FEMS Microbiology Letters 15 (1982) 7-17
Published by Elsevier Biomedical Press
7
Energetics of microbial growth: an analysis of the relationship
between growth and its mechanistic basis by mosaic
non-equilibrium thermodynamics
K.J. Hellingwerf, J.S. L o l k e m a , R. Otto, O.M. Neijssel a, A . H . S t o u t h a m e r b,
W. H a r d e r , K. van D a m c a n d H.V. W e s t e r h o f f c
Department of Microbiology, University of Groningen, Kerklaan 30, 9751 NN Harem ~ Department of Microbiology, University of
Amsterdam, Nieuwe Achtergracht 127, 1018 WS Amsterdam," b Department of Microbiology, Free University, De Boelelaan 1087, 1007
MC Amsterdam, and c Department of Biochemistry, University of Amsterdam, Plantage Muidergracht 12, 1018 TV Amsterdam, The
Netherlands
Received 8 December 1981
Accepted 20 April 1982
1. I N T R O D U C T I O N
Microbial growth-yield parameters are usually
analyzed in terms of an overall, phenomenological
description, first suggested by Pirt [1]. He proposed that for its survival a microbial cell must
carry out a number of processes unrelated to
growth. To generate the energy (ATP) required for
these processes, 'extra' catabolism at a rate independent of the growth rate is needed; this was
referred to as (growth-rate-independent) 'maintenance' catabolism. The extra substrate consumption connected with such maintenance catabolism
was shown [1] to affect growth yields so that they
become growth-rate-dependent. This notion, which
has been confirmed experimentally [1-3], modified
the concept of Monod [4], who had introduced
growth yield as a biological constant. Although
Pirt's description [1] allowed a fruitful rationalisation of microbial yield studies, research during the
past 10 years has led to several observations it
cannot account for. Two examples are: (i) the
differences between the experimentally obtained
Reprint requests to Dr. K.J. Hellingwerf at the first address.
maximal (i.e. corrected for growth rate-independent maintenance) growth yields of microbes and
the theoretical yields that can be calculated from
known biochemical pathways [5] and, (ii) the occurrence of overflow metabolism [2,6]. Phenomenologically, the deficit in maximal growth yield
can be accommodated in Pirt's analysis by postulating that the maintenance requirements are
growth rate-dependent [2,6].
However, with this modification this method of
describing growth becomes even more phenomenological. It just provides arithmetic expressions between parameters (growth rate-dependent maintenance coefficient and growth rate-independent
maintenance coefficient) that can be adjusted so as
to make the expressions fit the results of any
particular yield study. Although it thus serves the
important purpose of permitting the summary of
the results of growth experiments in terms of two
parameters only, thereby facilitating a comparison
of different growth studies, it cannot explain the
characteristics of a certain growth experiment in
terms of the molecular details of the underlying
biochemical and biophysical mechanisms. The need
for a description of microbial growth that enables
the prediction of a change in growth yield caused
0378-1097/82/0000-0000/$02.75 © 1982 Federation of European Microbiological Societies
by a certain change in stoichiometry, activity or
kinetic characteristics of one of the enzymes involved in growth, has become more pressing in
recent years, particularly in applied areas of microbiology (e.g. [7]). Such a refined description
should establish a link between yield studies and
biochemical/biophysical research at the level of
isolated enzymes or subcellular particles.
In bioenergetics a similar need for descriptions
of systems of enzymes that would take into account
the mechanistic structure of the system has arisen.
To satisfy that need 'mosaic non-equilibrium thermodynamics' (MNET) [10,11] has been developed
[8-10], and has turned out to be applicable to a
number of energy-transducing systems with various degrees of complexity, correctly predicting the
effect of changes in activity of the elemental
processes that characterized the mechanistic structure of the system (such as proton pumping, ,proton leakage), on macroscopically observable
parameters such as pH gradient [14], proton flux
[15], oxidation rate and rate of ATP synthesis [12].
The description has also proved useful in extracting information concerning the molecular details
of the energy transducing pathway from measurements of macroscopic parameters [11,12,16]. As
both bioenergetics and microbiology concern the
metabolic symbiosis of a number of independent
(enzyme-catalyzed) processes, it seemed likely that
an M N E T description of microbial growth would
prove equally informative. Moreover, microbial
growth can be considered to be itself an energytransducing system and as such it would be well
positioned in the sequence of biological energytransducing systems to which M N E T has been
applied. Until now, this approach, beginning with
the single energy-transducing enzyme bacteriorhodopsin reconstituted into liposomes [8,11,14,15]
and leading along to oxidative phosphorylation in
submitochondrial particles [16] and intact
mitochondria [8,12,13] has led to the description of
gluconeogenesis in rat liver cells [13]. The description of microbial growth extends these descriptions in the sense that it links anabolism to growth.
For these two reasons we have extended M N E T to
provide a description of microbial growth [17]. In
this paper we put forward the hypothesis that
M N E T correctly describes the relationship be-
tween microbial growth and microbial metabolism
(i.e. the underlying biophysical and biochemical
mechanisms).
2. T H E M O D E L
To illustrate the development of an M N E T
description of microbial growth a simple model
will be used (cf. Fig. 1). More complex models that
may more adequately describe microbial growth
can also be translated into quantitative M N E T
relations. However, such descriptions would be
mathematically more complex than is necessary to
illustrate our hypothesis. Therefore, we have modelled the process of microbial growth as consisting
of three elemental [10] processes which are assumed to be mutually dependent through the magnitude of the intracellular phosphate potential
(AGp) only. The first of these processes is catabolism, which we consider to be the conversion of the
growth-sustaining energy source (e.g. lactate) in a
particular microorganism into catabolic endproducts with the concurrent generation of energetic
intermediates (in this case ATP). The second elementary process is anabolism which consists of the
conversion of anabolic substrates (the substrates
providing carbon, nitrogen, etc.) via various monomers into biomass, using the high-energy intermediates generated in catabolism. The third process, leakage, contains all those processes that
consume ATP (or ATP equivalents) but do not
give rise to the formation of biomass. Passive
proton flux through the bacterial membrane that
~[IJCose ~ ,
?
coz/
biomass
~
Ic I
AOP
+~ja~ja
aGp~
~Ga
anabotic
"subsfrafe
Fig. 1. A simple model for microbial growth, suitable for an
M N E T description. For explanation see text.
results from a high protonmotive force, but is not
coupled to any biosynthetic (or transport) reaction, would be an example.
Growth of Eseherichia coli in a complex medium
(i.e., rich in monomeric precursors for anabolism)
in a glucose-limited chemostat or of Streptococcus
cremoris on a complex medium, plus lactose as the
energy source, are examples of microbial growth
modelled by Fig. 1 which also applies to lightlimited growth of phototrophic bacteria. The interrelating parameter can be either the phosphate
potential or the protonmotive force. For the description of growth under more complex growth
conditions, models are required that are also
mathematically more complex (see above).
For each of the three elemental processes MNET
describes relationships between the rate with which
that process proceeds (or the flux through that
elemental process) and the force(s) that drive(s)
the process (or reaction), namely the Gibbs freeenergy difference across that reaction. Anabolism,
for instance, is often (though not always [19])
uphill in the sense that the free-energy content
(chemical potential) per C-mol of biomass produced is higher than the free energy content per
C-mol of the anabolic substrates consumed (i.e.
AGa, the difference in free energy between biomass
and the anabolic substrates exceeds zero). It follows that the thermodynamic force of anabolism
A G a by itself tends to lead to degradation of the
produced biomass back to anabolic substrates
(positive J~ due to our sign convention) rather than
biomass synthesis (negative J~). That biomass is
synthesized is due to the pushing effect of the
phosphate potential (AGp) resulting from the enzymatic coupling of ATP hydrolysis to anabolism.
This pushing effect of the phosphate potential is
expected to be stronger as more ATP hydrolysis
steps are coupled to anabolism. The number of
ATP hydrolysis steps enzymatically coupled to the
synthesis of one C-mol of biomass (cf. 1/YA~~. theor
in [3,20]) is indicated by - n p . In summary, the
biomass synthesis ( - J a ) would be expected to be
driven by - n ~ times AGp, but counteracted by
AG~. Near-equilibrium thermodynamics [8,22] gives
this expectation a quantitative form by postulating
for the rate of biomass synthesis -J~:
-Ja-- La. ( - a c a + (-rip) • aGp} *
For clarity: - n ~ , AGa, AGp a r e usually positive, a
negative Ja signifies biomass synthesis. The coefficient La is always positive and constant in the
sense that L~ depends neither on AG,, nor on AGp.
Interestingly, however, L a is directly proportional
to the activity of enzyme(s) that catalyse anabolism. Consequently, the parameters L~ and ripa a r e
characteristic for the anabolic enzymic machinery.
Elemental flux/force relationships obtained on
the basis of the relationships between fluxes and
forces in enzyme kinetics have shown [9,12,16,21]
that for processes far from equilibrium relations
like Eqn. 1 are generally too simple to be correct.
In that case a modification is needed and Eqn. 1
should then read:
:a : La " ( ( AGa -- AG2 ) + ri; " yp " ( AGp -- AG~ ) )
(2)
Compared to Eqn. 1, this equation has two new
aspects. First, the AG terms have been replaced by
A G - AG*~. The alteration stems from both theoretical calculations [9,21] and experimental observations [21] indicating that for reactions far
from equilibrium the relationship between flux
and free-energy difference is not proportional (i.e.,
it is not a straight line through the origin as it
would be in near-equilibrium thermodynamics [22]
and consequently in Eqn. 1) but sigmoidal: at low
values of the free-energy difference (AG) the reaction rate (flux) hardly varies with the free energy
difference (the rate is close to a negative maximum
rate), at intermediate values of AG the rate of the
reaction increases sharply and almost linearly [21]
with AG until the reaction rate approaches its
maximum and relatively suddenly again becomes
* n is generally used as stoichiometric constant, np equals the
number of ATP molecules (or high-energy phosphate bonds)
hydrolysed to synthesize one C-mol of biomass, The sign
convention is such that all AG parameters are taken to be
positive. For actual synthesis of biomass Ja is taken to be
negative. The proportionality constant L a (relating the
anabolic flux to the force driving this flux) is positive but npa
is negative. Consequently, - - J . (i.e. biomass synthesis) is
driven by the phosphate potential (AGo) but counteracted by
the free energy difference between biomass and anabolic
substrates (AG,).
10
independent of AG. We choose to describe this
relationship between flux and AG by writing: J =
L . (AG--AG ~) [9,23]. Most evidently this equation describes the AG domain in which J depends
strongly on AG, because the constant AG e transforms the defective proportional relationship J =
L - A G of near equilibrium thermodynamics [8]
into a linear relationship. By itself the introduction
of the constant AG e does not result in the description of the two AG domains in which the flux
becomes independent of AG. This problem can be
solved by writing AG e as a function of AG [17].
For this paper, however, it will suffice to remember that in the AG domains where changes in AG
do not affect the flux (saturation), the effective
thermodynamic force AG--AG ~ is constant, and
that in the AG domains where changes in AG do
affect the flux, AG e is constant.
~,~ is a weighing factor for the relative effects of
the two thermodynamic forces ((2xG~- AGff) and
( A G p - AG~), which replace AGa a n d AGp, respectively) on the flow of anabolism. This factor is
introduced to indicate that, unlike the situation
near equilibrium, the relative sensitivity of the flux
to changes in the two forces may in theory be
unequal to the ratio of its stoichiometric numbers
(i.e., n~). An extreme case is the situation when a
reaction becomes saturated with respect to one
substrate/product couple, but not with respect to
another [13,23].
An example of this is found in mitochondrial
oxidative phosphorylation where at oxygen tensions around 0.2 atm, the oxidation rate does not
vary with oxygen tension (the K m for oxygen is
well below 1 matm), whereas it varies strongly
with the transmembrane electrochemical proton
gradient [16,23]. More generally, the dependence
of the rate of an enzyme catalyzed reaction on the
free-energy difference of a substrate/product couple does not only depend on the specific activity of
the enzyme. Allosteric effects of the substrate causing Hill coefficients to differ from 1 likewise affect
the dependence of reaction rate on AG. Such a
characteristic of an enzyme can be accounted for
by an altered value of the L-coefficient. In cases,
however, where there are two (or more)
substrate/product couples, the allosteric effects
caused by the one couple may differ from those
caused by the other. Accordingly, for every substrate/product couple a separate L would be
needed, e.g. for the present case of anabolism an
a
L a for the dependence on AQ and an ( - - n p times)
L~, for the dependence on AGp. To keep the effect
of a variation in the activity of an enzyme, due to
a variation in its concentration confined to one
parameter we prefer to write L ' a as ~ times L~.
Thus La contains the information concerning enzyme concentration as well as the information
concerning allosteric effects of AG~, ,/pd measures
the allostericity with respect to the AGp relative to
the allostericity with respect to AG~, -- n ap indicates
the relative stoichiometry with which the two reactants (ATP and C-tool anabolic substrate) react,
AG~ determines in which domain of AG~ the
anabolic reaction rate responds to changes in AGa,
whereas AGp~ determines in which domain of AGp
the reaction rate responds to changes in A@. This
somewhat lengthy discussion serves to show that
all these parameters are characteristic for the elemental reaction of anabolism. They could, in
principle, all be determined independently from a
growth experiment, by isolating the enzymes responsible for anabolism and then performing the
relevant kinetic experiments. Consequently, these
parameters are mechanistic and not phenomenological in nature [10].
The rate of ATP synthesis involved to support
biosynthesis equals the number of moles of ATP
mechanistically hydrolysed per C-mol biomass
produced in the biosynthetic reactions (-n~p) times
the rate of biomass production in C - m o l / m i n / g
dry weight (--Jd). Mathematically:
= ,,; . Z
(3)
Usually Jp" is positive, np and J~ being negative.
For catabolism the analogous relationships are
obtained:
Jc : L c . ( ( AG c -- A G ~ ) -~/'/;. ~ p - ( mGp -- AGp~ )}
J;=,pL
(4)
(5)
In these equations the index c stands for catabolism; the meaning of all other symbols is the same
as in Eqn. 2: - - n pc , the number of ADP molecules
mechanistically phosphorylated per C-mol of
II
catabolic substrate metabolised, AG~, the free-energy content (chemical potential) of the catabolic
substrate per C-mol minus the free energy content
per C-mol of the catabolic products, and Jc, the
rate of catabolism, are defined so that they are
positive under the usual conditions. Eqn. 4 states
that catabolism is driven by AGe and slowed down
by the phosphate potential AGp (np is negative). L c
is again independent of both AG~ and AGp, but
proportional to the activity of the enzymes catalysing the catabolic pathway. Eqn. 5 states that the
rate at which ATP is produced ( - J p ) by catabolism equals the number of molecules of ADP
phosphorylated per C-tool catabolised ( - n p) times
the rate at which the catabolic substrate is catabolised (J~).
The third flux to be considered is the rate of
uncoupled hydrolysis of ATP ( j l ) that occurs in
the cells. This ATP hydrolysis can be due to
proton permeability of the cytoplasmic membrane,
to the occurrence of cyclic metabolic reactions that
consume ATP, and also to processes more clearly
connected with maintenance, such as protein
synthesis to replace inactivated proteins, reuptake
of metabolites that escaped through leakage across
the cytoplasmic membrane etc. The rate of this
hydrolysis (j1) will increase with AGp, unless it
has reached its maximum rate. The rate will also
increase with the activity of the enzymes catalyzing the uncoupled hydrolysis, i.e. with L~p. Consequently the flux/force relationship of this ATP
hydrolysis is:
Jd = Lp' (
-- AG; ~)
(6)
in which the superscript 1 stands for leakage.
Generally, the intracellular pool of ATP in microbial cells turns over in less than 10s [24]. It
follows that the intracellular phosphate potential
AGp will adapt to a new metabolic condition within
1 rain. With respect to the AGp then a new steady
state has been reached in which there is no longer
any net change in intracellular ATP concentration.
The net rate of ATP synthesis ( - J p ) then equals
zero:
Jp:J;q-Jp q'-Jd:O
(7)
By use of this equation and Eqns. 2-6 we can
derive the following two relationships between the
rate of catabolism ( J c) and the rate of biomass
synthesis ( - J a ) :
Jc =
-7"nP 1 +
Lp
np
yp • (rip)2 • La
+{
"¢"} •
{L~:L_£o
+ e'
¢ 2
"(-Ja)
- o2)
(8)
}.(AG--AG,)(9)
+rp.(np)-co
In both equations the catabolic rate (Jc) equals the
sum of two terms. The first term expresses to what
degree J~ varies with the rate of biomass synthesis
(--Ja)" This expression (between the brackets) is a
mosaic of stoichiometric constants (ns), an allostericity factor (7) and specific enzyme-activity
parameters (the Ls). The second term relates the
flow of catabolic substrate(s) to the Gibbs free
energy of either Eqn. 8, the anabolic, or Eqn. 9,
the catabolic reaction. In both equations again the
proportionality constant between these two
parameters (the term between brackets) is composed of stoichiometric constants, a weighing factor and activity parameters. It is instructive to note
that in either equation the rate of catabolism J¢ is
written as a function of not only the growth rate
( - J ~ ) and a free-energy difference, but also of all
characteristics (Ls, ns, ys, AG#s) of the enzymes
catalysing the elemental processes. Clearly these
equations are not phenomenological but mechanistic in nature. It may be illustrative to read from
the above equations the effect of an increase in
ATP leakage (L~) on J~ at constant AG~, but
variable growth rate - J a .
The substance limiting growth in a chemostat
[25,26] can be uniquely the anabolic substrate
(carbon, or nitrogen limitation), the catabolic sub,
strate (energy limitation) or some essential cofactot. Also more than one substrate can be growthlimiting at the same time [2,27]. In this paper we
confine discussion to the first two possibilities.
12
When the anabolic substrate is uniquely growthlimiting changes in the concentration of the catabolic substrate do not affect the growth rate [27]:
in other words, (AGc -- AG~) is in the domain (see
above) in which it is effectively constant, due to a
saturation of the catabolic enzymatic machinery
with catabolic substrate (the Vmax domain of the
catabolic reactions [21]). Consequently, when
uniquely the anabolic substrate is growth-limiting,
Eqn. 9 writes Jc as a function of one variable
( - J a ) rather than as a function of two variables.
Hence Eqn. 9 becomes very useful. It predicts a
linear relationship between the rates of catabolism
and anabolism. By analogy, when uniquely the
catabolic substrate is growth-limiting, Eqn. 8 is
more useful.
In many experimental growth studies one substance serves both as catabolic (energy) and as
anabolic (carbon) substrate. In those cases one
often considers changes in the rate (Js) of consumption of this substance with growth rate. Since:
Js = Jc + (--Ja)
(10)
Eqns. 8 and 9 are transformed into:
/,/a
Js =
1 + ' pc "
/'/p
1
q-
Lp
(-s.)
LIp }. (AGa--AGa
+ np.np'yp
a(
1
+ n-y-p. 1
n;
+
Lp
G+
Lp_. L c
n p q1 - - ~ p - ( nc; )
)2
(ll)
)}
(-4)
(n; - Lo
• (AGc - AG~)
(12)
2 •n c
When the substrate yields only little Gibbs free
energy upon combustion there is energy limitation
and A G a - - AGff is effectively constant and Eqn. 11
is of great use. When the substrate is rich in
energy, growth is carbon-limited, AGc --AG~ is
effectively constant and Eqn. 12 is optimal for a
description of growth.
3. PREDICTIONS BY T H E M O D E L AS EXP L A I N E D BY T H E M N E T DESCRIPTION
Eqns. 8 and 9 (or, for single-substrate growth,
Eqns. 11 and 12) completely describe, and thus
predict, the growth rate and growth yield(s) of a
specific microorganism under a certain set of
growth conditions as a function of the rate of
catabolism and the mechanistic characteristics of
the elemental reactions of Fig. 1. Appropriate experiments will allow a complete determination of
the values of all parameters that are involved in
these two equations for that particular growth
condition. More generally, Eqns. 8 and 9 predict
which variation should occur in catabolic and
anabolic rates when one of the elemental processes
changes• In this section we will enumerate a series
of such predictions and in the next section a
comparison is made between these predictions and
experimental results that are available from the
literature•
(i) When either the catabolic (energy-yielding),
or the anabolic (carbon- or nitrogen-yielding) substrate is uniquely growth-limiting, a linear relationship between the rate of catabolism (Jc) (or
substrate consumption Js) and the rate of biomass
synthesis (--Ja) should exist•
(ii) Catabolism must occur even in the absence
of growth (if Ja ---~ 0, Jc is still positive)• Depending
on whether the catabolic or the anabolic substrate
limits growth, the magnitude of the growth rate-independent maintenance is given by the last term of
Eqn. 8 or 9, respectively•
(iii) Even when corrected for this growth rateindependent maintenance, conversion of catabolic
substrate into biomass is not according to the
theoretical stoichiometry (i.e. npCJc =fi:- npa Ja), unl e s s Lp1 and consequently ATP leakage is zero.
Apparently, ATP leakage alone (Lip = 0) is sufficient for growth rate-dependent maintenance
catabolism to occur.
(iv) The growth rate-independent maintenance
catabolism (i.e. the last term of Eqn. 8 or 9) differs
depending on whether the anabolic or the catabolic substrate limits growth• In either case it
increases with ATP leakage (L~). It can be calculated that the last term in Eqn. 9 always exceeds
the last term in Eqn. 8: under anabolic substrate
13
limitation the growth rate-independent maintenance catabolism is higher than under catabolic
substrate limitation.
(v) Also the growth rate-dependent maintenance catabolism depends on the nature of the
growth-limiting substrate. If the catabolic substrate uniquely limits growth, the last term in Eqn.
8 is constant• Hence the growth rate-dependent
maintenance catabolism is positive (more catabolism than accounted for by biomass production):
npa
LpI
" (--Ja)
,~ ( n ~,)2.L ~
n pc yp"
By contrast, if growth is limited by the anabolic
substrate the last term of Eqn. 9 is constant, so
that this equation can be used to predict maintenance catabolism. This maintenance is negative:
t/pa
--
--•
t pl
c
I
% G
c
(--Ja)
In either case the absolute magnitude of the growth
rate-dependent maintenance catabolism increases
with increasing ATP leakage (increase in Lap).
(vi) When the catabolic substrate is growthlimiting (Eqn. 8, (AGa--AGa~) being approximately constant), a n i n c r e a s e in the ATP/substrate stoichiometry of the catabolic pathway
( - n p ) would not affect the fraction of the maxim u m yield consumed by the growth rate-dependent maintenance, i.e.
,+
Lp
yp
It would, however, decrease the growth rate-independent maintenance catabolism:
Lp
a . ¢: . a
n p n p "yp
• ( AG. -- AGy)
When, on the other hand, the anabolic substrate is
growth-limiting the same increase in stoichiometry
(--n~) would decrease the fraction of the maximum yield produced by the growth rate-dependent
maintenance (cf. Eqn. 9):
L'
c.
yp
c) 2 • Lc
(rip
Maximum yield is defined as the yield (i.e. --Ja/Jc)
corrected for growth rate-independent maintenance and is therefore equal to the reciprocal of
the first term between brackets in Eqns. 8 and 9.
For single-substrate growth the 'yield' is more
often defined as --Ja/Js = --Ja/( Jc -- Ja )" Then the
maximum yield is given by the reciprocal of the
first term between brackets in Eqns. 11 and 12.
Qualitatively the effect of an increase in -npC
should be the same.
4. COMPARISON OF THE PREDICTIONS TO
EXPERIMENTAL RESULTS
Of the predictions summed up above, (i) [28]
and (ii) [5,29] confirm the classical observations in
chemostat cultures of microbes. The occurrence of
growth rate-dependent maintenance (cf. (iii)) was
postulated by Neijssel and Tempest [2] (see also
[30])• These authors also suggested that this form
of maintenance could depend on the nature of the
growth-limiting substrate and that it might be
negative (v). It has also frequently been reported
in the literature that the magnitude of the growth
rate-independent maintenance depends on which
of the substrates limits growth (e,g. [2,5,31])• The
effect of uncoupling on the growth rate-independent maintenance (iv) predicted here has been
demonstrated experimentally by Neijssel [32] and
changes in stoichiometry of ATP yield in catabolism (cf. (vi)) may result from alterations in the
H + / e - stoichiometry of the electron-transport
chains, such as the appearance/disappearance of a
complete energy-coupling site [34-39] (review in
[28]). Meyer et al. [39] found that disappearance of
energy conservation at site I in Paracoccus denitrificans led to a vast increase in growth rate-independent maintenance and a slight decrease in
the maximum yield. Since these results were obtained for growth on a single substrate as carbon
and energy source it is not clear whether the
experiments correspond to a case of catabolic substrate ('energy') limitation (Eqn. 11 with A G , AGa~ constant), or to a case of anabolic substrate
limitation (Eqn. 12 with AGc - AG7 constant). For
the growth rate-independent maintenance (the
right-hand term in these equations) this presents
14
no problem, as both equations predict it to increase when - npC decreases. This is actually observed. For the case of catabolic substrate limitation Eqn. 11 predicts a marked decrease in maximal growth yield (i.e. -Ja/Jc corrected for the
growth rate-independent maintenance found experimentally by extrapolating to infinite growth
rate). For the case of anabolic substrate limitation
Eqn. 12 predicts the maximum yield to equal
f
n c
Lp c
-t- P
np'yp .np. Lc n
a
c
ptf
// 1 -k- a -Lp
-np • yp •np. L c
n c
~--P
n
so that a decrease in --npc could either decrease or
increase the maximum yield depending on the
magnitude of the leakage coefficient Lp. The substrates used by Meyer et al. [39], gluconate and
succinate, are only slightly more oxidized than
biomass, so that growth limited by these compounds may correspond to a condition almost
halfway between catabolic substrate limitation
(energy limitation) and anabolic substrate limitation (carbon limitation) [17,19,40]. A decrease in
--npc is then predicted to result in a moderate
decrease in maximum yield. This is in line with the
experimental observations.
A conceptually important case is growth of
Lactobacillus casei on glucose [41]. As growth rate
decreases, this organism shifts from homofermentative growth (lactate being the end product) to
heterofermentative growth (acetate being pro"
c
duced in addition to lactate). Consequently rtp
becomes more negative with growth rate. This has
an important consequence for prediction (i): a
linearity between catabolic rate and growth rate is
only predicted by Eqns. 8 and 9 when the terms in
brackets, which contain the parameters reflecting
the properties of the enzymes, are constant. Now
that - n p decreases with -J.,, the relationship
between catabolism and growth rate is predicted
to be nonlinear (actually: sigmoidal). Such a relationship was indeed observed [41]. Knowing the
production rates of lactate and acetate at every
growth rate, the dependence of - rtpC on growth
rate can be calculated. De Vries et al. [41] carried
out this calculation and then considered - J ~ =
-- ?/pC . Jc('qATP') as a function of growth rate -a(~.
For cases of catabolic substrate limitation the rela-
tionship between qATP ( - n p "Jc) and growth rate
is indeed expected to be independent of - n p12, so
that np. Jc should vary linearly with -a~a. De Vries
et al. [41] observed a linear dependence of qATP on
growth rate. It seems noteworthy that for cases of
anabolic substrate limitation a completely linear
relationship between qATP ( - - n ; "J c) and growth
rate ( - J a ) is not expected (Eqn. 9) when such a
shift in catabolism would occur.
Thus, the M N E T description of bacterial growth
indicates the relevance of the observation of 'nonMitchellian' I 4 + / e - stoichiometries for microbial
growth [42]. The yield characteristics of bacterial
growth depend on the exact ATP/catabolic substrate stoichiometry and the M N E T description
gives a full account of the effects of a variation in
that stoichiometry.
5. DISCUSSION
The available experimental observations fit into
the M N E T description of microbial growth.
Accordingly, these observations have now acquired
a mechanistic interpretation. It is part of our hypothesis that additional examples, put forward by
the reader, will also fit into the M N E T description
(or its more elaborate relatives, for instance the
one discussing futile cycling [17]).
At this point we wish to take away possible
misinterpretations that could lead to erroneous
applications of the M N E T description of microbial growth. First, a mathematical characterization
of a specific case of bacterial growth in terms of
quantitation of J, L, n, 7 and AG # values is
uniquely linked to that case. One should be extremely cautious when transposing an L value
determined under one condition to a culture under
a different condition, unless it is certain that the
specific activity of the corresponding elemental
process is identical. Second, we pictured anabolism and catabolism as two completely separate
metabolic pathways. Thus most cases of microbial
growth are formally excluded from direct application of the present description, because catabolic
and anabolic pathways commonly overlap. Usually (cf. [17]) a slightly modified version of the
present description can again be applied, but such
modifications should be made with great caution.
15
Third, although overflow metabolism has things in
common with ATP leakage, it cannot be described
by an increase in Lp. Similarly futile cycles cannot
be described by an increase in Lp1 only [17]. Although this paper does not show how these two
interesting elemental processes affect growth macroscopically, we hope to have conveyed the message that the appropriate M N E T descriptions can
be (and for futile cycling have been [17]) developed.
In a more extensive account of non-equilibrium
thermodynamic descriptions of microbial growth
[17] we conclude that to our knowledge no published microbial growth experiment is incompatible with an M N E T description of microbial growth.
Strictly speaking, the status of this experimental
confirmation of Eqns. 8 and 9 (or 10 and 11) is
unclear, because the basis of these equations is
twofold. First, the equations assume the validity of
the scheme of microbial growth given in Fig. 1.
Second, they are based on the hypothesis put
forward in this paper, i.e. the ability of M N E T to
quantitatively transform such a scheme into relationships between rate of catabolism and growth
rate (Eqns. 8 and 9). We do not wish to contend
now that both Fig. 1 and the M N E T approach can
be concluded to be valid. Shortcomings of the
scheme of microbial growth in Fig. 1 may have
been compensated by shortcomings in M N E T to
relate biochemical detail to macroscopic growth
behaviour. Or, more likely, the experiments documented in the literature may not yet have been
sufficiently aimed at testing either validity. What
we do wish to convey is that every specific scheme
of microbial growth (such as Fig. 1, or more
elaborate schemes including futile cycles [17], or
overflow metabolism) gives rise to its own mosaic
in the terms between brackets in Eqns. 8 and 9
and that therefore these equations, and their analogues for the other schemes, may enable us to
devise so-called analytic experiments, the results of
which will falsify one scheme in favour of another.
Thus our hypothesis is that M N E T allows us to
translate proposed metabolic schemes into testable
macroscopic consequences in terms of growth behaviour.
The test of the hypothesis itself calls for so-called
synthetic experiments. Such experiments differ
from those aimed at finding the correct growth
scheme for a particular microbial culture. They
have to be carried out with microbial cultures for
which the metabolic scheme is known and must
involve varying as many of the elemental parameters (L, y and n) as possible to see whether the
effects of these variations on the relationship between catabolism and anabolism are indeed predicted by (the analogues of) Eqns. 8 and 9. This
paper is meant to be an invitation to carry out
both the analytical and the synthetic experiments.
The latter may or may not falsify the hypothesis
put forward in this report.
The unique contribution of M N E T to the study
of microbial growth lies in its property of predicting the effects of changes in the catalytic characteristics of microbial metabolism on growth yield
and related parameters. Consequently the large
amount of knowledge of microbial biochemistry
and biophysics and their regulation may now be
recruited to manipulate relevant growth or production yields. More than by indicating which enzymes should and which should not be present for
a certain product to be made, we believe that
M N E T descriptions may help to optimize microbial product formation by allowing fine tuning of
all parts of microbial metabolism to the desired
microbial performance. In this respect it may be
relevant that the M N E T description also allows us
to understand the usefulness of phenomena such
as uncoupling and futile cycling [17,19]. Consequently no longer must we be satisfied to qualitatively manipulate the microbe's metabolism in a
microbial culture, as it may well prove fruitful to
try and regulate the activities of uncoupling and
futile cycling devices to the (non-zero!) level that is
optimal for the formation of the desired product.
Similarly, aberrant or unwanted growth characteristics can be traced back to their relevant metabolic causes which may be of interest for microbiology as a discipline, but also for continuity in
microbial production lines.
ACKNOWLEDGEMENT
We thank Douglas Kell and Wil Konings for
discussions.
n;
a
t/p
LI
L~
L~
J~
J;
L
J.
AGp
AG~
AGc
AGa
AG #
7p
c
7p
AG
a
Symbol
differential allostericity of anabolism
differential allostericity of catabolism
Gibbs free energy (free
enthalpy) of reaction
a constant fixing the range
of AG where J varies
Gibbs free energy of biomass
synthesis (anabolism); chemical
potential of biomass minus chemical
potentials of anabolic substrates
a constant belonging to anabolism;
determining the range of AGa
where Ja varies with AGa
Gibbs free energy of catabolism;
chemical potential of catabolic
substrate minus chemical potentials
of catabolic products
a constant belonging to catabolism;
determining the range of AGc where
Jc varies with AGe
phosphate potential~ AGpo
+ RTln ([ATP]/[ADP]. [Pi ])
a constant belonging to phosphorylative systems determining where
AGp influences the flows
(minus the) rate of biomass
synthesis (anabolism)
rate of catabolism
total rate of ATP hydrolysis
rate of ATP hydrolysis coupled
to biomass synthesis
(minus the) rate of ATP synthesis
coupled to catabolism
rate of uncoupled ATP hydrolysis
rate of total substrate utilisation
specific activity of anabolic enzymes
specific activity of catabolic enzymes
specific activity of uncoupled ATPase
(minus the) theoretical number of moles ATP
hydrolyzed per C-tool biomass produced
(minus the) number of moles ATP produced per C-mol catabolic substrate consumed
Meaning
APPENDIX - - GLOSSARY
1
6
1
4
6
10
4
1
2
First use (Eqn.)
1
1
+
+
k J/C-tool
kJ/C-mol
kJ/C-mol
+ (energy-poor)
- (energy-rich)
(substrates)
+
+
+
+
+
+
+
1
8,17
8, 17
17
8,17
8,17
8,17
8,17
17
n m o l / m g dry w t / m i n
n m o l / m g dry w t / m i n
n m o l / m g dry w t / m i n
k J - i Cmol.nCmol/mg dry w t / m i n
kJ t Cmol.nCmol/mg dry w t / m i n
kJ i mol.nmol/mg dry w t / m i n
17
8,17
17
n C - m o l / m g dry w t / m i n
nC-mol/mg dry w t / m i n
n m o l / m g dry w t / m i n
n m o l / m g dry w t / m i n
17
kJ/mol
+
+
0
+
10, 16
kJ/mol
9,12
10, 17
17
10, 17
17-19
10, 12, 23
8
12, 16
12, 16
Further ref.
+
kJ/C-mol
kJ/mol
+
kJ/mol
Unit
Usual sign
17
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