FEMS MicrobiologyEcology73 (1990) 203-210
Published by Etseviex
203
FEMSEC 00251
Maintenance requirements: energy supply from simultaneous
endogenous respiration and substrate consumption
H . H . Beeftink, R.T.J.M. v a n dcr H¢ijdcn a n d J.J. H e i j n e n
Departmem ol Biochemical Engineerin& Delft Uniocr$ity of Technology, Delft. The Nedcerlands
Received20 April 1989
Revision receivedand accepted 12 Oclobcr 1999
Key words: Maintenance model, Pitt model; Herbert model
1. S U M M A R Y
2. I N T R O D U C T I O N
A model is presented that describes energy for
maintenance purposes (ATP) as being obtained
simultaneously from biomass degradation as well
as from substrate degradation in excess of growth
requirements. The ratio between both catabolic
processes was takers, to be growth rate dependent.
As such, this approach is intermediate between
established models; its stgnificant features are
negative growth and the absence of substrate consumption at zero substrate concentration, and the
attainability of the maximum specific growth rate
(the model parameter ~ , ~ ) at elevated substrate
concentrations. As a simple case, the amounts of
A T P obtained from direct substrate catabolism or
from the degradation of an equivalent amount of
biomass were taken as identical. Also, the maintenance demand in terms of ATP per unit time and
biomass was taken to be constant. True growth
rate dependency of maintenance can be implemented by relaxing either of these assumptions.
Mathematical expressions for the maintenance
requirements of microorganisms are commonly
formulated in two alternative ways. Herbert [;t]
modified a common balance equatiolx on biomass
reactions so as to contain a maintenance term in
addition to a strictly growth-associated term. As
an alternative, a conunon balance on substrate
reactions was modified by Pitt [2]. The present
contribution combines these approaches, and postulates simultaneous modifications for both the
balances on biomass and on substrate. A structured metabolic mechanism is supplied to support
the model equations. Although maintenance demands for reducing equivalents or for mass have
been reported [3,4], the present model ¢onsid~s
energy requirements only.
The concept of Herbert [1], which was further
elaborated upon by Man" et al., [5], assumes two
fundamental reactions involving biomass and one
reaction involving substrate. Although this shall
be modified below, all reaction rates may be functions of the substrate concentrations s .
Correspondence to fPresent address): H.H. lteeftink, Department of Food Science, Food and Itioprocess Engineering
Group, Agricultural UniversityWageningen,Bomenweg2, 6703
riD Wa$cningen, The Netherlaad:.
~,~,(s) = p ~ e ( s ) -- p c ( s )
(1)
qo (s) = q (s)
(2)
Observed growth at a specific rate Pot~(s) thus
0168-6496/90/$03.50 © 1990 Federation of European MicrobiologicalSocieties
204
results from true growth at a specific rate/~t~(s),
and from negative 'endogenous' growth or biomass degradation at a specific rate ge(s). The
latter decay process is thought to supply the energy (ATP) required for maintenance purposes. As
opposed to the Pirt approach (see below), observed consumption of substrate at a rate qob~(s)
results only from growth-related consumption at a
rate qgr(S), reflecting anabolism and growth-associated catabolism.
Pitt [2] presents a modification of the balance
equation for the carbon and energy substrate. In
agreement with earlier concepts [6,7], he proposed
one fundamental reaction for biomass and two for
substrate (again, these rates are potentially substrate-dependent):
~,o~,(s) = ~,,..o(s)
(3)
qo~,(s ) = qv(s) + m,(s)
(4)
Here, snbstrate is consumed at a rate q~(s) for
growth requirements proper (anabolism and
growth-associated catabolismL but in addition at
a rate ms(s) to supply catabolic energy for maintenance requirements. Contrary to the Herbert
approach there is no biomass degradation, and the
observed growth rate ~o~(S) equals the true
growth rate gtr~e(s).
Both models consist of two independent equations for the dependent variables /~o~(s) and
qo~(S), involving three unknowns, namely qg(s),
/tt,~(s), m~(s) in the Pirt model, and q~(s),
#,~¢(s) and /~e(s) in the Herbert model. In both
models, one of these unknowns is eliminated by
postulating a fixed ratio between true growth and
growth-associated substrate consumption, called
the true yield of biomass on substrate:
~,x = ~,,~o(s)/q,~(s)
(5)
Additional information on two of the remaining
variables is required to make equations (1)-(4)
meaningful Therefore, some form of substrate
dependency (generally saturation or Monod kinetics) is assumed to apply to either the true specific
growth rate gwd¢(s) or the growth-related substrate consumption qg(s). As may be judged from
equation (5), these alternatives are equivalent. In
addition, the maintenance rates (/z¢(s) and m~(s),
respectively) axe taken to be independent of substrate concentration. The Final Herbert model is
thus represented by the following equations:
~,o~(,)
=
~%~K , s+
s - ~°
(6)
$
qobs(s) = t~tn~ ys.~( K s + s)
(7)
Alternatively, for the Pirt model:
~o~,(~)
(8)
= ~!".~ K , ~+ ~
max
S
qo~(s) = t,,,,~ r , , , ( r , + s) +m~
(9)
The general characteristics of these sets of
[I"~X
,Z,~true
/'~lrte
o
,
,
--
o
Herber+
P~rt
I
oKs
I
sL~strate
concentration
oKs
Substrate concentration
Fig. 1. Comparisonof maintenancemodels:specificrate of overallsubstrate consumptionand growth accordingto H~rbortand Pitt.
Symbols. s¢¢ text.
205
equations are illustrated in Fig. 1. Often, the models of Herbert and Pitt are regarded equivalent.
This is not the case, however, as both have their
idiosyncrasies.
The Herbert model allows for negative net
growth: at s = 0 , ~ o ~ ( 0 ) = - / ~ , and a decrease in
biomass for a closed system can thus descriLed.Th¢
model, however, features a maximum specific
growth rate t z ~ that is less convenient from an
empirical point of view, as it cannot be observed
or measured directly: at s = o0 the observed
specific growth rate /%b,(00) equals ~ q m ~ _ ~ .
Also, it would be conceptually more attractive if a
microorganism would not cover its maintenance
requirements from bioraass in the presence of
excess substrate.
In the Pitt model observed growth is always
positive, which is not in agreement with experimental observations. Another peculiar aspect is
due to the postulated substrate independence of
the maintenance. Although it was originally formulated for non-zero substrate concentrations, the
Pitt model describes consumption of substrate
even in its absence: at s = 0, qob~(0) = m~.
The present contribution attempts to reconciliate both approaches, and views maintenance
energy as being supplied from either substrate or
biomass, depending on environmental conditions.
A model formulation is put forward that contains
the required features of:
- negative growth at s = 0
- no substrate consumption at s = 0
- no endogenous decay at s -_
/toby(do ) =//max
3. M O D E L D E V E L O P M E N T
A new set of equations is proposed that combines attractive feature~" of previous models. The
basis of these equations is the assumption of four
fundamental rates (instead of three for the models
discussed above). Thus, the biomass balance (10)
contains, besides an observed rate, a true growth
rate and a maintenance rate; it thus follows the
approach of Herbert [1] (cf. equation 1). In addition, however, the substrate balance (11) contains terms for observed consumption, growth-re-
lated consumption, and maintenance consumption; as such it follows the approach of Pirt [2] (cf.
equation 4).
la,b~(x)--I~"~-Mo(s)-M.(1 -Mo(s))
(10)
qobs(S) = (prnax/y~.g)- Mo(s) + ( M/Ys.~)" Mo(s)
(11)
Here Mo(s) represents a substrate function such
that Mo(O)= 0 and Mo(ov)= I. By the second
terms on the right-hand sides of equations (10)
and (11), two separate maintenance processes are
postulated; both are substrate-dependent and are
represented by a constant M and a substrate-dependent factor involving Mo(s).
During feast conditions (i.e. at s = oo and hence
Mo(oo)--1), the observed growth rate actually
approaches #ma~ while the microorganism covers
its maintenance ATP requirements entirely from
non-growth-related substrate consumption (the
Pitt situation).
Famine, on the other hand, forces it into endogenous respiration: at s ffi 0 and Mo(O) ffi 0 there
is no observed substrate consumption and maintenance ATP is obtained from biomass degradation
(p,b~(s) < 0; i.e. the Herbert situation).
At intermediate substrate concentrations, the
mechanism of maintenance energy generation
gradually shifts between these extremes. This shift
is brought about by the factors ( 1 - Mo) and
(Mo) in the biomass and substrate balances (10)
and (11).
Fig. 2 outlines this gradual shift in mechanism,
by comparing the proposed equation to the Herbert and Pirt equations for net substrate consumption and net biomass growth. In addition, it is
important to realize that equations (10) and (11)
may be combined to give the well-known relation
between Pob~(S) and qobs(s):
qob~(s) =
(po~(s)/Y~.,) +
(M/Y~.~)
(12)
This relation also holds for the models of Herbert
and Pirt, as it may be obtained from equations (¢~)
and (7) or (8) and (9), with g.JY,.x or m~ replacing
M/Y~.~. Also, the parameter Ys.x is seen to have
the same meaning for all three models.
The equations proposed here may be substantiated by mechanistic considerations; a general out-
206
,ra.
IIk'l R
"/'g tr~
I
Pirt _
_
~
....... . ~ - ~ . ................................
/f//f
0
f/
t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
O
growth
....
I-~rbert
............................................
L
con,~um0tion
I
o Ks
o Ks
substrate coracentration
substrate concentration
Fig. 2. Comparison of shifting-mefhanism model with Herbert and Pitt equations. Symbols: s¢¢ text,
line of the basic mechanism is given in Fig. 3.
Analogous to the Herbert model, microbial growth
is taken to consist of two parts, according to:
Using the well known concept of a constant true
yield of biomass on substrate from equation (5),
equation (14) may be rewritten as:
,%~(s) = pt~¢(s) - pc(s)
q~(s) = (~,~,(,)/v~..)
(13)
Analogous to the Pitt model, the observed s u b
strate consumption rate qot~(s) is taken to consist
of anabofic substrate consumption (i.e. as Csource), consumption tbr growth-related catabolism, and consumption for maintenance-related
catabolism, according to:
qo~(~) = #g,(s) + qo=.,.(~)
= qo,(~) + q.,.,~ (s) + q~,.~(,)
(14)
q ..,(s)
-as~oc~ateO
c~tabol,~
(15)
Both catabolic processes in equation (14) generate
ATP according to an identical and constant
stoichiometry, resulting in the ATP-flows . ( s )
and 18(s):
a(s ) = v=,.A~p, q,,.~ (s)
(16)
,e( s) = t'=,.A.,-,, . qo~,.,,,( s )
07)
ATP formed by growth-associated catabolism at
rate a(s) is used entirely in the formation of
-A
°'---
+ q~,.=(s)
,u, .,,.~(s)
- - ~
~-[ b i o m a 8 8
:
product
q c=t.,,{S)
Fig. 3. UtlriTAdon of substrate and formation of biomass as interrelaugl by A'IT productio'll and consmnption~ Symbols: ~:c text,
207
biomass by true growth, which proceeds at a rate
/ ~ e ( s ) . It is assumed that, besides growth-related
ATP consumption a(s), there is a substrate-independent and hence constant specific rate # of
ATP consumption for maintenance purposes. This
maintenance ATP requirement is provided for at
rate fl(s) from maintenance-associated substrate
catabolism (proceeding at rate qcacm(S)), and,
simultaneously, at a rate ~,(s) by means of biomass decay (proceeding at a rate p~(s)). The assumption of a pseudo-steady state for the intracellular ATP pool requires:
=
fl(s)
+ y(s)
(18)
The central assumptions of me present model
concern the constancy of maintenance in ATP
terms (i.e. the constant 8), and the modulation
between the two constituent rates fl(s) and ~,(s).
While the Herbert and Pitt models would regard
¥(s) or fl(s) as constants, respectively, they are
formulated here as substrate-dependent but complementary fractions of the constant &
As a final assumption it is postulated that biomass
synthesis is subject to saturation ldnet;c~:
i ~ ¢ ( s ) = ~m~. M o ( s )
(26)
which leads to:
~obs(S) = ptm~ " M e ( s )
- ($/Y~.AxP)" (1
qo~,(~) = ( ~ ' / K , . ) -
-
-
Mo'(s))
(27)
Mo(s)
+ ( , / v , ~ , ~ , ) . Mo'(s)
(2s)
Exactly the same result can be obtained assuming
qobs(s) = q " ~ " Me(s).
In order to keep the number of parameters
limited, the functions M e ( s ) and Mo'(s) can be
equated. If it is furthermore assumed that there
are no ene~'gy losses, i.e. the amount of ATP
produced from substrate via biomass equals the
ATP production by direct substrate catabolism:
~,~" K.^'rP = Y.t.^'rP
(29)
~ ( s ) = 8. ,Vo'(s)
(19)
equations (27) and (28) reduce to (10) and (11),
with:
),(s) ----* . (1 - M o ' ( s ) )
(20)
M
Like Mo(s), Mo'(s) is some substrate function
with the property that at s = 0, Mo'(O) = 0, while
34o'(0o) = 1 at s----oo. This way, y ( 0 ) = 6 and
fl(0) = 0 at zero substrate concentration (as Mo"
-- 0), while under substrate sufficiency (i.e. Mo" =
1) f l ( ~ ) = $ and V(oo)= 0.
Defining:
v(s) = r~,,,Tp-~,o(s)
(21)
and combining equations (17), (19), (20), and (2I)
yields expressions for the supply of maintenance
ATP by substrate and biomass combustion:
I%(s)
(~/Yx~,Te)" (1 -- M o ' ( s ) )
=
q.,.~(~) = (81 v.,.^~p), mo'(s)
(22)
(23)
Substitution of these expressions in the substrate
and biomass balances (14) and (11) yidds:
~,o~,(s)
=
~,~o(s) - (*/V,,^T,)" (i -mo'(.))
(24)
qo~(S) - O,,~o(s)/~.~) + (~IY~,,A.~)- Mo'(s)
(25)
----
~/Yca|.ATP
(30)
4. DISCUSSION
The present model regards maintenance ATP
requirements by microorganisms as being constant; they are supplied by both substrate catabolism and biomass degradation in substrate-dependent ratios. As such, it is a compromise between
earlier concepts, combining aspects from the Herbert and Pitt models. In particular, the model
predicts negative observed growth rates at zero
substrat¢ concentrations; these negative rates are
the result of biomass decay under these conditions. On the other hand, the maximum of the
observed growth rate as s = o¢ equals the maximum rate of the true growth process (i.e. ~mat)
due to the absence of biomass degradation under
these conditions. Apart from the non-zero substrate concentrations at zero dilution rates, all
three models discussed here would describe
steady-state continuous-culture data equally well
(as negadve overall growth is excluded by defmi-
208
finn). The model presented here, however, should
be preferable if transient and periodic reactions
are to be described.
Seemingly, the present model equations (10)
and (I1) may be derived directly from Herbert's
equations (6) and (7) by simple rearrangement and
a redefinition of kinetic parameters. While the
result thus obtained and equations (10) and (11)
are identical in a mathematical sense, there remains an essential difference in interpretation.
Herbert's model (and any derivation thereof) rests
on the assumption of three fundamental rates
(equations (1) and (2)); therefore it cannot be
interpreted in terms of the assumptions underlying
the model presented here (i.e. four fundamental
rates; eL equations (13) and (14)).
It should be stressed that our model features a
total demand for maintenance energy in ATP
terms that is a constant (6). The relative contributions to this overall demand from hoth ATP
supplying routes, however, are substrate-dependent; maintenance from biomass degradation in
itself is growth-rate dependent, as is the supply
from substrate catabolism. This flexibility has been
obtained without the addition of extra parameters
as compared to the Herbert and Pitt models. By
means of a mechanistic basis for the final model
equations, a relation is obtained between fundamental parameters (ATP-yietds, ~$, M and Y~.x).
Further modifications of maintenance concepts
may be introduced. Although it is relatively simple
to introduce a growth rate dependency for maintenance for either the Herbert or the Pirt model [8],
the present model offers the opportunity to do so
on a mechanistic basis (e.g. by releasing the assumption of a constant 8). Further modifications
include the incorporation of energy losses in binmass turnover, and alteration of the ratio between
the pathways that supply maintenance ATP (i.e.
Me' ~ Me).
APPENDIX
Symbols
s:
broth substrate concentration (mol • 1-1)
K~:
kinetic constant (mol- 1- t)
M:
specific maintenance factor: 6/Y~ot~ATP
(h -1)
Me:
..,.'yls ."
qan :
eta !.gt :
qc~t.m:
q~r:
qmaX:
qobs:
Yx.ATp ~"
Ycat,A'rp:
Ys.x
or:
P:
y:
&
/'~ohs:
saturation term ( - )
specific maintenance coefficient on substrate (reel- kg- 1. h - 1)
specific substrate consumption rate for
carbon in biomass production (reel. kg- t
.h-I)
catabolic growth-related specific substrate consumption rate (reel. kg- ~ • h 1)
catabolic maintenance-related specific
substrate consumption rate (reel. k g - t
h -1)
growth-related specific substrate consumption rate. (reel. kg-1. h - l )
maximum specific substrate consumption
rate (mol. kg- 1 h - 1)
observe~d specific substrate consumption
rate (reel. k g 1. h - 1)
ATP yield on biomass (reel- kg- I)
ATP yield on catabolic substrate ( - )
true biomass yield on growth-related substrafe consumption (kg/mol)
growth-related specific ATP production
rate (mol. kg- i . h - 1 )
maintenance-related specific ATP production rate from substrate degradation
(mol. kg- 1 . h - 1 )
specific ATP production rate from binmass decay (mol. kg - I . h - t )
maintenance-related specific ATP consumption rate (mol. kg- t . h - t )
specific biomass degradation rate (h-1)
maximum specific growth rate (h l)
observed specific growth rate (h - t )
true specific growth rate (h - t )
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