eqn. 54a,

two degrees of coupling have to be defined (cf.

Eqn. 14 and Ref. 54):

Vlll. Coupling, stoicheiometry and efficiency in the

mosaic non-equilibrium thermodynamic interpreta-

tion

The mosaic non-equilibrium thermodynamic

description considers a biological energy converter

not as a black box, but as a set of mutually

independent processes. In our simple model of

microbial growth, these processes are catabolism,

anabolism and the leakage reaction (uncoupled

ATPase). For the mechanism of uncoupling pro-

posed in Fig, 7 (ATP leak only), one can de-

termine the effect on the degree of coupling q and

the phenomenological stoicheiometry Z by ex-

pressing the parameters of the phenomenological

non-equilibrium thermodynamics in those of the

mosaic non-equilibrium thermodynamic descrip-

tion. Using Eqns. 10, 28-30 and 34 one obtains:

(

Lcc

L~ L ,

i 2 I c a c

L t i, a c c 2 , 1

-ypnpnpLaL c ( np) ,pLaL c+ LpL a

with:

(54a)

de,{ O J+

(t )()

=-- ,p/ 1 + Lp 1 + Lp

c 2 ,

a ~ a

(.p) rp< (.;) ,;<

~s4f)

Symmetrically:

q~ ~Tq~a (54g)

Zl c

Here symmetry would imply that ~,p and "yp are

equal.

From these equations a number of conclusions

must be drawn.

(i) Since (at least in the model of Fig. 7, for

other cases see Ref. 58a) uncoupling is brought

~" cf. Eqn.

about by an increase in ATP leak (i.e., Lv,

30), in principle at constant values of the other

microscopic parameters, Z will generally vary with

q (unless yp(np)2Lc = a a 2 "yp( Hp) La). Hence, a degree

of coupling different from unity will generally

imply a deviation of the phenomenological

stoicheiometry from the theoretical stoicheiometry

(Zr~ d =*= 1). Surprisingly, in those cases Z can also

exceed the theoretical s toicheiometry (np/n:~).

(ii) At constant ATPase leak (Lip) the degree of

coupling q changes when one of the other micro-

scopic parameters is varied; activation of the

anabolic pathway (increase in La) would increase

q. This phenomenon reflects the intuitive idea that

the leak becomes less important in a relative sense.

We stress that an organism cannot only vary the

degree of coupling of its growth to its catabolism

(and its efficiency) by varying a 'leak' (L~), but

also by varying the activity of the rest of its

metabolism. Also, an organism could compensate

for leak induced by the environment, by accelerat-

ing its catabolism, or anabolism. Possibly, such a

shift in metabolism occurred when Neijssel [73]

studied the effect of 1 mM 2,4-dinitrophenol on

growth of K. aerogenes.

(iii) Symmetry of the phenomenological equa-

tions (Eqn. 10) and of the matrices in Eqn. 54a is

obtained if, but not only if, the elemental flow-force

relations (e.g., Eqn. 26 plus Eqn. 28) are symmetri-

cal (i.e., 7p~-- 7p~= 1). The symmetry is also ob-

tained if 7¢ = 7p. Hence, the possibility that the

phenomenological equations for microbial growth

are symmetrical is not necessarily hurt by the

considerations implying elemental flow-force rela-

tions to be asymmetrical [51,58-60,69a,86]. Con-

versely, such symmetry would not necessarily im-

ply that the 7 terms equal unity.

When considering the efficiency of microbial

growth, the efficiency of each of the elemental

processes in Fig. 7 may be considered separately.

The following equations give the efficiencies of the

three elemental processes of growth (cf. Fig. 7):

def ( -- Jp) AGp c AGp

~ ;~.,aG~ ( - ";) Ta~ (55)

def - - gaAGa 1 AG a

"% = Jp"AGp - ( - np) AGp (56)

def 0

o (57)

TI I J~AGp

The efficiency of the leak process equals zero. It is

interesting to consider the relation between the

overall efficiency (Eqn. 9) and the efficiencies of

the partial processes:

J~

,,o, = ~.,. (_ j~) (58)

205

If the leakage reaction equals zero, the two ATP

flows are equal in magnitude. Then the efficiency

of the overall process is the arithmetical product of

the efficiencies of the two remaining elemental

processes. It may be useful to stress the point that

the efficiency of growth has to be lower than

100%, even if no leakage occurs in the growth

processes (i.e., at complete coupling of the anabolic

reactions to the catabolic reactions). This can be

concluded from either Eqn. 55 or Eqn. 56. For

growth to occur, AG~ must exceed -npAGp and

-napAGp must exceed AG a. Consequently, ~ and

~a must be less than unity.

Generally, some leak reactions occur (see

above), so that the overall efficiency is lower than

the arithmetical product of the efficiency of the

catabolic pathway times the efficiency of the

anabolic pathway. Using Eqns. 33, 53 and 58, the

following expression for the overall efficiency can

be derived:

Y YATP

(59)

7~tot = "~a~c~ = ~a~c yA~at~.theor

The product of the two efficiencies in this equation

equals the efficiency the process would have, if at

the same free enthalpies of catabolism and anabo-

lism the overall process were to be fully coupled:

aG,/,;

rla'qc = "q,o,( q = 1) = - - (60)

Consequently:

Y }rATe

"q~ot = ~t ot (q = 1) .-7~22~.~ = ~to~(q = 1) (61)

rA%X,heor It heor

In short, an efficiency lower than 100% is the

result of free enthalpy sacrificed to enable the

system to grow at a non-zero rate, and a free

enthalpy loss due to free enthalpy leakage (arising

from uncoupling). The latter tem only is reflected

in the y/ymax ratios.

/ J theor

IX. The interpretation of yield deficiencies

Eqn. 59 shows the bearing of the Y values on

the overall efficiency of microbial growth: in mi-

crobial growth deviations from 100% efficiency are

caused by inefficiency in fully coupled catabolism,

206

inefficiency in fully coupled anabolism and by free

enthalpy leaks ('uncoupling'). The tempting as-

sumption that fully coupled processes are optimal

when their efficiency is 100% has been shown

[38,55] to be incorrect. For the fully coupled

catabolic process, the efficiency can be rewritten in

terms of the force ratio (Eqn. 55). At 100% ef-

ficiency a fully coupled process cannot proceed, as

the output force just matches the input force. For

a process to proceed at a non-zero rate some of its

efficiency must be sacrified by lowering the output

force. Thus, with respect to Eqn. 59, it could be

said that rio measures the efficiency sacrificed to

keep the catabolism going and "Oo the efficiency

sacrificed to keep the anabolism going, whereas

the efficiency loss due to slip only occurs in the

deviation of Y from ym~,~. The deviation of Y

from Y~h~or does not function in keeping the pro-

cess of biomass synthesis going. Consequently, it is

not involved in an optimization of the system

towards maximum output.

To illustrate how important this conclusion is,

we turn to an earlier article by Harder et al. [77].

In this article two potential causes of low effi-

ciency for microbial growth are mentioned. One is

that reactions (even fully coupled ones) are rela-

tively useless when they proceed at 100% effi-

ciency, because such a high efficiency necessarily

implies that their rate is zero. For a reaction to

have maximal output power it is best to have an

efficiency of only 50%. Here the authors consider a

fully coupled process and the low efficiency is

brought about by placing a load onto the system

so that the output force is decreased to 50% of its

maxi mum value (see subsection VB). The other

cause of lowered efficiency considered by these

authors consisted of slip reactions, incomplete

coupling therefore. However, the authors state:

"The difference between YA~.~;~ and YATP (when

corrected for maintenance free enthalpy require-

ment) is slightly more than a factor of 2, indicating

that the efficiency of energy transduction in micro-

organisms may be close to 50%". Next, they refer

to the fact that this 50% agrees with the calcula-

tion that the output power for a (fully coupled)

system is maximal when its efficiency is 50%. Here

an erroneous interpretation is suggested, as the

authors attribute the efficiency loss measured as

]/ATe//Y~!~e'~t ~ ..... and therefore arising from leaks to

the efficiency loss that would occur in the fully

coupled process when it would optimize for maxi-

mum output power. Eqn. 59 suggests that the

efficiency loss that arises because the system tries

to optimize for maximal output power will prim-

arily appear in the two rl terms and not in the Y

term. To rephrase this in terms of Eqn. 59, the

authors identify the efficiency loss in the term

YA.rp/Y~?~t h .... as an efficiency loss in terms of the

two elemental ~ values.

Our above equation, however, shows that

YATP/Y~,h .... is not equal to the overall effi-

ciency, and even that a very large part of the

efficiency loss may be caused by reactions not

involving YAVP/Yf,'~'~P'~th,,or. It follows that Harder et

al. [77] erroneously concluded from the fact that

YATP is usually only about 50% of Y~n-}apXth .... that

the efficiency of the process amounts to 50%.

Consequently, also their suggestion that microbes

may have optimized with respect to output power

loses its experimental support.

This then leaves us with the problem of inter-

preting the general (e.g., see Ref. 21 and 77) ob-

servation that Y},~"(,'/)A~~,h .... is only about 0.5. To

that purpose it will prove useful to borrow the

notion 'level flow' from non-equilibrium thermo-

dynamics. Level flow has been defined as the

condition [38,55] in which the output force of the

system is equal to zero. In our description this

would mean that AG~,- AG~ would equal zero.

Although in actual systems of microbial growth

(with the possible exception of anabolic substrates

that are at least as rich in free enthalpy content as

glucose [46]) this condition may never be reached,

one can extrapolate towards it. Eqn. 23 shows that

under level-flow conditions the yield is identical to

y ..... = l/a. Combination of Eqns. 49 and 50

shows that also:

( )ATP),evemo,, = y,~r]!~ (62)

Eqns. 33 and 52 now allow us to understand the

relationship between the ratio i nput/out put flow

at level flow and ymax/yma× .

ATP / ATP.t heor"

y ..... ",,Z= j

&TP.t heor level flo~ P ] level flow

(63)

Using Eqns. 13 and 54d under the level-flow con-

207

dition (X = 0) one obtains:

yh~ap "

qZ~ d (64)

It turns out that when the observation that Y~"+~x is

only half its theoretical value is translated into

non-equilibrium thermodynamics, it indicates that

q times Zre d is only 0.5. The value of Z is un-

known, except when q equals 1, when Zr~ a also

equals 1 (Ref. 38, cf. Section VIII). Some authors

(e.g., see Ref. 56) have implictly assumed that Z~ d

equals 1, even when q does not. If this assumption

were correct for microbial growth, then it could be

concluded that q equals approx. 0.5. Comparison

of this value with the q values obtained after any

one of the four optimization schemes (each for a

different output function, cf. Table II) shows that

this value of q is lower than any one of the values

predicted.

Van Versseveld [76] also assumed that Z~¢ d

would equal 1. He compared the theoretical results

obtained by Stucki [55,78] to experimental results

obtained with P. denitrificans growing either under

succinate, or under sulphate limitation. Under

these growth conditions, the P/O values found

were only half those theoretically expected, or, in

other words, Y~"~ was only 50% of Y~"}~h¢or. In

comparing this value of 0.50 to the flow ratios

predicted by Stucki [55] for four different optimi-

zation patterns (see left-hand column of Table II),

he found that this ratio was close to the flow ratio

of 0.49 predicted for the case of optimization of

output flow under simultaneous optimization of

the efficiency. However, the flow ratio as defined

by Stucki was Joutput divided by "]input' divided by

the phenomenological stoicheiometry Z. Conse-

quently, Van Versseveld's conclusion that P. de-

nitrificans has optimized for maximal rate of bio-

mass synthesis at optimal efficiency is based on

the implicit assumption that Z equals the theoreti-

cal stoicheiometry (/red = l). There is no a priori

reason for this assumption.

In general, the uncertainty about the value of Z

precludes comparisons of the results of the optimi-

zation theories [38,55] with experimental values,

although this point has often been overlooked

(e.g., see Ref. 55). To eliminate this uncertainty it

will be necessary to establish how Z varies with

the variation of q. It has been shown (Ref. 58; cf.

Section VIII) that the relation between q and Z

depends on the actual mechanism of uncoupling,

as well as on the relative activities of the catabolic

system with respect to the anabolic system (i.e.,

LJLa). Clearly, more work is needed here before

more definitive conclusions can be drawn from the

ratio of Y .... /v .... An upper limit for Iq] can,

ATP / l ATP.theor"

however, be derived. When one assumes propor-

tionality and symmetry:

Iql <~ Z,ea ~ l/Iql (65)

In combination with Eqn. 64 this implies that:

q2 ~< yA~a~/yA~al~.theor ~< ] (65a)

Hence, the observation that g/~TpmaX =0.5 ymaxATP.th ....

would indicate that Iql would be lower than 0.8.

About the efficiency one additional statement

can be made. Through the manner in which

YAvP/yAn~a~.theor appears in Eqn. 59, it can be con-

cluded that the efficiency of microbial growth

must be lower than 50% at the generally observed

values of YA~ x (usually YaTp is lower than Y,~Pr~).

It may be necessary to comment upon the point

that the interpretation of many of the reported

(e.g., see, Ref. 21) Y~,x/Y/~ar~theo r values in terms

of the present theory is restricted by the fact that

many of those studies have been carried out with

single substrates for both anabolism and catabo-

lism. Although such systems formally do not fit in

the scheme of Fig. 1, we shall show below that the

interpretation of YATP remains essentially unal-

tered. As to the efficiency of growth on single

substrates reported thus far (e.g., see Ref. 35), it

will be shown that they cannot be compared with

any of the efficiency values mentioned up to now.

After recalculation of those efficiency values we

shall be able to draw conclusions about the

criterium according to which microbial growth has

been optimized (cf. Section XI).

X. Futile cycles

Up till now (cf. Ref. 91a) the mosaic non-equi-

librium thermodynamic description of microbial

growth was limited to those cases that conform to

the scheme given in Fig. 7. Consequently, incom-

208

plete coupling has been assumed to arise from an

ATP leak describable by Eqn. 30 only. Although

such a mechanistic basis for uncoupling (or in

analogy, i.e, proton permeability) of the energy-

coupling membranes [79,96-98]) may indeed oc-

cur, other metabolic features that cause anabolism

to be decoupled from catabolism may exist. In rat

liver mitochondria some evidence for occasionally

'slipping' respiration (i.e., respiration without con-

comitant proton pumping) has been produced

[98a]. In liver cells gluconeogenesis is at least

partially controlled by the flux 't he other way'

through pyruvate kinase [28,98b]. In general, futile

cycles can occur when there are two or more

pathways for interconversion of a substrate S into

a product P with different amounts of energy-car-

rying coenzymes (ATP or A/2H, ) being converted

along the way (e.g., hexokinase and glucose-

6-phosphatase). When in practice the fluxes

through two such pathways flow in opposite direc-

tions, the net result can be written as a net flux

(equal to the absolute difference of the two fluxes)

plus a cyclic flux (equal to the smallest of the two).

The cyclic flux will consume energetic equivalents

for every turn of the cycle. When the two fluxes

flow in the same direction, in principle a similar

construction can be made, i.e., a net flux equal to

the sum of the two fluxes and a cyclic flux equal to

the smallest of the two fluxes, also this case can be

considered as futile cycling, although one usually

speaks of slip, the flux generating most ATP occa-

sionally 'forgets' to phosphorylate an ADP mole-

cule.

The effects of futile cycles in anabolism on the

thermodynamic description of liver cell gluconeo-

genesis have been discussed [45,95,96]. The effects

of the futile cycles in the anabolic and catabolic

pathways (e.g., see Refs. 25, 26 and 74) of micro-

bial organisms may be expected to be similar.

Here we shall treat the example of a futile cycle in

the anabolic pathway. Fig. 9 presents a scheme of

such a futile cycle. In this scheme the anabolic

reaction pathway of Fig. 7 is replaced by two

parallel anabol i c pat hways with different

stoicheiometries of ATP consumption. Conse-

quently, Eqn. 29 is replaced by the following set of

rate equations:

ja=Lal{(aGa__AG,r) a] al

+npyp (AGp-AG~)} (66)

ADP+P~

I

Jp

ATP

[

biomass

anabolic subsfrafe

Fig. 9. Model of microbial growth with a futile cycle in the

anabolic pathway.

and:

Z,2 = L.2{(AOo - aa2) + n;2r;2(~Cp - JC;')} (67)

Assuming in the individual reactions strict cou-

pling between rate of ATP consumption and rate

of biomass production:

jpd : .pi Ll (68)

and:

jpa2 = n p2]a 2 (69)

The total rate of ATP consumption in the anabolic

reaction (jptot) is the sum of these two rates:

J;' "" = J;~ + J;°~ (70)

We define the overall stoicheiometry at which

ATP is used (np) in anabolism as a weighted

average:

a 2

afcdef n~plL,,t + rtp L~2 (71)

rtp Lal + La2

Furthermore:

L~,~~fL, I + La2 (72)

and:

~,r. def .;'y;"c~, +, ;-" ~;-%2

(731

yp - n~p)L.) + n~p2L~e

If also the futile cycle is introduced:

LpfC def Lal La2 a) al

L~, + L. ( np'- nf np -'yp2n"f) (74)

then for the total rate at which ATP is consumed

in the anabolic reactions it follows that:

., ..... L.fc

a.tot = rip J~ + p ( AGp

(75)

The rate of anabolism is:

j~:ot = L~C{(AGa _ AGa~)+ n~fC.,lpfC( AGp _ AGp) ) (76)

The important point is that the general appearance

of these equations is similar but not identical to

that of Eqns. 27, 29 and 30. Only the values of the

parameters 7p, np and L a are different. The

stoicheiometry (np fc) at which ATP is used in

anabolism is still a Constant in the sense that it is

independent of the forces across the pathways.

The stoicheiometry will vary, however, when the

relative activities of the two pathways are varied.

aft

In Eqn. 75 a leakage term Lp appears. Obviously

this leakage term will be zero when both anabolic

pathways have the same stoicheiometry.

All the calculations in the preceding sections

could now be repeated, taking the effects de-

scribed here of the futile cycles into account. Such

an exercise is unnecessary, however, as it will be

clear that the results can be obtained quicker by

substituting Eqns. 71-73 for np, L~ and 7p', and by

adding to the Lp term the extra term defined by

Eqn. 74. Consequently, the conclusions reached in

the above sections will qualitatively remain valid.

It will be clear how to evaluate the effect of futile

cycling in the anabolic pathway on the relation-

ship between the experimentally determined 1/a

( y max ) and Y max

theor"

It may be instructive to consider the clearest

case of futile cycling in more detail. This is the

case in which the stoicheiometry of one of the two

anabolic pathways is equal to zero (in this exam-

ple, np 2 =0). the overall stoicheiometry of the

anabolic pathway is then reduced by a factor 4b~:

n p2 = 0

"f~ - np'(1-~,21 ) (77)

F/p

in which:

def La2

02, (78)

Lal + La2

As a result of the operation of the futile cycle, the

209

1 (Eqn. 30) is increased: leakage t er m Lp

ltot I a, yal/ al ) 2

Lp =Lp+v21 p \np Lal (79)

Finally, substitution of the modified stoicheiome-

a tl

try n p, Yp, and leakage term in Eqns. 39-44 shows

that YATV is decreased by futile cycling both

through an effect on the growth-independent and

through an effect on the growth-dependent main-

tenance coefficient, the latter effect being identical

to a decrease in the ymax observed. The exception

is again the growth rate-dependent maintenance

coefficient in the case of growth limitation by the

anabolic substrate, which is expected to decrease

with futile cycling.

XI. Efficiency and optimization of microbial growth;

alternative growth models

Even with the extension of the potential futile

cycles in the anabolic or catabolic pathways, the

model for microbial growth we have used until

now is limited in its correspondence to actual

microbial systems. An important limitation is the

assumption that anabolism, leakage and catabo-

lism are mutually independent. Independence of

catabolism and anabolism can only be assumed

for some organisms.

In an important group of microorganisms, there

is a relatively strong relationship between the

catabolic reactions and the leakage reactions. These

are the organisms in which an important fraction

of the ATP that is synthesized in catabolism is

synthesized by the proton-translocating ATPase.

The free enthalpy source for this is the proton-mo-

tive force across the membrane that is in its turn

generated by redox reaction-driven proton pumps

[97,98]. Although, indeed, this case does not ex-

plicitly fall into the category of cases modelled by

Fig. 7, it can still be described by the equations

that were derived above. An example is the case of

microorganisms that use light as their energy

source. Most converters of light energy into bio-

logically useful energy convert the energy of the

photon into an electrochemical potential dif-

ference for protons across a membrane, a so-called

proton-motive force. In this case, the theory pre-

sented in the above sections can be applied when

the phosphate potential and the ATP flows are

210

(A)

Js

JJ b

Jpr Jb

(B)

JsI JS2

,Us[ ~s~jj

JLlpr b

Jpr Jb

(C)

Jc Ja

JE Ja

Fig. 10. Three equivalent formulations of microbial growth on

a single substrate. For explanation see text.

replaced by the proton-motive force and proton

flow across the membrane, respectively. The

leakage reaction J~ (cf. Eqn. 30) should then, for

instance, be identified with the passive proton

back-leakage across the membrane.

Even so, it may seem that a great number of

cases of microbial growth described in the litera-

ture cannot be treated by the theory presented in

this paper. This is because they do not correspond

to Figs. 1 and 7. These are the cases in which the

substrate for anabolism is identical to that for

catabolism. Fig. 10A (cf. Ref. 74) illustrates this

situation. The free enthalpy contents of the flow-

ing substances are indicated in this figure. As an

example one may consider aerobic growth of E.

coli on glucose only. In the following we shall

confine the discussion to the flow of the carbon

atoms. Expressing all flows in C-moles, one can

apply the conservation condition [20,35,39] for this

element:

J, = Jh + Jpr (80)

In cases of highly reduced substrates and aerobic

growth, carbon dioxide reduction may or may not

occur, leading to negative or zero Jpr boundaries.

Fig. 10B shows a modification of Fig. 10A in

which substrate flow has been subdivided into two

parts. One part of the substrate is degraded to the

products, the other part is converted to biomass.

Eqn. 80 allows the definitions:

def

42 = Jb = -- Ja ( 81)

and:

def

']st = Jpr = J,, ( 82)

Here Jb and Jpr have been identified as the anabolic

and the catabolic flux, in view of the identity of

Fig. 10B and C and therefore Fig. 1. Also, the

input and output forces in Fig. 10C can be de-

fined:

and:

~Gc = ;i - #pr (84)

The ease with which this definition is achieved is

not trivial. It is difficult to define the input free

enthalpy in Fig. 10A in absolute terms, because

the chemical potentials have values only with ref-

erence to a standard state. One way to define the

input free enthalpy in Fig. 10A is to define it

relative to the chemical potential of the elements.

Although this is customary in thermodynamics, in

microbial growth kinetics a different standard state

is usually taken [32,35,37]: the state of complete

combustion. In this case, the input free enthalpy

for a substrate like glucose is taken as the free

enthalpy difference between 1/6 mol glucose plus

1 mol oxygen, and 1 tool carbon dioxide plus 1 tool

water.

The schemes of Fig. 10A and C are also differ-

ent in another important aspect. They suggest

different interpretations for input and output flow.

By Fig. 10A, J~ and Jb appear to be suggested as

input and output flows, respectively. Fig. 10C

seems to suggest Jpr and Jb"

An important consequence of these two differ-

ent interpretations of growth on a single substrate

is that also the definitions of efficiencies differ.

Fig. 10A suggests the definition [36]:

ref def Jb ~tb //'~f

,7,h = - - - (85)

j, # /r~r

s- - s

whereas Fig. 10C suggests (see Eqn. 9) the defini-

tion:

ff def Jb ,at,-- #~2 - J,, AG~,

~i ~ = j,~ ~,1_~,~ g. aG~ n .... (86)

where the superscript ss reminds one of the single

(carbon) substrate growth condition. The former

definition has the disadvantage that the efficiency

of growth would depend on the standard state

used by the observer. Roels (Ref. 35, in contrast to

Ref. 36) defined:

rssdef Jb ( Js ) (8'7)

g]t h- JZ ~b equilibrium

Elaboration of the equilibrium condition shows

(provided that catabolism produces only one

carbon-containing substance):

Jb /~b -- Ppr (88)

Thus, this definition is identical to the definition

in Eqn. 85 only if the metabolic state of the

product is used as the reference state. Even then,

however, a difference between "qt"~ f and the other

efficiencies remains:

=-Ja{ Jc

J~ \ - A !eq.ibb~, ....

-1

,Jb]

J,] L

(89)

It should be noted that this equation is valid only

for growth on one single substrate for catabolism

and anabolism (see Eqn. 91, below): the super-

script 'ss' is meant to signify this. Moreover, the

equation has to be modified (see Appendix C)

whenever the assimilation equation contains more

carbon substrates than the general single substrate

only (e.g., bicarbonate).

Readers who are interested in comparing differ-

ent published efficiency values should note that

Roels [35,36] uses two definitions of ~,h that can

be inconsistent under quite a few growth condi-

tions. This becomes clearest as Roels redefines

(Ref. 35, p. 2495): "The thermodynamic efficiency

of anaerobic growth will be defined analogous to

that of aerobic growth, the free energy of combus-

211

tion conserved in the biomass relative to the sum

of the free energy of combustion conserved in the

biomass and the dissipation in the process". With

this definition he means (which can be checked by

calculating the efficiencies in his Table XII):

r' Jb( ~ b -/~'~'mb )

~qth - (89a)

where the symbols of Fig. 10A are used and fib --

~comb

/% designates the free enthalpy of combustion

of the biomass. Apart from the fact that it would

seem more correct to use (J, - J b) in place of ~ in

this equation, one may wonder why the fully com-

busted state should be taken as the reference value

for the free enthalpy of the biomass and not

another reference state; this definition makes ~,'h

dependent on the reference state selected. The

latter problem could be circumvented when ~-

~pr is replaced by/~+ - ~o,,,b and this may explain

(as then /~pr ~ .... b- = /,t ) why in aerobic growth the

two efficiency definitions of Roels (i.e., Eqns. 87,

88 and 89a) tend to give similar results (especially

when J+ >> Jb). However, for anaerobic growth the

two definitions give highly different results (see

also Appendix B):

~ -comb

~b r' -- r t"b

n,h - n,. ~b - r*~ ~

1+

Jb ~b -- Pcb 'mb

J~ P~- ~p,

(89b)

In Appendix B we present a numerical example.

Using Eqn. 87, Roels [35] estimated the efficien-

cies of a number of growth studies published in

the literature. Although he erroneously used the

standard free enthalpies rather than the actual free

enthalpies in his estimations, his values for aerobic

growth will still approach the actual ~q~,~' values.

Table III (cf. Ref. 98c) shows the values he calcu-

lated as well as the values of ~h ff'ss we calculated

through the use of Eqn. 89.

A number of interesting conclusions can be

drawn from Table III. The first is the fact that

rl't'~[ f~ is always lower than O~h ~ (see also Eqn. 89).

The differences between the two types of efficien-

cies are quite significant.

The second point is that, especially at higher

degrees of reduction of substrate, vl,~ r~ becomes

negative. This effect is essentially the result of the

free enthalpy of biomass synthesis becoming nega-

212

TABLE Ill

'EXPERI MENTAL VALUES OF FREE ENTHALPY EFFI CI ENCY OF AEROBI C GROWTH: fifth AND rt~'h rt

The values of the st andar d chemi cal pot ent i al of the subst r at e (fi°), the degree of r educt i on of the subst r at e (y~), the si ngl e-subst rat e

gr owt h yield (in C- mol/C- mol ) and the t her modynami c effi ci ency as defi ned by Roel s (~t[,) for gr owt h of the i ndi cat ed or gani sm on

the i ndi cat ed single subst rat e, are as given by Roel s [35]. Fr om these values we cal cul at ed (see Appendi x A) rl ~ f by maki ng use of Eqn.

89.7, is the degree of r educt i on of the subst rat e, every car bon, hydr ogen and oxygen at om count s for 4, I and - 2, respectively. The

obt ai ned sum is di vi ded by the number of car bon at oms. For bi omass y~ is about 4 [35]. All gr owt h is with ammoni a as ni t rogen

source.

Or gani sm Subst r at e ~ o Y, Jh/J~ 7/~" 71 t~l r~

( kJ/C- mol )

Pseudomonas oxalaticus oxal at e 337 1.00 0.07 0.24 0.20

f or mat e - 335 2.00 0.18 0.34 0.20

Pseudomonas sp. 0.18 0.34 0.20

Pseudomonas denitrificans ci t rat e - 195 3.00 0.38 0.54 0.26

A. aerogenes 0.34 0.49 0.23

Ps. denitrtficans mal at e 212 3.00 0.37 0.52 0.24

Ps. fluorescens 0.33 0.46 0.19

Ps. denitrificans f umar at e - 151 3.00 0.37 0.53 0.25

Ps. denitrificans succi nat e - 173 3.50 0.39 0.49 0.16

Pseudomonas sp 0.41 0.52 0.19

A. aerogenes l act at e - 173 4.00 0.32 0.34 0.03

Ps. fluorescens 0.37 0.40 0.05

Pseudomonas sp acet at e - 186 4.00 0.44 0.49 0.09

Candida utilis 0.42 0.47 0.09

Ps. fluorescens 0.32 0.36 0.06

C. tropicalis 0.36 0.40 0.06

S. cerevisiae gl ucose - 153 4.00 0.59 0.59 0.00

S. cerevisiae 0.57 0.57 0.00

E. cob 0.62 0.62 0.00

Penicillium chrysogenum 0.54 0.54 0.00

A. aerogenes gl ycerol 163 4.67 0.66 0.57 - 0.26

C. tropicalis et hanol - 91 6.00 0.61 0.44 - 0.44

(~ boidinii 0.61 0.44 - 0.44

C. utilis 0.61 0.44 - 0.44

Ps. fluorescens 0.43 0.23 - 0.35

C. utilis 0.55 0.39 - 0.36

C. brassicae 0.64 0.46 - 0.50

(" boidinii met hanol 176 6.00 0.52 0.36 - 0.33

Klebsiella sp 0.47 0.32 - 0.28

M. methanolica 0.60 0.41 - 0.48

Candida N- 17 0.46 0.31 - 0.28

H. polvmorpha 0.45 0.31 - 0.25

Pseudomonas C 0.67 0.46 - 0.64

Pseudomonas EN 0.67 0.46 - 0.64

Torulopsis 0.70 0.48 - 0.73

Metfzvlornonas sp. 0.50 0.34 - 0.31

M. methanolica 0.64 0.44 - 0.56

C. tropicalis hexadecane + 5 6.13 0.56 0.41 - 0.34

C. lipolytica dodecane + 4 6.17 0.41 0.30 - 0.19

Job 5 pr opane 8 6.67 0.71 0.48 - 0.79

et hane 16 7.00 0.71 0.46 - 0.82

M. capsulatus met hane - 51 8.00 0.63 0.37 - 0.70

M. methanooxidans 0.68 0.40 0.88

tive above a certain degree of reduction of the

substrate [46,98c,104]. For such substrates, there is

no real need for a catabolic generation of free

enthalpy. The problem of these growth systems is

to get rid of their excess free enthalpy, rather than

to be economical with it [21,25,26,32,35,40,

84,99,100]. Due to the identity of the substrate for

carbon in the biomass and the substrate for free

enthalpy the amount of carbon rather than the

amount of enthalpy becomes growth limiting.

The third point in Table III to which we wish to

draw the readers' attention is the value of the

efficiency which the systems tend to have that

possess substrates lacking the problem of carbon

limitation (i.e., those with a degree of reduction

significantly under 4): some 20%.

Most studies of non-equilibrium thermody-

namics and especially those concerning the effects

of optimizations in systems have used a system

such as that of Fig. 10C rather than Fig. 10A as

the model system [38,55]. As a consequence, the

efficiency definitions used in those studies are

identical to ~t'~ f~ and not to r~

Y~th"

Therefore, we should compare the values of

~,"~f~ (and not those of ~/rt~ ) to the efficiency

values obtained in non-equilibrium thermody-

namics.

Here, it becomes relevant to recall that if a fully

coupled system were to optimize for maximal

power output, the efficiency would be 50%. At any

fixed degree of coupling, optimization for output

flow leads to an efficiency of zero, or, if such

values exist, as negative as possible. Optimization

for output flow, or output power, whilst keeping

the efficiency as high as possible by adjusting the

degree of coupling, would result in efficiencies of

24 or 41%, respectively (cf. Table II).

Against the background of these theoretical re-

sults, the ~t~ f~ values in Table III lead to the

conclusion that when the substrate is significantly

more oxidized than the biomass, the efficiency of

microbial growth tends to be about 24%: counting

substrates with degrees of reduction up to 3.3 the

efficiency is 22.4% (_+0.9 S.E., n = 10), including

degrees of reduction of 3.5, the efficiency is 21.0%

( _+ 1.0 S.E., n = 13). This would be consistent with

a maximization of the rate of biomass synthesis,

all the time keeping the efficiency as high as

possible, as the principle for optimization.

213

This principle seems to disappear for substrates

with degrees of reduction comparable to or higher

than the degree of reduction of biomass. Then the

efficiency drops rapidly with the degree of reduc-

tion and becomes negative, consistent with optimi-

zation towards maxi mum rate of biomass synthesis

only. Free enthalpy efficiency then seems to lose

its relevance. This is in line with the fact that in

these cases lack of carbon rather than lack of free

enthalpy becomes the problem [99].

Especially with respect to the results obtained

for methane in Table III, it may be proposed that

the low efficiency obtained with this substrate is

due to the absence of NAD +- or FAD-met hane

oxidoreductases. Only mixed-function oxidases ex-

ist for this substrate. Although there might be a

mechanistic impossibility for redox reactions be-

tween methane and NAD + in the absence of a

second reductant, the absence of an enzyme

catalysing the reaction could also be visualised as

the evolutionistic result rather than mechanistic

cause of the low growth efficiency on highly re-

duced substrates. Similarly, the fact [93] that

methanol dehydrogenase donates its electrons to

the P. denitrificans respiratory chain at a site

downstream from site 2 may find its explanation

here.

It may be important to stress that this analysis

yields conclusions that are in contradiction with

earlier contentions by different authors [77]. They

concluded that microbial systems were optimized

with respect to power output. In Section IX, it has

been indicated that the basis for these earlier

conclusions was incorrect. On the other hand, our

conclusion seems to be similar to the conclusion

concerning growth of P. denitrificans with suc-

cinate or sulphate as the growth-limiting substrate

[76]. Indeed, succinate is among the substrates for

which we calculated an efficiency of about 20%

(cf. Table III). The conclusion of Van Versseveld

[76] was, however, based on the implicit assump-

tion that Zr~ a = 1 (cf. Section IX), for which no a

priori justification existed. The correspondence be-

tween Van Versseveld's [76] and our conclusion

can therefore only be taken as a justification a

posteriori that Zre d -- l in the case studied by Van

Versseveld.

Table III only cites cases of growth with a

single growth-limiting carbon substrate both for

214

catabolism and anabolism. Therefore, strictly

speaking the above conclusions concerning ~1'~ f are

also limited to this case. We shall now show that

the above conclusions may well apply to cases of

separate substrates for catabolism and anabolism.

We used data on growth yields of P. denitrificans

on different substrates obtained by Van Versse-

veld [93] to evaluate the thermodynamic efficiency

of these growth processes. With succinate or malate

as the substrates, P. denitrificans has only one

substrate both for catabolism and for anabolism.

Therefore, for these two substrates the calculations

can be carried out exactly as described above.

Methanol and formate, however, are not used as

substrate for anabolism. Rather, CO 2 is fixed

through the ribulose bisphosphate cycle. For such

a dual-substrate case the definition of ~'~h ff be-

comes:

,qthff ds Jb fib -- ~sa -- Ja aGa

= J* /~++-fipr =~-~ " AG~=rt .... (90)

Here ds, sa and sc denote dual substrate, anabolic

substrate (in this case CO2) and catabolic sub-

strate, respectively. Consequently, Eqn. 89 is re-

placed by:

~f xl s ~ r.d+def Jb { J~c)

=Z Z °++,++o,o

(9l )

In this case of a difference between anabolic and

catabolic substrate, the efficiency as calculated by

Roels [35] would correspond to the efficiency used

in the calculations of Stucki [55].

Table IV shows the results of the calculations

for P. denitrificans. It turns out that also in this

case of separate substrates for catabolism and

anabolism, a substrate with a lower degree of

reduction than biomass (formate) has a growth

efficiency slightly exceeding 20%. Of the other

substrate that had a much lower degree of reduc-

tion than the biomass (3.0 vs. 4.1), malate, the

growth efficiency is also close to the 24% that

would be indicative of optimization of the rate of

biomass synthesis at optimal efficiency. Also, the

succinate result is in line with the results presented

in Table II. It is striking to note the unusually high

thermodynamic efficiency of 33% reached by

Paracoccus when growing on methanol. Above, we

TABLE IV

THE FREE ENTHALPY EFFICIENCY OF GROWTH (rhi~ r)

OF PARACOCCUS DENI TRI FI CANS ON FOUR SUB-

STRATES OF DI FFERENT DEGREES OF REDUCTION

USING A GIBBS FREE ENERGY OF BIOMASS OF 67.1

kJ/C-mol [35] AND A BIOMASS COMPOSITION FOR-

MULA OF C~,HI0.~NI 5P29 [93]

Growth yields given by van Versseveld [93], Y" (tool carbon in

biomass produced divided by amount of carbon in the sub-

strate+ not CO~, consumed) and ~h were calculated as indi-

cated in Appendix A. Then for succinate and malate Eqn. 89,

but for formate and methanol Eqn. 91 was used to calculate

~'~h fr, the free enthalpy efficiency that can be compared with the

efficiency values calculated by Stucki [55].

Substrate ~, Y" ~ Jb/Jq ~J~h ~/tl~ "f

(%) (%)

Formate ~ 2 0.12 21 21

Malate h 3 0.42 57 26

Succinate h 3.5 0.40 48 14

Methanol" 6 0.54 33 33

" The substrate is oxidized to carbon dioxide; carbon dioxide is

assimilated to biomass.

h No carbon dioxide assimilation takes place.

observed the phenomenon that with highly re-

duced substrates optimal efficiency is dropped as a

criterion for optimization, probably because the

energy content of the substrate is more than suffi-

cient to convert the existing carbon atoms to bio-

mass. When, however, such a bacterium is able to

assimilate carbon dioxide, then one might expect

again maximization of the rate of biomass synthe-

ms at optimal efficiency. The observation of 33%

efficiency would mean that the bacterium goes

even further: its optimization aim seems to lie in

between maximization of the rate of biomass

synthesis and maximization of the rate of synthesis

of Gibbs free energy in the form of biomass, both

at optimal efficiency. The reason for this unique

(?) optimization criterion is not yet fully under-

stood, but further research will hopefully show

whether this high growth efficiency is a common

property of all bacteria growing on the combina-

tion of highly reduced methanol and highly

oxidized carbon dioxide. A similar effect has been

demonstrated to occur when P. denitrificans is

grown on a mixture of mannitol and methanol

[92,93]: the growth yield (corrected for growth

rate-independent maintenance) on the basis of

oxygen consumption is higher at a 1:1 molar

mixture of the two substrates than when either of

the substrates is present alone. Evidently, for the

use for production of single cell protein, the growth

of P. denitrificans on methanol and CO 2 is the

most efficient method.

XII. Discussion

In this paper we have made an attempt to

describe microbial growth quantitatively in terms

of non-equilibrium thermodynamics. Relation-

ships between the rates of processes that play a

central role in microbial growth have been ex-

amined, also with respect to the way in which they

are determined by the characteristics of metabo-

lism. It has been shown that the derived relation-

ships are in agreement with many of the experi-

mental results that have appeared in the literature.

The derived relations specify the influence of the

different metabolic processes that are connected

with growth (e.g., uncoupling, futile cycling, ef-

ficiency loss in order to have high velocities of

reaction, the values of stoicheiometric numbers)

on the overall behaviour of the system. We showed

that mosaic non-equilibrium thermodynamics

makes it possible to interpret quantitatively yield

values whilst embracing the criteria summarized

by Tempest and Neijssel [74]: no strict coupling

between anabolism and catabolism, carbon sub-

strate growth is not necessarily energy limited

(catabolic substrate limited), the effects of a varia-

tion in the number of energy-conservation sites

can be taken into account, maintenance varying

with growth rate and essential differences between

anabolic and catabolic substrate limitation. Also,

conclusions could be drawn about the criteria

according to which microbial growth may have

been optimized. For this it was necessary to point

out a difference between two definitions of ther-

modynamic efficiency (i.e., one used by Roels [35]

and the other used by us [98c]). It turned out that

only the definition of efficiency used by us allows

comparison of efficiency with the efficiency values

that have been predicted by non-equilibrium ther-

modynamics.

Microbial growth seems to have been optimized

with respect to maximum growth rate. Only in the

215

cases of degrees of reduction of the (single) sub-

strate that are significantly lower than the degree

of reduction of biomass, does the optimization

procedure seem to have been carried out following

a course in which the efficiency was all the time

kept as high as possible. These conclusions are in

contrast to [77] or in keeping with [76] earlier ill

based conclusions about optimization of microbial

growth.

In order not to make the calculations too com-

plicated, the derivations have been carried out for

a model cell in which several simplifications were

made. However, we do not feel this as a serious

drawback, since the derivations will not be essen-

tially different for many, more sophisticated, mod-

els of microbial cells (cf. Section X). We shall now

discuss this point at greater length.

The first limitation of the system as presented

in Fig. 7 is that only free enthalpy transduction via

the intracellular phosphate potential is considered.

This would, for instance, mean that aerobic growth

with respiratory chain-linked oxidative phosphory-

lation would be excluded from the description.

This is not the case. If in all the derivations the

phosphate potential is replaced by the proton-mo-

tive force across the plasma membrane of the

organism, and the ATPase, or ATP-synthase fluxes

are replaced by the corresponding proton fluxes

across the membrane, then the description is fully

analogous. The synthesis of ATP from the

proton-motive force would then be considered to

be part of the anabolic pathway. Aspecific ATP

hydrolysis would be described as futile cycles in

the anabolic pathway with concomitant increase in

the leakage coefficient and change in the

stoicheiometry H+/bi omass of the biosynthetic

reaction. This replacement of the phosphate poten-

tial by the proton-motive force is straightforward,

when substrate level phosphorylation is absent (as

is the case for some phototrophic and chem-

olithotrophic and a few chemoorganotrophic

organisms [74]). When free enthalpy needed for

the biosynthetic reactions is derived both from

respiratory chain-linked oxidative phosphorylation

and from substrate level phosphorylation, and/or

when anabolism is driven by both the proton-mo-

tive force (transport) and the phosphate potential

(as is the case for many chemoorganotrophic

organisms with an active respiratory chain), the

216

description becomes more complicated. Mosaic

non-equilibrium thermodynamics is also applica-

ble to these systems, be it that the necessary equa-

tions will be more complex.

Another case in which there is more than one

link between catabolism and anabolism is that in

which the catabolic substrate also serves as the

carbon source. Growth of chemoorganotrophic

organisms on mineral salts and glucose is an exam-

ple. Such systems have been discussed in Section

XI. It could be shown that they can easily be

treated by the theory developed in this paper.

After this discussion of the limitations of the

theoretical approach presented in this paper, we

shall briefly indicate some of the future applica-

tions that we consider promising:

(1) Further study of the criteria according to

which specific species of microorganisms may have

optimized their growth with special reference to

species competing for the same niche.

(2) To treat problems like the evolution of

volatile end products of glucose metabolism that

in some cases evolve at slow growth (Ref. 71, cf.

Ref. 39).

(3) To analyze 'overflow metabolism' [25] in

terms of the effect on growth behaviour. To analyze

the different effects of different forms of uncou-

pling, futile cycling, slip, leak.

(4) To analyze the effects of increments in the

activities of certain enzymes on the growth be-

haviour.

(5) To analyze the effects of changes in reac-

tion stoicheiometries [21,91,91 a, 101,102].

(6) To analyze efficiencies of microbial growth

and devise tricks to make the 'bugs' optimize for

what we want.

All of this will hopefully lead to an increased

insight into the relation between microbial

metabolism and microbial growth. In the end it

may help in directing man in interfering with

microbial metabolism in order to adjust microbial

production to his needs. We conclude that mosaic

non-equilibrium thermodynamics and microbial

growth studies are ready for a symbiosis [91a, 98c,

104]. Let us continue to work on it.

Appendix A. Calculation of efficiencies from growth

yields

A 1. Calculation of the carbon / carbon yield at equi-

librium

For biomass of the formula CH/tNNOS , and a

(single) substrate of the formula CHhNnO[ -, the

following formul a was used to calculate

(Jb/Js)equilibrium (the carbon in the biomass pro-

duced divided by the carbon consumed as (non-

CO2) substrate at equilibrium); assuming NH~ as

the nitrogen substrate:

= +( v +,- 1) a,.

Z equilibrium

+(2+3n2 h--P)~r'~°--~Hc°;} /

{--N~NHg +(N-1)/~+ +(3N+2-2 H)fiH,o

+/~bi ........ - ~Jlco;}

/~NH~ ( - 79.6 kJ/mol ),/~H' ( - 40.5 k J/tool),/XH,_o

( - 238 kJ/mol ), /Xnco, ( - 588 kJ/mol ), /2hi .........

( - 67.1 k J/C-t ool, taken from Roels' results, al-

though it is slightly erroneous: should be -69.1 as

Morowitz' ammonia was aqueous) and /~ were

given the values compiled by Roels [35]. As a

control all values of (Jb/Js)equilibri .... were recalcu-

lated and found to equal the values given by Roels

[35]. We used H = 1.8, N = 0.2 and O = 0.5. The

basis of the above equation is that for growth with

yield Jb/Js the overall reaction equation can be

written as:

+ 20+3N-H-4+ (4+h+p-3n-2o) ~O 2

,)HCO

Replacing the formula of each compound with its

chemical potential (~) and ~ with =, one ob-

tains the above f or mul a: ( AG) e q = 0.

A2. Calculation of the thermodynamic efficiencies

To obtain the t hermodynami c efficiency as de-

fined by Roels [35] (~h), we divided the growth

yields (in C-mol biomass per C-mol substrate in

cases of single substrate growth and in C-tool

biomass per C-mol non-carbon dioxide substrate

in cases of carbon dioxide fixation) by (Jb/Js)eq.

To obtain the t hermodynami c efficiency as de-

fined by us we used Eqn. 89 in cases of single

substrate growth and Eqn. 91 in cases of dual

(carbon-) substrate growth.

Appendix B. Numerical examples of the calculation

of efficiencies

B1. Anaerobic growth on methanol

The anaerobic dissimilation reaction equation

is:

CH3OH ---, ~4HCO; + 34CH 4 + ~4H + + ] H20

AG~ = ( - 176.5) - ¼( - 588) - 3( - 50.9) - ¼( - 40.5)

- 1( - 238) = 78.3 kJ/C- mol, the values between

brackets being the respective standard free en-

thalpies [35].

The assimilation reaction is:

0.2NH~ +0.7CH3OH+0.3HCO 3 +0.I H + ~ CH1.800.sN0.2

+ 1.1H20

AG~ = - 0.2( - 79.6) - 0.7( - 176.5) - 0.3( - 588) -

0.1( - 40.5) +( - 67.1) + 1.1( - 238) = - 9.0 kJ/C-

mol (downhill growth).

The free energy of combust i on of biomass is

taken as 560 kJ/C- mol [35] and an experimental

yield for this growth process of 0.219 =Jb/J, is

cited I35].

Appl yi ng Eqn. 89a, one finds:

,[; 560 =0.61

560+( 78.3/0.219)

This is the value calculated by Roels [35].

To calculate ~/~h (Eqn. 87) we first calculate

(JJJb)~q" Not e that J~ is defined in terms of C-mol

217

methanol, hence J~2 = 0"7Jb, therefore,

( J~ - 0.7,/h )78.3 + Jb9.0 = eq0

Hence ~r,h = 0.219 " 0.58 = 0.13.

To calculate ~th ~f we establish that

07)

B~f Jb~G a 1 - 9.0 = _ 0.03

J~AG~, ( 1/0.219) - 0.7 78.3

Application of the modified (see Appendi x C)

version of Eqn. 89, with ~s~ = 0.70, gives the same

result:

0.13 - 0.7.0.219

~f = - - 0.03

1- 0.7.0.219

B2. Aerobic growth on methanol

The dissimilation reaction is:

CH3OH+ a202 ~ HCO 3 +H + +H20

AG~ = - 176.5 - ( - 588) -- ( - 40.5) - ( - 238) =

690 k J/C- t ool

The assimilation reaction is:

CH3OH + 0.2NH~ + 0.4502 --* CHl.800.sN0. 2

+0.2H + + 1.4H20

AG~ = - 67.1 - ( - 176.5) - 0.2( - 76.6) +

0.2( - 40.5) + 1.4( - 238) = - 217 kJ/C- mol

A reported actual growth yield was, for ins-

tance, 0.52 C-mol biomass per C-mol methanol

[35]:

560

<; - = 0.30

560+ (690/0.52)

The equilibrium condition is obtained as follows:

( J~- Jb) 690+ Jb217 = 0

218

Hence, ~/~h = 0.69 0.52 = 0.36.

This is equal to the value calculated by Roels

[35]. Note, therefore, that for aerobic growth Roels

calculates ~q~o,, but for anaerobic growth "9~tl,.

For ~'~f we obtain:

wff 1 - 217

"0,h -- (1/0.52)--1 690

= - 0.34

and by use of Eqn. 89 (Oh+ = 1):

~rff 0.36 0.52

0.33

+th 1 --0,52

Appendix C. Derivation of Eqn. 89

Carbon conservation: J~ = J~l + J~2 = J~ + ~shJb -

Def i ni t i on: Jb = - J a-

Here, q'~b is a conversion factor from the units

C-tool biomass to the units C-mol substrate. Often

'~b = 1, but, e.g., for the case of anaerobic growth

on methanol the assimilation equation is:

0.2NH ~- + 0.7CH 3OH + 0.3HCOf

+0.1H ÷ -, CHI ~O0.sNo 2 + I.I H20

so that, in this case ep~h = 0.7. It follows that:

J~ i

- - - - ~sb

J~,

With the above definition of ~'i)1 Eqn. 89 is found,

but only if gP~b = 1.

Acknowledgements

We are indebted to Drs. Harder, Konings,

Neijssel, St out hamer and Van Dam for critical

remarks, helpful suggestions and continuous inter-

est. We also thank Mrs. M. van der Kaaden and

Mr. H. Laloli for expert librarian help and Mrs.

M. Pras and M. Broens-Erenstein for expert typ-

ing.

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