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Bioehimica et BiophysicaActa, 683 (1982) 181-220 181
Elsevier Biomedical Press
" 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)
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
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
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
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
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
A 2
V' 6
,#,, 5
grad ( O~ Ox, O~ Oy, O~ & )
the negligible part of  J-h I = W/3600
volume-integrated free enthalpy dissipation J.h ~ = W/3600
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
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
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
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-
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
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
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-
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
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
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
a free enthalpy of biomass synthesis can in princi-
ple be defined:
AG, = AG~ + RT ln([biomass]/[anabolic substrate]) =/i b -/~.~
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
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
cb = J,.AG,. + J AG~ + f/odV
+o = ZJ,~7( - ~) + ZJ,.h ...... Ar +CJ;/T) wC- T)
i r
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
q~o- ( J~/T) V( - T) = ~ - ( dci/dt ) ~ ~, (7)
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:
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
J¢ = Lcc( AG,: - AGc~)+ Lca(AGa -- AGa~ )
Ja = Lac( AG¢ -- glGc~) + Laa( AGa - AGa~)
in which the parameters carrying the symbol
are constants (i.e., independent of AG~ and AGe).
(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.
[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
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]
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)
q = LJCL,,L~c (la)
z : f~,,,/c~c (is)
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
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
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
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
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
Lc ~ L~a )
Using the shorthand notation of Eqn. 22:
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
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).
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 )
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'
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'
~/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
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).
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
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
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
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
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
- I
- 1
- I
- 1
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.
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-
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
COz ~
Fig. 7. Model of microbia! growth (see text).
/....~ biomass
consider this simple model system and then later
(see Sections X and XII) add some modifications
and analyze how these modifications affect the
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-
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
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.
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 )+ -
= L,{(Ac. - ac:) - acg) )
= 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-
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
+ 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.
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:
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:
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
( ,~G, -
Jc=( nap/np) 1+ a, ~2 ( - Ja) + P , AG~?
yp( Hp) L a J t/pHp~p
Using Eqn. 28 instead of Eqn. 29, we obtain:
( s,)
+ (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.
Jc = °~csl(- Ja ) -¢-/~cs[ (37)
and, analogously, for Eqn. 36:
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:
__ 1 a a c
fl¢~ - (Lp/(ypnpnp))(AV.- AG a )
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
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).
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-
flagdd I def -- Lp ( 44)
I c 2
Lp + ~;(.;) L~
Defining the theoretical yield 1/a o
1/% = G/n;
a= ao + flgd (46)
so that:
L = "o( - Jo) + t~ + #~( - Jo)
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
=(l/aa~')/( 1+ y~(np)L~)c 2L~
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
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
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
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:
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
.half ( AGa - AGa~ )
- -a.c~ = (5 r )
It would be interesting to see whether indeed
uncoupler decreases y max and increases the
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
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,
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
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
YATP//YAnq'al~.lheor =
I 1
,;(.;)~c. 'p"~ 7;
/ L. L'~ + ,;(.;)%
]// l a at 2
Lc Lp + ,p( Hp) L a
Zm, = Znp/np (54d)
V I ' a 2 aL
,; , + ,.
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-
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:
L~ L ,
i 2 I c a c
L t i, a c c 2 , 1
-ypnpnpLaL c ( np) ,pLaL c+ LpL a
de,{ O J+
(t )()
=-- ,p/ 1 + Lp 1 + Lp
c 2 ,
a ~ a
(.p) rp< (.;) ,;<
q~ ~Tq~a (54g)
Zl c
Here symmetry would imply that ~,p and "yp are
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)
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:
,,o, = ~.,. (_ j~) (58)
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:
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:
rla'qc = "q,o,( q = 1) = - - (60)
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,
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
Using Eqns. 13 and 54d under the level-flow con-
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
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 ....
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-
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-
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)
anabolic subsfrafe
Fig. 9. Model of microbial growth with a futile cycle in the
anabolic pathway.
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)
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
a 2
afcdef n~plL,,t + rtp L~2 (71)
rtp Lal + La2
L~,~~fL, I + La2 (72)
~,r. def .;'y;"c~, +, ;-" ~;-%2
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
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.
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
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)
in which:
def La2
02, (78)
Lal + La2
As a result of the operation of the futile cycle, the
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
JJ b
Jpr Jb
,Us[ ~s~jj
JLlpr b
Jpr Jb
Jc 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
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:
42 = Jb = -- Ja ( 81)
']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-
~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
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-
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~, ....
J,] L
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-
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 --
/% 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*~ ~
Jb ~b -- Pcb 'mb
J~ P~- ~p,
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-
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
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~
Therefore, we should compare the values of
~,"~f~ (and not those of ~/rt~ ) to the efficiency
values obtained in non-equilibrium thermody-
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.
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
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
Table III only cites cases of growth with a
single growth-limiting carbon substrate both for
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-
,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
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-
Microbial growth seems to have been optimized
with respect to maximum growth rate. Only in the
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
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
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-
(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
A 1. Calculation of the carbon / carbon yield at equi-
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
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
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
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
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
0.13 -
~f = - - 0.03
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
<; - = 0.30
560+ (690/0.52)
The equilibrium condition is obtained as follows:
( J~- Jb) 690+ Jb217 = 0
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
+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
With the above definition of ~'i)1 Eqn. 89 is found,
but only if gP~b = 1.
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-
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